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# GEOSTATISTICS

## A.G. JOURNEL AND

Ch. J. HUIJBREGTS
.

Mining Geostatistics
A. G. JOURNEL

and
CH. J. HUIJBREGTS
Centre de Geostatisrique
Fontainebleau, France

Harcourt Brace Jovanovich, Publishers

## London . San Diego . New York Berkeley

Boston . Sydney . Tokyo . Toronto

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London NWl 7DX
San Diego, CA 92101

Reprinted with corrections 1981
Third Printing 1984
Fourth Printing with corrections 1989

No part of this book may be reproduced in any form by photostat, microfilm,
or any other means, without written permission from the publishers

## Library of Congress Catalog Card Number: 77-92823

ISBN Hardback: 0-12-391050-1
ISBN Paperback: 0-12-391056-0

## Printed in Great Britain by St Edmundsbury Press Limited

Bury St Edmunds, Suffolk

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Foreword
by G. MA THERON

## The distribution of ore grades within a deposit is of mixed character, being

partly structured and partly random. On one hand, the mineralizing process
has an overall structure and follows certain laws, either geological or
metallogenic; in particular, zones of rich and poor grades always exist, and
this is possible only if the variability of grades possesses a certain degree of
continuity. Depending upon the type of ore deposit, this degree of continuity
will be more or less marked, but it will always exist; mining engineers can
indeed be thankful for this fact because, otherwise, no local estimation and,
consequently, no selection would be possible. However, even though
mineralization is never so chaotic as to preclude all forms of forecasting, it is
never regular enough to allow the use of a deterministic forecasting technique . This is why a scientific (at least, simply realistic) estimation must
necessarily take into account both features - structure and randomness ~
inherent in any deposit. Since geologists stress the first of these two aspects,
and statisticians stress the second, I proposed, over 15 years ago, the name
geostatistics to designate the field which synthetizes these two features and
opens the way to the solution of problems of evaluation of mining deposits.
Practicians are well aware that such problems are numerous and often
very complex. They sometimes infer from this that the corresponding
mathematical theory is necessarily very complicated. Actually, they should
set their minds at ease, for this is luckily not the case . The mathematical
formalism necessary for geostatistics is simple and, in some cases, even trite.
The user of A. Journel and Ch. Huijbregts' book will come across nothing
more than variances and covariances, vectors and matrices; in short, only
linear calculations comparable to those which any computer user manipulates every day. After the preliminary effort of familiarizing themselves with
the notations, they will note, possibly to their surprise, that they can follow
all of the authors' steps withou t difficulty.
I might address a symmetric warning to statisticians. Those unfamiliar
with the complexity and specificity of mining problems might find this
v

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VI

FOREWORD

treatise banal and of little interest. I feel that they would be mistaken. At this
point, a comparison with geophysics could be instructive (besides, the very
term "geostatistics" was formed after the model of "geophysics" or of
"geochemistry' ').
If a theoretical physicist were to read a treatise on geophysics, he would
risk getting the disappointing impression that it is but a collection of
particular cases of the theories of potential and of wave propagation: a bit of
gravitation and magnetism, quite a lot of electromagnetism, acoustics, etc.,
nothing very original, really. And yet, without special training, that very
physicist, be he a Nobel prize winner, would admit to be unable to interpret
the results of a seismic prospection campaign.
In the same way, a specialist in mathematical statistics, having paged
through the present book, would not find much more than variances,
covariances, optimal linear estimators, etc., and would find no justification
in this for creating a separate field. However, let us submit to this specialist
the following problem, which is quite a standard one for the mining engineer
(in geostatistical terminology, it is the problem of the estimation of recoverable reserves).
"We have, at a certain depth, an interesting copper deposit. The geologists who have studied the deposit have arrived at certain conclusions about
its genesis and structure, which you will find in this report. The available
numerical data consist of a square grid of drill -holes 200 m apart. The core
sections were analysed metre by metre, and, thus, we have a total of some
5000 chemical analyses. We are considering open-pit selective mining. This
means that, during the mining of our approximately 200 Mm 3 deposit, which
has been mentally divided into 50 000 blocks 20 m x 20 m x 10m, we have
the choice of sending each newly excavated block either to waste , or to the
processing plant. Taking account of an economic pre-feasibility study, we
have decided that the blocks with a mean ore grade greater than x = 0.4%
Cu should be sent to the plant, and that the rest should be set aside with the
waste . This cut-oft grade x is subject to revision, since the market price of
copper as well as the mining and processing costs can vary. x should be
considered as a parameter, and the evaluations should be made under
various hypotheses, i.e., with x =0.3%,0.4%,0.5% , for example.
"Moreover, we must consider the fact that our block by block decision is
based, not on the unknown true block grade but on an estimator, this
estimator being based on information which will be much richer than that
now available (core drillings on a 200 ill X 200 m grid): at the time of
selection, we shall have the results of the chemical analyses of blast holes on
a 5 m x 5 m grid. Based upon this Anal information, we shall build an
estimator z 1 for each of our blocks, and we shall send this block to the plant
only if its estimated grade z i is greater or eq ual to the cut-off grade .r.

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FOREWORD

VII

## "We would like to know, as a function of x, the tonnage T(x)ofthe finally

selected blocks, and their mean grade m (x); this time, of course, we are
referring to the mean value of the true mean grades of the selected blocks,
and not of their estimated mean grades z": On top of this, the evaluation,
based upon the rather poor information we have now (drilling on a 200-m
spacing), will inevitably be erroneous. So we shall also want to know the
orders of magnitude of the errors committed in the evaluation of T(x) and
m (x), for example through their estimation variances."
In the very same way our Nobel prize physicist above admits being unable
to interpret the results of a seismic prospection campaign, I think that a
statistician not versed in mining practice would honestly have to admit his
inability to give a realistic solution to the above problem. Actually, this is a
rather simple and favourable case: only one metal, a well-explored deposit
with a regular sampling grid, no missing data points nor preferential or
biased data, etc. One should also note that the mere statement of the
problem itself takes up about a page. A statistician who is not familiar with
mining may well be discouraged before he can even get a good idea of the
problem at hand.
In fact , the difficulties are several. First of all, there is a violent contrast in
masses: even though the experimental data are numerous (5000 samples),
they represent but a minute part of the total mineralized mass (about SO
tonnes out of hundreds of millions). Then there is a volume (or support)
effect: the 20 m x 20 m x 10m blocks are not at all distributed (in the spatial
sense as well as in the statistical sense) in the same way as the kilogram-size
core sections. Finally, there is the fact that the criteria to be used for the
future selection of blocks will not be based on the true grades of the blocks,
nor upon the data now at hand, but on an intermediate level of information
(the blast-holes). One can, thus, understand why the need to solve such
problems would instigate, over the years, the creation of a field as specialized
as geophysics, with its own terminology and special techniques - a field
which carries the name of geostatistics.
Andre Journel and Charles Huijbregts, who have worked at the Centre de
Geostatistique at Fontainebleau since its creation in 1968, are surely among
the persons who have contributed the most to elevating this field to its
present level of practical and operational efficiency. They have personally
achieved studies and estimations upon more than 30 ore deposits of the most
diverse types. The vast experience which they thus accumulated, and the
dose and permanent contact they maintained with the everyday realities of
mining, account for the primarily operational and concrete nature of this
book. Here is a practical treatise , written by practising mining engineers, for
the benefit of other practicians, The authors have deliberately ~ and justly
so - simplified to the utmost the purely mathematical part of their expose,

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viii

FOREWORD

## while remammg as rigourous as possible. They have concentrated their

efforts upon two main points: first of all, clearly define the problems at hand,
and we have seen that this is no mean task; second, show how one should go
about solving them in practice. Practicians will appreciate this double
approach: clarifying the objectives and efficient use of the techniques.
Moreover, the mathematical formalism is laced with concrete comments
which make intuitive sense to the practician. Each time a new notion is
introduced, its definition is immediately followed by examples that illustrate
its meaning.
This book is structured according to the most typical problems which
occur in practice, progressing from the simplest to the most complicated:
structural analysis, estimation of in situ resources, estimation of recoverable
reserves, simulation of deposits. In each instance, several examples of
applications to actual deposits of the most diverse types are given: the
practician will be led by the hand through the labyrinth and forewarned of
traps and accidents which lurk at each corner.
In Chapter VII, devoted to conditional simulations, the strictly "linear"
code of the first six chapters is dropped. The Object of such a simulation is to
build a numerical model which has the same structure as the real deposit.
More precisely, the numerical model must have the same histogram, variogram and cross-correlations as the real deposit, and must coincide with the
experimental data values at the locations where they were taken. The
authors describe the novel technique developed by the Centre de Geostatistique. These conditional simulations will be used more and more often to
solve difficult non-linear problems which defy the use of any other technique
(e.g., forecasting the fluctuations of ore characteristics at the mill feed,
homogenization obtained after judicious ore blending and/or stockpiling,
etc.). Here, geostatistics no longer confines itself to predicting once and for
all the recoverable reserves, but helps in optimal planning and control of the
mining operations, at the daily, monthly or yearly scale. The last chapter
gives general indications about "non-linear" geostatistics, which is still
under elaboration.
In summary, A. Journel and Ch. Huijbregts' book constitutes a systematic
state-of-the-art account of geostatistics and traces its progress over the last
ten years (progress to which the authors have largely contributed). This
treatise, intended for practicians, is surely destined to become the reference
text for the field during the 1980's.

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Contents
Foreword by G. Matheron
Contents

xi

INTRODUCTION

## I. GEOSTATISTICS AND MINING APPLICATIONS

Summary
LA. The steps involved in proving a deposit
I.E. The geostatistical Ianguage
I.e. Some typical problems and their geostatistical
approach
II. THE THEORY OF REGIONALIZED VARIABLES
Summary
II.A. Regionalized variables and random functions
(I.B. Estimation variance
I1.c. Dispersion variance
I1.D . Regularization
I1.E. Calculation of the mean values y

7
7
7
10

18
26
26
27
48
61
77
95

## III. STRUCTURAL ANALYSIS

Summary
liLA. Nested structures and the nugget effect
I1LB. Models of variograms
IILe. The practice of structural analysis

148
148
148
161

## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

Summary
IV.A. Fitting models
IV.B. Hole effect and proportional effect
IV.C. Coregionalizations
IV.D. Anisotropies

236
236
236
247
253
262

IX

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195

CONTENTS

## IV.E. Various structural analyses

V. KRIGING AND THE ESTIMATION OF IN SITU
RESOURCES
Summary
V.A. Theory of kriging
V.B. The application of kriging
V.c. Variance of global estimation
VI. SELECTION AND ESTIMATiON OF RECOVERABLE
RESERVES
Summary
VI.A. In situ resources and recoverable reserves
VI.B. Permanence of distribution
VII. SIMULATION OF DEPOSITS
Summary
The objectives of simulation .
VILA. The theory of conditional simulations
VILB. The practice of simulations

271
293

303
303
304
343
410

444
444
444

466
491
491
491
494
517

## VIII. INTRODUCTION TO NON-LINEAR GEOSTATISTICS

Summary
Why non-linear geostatistics?
VIII.A. Kriging in terms of projections
VIII.B. Disjunctive kriging and its applications

555
555
556
557
573

Bibliography

581

591

595

## Index C. FORTRAN programs

596

Index D. Notations

597

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Introduction
This book is an attempt to synthetize the practical experience gained in
mining geostatistics by the researchers from the Centre de Morphologic
Mathernatique in Fontainebleau, France t and by mining engineers and
geologists around the world who were kind enough to let us know of their
experience. This book is designed for students and engineers who wish to
apply geostatistics to practical problems, and, for this reason, it has been
built around typical problems that arise in mining applications. Several
FORTRAN programs are given, and the techniques developed are illustrated by a large number of case studies. A modest knowledge of integral
calculus, linear algebra, statistics, notions of stochastic processes, variance,
covariance and distribution functions is required, as well as a geological
and mining background. With this background, the elements of mining
geostatistics can be understood and the approximations necessary for an
experimental approach can be accepted.
Definition and basic interpretation of geostatistics
Etymologically, the term "geostatistics" designates the statistical study of
natural phenomena. G. Matheron (1962) was the first to use this term
extensively, and his definition will be retained: "Geostatistics is the application of the formalism of random functions to the reconnaissance and
estimation of natural phenomena".
A natural phenomena can often be characterized by the distribution
in space of one or more variables, called "regionalized variables". The
distribution of grades in the three-dimensional space, for example,
t Namely and particularly 1. Serra, A. Marechal, J. Deraisme, D. Guibal. Special thanks to P.
A. Dowd who contributed extensively to the English version from the original French draft.

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MINING GEOSTATISTICS

characterizes at least some of the aspects of a mineralization. The distribution of altitude in a horizontal space characterizes a topographical
surface.
Let z (x) be the value of the variable z at the point x. The problem is to
represent the variability of the function z(x) in space (when x varies). This
representation will then be used to solve such problems as the estimation of
the value z(xo) at a point Xo at which no data arc available, or to estimate
the proportion of values z (x ) within a given field that are greater than a
The geostatistical solution consists of interpreting each value z (Xi) (z in
lower case) as a particular realization of a random variable Z(Xi) (Z in
upper case) at the point Xi. The set of these auto-correlated random
variables {Z (x), when x varies throughout the deposit or domain D}
constitutes a random function. The problem of characterizing the spatial
variability of z (x) then reduces to that of characterizing the correlations
between the various random variables Z (Xi), Z (Xi) which constitute the
random function {Z (x), xED}. This fundamental interpretation is justified
a posteriori if it results in coherent and acceptable solutions to the various
problems encountered in practice.

## Geostatistics, and the probabilistic approach in general, is particularly

suited to the study of natural phenomena. Among the many techniques and
mathematical models available to the practitioner of geomathematics, a
particular technique or approach will be chosen if, from experience, it is
known to be best suited to the problem concerned. A Fourier analysis will
suit one problem. while, in other cases, geostatistical tools are best suited;
in some cases, a mathematical formulation may even be rejected for simple
qualitative experience.
In the mining field, geostatistics provides a coherent set of probabilistic
techniques, which are available to each person (geologist, mining engineer,
metallurgist) involved in the mining project; the proper use of these techniques depends on these people. Just as the efficiency of the axe and the
saw depends on the arm of the lumberman, so the efficiency of any
geostatistical study depends on the liaison between the various people and
departments involved. The proper conclusions of an analysis of spatial
variability will be reached only if there is a close contact with the geologist
or, better still, if the analysis is carried out by the geologist himself. The
evaluation of recoverable reserves has no meaning unless technically
feasible processes of selection and mining are clearly defined; in other

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INTRODUCTION

words, the mining engineer must be involved. Outside the mining field,
geostatistics applied to forestry requires the ability to interpret spatial
correlation between the densities of growth of different species of trees;
geostatistics applied to bathymetry must take into account all the problems
related to the localization of the hydrographic survey ship at sea. Thus, a
book on geostatistics cannot be dissociated from its practical context, in
this case mining.
This instrumental nature of geostatistics both opposes and completes the
explanatory aspects of the standard, deterministic, geological approach .
For example, even if a metallogenic study is sometimes successful in
explaining the overall genesis of a given deposit , such a deterministic
approach wil1 be clearly insufficient on a smaller scale, such as that of metal
concentrations. In such a case, it is said that the grades are "randomly"
distributed within the defined genetic outline.
But nothing is more precise than this "random" distribution. It will differ
from a gold placer deposit to a porphyry-copper deposit, and the random
distribution of the grades within the deposit will condition its survey and its
selective mining, which represent considerable investments. On the smaller
scale of mining, the probabilistic approach can relay the deterministic and
genetic approaches by which the deposit was delineated on a larger scale.
Our lumberman is not a botanist; similarly, the mining geostatistician
accepts the reality imposed by nature, and is content to study the
consequences of this reality on the mining project he must prepare. A
probabilistic approach can, of course, be guided by genetic considerations,
and the best mining geostatistician is the one who is able to combine his
geological knowledge and his technical mining skill with a good use of the
probabilistic language.

## Plan of the book

The acceptance of geostatistics, and mining geostatistics in particular, rests
on the coherence and effectiveness of the solutions it provides to the
various problems encountered in practice. Thus, the primary aim of this
book is to convince the various people involved in a mining project of the
practical interest of geostatistics, An engineer will not be convinced in the
same manner as a mathematician or a researcher; he will first want to know
how the techniques have fared in practice. For this reason, before beginning any theoretical development, a practical presentation of geostatistics,
by way of its mining applications, is given in Chapter 1. This chapter can be
read separately from the rest of the book, and is intended as a resume for
readers who are pressed for time, and anxious to make a quick decision on
the usefulness of the proposed techniques. In the remainder of the book,

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MINING GEOSTATISTICS

## each new concept is introduced by a practical example drawn from the

study of a real deposit.
The theoretical structure of the book begins with Chapter It which
presents the theory of regionalized variables and the main tool of geostatistics - the variogram - and the two variances of estimation and dispersion.
Any geostatistical study begins with a structural analysis, which consists
of modelling the in situ spatial variabilities of the parameters studied, e.g.,
grades, thicknesses, accumulations. Chapters III and IV show how a structural analysis is carried out in practice from the construction of variograms
to their interpretation and fitting by models. Both chapters insist on the
"skilled" nature of the structural analysis. Even though many computer
programs have been written for this purpose, a structural analysis must
never be carried out blindly; it must integrate the qualitative experience of
the geologist and carry it through to the subsequent resource-reserves
estimation and the eventual planning of the mining project.
It is obvious that a massive porphyry-copper deposit cannot be estimated
in the same way as a sedimentary phosphate deposit. The estimation
procedure must take into account both the structure of the spatial variability peculiar to each deposit, and the particular manner in which the
deposit was surveyed. It is only by taking these peculiarities into account
that a confidence interval can be ascribed to each estimated value. Kriging
takes these factors into account, and is presented in Chapter V, along with
the geostatistical approach to the estimation of the in situ resources.
The next step is to evaluate the proportion of these in situ resources that
can be recovered within a given technological and economical framework .
This estimation of recoverable reserves must take the nature of the proposed selection and mining methods into account. Chapter VI presents the
geostatistical formalization of these problems of selection and estimation
of recoverable reserves.
Since geostatistics interprets the regionalized variable z (x) as a particular realization of a random function Z (x), the possi bility arises of drawing
other realizations of this same random function, in much the same way as
casting a die several times. This new, simulated realization represents a
numerical model of the real spatial variability z (x). Within certain limits,
the consequences of various planned mining methods can then be
examined by simulating these methods on the numerical model. The techniques for simulating deposits and the practical uses of these simulations
are given in Chapter VII.
Each chapter begins with a summary of its contents, so as to allow the
reader to sort out items of particular interest. For this purpose, the reader
can also make profitable use of the indexes of geostatistical concepts,

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INTRODUCTION

## quoted deposits and FORTRAN programs. At a first reading, the reader is

advised to omit the various theoretical demonstrations. These demonstrations are necessary for the logical coherence of the book, but not for a
preliminary overall understanding of geostatistics.
First notations
For the uninitiated, one of the main difficulties of geostatrsncs is the
notation. These notations are precisely defined as they enter into the work,
and are summarized in Index D. However, it may prove helpful at this
stage to introduce the following basic notations.
A deposit is always a regionalized phenomenon in the three-dimensional space. To simplify notation, the coordinates (xu, xv, x.,) of a point are
usually denoted x. Thus, the integral (extended over the three-dimensional
space) of the function [(Xu., xv, x.,) is written as

+OO

-co

I(x) dx =

-co

dx,

f+>
-:x;l

dx,

r.
-co

[(x"'

Xli'

x.,) dx w

with a modulus

## and direction Cf. Note that a direction in a three-dimensional space is

specified by two angles, its longitude and its latitude.
Lower-case letters, z or y, are reserved for real values as, for example,
the grade z(x) at point x or the density y (x) at the same point. The
corresponding upper-case letters, Z or Y, denote the interpretation of this
real value as a random variable or function, as, for example, the random
function Z(x) with expectation E{Z(x)}. The real value z(x) is said to be a
particular realization of the random variable or function Z (x).
An estimated value is distinguished from a real value. known or
unknown, by an asterisk *. Thus, z *(x) represents the estimated value of
z(x), and Z*(x) represents the interpretation of this estimator as a random
variable or function.

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Applications
SUMMARY

## This chapter is intended to be both an introduction to, and an overview of,

geostatistics as applied to the mining industry. Each successive problem encountered from the exploration phase to the routine operating phase is presented along
with its geostatistical approach. The problems of characterizing variability structures of variables of interest within the deposit are solved with the "variogram",
Optimal local estimates with corresponding confidence intervals are found by
"kriging". The choice of various working techniques (machinery, need for stockpiling or blending) can be facilitated by simulating them on a geostatistically
determined numerical model of the deposit.
The main steps in proving a deposit are outlined schematically in section LA
along with some typical evaluation problems arising from such steps.
Section LB introduces the geostatistical terminology from a practical and intuitive point of view. Such key notions as regionalized variables, variogram, estimation and dispersion variances, are presented.
The geostatistical approach to several of the problems encountered in section LA
is given in section I.e, which also outlines the sequence of steps in a geostatistical
study.

## I.A. THE STEPS INVOLVED IN PROVING A DEPOSIT

In this section, we shall consider the various steps involved in proving a
deposit from prospection to production. The sequence of these steps and
the critical nature of certain problems will obviously depend on the type of
deposit.
7

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MINING GEOSTATISTICS

Geological surveys
A preliminary survey of a province or favourable metallogenic environment, by such various means as geological mapping, geochemistry or
geophysics, will usually result in the location of a certain number of
prospects. If one of these prospects locates a mineralized zone, the next
step is to determine its value and extent by a more detailed survey involving, for example, drilling, trenches or even, sometimes, drives. The
locations of these samples are generally not regular, but are based on
geological considerations such as the need to verify the existence of the
mineralization and to provide a qualitative description of it (e.g.,
mineralogy, tectonics, metallogenic type). The results of such sampling
may be-a proper basis on which to plan further reconnaissance, but they are
not entirely suited for quantitative estimations of tonnage and yardage.
The first estimations will tend to be more qualitative than quantitative , as
will be the- geologist's image of the deposit.

## Systematic survey on a large grid

The second stage of the survey usually consists of drilling on a large and
more or less regular grid. In general, this grid will cover what is considered
to be the mineralized zone and should be sufficient to evaluate the overall
in situ resources (tonnages and mean grades). If promising results are
obtained from this second stage, the survey may be extended to some in-fill
drilling or by drilling in adjacent zones. At this stage, the geologist and
project engineer are faced with two fundamental quantitative problems.
The first one is how to deduce the required global estimations from the
available information, which consists of qualitative geological hypotheses
and quantitative data from core samples, trenches, pits, etc. In general,
estimations of a geometric nature such as mineralized surface and yardage
will make more use of qualitative information than estimations of a
"concentration" nature such as mean grades and densities.
The second problem is to express the confidence that can be given to
these estimations. In practice, it is common to classify the in situ resources
in different categories such as, for example, those recommended by the US
Bureau of Mines: measured ore, indicated ore and inferred ore. The
boundaries between these categories do not always appear clearly. The
geologist knows from experience that the precision of an estimation, such
as mean grade, does not only depend on the number of the available assays
but also on the in situ variability of the grade in the deposit and the manner
in which this variability has been sampled. He also knows that the concepts
of continuity and zone of influence, which are used in the evaluation of

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## 1. GEOSTATISTICS AND MINING APPLICAnONS

resources, depend on the particular deposit and the type of are considered.
And, finally, he knows that for a given ore the variability of assays of core
samples, cuttings or channel samples can be markedly different.
In this chapter, it will be shown how geostatistics takes these various
concepts into account by means of the variogram and structural analysis.
Sampling on a small grid

Once the deposit has been deemed economically mineable, the next step is
to define the technological and economical framework for its exploitation.
The total in situ resources of a deposit are seldom entirely mineable,
both for technological reasons (depth, accessibility) and for economic
reasons (mining cost, cut-off grade). To define recoverable reserves within
a given technological and economic context, or, conversely, to define the
type of mining which will result in maximum profit from the deposit it is
necessary to have a detailed knowledge of the characteristics of this
deposit. These characteristics include the spatial distribution of rich- and
poor-grade zones, the variability of thicknesses and overburden, the
various grade correlations (economic metals and impurities).
To determine these characteristics, it is usual to sample on a small grid,
at least over the first production zone. The results of this sampling
campaign are then used to construct block estimations within this zone.
These local estimations set the usual problem of determining estimators
and evaluating the resulting estimation error. There will also be problems
associated with the use of these local estimations: the evaluation of
recoverable reserves depends on whether the proposed mining method is
block-caving with selection units of the order of 100 m x 100 ill X 100 m , or
open-pit mining with selection units of the order of 10 m x 10 ill xl 0 m.
From experience, the mining engineer knows that, after mining, the
actual product will differ from the estimations of the geologist. In general,
he knows that tonnage is underestimated and, more importantly, quality is
overestimated. In practice, this usually results in a more or less empirical
"smudge factor" being applied to the estimation of recoverable reserves,
which may reduce the estimated quantity of recovered metal from 5 to
20%. This is not a very satisfactory procedure, because it does not explain
the reasons for these systematic biases between the geologist's estimations
and actual production; in addition, it is far too imprecise when evaluating a
marginally profitable new deposit.
Given the size of present day mining projects and the cost of their
investments, the risks of such empirical methods are no longer acceptable,
and it is imperative to set and rigorously solve such problems as those given
below.

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MINING GEOSTATISTICS
(i) The definition of the most precise local estimator and its confidence

## interval, taking account of the peculiarities of the deposit and the

ore concerned. It is obvious that a gold nugget deposit cannot be
evaluated in the same way as a sedimentary phosphate deposit.
(ii] The distinction between in situ resources and mineable reserves. It
should be possible to model rigorously the effect of the mining
method on the recovery of the in situ resources.
(iii) The definition of the amount of data (drill cores, exploration drives)
necessary for a reliable estimation of these recoverable reserves.
Mine planning

## Once the estimations of resources and recoverable reserves have been

made, the next step is to plan the mining project itself. This involves
choosing mining appliances, type of haulage, requirements of stockpiling
and blending, mill planning, etc. Some of these choices are conditional, in
all or part, upon the structures of the variabilities of the are characteristics,
either in situ or at any of the various stages of mining, haulage, stockpiling
and mill feeding. It will be shown later how geostatistics can be used to
provide representations of these variabilities by means of numerical models
of the, deposit. The effects of various extraction and blending processes can
then"be tested by simulating them on these numerical models.

## LB. THE GEOSTATISTICAL LANGUAGE

I.B.1 Regionalized variable and random function

## A mineralized phenomenon can be characterized by the spatial distribution

of a certain number of measurable quantities called "regionalized variables". Examples are the distribution of grades in the three-dimensional
space, the distribution of vertical thicknesses of a sedimentary bed in the
horizontal space, and the distribution of market price of a metal in time.
Geostatistical theory is based on the observation that the variabilities of
all regionalized variables have a particular structure. The grades z (x) and
z (x + h) of a given metal at points x and x + h are auto-correlated; this
auto-correlation depends on both the vector h (in modulus and direction)
which separates the two points, and on the particular mineralization
considered. The variability of gold grades in a placer deposit will differ
from that of gold grades in a massive deposit. The independence of the two
grades z (x) and z (x + h) beyond a certain distance h is simply a particular
case of auto-correlation and is treated in the same way.

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11

## The direct study of the mathematical function z (x) is excluded because

its spatial variability is usually extremely erratic, with all kinds of discontinuities and anisotropies. It is equally unacceptable to interpret all the
numerical values z (x), z (x') as independent realizations of the same
random variable Z, because this interpretation takes no account of the
spatial auto-correlation between two neighbouring values z(x) and z(x +
h). Both the random and structured aspects of the regionalized variable are
expressed by the probabilistic language of "random functions".
A random function Z(x) can be seen as a set of random variables Z(Xi)
defined at each point Xi of the deposit D: Z(x)={Z(Xi)' \;fXiED}. The
random variables Z (x,) are correlated and this correlation depends on both
the vector h (modulus and direction) separating two points Xi and Xi + h
and the nature of the variable considered . At any point Xi, the true grade
z(Xj), analysed over a core sample, for example, is interpreted as one
particular realization of the random variable Z (Xi)' Similarly, the set of
true grades {Z(Xi), \iXi E D} defining the deposit or the zone D is interpreted as one particular realization of the random function {Z (Xi), 'rIXi E
D}.
The next section will deal with the operational aspect of this formalization by describing how the spatial variability of the regionalized variable
z(x) can be characterized.
1.8.2. The variog ram

Consider two numerical values z (x) and z (x + h), at two points x and x + h
separated by the vector h. The variability between these two quantities is
characterized by the variogram function 2 y(x, h), which is defined as the
expectation of the random variable [Z(x)-Z(x+h)J 2 , i.e.,
2)'(x, h)

= E{[Z(x)-Z(x +h)J2}.

(1.1 )

## In all generality, this variogram 2'}'(x, h) is a function of both the point x

and the vector h. Thus, the estimation of this variogram requires several
realizations, [zk(x),zdx+h)], [zk(x),zdx+h)], ... , [Zk"(X),Zk,,(x+h)),
of the pair of random variables [Z(x), Z (x + h)]. Now, in practice, at least
in mining applications, only one such realization [z(x), z(x+h)] is available and this is the actual measured couple of values at points X and X + h.
To overcome this problem, the intrinsic hypothesis is introduced. This hypothesis is that the variograrn function 2y(x, h) depends only on the
separation vector h (modulus and direction) and not on the location x. It is
then possible to estimate the variogram 2y(h) from the available data: an
estimator 2'}'*(h) is the arithmetic mean of the squared differences between

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MINING GEOSTATISTICS

two experimental measures [z (X;), Z(Xi + h)] at any two points separated by
the vector h ; i.e.,
(1.2)
where N(h) is the number of experimental pairs [z (Xi), Z(X; + h)] of data
separated by the vector h.
Note that the intrinsic hypothesis is simply the hypothesis of secondorder stationarity of the differences [2 (x) - 2 (x + h)]. In physical terms,
this means that, within the zone D, the structure of the variability between
two grades z (x ) and z (x + h) is constant and, thus, independent of x; this
would be true, for instance, if the mineralization within D were homogeneous.
The intrinsic hypothesis is not as strong as the hypothesis of stationarity
of the random function 2 (x) itself. In practice, the intrinsic hypothesis can
be reduced, for example, by limiting it to a given locality; in such a case, the
function y(x, h) can be expressed as two terms: )I(x, h) = I(x) . )lo(h),
where yo(h) is an intrinsic variability constant over the zone D and [(x)
characterizes an intensity of variability which depends on the locality x, d.
the concept of proportional effect, treated in section IILB.6.

## The variogram as a tool for structural analysis

"Structural analysis" is the name given to the procedure of characterizing
the structures of the spatial distribution of the variables considered (e.g.,
grades, thicknesses, accumulations). It is the first and indispensable step of
any geostatistical study. The variagram model acts as a quantified summary
of all the available structural information, which is then channelled into the
various procedures of resource and reserve evaluation. Thus y(h) can be
regarded as injecting extensive geological experience into the sequence of
studies involved in a mining project.
In the proving of any mining project, it is imperative to have good
communication between all the departments involved (geology, mining
techniques and economy, processing). It is a waste, for instance, to have a
geologist carry out an excellent study of geological structures, only to be
followed by a project engineer evaluating resources by such arbitrary
procedures as polygons of influence or inverse-squared distance weighting.
Such evaluations are independent of the deposit and its geology, and make
no distinction, for example, between a gold placer deposit and a porphyrycopper deposit. The variogram model 2y(h) can help in this essential
communication between the geologist and the project engineer.

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## I. GEOSTATISTICS AND MINING APPLICATIONS

13

Continuity
In the definition of the variograrn 2y(h), h represents a vector of modulus Ih I
and direction a. Consider a particular direction a. Beginning at the origin,
y(O) = 0, the variogram increases in general with the modulus Ih I. This is
simply an expression of the fact that, on average, the difference between
two grades taken at two different points increases as the distance Ih I
between them increases. The manner in which this variogram increases for
small values of Ih I characterizes the degree of spatial continuity of the
variable studied. Figure 11.10 shows profiles of four piezometer readings,
giving the height of a water table, measured at four different points
(denoted by 3, 4,33 and 18), as a function of time. The four corresponding
semi-variograms have markedly different shapes, from which a geostatistician would be able to deduce the degree of continuity of the four
piezometer profiles.
Anisotropies and zone of influence
In a given direction a, the variogram may become stable beyond some
distance Ih I= a called the range, cf. Fig. 1.1. Beyond this distance a, the
mear. square deviation between two quantities z(x) and z(x +h) no longer
depends on the distance Ih I between them and the two quantities are no
longer correlated. The range a gives a precise meaning to the intuitive
concept of the zone of influence of a sample z (x). However, there is no
reason for the range to be the same in all directions a of the space. In Fig.
1.1, for instance, the vertical range characterizing the mean vertical
dimension of the mineralized lenses differs from the horizontal ranges. For
a given distance Ihl, the horizontal variograrn presents a weaker variability
than the vertical variograrn: this expresses the horizontal sedimentary
character of the phenomenon considered.

- - - Horizontal

C7
Ihl

FIG. 1.1.

## Structural anisotropy expressed by the variogram.

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MINING GEOSTATISTICS

14

## 1.8.3. The estimation variance

The variogram defined in (1.1) can be viewed as an estimation variance, the
variance of the error committed when the grade at point x is estimated by
the grade at point x + h. From this elementary estimation variance, 2y(h) =
E{[Z (x) - Z (x + h )]2}, geostatistical techniques can be used to deduce the
variance of estimation of a mean grade Zo by another mean grade Zu. This
estimation variance is expressed as
(1.3)

The mean grades Z; and Zv can be defined on any supports, e.g., V may
represent a mining block centred on the point y, and v the set of N drill
cores centred on points {z, i = 1 to N}. The mean grades are then defined
by
1 N
I(
Zv = V )
Z(x)dx and z; = - I Z(x;).
N ;=1
y v(y)

The notation H V, v), for example, represents the mean value of the
elementary semi-variogram function y(h) when the end-point x of the
vector h = (x - x') describes the support V and the other end-point x'
independently describes the support v, i.e.,
1
HV, v)=NV

,IN
,~l

y(x;-x)dx.

v(y)

The general formula (1.3), shows that the estimation variance, i.e., the
quality of the estimation, depends on all four of the following.
(i) The relative distances between the block V to be estimated and the
information v used to estimate it. This is embodied in the term
y(V, v).
(ii) The size and geometry of the block V to be estimated. This is
embodied in the term y(V, V).
(iii) The quantity and spatial arrangement of the information v, which is
embodied in the term
v).
(iv) The degree of continuity of the phenomenon under study, which is
conveyed by its characteristic serni-variogram y(h).

-rev,

Thus, the theoretical formula (1.3) expresses the various concepts that
we intuitively know should determine the quality of an estimation.
Confidence interval

## The estimation variance itself is not enough to establish a confidence

interval for the proposed estimate, the distribution of the errors must also

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## be known. In mining applications however, the standard 95%

Gaussian confidence interval [ 2ifEl is used, where (T~ is the estimation
variance.
Experimental observation of a large number of experimental histograms
of estimation errors has shown that the Gaussian distribution tends to
underestimate the proportion of low errors (within O"E/2) and of high
errors (outside the range 30"E)' However, the standard Gaussian
confidence interval r20"E] in general correctly estimates the 95 %
experimental confidence interval. Figure II.17 shows the histogram of
estimation errors of 397 mean block, grades from the Chuquicamata
deposit in Chile. The arithmetic mean of the errors is O.Ol % Cu for a
mean grade of 2% Cu. The experimental variance of the 397 errors is
equal to 0.273 (% CU)2, which is close to the geostatistically predicted
value of 0.261. The predicted 95% confidence interval is, according to the
Gaussian standard, [2.J(O261)] = [102% CuI. In fact, 96% of the
observed errors fall within this interval.
Change of support (volume)

## The elementary variogram 2y(h), estimated from experimental data, is

defined on the support of these data, e.g., core samples of given crosssection and length. By means of formula (1.3), the variograrn 2yv(h)
defined on any other support V can be deduced from the variogram 21'(h),
e.g., V may be the support of the mining block. In fact,
2yv(h)= E{[Zv(x) - Zv(x + h )]2}

## appears as the variance of estimation of the mean grade of the block V

centred on point x by the mean grade of the block V centred on the point
x + h. This formalism for the change of support has great practical applications. Since future mining will be carried out on the basis of blocks and
not drill cores, it is of great benefit to be able to evaluate the spatial
variability of the mean grades of these blocks by means of the variogram
2Yv(h) defined on the block support.
1.8.4. The dispersion variance

There are two dispersion phenomena well known to the mining engineer.
The first is that the dispersion around their mean value of a set of data
collected within a domain V increases with the dimension of V. This is a
logical consequence of the existence of spatial correlations: the smaller V,
the closer the data and, thus, the closer their values. The second is that the
dispersion within a fixed domain V decreases as the support v on which

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MINING GEOSTATISTICS

each datum is defined increases: the mean grades of mining blocks are less
dispersed than the mean grades of core samples.
These two phenomena are expressed in the geostatistical concept of
dispersion variance. Let V be a domain consisting of N units with the same
support v. If the N grades of these units are known, their variance can be
calculated. The dispersion variance of the grades of units v within V,
2
written D (v/ V), is simply the probable value of this experimental variance and is calculated by means of the elementary variogram 2y(h)
through the formula
D 2(v/V)= i'(V, V)-y(v, v).

(IA)

## Taking the generally increasing character of the variogram into account,

it can be seen that D 2(v/V) increases with the dimension of V and.
decreases with the dimensions of v.
This formula can be used, for example, to calculate the dispersion
variance of the mean grades of production units when the size of the units
(v) varies or when the interval of time considered varies (i.e., V varies).
1.8.5. Coregionalizations

The concepts of variogram and various variances, which so far have been
defined for the regionalization of one variable, can be generalized to spatial
coregionalizations of several variables. In a lead-zinc deposit, for example,
the regionalizations of the grades in lead, ZI (x), and in zinc Zz(x), are
characterized by their respective variograms 2YI(h) and 2Y2(h). However,
these two variabilities are not independent of each other and we can define
a cross-variograrn for lead and zinc:

[

"( 1

"(IZ],

"(12

"(2

## called the coregionalization matrix. This coregionalization matrix can then

and zinc data.
Sometimes, two different types of data relating to the same metal can be
used in block estimation. One set may be the precise data taken from core
samples, while the other may be imprecise data taken from cuttings and
blast-holes. A study of coregionalization of these two types of data will
determine the proper weights to be assigned to them during evaluation.

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## The regionalized variable z (x), which is the subject of the geostatistical

structural analysis, is interpreted as one possible realization of the random
function Z (x). It is possible to generate other realizations zs(x) of this
random function Z(x) by the process of simulation. This simulated realization zs(x), will differ from point to point from the real realization z (x),
but, statistically, will show the same structure of variability (same histograms and variograms); moreover, at the experimental data locations Xi, the
simulated values will be equal to the true values, i.e., Zs(Xi) = Z(Xi), "tIx; E
data set. The simulated and actual realization will, thus, have the same
concentrations of rich and poor values at the same locations. The two
realizations can be viewed as two possible variants of the same genetic
phenomenon characterized by the random function Z (x ). The simulated
realization, however, has the advantage of being known at all points and
not only at the data points Xi.
Various methods of extraction, stoping, blending, stockpiling, etc., can
then be simulated on this numerical model {z.x), xED} of the deposit or
zone D. Using feedback, the effect of these simulated processes on recovery, fluctuations of ore characteristics, and on the running of the mine and

100

50

Rain - go uge

## FIG. 1.2. Locations of the rain gauges.

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MINING GEOSTATISTICS

## mill in general can be used to modify certain parameters of the mining

project.
The techniques for the simulation of a regionalization z (x) can be
extended to simulations of coregionalizations, i.e., to the simulations of the
simultaneous spatial distributions of the various characteristic variables
(thicknesses, grades, density) of a mineralization.

variant I

50

Variant 2

Voria nt 3

## FIG 1.3. Three simulated rainfalls.

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19

## This was a study of the regionalization of water heights (measured in rain

gauges) after rain showers, but it could equally well apply to the regionalization of the vertical thickness of a sedimentary bed.
Figure 1.2 shows the location of rain gauges in the Kadjerneur basin in
Tchad (Central Africa). After one particular rainfall, the height of water in
each rain gauge was measured. These data were then used to construct an
experimental semi-variogram to which a linear model was fitted. Figure 1.3
shows three simulated rainfalls, each of which has the same linear variogram as the real rainfall and the same water heights at the actual rain gauge
locations. At points other than these rain gauge locations, the three simulations can differ markedly from each other. This is noticeably evident in
the extreme right-hand corner of the basin, where there are no rain
gauges.

APPROACH

## As previously mentioned, the initial geological surveys of a deposit result

in essentially qualitative information, and any quantitative information will
generally be on a non-systematic grid. In general, it is not possible to
construct sufficiently representative histograms and variograms on the
basis of such information. However, even if it is of a purely qualitative
nature (e.g., structural geology) this information can be used during the
geostatistical structural analysis, which will later be carried out on the data
obtained from the first systematic sampling campaign. The geology is used
as a guide in the construction and interpretation of the various experimental variograms. These variograms then channel this geological knowledge
into the various studies involved in a mining project.
l.C.}. Global estimation

## Once the first systematic campaign is completed, it is usual to proceed

to estimations of global in situ resources, such as tonnages of are and
overburden, quantities of metals and mean grades. These estimations must
also include confidence intervals which will determine the point at which
the in situ resources are sufficiently well estimated to proceed to the next
stage of evaluation, i.e., that of the recoverable reserves.
At this stage of global evaluation of resources, there are no specific
geostatistical methods for determining estimators. For example, a
mineralized surface may be estimated by interpolation between positive

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MINING GEOSTATISTICS

## and negative drill-holes. Mineralized thickness or mean grade may be

estimated by the arithmetic mean of the positive data on a pseudo-regular
grid. or by any of the traditional methods of extension of profiles or
cross-sections. However, through the estimation variance. geostatistics can
quantify the reliability of each of these traditional methods and so provide
a criterion for choice between them. The estimation variance can also be
used to objectively classify resources and to predict the number, location
and distribution of additional information required to improve the quality
of estimations. e.g., to move a given tonnage from the "possible" to the
"proved" category, d. A. G. Royle (1977).
I.C.2. Local estimation

Once the deposit has been deemed globally mineable, the next phase is
local, block by block estimation. This local estimation gives an idea of the
spatial distribution of the in situ resources. which is necessary for the
evaluation of the recoverable reserves. This local estimation is usually
carried out on the basis of samples obtained from a smaller sampling grid.
e.g., through in-fill drilling. The estimation variance of a given panel, d.
formula (1.3), can be calculated as a function of the grid size and, thus.
indicates the optimal sampling density to be used for the local estimation.
Once this second, denser sampling campaign is completed, the next
problem is to determine the best possible estimate of each block. This
amounts to determining the appropriate weight to be assigned to each
datum value, whether inside or outside the panel. The determination of
these weights must take into account the nature (core samples, channel
samples) and spatial location of each datum with respect to the block and
the other data. It must also take into account the degree of spatial
continuity of the variable concerned. expressed in the various features of
the variograrn, 2y(lhl. a).
For example, suppose that the mean grade Zv of the block V is to be
estimated from a given configuration of N data Z; as shown on Fig. IA.
The geostatistical procedure of "kriging" considers as estimator Z t of this
mean grade, a linear combination of the N data values:
N

zt= L
j~

AjZj

(1.6)

## The kriging system then determines the N weights {Ai, i = 1 to N} such

that the following hold.
(i) The estimation is unbiased. This means that) on average, the error
will be zero for a large number of such estimations, i.e .
E{Zv -zt} =

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21

## (ii) The variance of estimation u~ = E{[Zv - zt ]2} is minimal. This

minimization of the estimation variance amounts to choosing the
(linear) estimator with the smallest standard Gaussian 95%
confidence interval (2UE).
Consider the example shown on Fig. 1.4, in which structural isotropy is
assumed, i.e., the variability is the same in all directions. Provided that all
the N = 8 data are of the same nature, and all defined on the same support,
kriging gives the following more or less intuitive results:
(i) The symmetry relations, "\2 = "\3, As = "\6;
(ii) The inequalities, Al ~Ai, Vi '1'-1; A7 :os ;\4, i.e., the datum Z4 screens
the influence of Z7; As ~ ;\4, i.e., part' of the influence of Z4 is
transferred to the neighbouring data Z5, Z6 and Z7.

## I.C.3. Selection and recoverable reserves

The terms "selection" and "recoverable reserves" have little meaning
unless they are used within the context of precisely defined technological
conditions for recovery, From experience, the mining engineer knows that
the quantity and quality of recovery improves as:
(i) The mining selection unit becomes smaller (although this intuition is
true only for highly selective mining);
(ii) The information available at the time of selection improves, to allow
better discrimination between rich and poor units.

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MINING GEOSTATISTICS

following way.

## 1. Recovery depends on the support of the selection unit

Consider the two histograms shown on Fig. 1.5(a). The first one represents
the distribution of the mean grades of 5 m x 5 m x 5 m units v in the deposit
D. These units may represent, for example, selective underground mining
by small blocks. The second histogram represents the distribution of the
mean grades of 50 m x 50 m x 50 m units V in the same deposit D, which
may represent, for example, mining by block-caving. It is obvious that the
first histogram (v) will be much more dispersed than the second one (V),
(0 )

FIG 1.5. Recoveries estimated from cut-off grades applied to histograms. (a)
Influence of the support; (b) influence of the standard of information.

and that the same cut-off grade Zo applied to both histograms will result in
very different recoveries. These different dispersions are characterized by
2(V/D)
the two dispersion variances D\v/D) and D
with D\vID
D 2( VI D). By assuming some hypotheses related to the conservation of the
type of distribution law, it is possible to deduce these two histograms from
the experimental histogram defined on the support C of core sample
lengths. For any given selection unit size, the estimated histogram can then
be used to determine the recoveries corresponding to different cut-off
grades zoo These recoveries are then used to build the so-called "gradetonnage" curves.
2. Recovery depends on the standard of available information
The preceding histograms concerned the real grades of units v and V. In
practice, however, it seldom happens that the true grades are available at

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## L GEOSTATISTICS AND MINING APPLICATIONS

23

the time of selection. Selection is made on the basis of the estimated grades
of these units, which will inevitably result in a recovery different from the
optimum recovery based on the true grades.
Figure 1.5(b) shows the histogram of the true grades of the units V
superimposed on the histogram of their estimated values. As actual selection will be made on the basis of these estimated values, the proportion of
recoverable units V must be read from the histogram of the estimated
grades (dotted area on Fig. 1.5(b)) and not from the histogram of the true
grades (hatched area on Fig. 1.5(a)).
If the estimated grades are obtained by kriging, then the dispersion
variance D 2 ( V* I D) of the histogram of the estimated grades can be calculated from a relation known as the "smoothing effect" due to kriging:
(I. 7)

## where D 2 ( VID) is the dispersion variance of the true grades given by

formula (1.4), and (T~ is the estimation variance given by the kriging
system.
2
Knowing the two dispersion variances, D 2 ( V* / D) and D ( VI D), the
estimates of the two histograms on Fig. 1.5(b) can be constructed and the
effect on recovery of selection, based on estimated grades rather than real
ones, can be evaluated. A balance can then be made between the profit in
improving the recovery and the cost of increased and better information
for selection.
The preceding recoveries, estimated from the histograms, take no
account of the spatial locations of the recovered units. It is weI! known that
a rich unit (with a grade greater than the cut-off zo) in a poor zone will not
be mined, and a number of poor units in a rich zone will be mined. Thus, a
precise evaluation of recoverable reserves should take into account all the
technological parameters of the envisaged mining method, e.g., pit
contours, conditions and cost of accessibility, haulage, etc. The geostatistical solution to this problem is to work on a numerical model of the deposit.
This model will have the same spatial distribution of grades, and, thus, can
be considered as a deposit similar to the real one. The simulation of the
foreseen mining process on this model will, thus, provide the required
evaluation of recoveries and "smudge" factor.
I.CA. Working out the mining project

Once the general framework of the mining method has been defined and
the corresponding recoverable reserves have been estimated, the next
problem is a more precise determination of the parameters of the mining

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24

## project. Some of these parameters depend on the variability of the are

characteristics. Examples are as follows:
(i) The number and size of simultaneous stapes, and whether or not
blending of the production from these stapes is required. These
choices will of course affect the trucking equipment.
(ii) The need for stockpiling, for control or blending, and the characteristics of the stockpiles.
(iii) The flexibility of the mill, to allow absorption of the fluctuations in
quantity and quality of the recovered are.
The choice of such parameters involves a study of the fluctuations over
time of the mined ore after each of the mining phases, trucking and
stockpiling. The structure of the in situ variabilities will be partially
destroyed after mining and haulage, and the modified structures must be
known to determine the blending procedures necessary to meet the mill
feed constraints. It may be possible to make such choices from experience
gained on similar already mined deposits, but the analogy can sometimes
be limited.
When there is no such "similar" deposit, the geostatistical approach
consists in building a numerical model of the deposit on which the various
processes of mining, trucking and blending can be simulated: their
consequences on the production and the mill feed can then be observed,
and, by feedback, the parameters adopted for the various mining processes
can be corrected. These simulations are, of course, only as reliable as the
numerical model of the deposit. This model must faithfully reproduce all the
features of the spatial distribution of the characteristic variables (grades,
thicknesses, accumulations, but also morphology and tectonics) of the real
deposit. The better the characteristics of variability and morphology of the
real deposit are known, the better the numerical model will be. Modelling
does not create information, it can only reproduce the available structural
information in a simple numerical manner, easy to handle.

## l.e.5. Mine planning in the routine phase

Once routine mining of the deposit has begun, the information about the
deposit becomes increasingly more detailed, both qualitatively (e.g., the
working faces can be seen) and quantitatively (data from blast-holes, mill
feed grades, etc.), Using these data, the estimations can be constantly
updated to improve overall operation. These estimations may be the basis
for short- or long-term mine planning. For example, for a given period of
production, these estimations may be used to determine hanging-wall and

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## foot-wall outlines in a subhorizontal sedimentary deposit, or to carry out

selection that satisfies the various constraints on tonnage and quality.
Mine-mill production schedules can be made more reliable by specifying
confidence intervals for the mean grades of ore sent for milling.

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II

Variables
SUMMARY

## This fundamental theoretical chapter gives the conceptual basis of geostatistics,

then introduces, successively, the three main geostatistical tools: the variogram, the
estimation variance and the dispersion variance.
Section II.A gives the interpretation of a regionalized variable z (x ) as a realization of a random function Z (x ), The spatial correlations of this random function,
i.e., the variability in space of the variable being studied, are characterized by the
covariance function C(h) or by its equivalent, the semi-variogram function defined
by relation (JIA):
y(h) = 1E{[Z(x + h)- Z(X)]2}.

## Section II.B introduces the distribution function of the error of estimation. If a

normal distribution is assumed, this error is then completely characterized by its
expectation (or estimation bias) and its variance (or estimation variance). For the
important class of linear estimators, the estimation variance of any domain V by
any other domain v is given by a combination of mean values of the semi-variogram
y(h), d. formula (11.27):
(T~(v, V)= 2-y(V, v)- )I(V, V)- Y(v, v).

Section II.C introduces a second variance, the dispersion variance D 2 (v/ V),
which characterizes the dispersion of the grades of mining units v within a stope V,
for example. This dispersion variance is written as a combination of mean values of
the semi-variogram 'Y(h), cf. formula (11.36):
D 2 (v/ V) =)i( V, V) - )I(v, u),

## Section II.D introduces the important notion of regularization (taking the

average value in a certain support v). If Z (x) is the point grade with the point
serni-variogram vin). the mean value Z,,(x) in the volume v is then characterized
by a regularized semi-variogram y,,(h), which is written as a combination of mean
values of the point serni-variogram, d. formula (11.41):
'Yv(h}=y(v, Vh)'--Y(V, v}.

26

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## II. THE THEORY OF REGIONALIZED VARIABLES

27

Two sets of information defined on two different supports v and v' and characterized by the two regularized semi-variograms Y~ and Yv' can then be compared.
As all the previous geostatistical operations make continuous use of mean values
y of the point serni-variogram y(h), section II.E is devoted to the practical
calculation procedures of those mean values y. Either a direct numerical calculation
or the use of appropriate auxiliary functions and charts is possible.

## 11.A. REGIONALIZED VARIABLES AND RANDOM FUNCTIONS

II.A 1. Regionalized variables and their probabilistic representation

## When a variable is distributed in space, it is said to be "regionalized". Such

a variable is usually a characteristic of a certain phenomenon, as metal
grades, for example, are characteristics of a mineralization. The
phenomenon that the regionalized variable is used to represent is called a
"regionalization", examples of which are:
(i) the market price of a metal which can be seen as the distribution of
the variable price in time (one-dimensional space);
(ii) a geological phenomenon such as the thickness of a subhorizontal
bed which can be regarded as the distribution in the two-dimensional space of the variable thickness.
(iii) a mineralized phenomenon can be characterized by the distribution
in the three-dimensional space of variables such as grades, densities,
recoveries, granulornetries etc.

## Regionalized variables are not restricted to mining, and examples from

other fields include population density in demography, rainfall measurements in pluviometry and harvest yields in agronomy. In fact, almost all
variables encountered in the earth sciences can be regarded as regionalized
variables.
The definition of a regionalized variable (abbreviated to ReV) as a
variable distributed in space is purely descriptive and does not involve any
probabilistic interpretation. From a mathematical point of view, a ReV is
simply a function f(x) which takes a value at every point x of coordinates
(xu, xv, x w ) in the three-dimensional space. However, more often than not,
this function varies so irregularly in space as to preclude any direct
mathematical study of it. Even a curve as irregular as that of the nickel
grades of core samples shown in Fig. ILl underestimates the true variability of the point grades due to the smoothing effect of taking average
values over the volume of the core. However, the profiles of the nickel
grades from three neighbouring vertical bore-holes have a common
general behaviour; a relatively slow increase of the grade values followed
by a rapid decrease, and, then, at the contact with the bedrock, a further

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28

MINING GEOSTATISTICS

## increase. This behaviour is characteristic of the vertical variability of nickel

grades in the lateritic type deposits of New Caledonia, (cf. Ch. Huijbregts
and A. Journel, 1972).
f.--------~400 m

--------+-1
....---- 20 m --..........,~

Ni

0-8

0-8

Ni

0-8

10
s:

ii.
Q)

20

30

Poor
zone

x
FIG.

## Il.l. Random and structured aspect of a ReV.

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Ni

29

## In practically all deposits, a characteristic behaviour or structure of the

spatial variability of the ReV under study can be discerned behind a locally
erratic aspect. In almost all deposits, there exist zones which are richer than
others; samples taken in a rich zone will be, On average, richer than those
taken in a poorer zone, i.e., the value of the ReV I(x) depends on the
spatial position x, see Fig. II.2.
Thus a ReV possesses two apparently contradictory characteristics:
0) a local, random, erratic aspect which calls to mind the notion of

random variable;
(ii) a general (or average) structured aspect which requires a certain
functional representation,
A proper formulation must take this double aspect of randomness and
structure into account in such a way as to provide a simple representation
of the spatial variability and lead to a consistent and operational approach
to the solution of problems. One such formulation is the probabilistic
interpretation as provided by random functions.

## The concept of a random function

A random variable (abbreviated to RV) is a variable which takes a certain
number of numerical values according to a certain probability distribution.
For instance, the result of casting an unbiased die can be considered as a
RV which can take one of six equally probable values. If one particular cast
result is 5, then, by definition, we shall say that this value 5 is a particular
realization of the RV "result of casting the die".
Similarly, let us consider the grade z(xd= 15% Cu at a particular point
x I in a copper deposit. This grade can be considered as a particular
realization of a certain RV Z(Xl) defined at point XI. Thus, the set of
grades z(x) for all points x inside the deposit, i.e., the regionalized variable
z (x), can be considered as a particular realization of the set of RV {Z (x),
X E deposit}. This set of RV is called a random function and will be written
Z (x ).1
This definition of a random function (abbreviated to RF) expresses the
random and structured aspects of a regionalized variable:
(i ) locally, at a point x I, Z (x i) is a random variable;
(ii) Z(x) is also a RF in the sense that for each pair of points XI and
Xl + h, the corresponding R V's Z (x I) and Z (x 1 + h) are not, in
In order 10 distinguish clearly between observations and a probabilistic model.
the regionalized variable or the realizations of the RV or RF are written in lower-case
letters, thus: y(x), z (x ), t(x), while the RV or RF themselves are written in upper-case
letters, thus: Vex), Z(x). T(x).

1- Notation

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30

MINING GEOSTATISTICS

## general, independent but are related by a correlation expressing the

spatial structure of the initial regionalized variable z (x).

Statistical inference
The probabilistic interpretation of a ReV z (x) as a particular realization of
a certain RF Z (x) has an operative sense only when it is possible to infer all
or part of the probability law which defines this RF in its entirety.
Obviously, it is not rigorously possible to infer the probability law of a
RF Z (x) from a single realization z (x) which is, in addition, limited to a
finite number of sample points Xi- It is impossible, for instance, to determine the law of the RV "the result of casting the die" from the single
numerical result 5 resulting from a single cast of the die, and, in particular,
it is not possible to determine whether or not the die is biased. Many casts
of the die are necessary to answer such a question. Similarly, many realizations z 1(x), Z2(X), ... , zdx) of the RF Z (x) are required in order to infer
the probability law of Z (x). Since, in practice we shall be limited to a single
realization {z (Xj)} of the RF at the positions Xj, we appear to have come to a
dead end and, in order to solve it, certain assumptions are necessary. These
assumptions involve various degrees of spatial homogeneity and are introduced under the general heading of the hypothesis of stationarity.
In practice, even if only over a certain region, the phenomenon under
study can very often be considered as homogeneous, the ReV repeating
itself in space. This homogeneity or repetition provides the equivalent of
many realizations of the same RF Z (x) and permits a certain amount of
statistical inference. Two experimental values z (xo) and z (xo + h) at two
different points Xo and Xo + h can, thus, be considered as two different
realizations of the same RV Z (xo). This type of approach is not peculiar to
geostatistics, it is used to infer the distribution law of the RV Z (x) from a
histogram of data values {z(Xj)}, or more simply, to infer the mathematical
expectation E {Z (x)} from the arithmetic mean of the data.

## II.A.2. Moments and stationarity

Consider the RF Z (x). For every set of k points in R" (n -dimensional
space), x h X2, , xi, called support points, there corresponds a kcomponent vectorial random variable

## This vectorial RV is characterized by the k -variable distribution function

FXIX;:_ ..

X~ (ZI,

Z2, . , Zk)=

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Prob {Z(Xl)<

ZI, . . . ,

Z(Xk)<

zd.

## II. THE THEORY OF REGIONALIZED VARIABLES

31

The set of all these distribution functions, for all positive integers k and for
every possible choice of support points in R", constitutes the "spatial law"
of the RF Z(x).
In mining applications, the entire spatial law is never required, mainly
because the first two moments of the law are sufficient to provide an
acceptable approximate solution to most of the problems encountered.
Moreover, the amount of data generally available is insufficient to infer the
entire spatial law. In geostatistics (more precisely, linear geostatistics), only
the first two moments of the RF are used; in other words, no distinction is
made between two RF's Z1(X) and Z2(X) which have the same first- and
second-order moments and both functions are considered as comprising
the same model.

## Mathematical expectation, or first-order moment

Consider a RV Z(x) at point x. If the distribution function of Z(x) has an
expectation (and we shall suppose that it has), then this expectation is
generally a function of x, and is written
E{Z(x)}

= m(x).

Second-order moments
The three second-order moments considered in geostatistics are as follows.
(i) The variance, or more precisely the "a priori" variance of Z(x).
When this variance exists, it is defined as the second-order moment
about the expectation m(x) of the RV Z(x), i.e.,
Var {Z (x)} = E{[Z(x) - m (x )]2}.
As with the expectation m (x), the variance is generally a function of

x.
(ii) The covariance. It can be shown that if the two RV's Z(XI) and

Z (X2) have variances at the points XI and X2, then they also have a
covariance which is a function of the two locations x I and Xl and is
written as

## (iii) The uariogram. The variogram function is defined as the variance of

the increment [Z (Xl)- Z (X2)], and is written as
2y(XI, X2) = Var {Z(XI)- Z(X2)}'

## The function Y(X1' X2) is then the "serni-variogram".

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MINING GEOSTATISTICS

## The hypothesis of stationarity

It appears from the definitions that the covariance and variogram functions
depend simultaneously on the two support points Xl and X2. If this is indeed
the case, then many realizations of the pair of RV's {Z(xd, Z(X2)} must be
available for any statistical inference to be possible.
On the other hand, if these functions depend only on the distance
between the two support points (i.e., on the vector h = x, - X2 separating Xl
and X2), then statistical inference becomes possible: each pair of data
[z (Xk), z (xd} separated by the distance (Xk - xd, equal to the vector h, can
be considered as a different realization of the pair of RV's {Z(x,}, Z(X2)}'
It is intuitively clear that, in a zone of homogeneous mineralization, the
correlation that exists between two data values Z(Xk) and Z (XI,,,) does not
depend on their particular positions within the zone but rather on the
distance which separates them.

## Strict stationarity A RF is said to be stationary, in the strict sense, when

its spatial law is invariant under translation. More precisely, the two
k-component
vectorial
RV's
{Z(x,), ... ,Z(xd}
and
{Z'(x, +
h), ... , Z (x, + h)} are identical in law (have the same k -variable distribution law) whatever the translation vector h.
However, in linear geostatistics, as we are only interested in the previously defined first two moments, it will be enough to assume first that these
moments exist, and then to limit the stationarity assumptions to them.
Stationarity of order 2

## A RF is said to be stationary of order 2, when:

0) the mathematical expectation E{Z (x)} exists and does not depend
on the support point x; th us,
E{Z(x)} = m,

(11.1 )

Ir/ x;

(ii) for each pair of RV's {Z (x ), Z (x + h)} the covariance exists and
depends on the separation distance h,
C(h)=E{Z(x +h) Z(x)}- m;"

(II. 2)

't/x,

## where h represents a vector of coordinates (h u ) h.; h w ) in the threedimensional space.

The stationarity of the covariance implies the stationarity of the variance
and the variograrn. The following relations are immediately evident:
Var{Z(x)} = E{[Z(x) - m]2} = C(O),

## y(h) =!E {[Z (x + h)- Z (x )]2} = C(O)- C(h),

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"Ix

(11.3)

Vx.

(IL4)

33

## Relation (11.4) indicates that, under the hypothesis of second-order

stationarity, the covariance and the variogram are two equivalent tools for
characterizing the auto-correlations between two variables Z (x + h) and
Z (x) separated by a distance h. We can also define a third tool, the
correlogram:
p

C(O)
C(O)

(11.5 )

## The hypothesis of second-order stationarity assumes the existence of a

covariance and, thus, of a finite a priori variance, Var {Z(x)} = C(O). Now,
the existence of the variogram function represents a weaker hypothesis
than the existence of the covariance; moreover, there are many physical
phenomena and RF's which have an infinite capacity for dispersion (ct.
section II.A.S, Case Study 1), i.e., which have neither an a priori variance
nor a covariance, but for which a variogram can be defined. As a
consequence, the second-order stationarity hypothesis can be slightly
reduced when assuming only the existence and stationarity of the variogram.
Intrinsic hypothesis

## A RF Z(x) is said to be intrinsic when:

(i) the mathematical expectation exists and does not depend on the

support point x,
E{Z(x)}=m,

Vx;

Var {Z(x

Vx.

(11.6)

## Thus, second-order stationarity implies the intrinsic hypothesis but the

converse is not true: the intrinsic hypothesis can also be seen as the
limitation of the second-order stationarity to the increments of the RF
Z(x).

## Quasi -stationarity In practice, the structural function, covariance or

variogram, is only used for limited distances Ih 1:0::::; b. The limit b represents,
for example, the diameter of the neighbourhood of estimation (i.e., the
zone which contains the information to be used, cf. Fig. II.3). In other
cases, b may be the extent of an homogeneous zone and two variables
Z(xd and Z(Xk + h) cannot be considered as coming from the same
homogeneous mineralization if Ih I> b.
In such cases, we can, and indeed we must, be content with a structural
function C(x, x + h) or y(x, x + h), which is no more than locally stationary

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34

MINING GEOSTATISTICS

(for distances Ih I less than the limit b). This limitation of the hypothesis of
second-order stationarity (or the intrinsic hypothesis if only the variograrn
is assumed) to only those distances Ih I~ b corresponds to an hypothesis of
quasi-stationarity (or to a quasi-intrinsic hypothesis).
In practical terms, we can define sliding neighbourhoods inside of which
the expectation and the covariance can be considered as stationary and the
data are sufficient to make statistical inference possible.
The hypothesis of quasi-stationarity is really a compromise between the
scale of homogeneity of the phenomenon and the amount of available data.
In fact, it is always possible to produce stationarity by considerably reducing the dimension b of the zones of quasi-stationarity, but then most of
these zones would not have any data in them, and, thus, it would be
impossible to infer the quasi-stationary moments in these zones.
From a data grid of 1 to 3 m, a 200-m-Iong hillside would appear as a
quasi-stationary model, but would require a non-stationary model
(because of the increasing altitude) for a grid of 20 to 30 m, cf. Fig. 11.3. If
the data grid is of the order of 200 m, then a certain stationarity again
becomes evident: at the scale of the peneplanar basin the entire hillside
appears as a local variability of the stationary RF "altitude". This example
shows, incidentally, that it is not possible to devise a theoretical test for the
a priori verification of the hypothesis of stationarity using a single realization of the RF. It is always possible to find a particular realization of a
stationary RF which would display a systematic and continuous change in
value (apparent non-stationarity) over a limited distance, so that a
theoretical test could never refute the hypothesis of stationarity.

I b=IO m I

~~

x
x

I'
I <,
<,
<,

z(xo}

<,

..

x
200

## 11.A.3. Properties of the covariance and the variogram

The following properties are classical results of the theory of stochastic
processes, cf., in particular, J. Bass (1974) or J. L. Doob (1953).

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35

## Their proof is of no interest in the application of geostatistics; we shall

simply state the properties and comment on them.

## Positive definite conditions

Let Z(x) be a stationary RF of expectation m and covariance C(h) or
semi-variogram l'(h). Let Y be any finite linear combination of the type
y=

L"

(II.7)

AiZ(Xi)

i=l

## for any weights Ai.

This linear combination is a RV and its variance must never be negative,
Var {Y} ~ O. Explicitly, this variance is written
Var {Y}=

L L AiAiC(Xi -Xj)~O.
i

(11.8)

The covariance function must be such that it ensures that the previous
variance will always be positive or zero. By definition then, the function
C(h) is said to be "positive definite".
Thus, not any function g(h) can be considered as the covariance of a
stationary RF: it is imperative that it be positive definite.
Taking account of relation (11.4), expression (11.8) can be written in
terms of the serni-variogram:
Var {Y} = C(O)I AiL Aj

LI

## AiA j l' (Xi - x,).

(11.9)

iii

In the case where the variance C(O) does not exist, only the intrinsic
hypothesis is assumed, and the variance of Y is defined on the condition
that

Var{Y}=

## -LLA iA l' (Xi j

X j)

(11.10)

The variogram function must be such that Var {Y} is positive or zero, with
the condition that It Aj = O. By definition, -l'(h) is said to be a "conditional
positive definite" function.

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MINING GEOSTATISTICS

36

## Authorized linear combinations

When only the intrinsic hypothesis is assumed, y(h) exists but C(h) may
not exist; the only linear combinations that have a finite variance are those
verifying the weights condition L Ai = O. Thus, if only these authorized
linear combinations are considered (which is the case in linear geostatistics), there is no need to calculate the a priori variance C(O), nor to know
the expectation m = E{Z (x)} or the covariance C(h); it is enough to know
the serni-variogram.
Note that, for the authorized linear combinations, formula (II.IO) is
simply deduced from formula (11.8) by substituting (- y) for C. This rule
for passing from a formula written in terms of covariance to the corresponding formula written in terms of semi-variogram is genera! in linear
geostatistics.

## Properties of the covariance

The positive definite character of the covariance function C(h) entails the
following properties:

## C(O) = Var {Z(x)} ~ 0,

C(h) """ C(-h),
IC(h)1 =:;; C(O),

## the covariance is an even function;

(II.ll)

Schwarz's inequality.

## Since the degree of correlation between two variables Z (x) and Z (x + h)

generally decreases as the distance h between them increases, so, in
general, does the covariance function, which decreases from its value at the
origin C(O). Correspondingly, the semivariogram y(h) """ C(O)- C(h)
increases from its value at the origin, 1'(0) = 0, see Fig. 11.4.

Absence of correlation
Very often, in practice, the correlation between two variables Z (x) and
Z (x + h) disappears when the distance h becomes too large:
C(h)~O,

when Ihl~co,

## and, in practice, we can put C(h) = 0, once Ih 1~ a. The distance a beyond

which C(h) can be considered to be equal to zero is called the "range" and
it represents the transition from the state in which a spatial correlation
exists (Ih I < a) to the state in which there is absence of correlation (Ih j > a),
d. Fig. IIA.

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37

## The definition of the variogram as the variance of increments entails the

following properties:
y(h)=y(-h);::.O.

1'(0) = 0,

(II, 12)

## In general, but not always, as h increases, the mean quadratic deviation

between the two variables Z (x) and Z (x + h) tends to increase and so y(h)
increases from its initial zero value.

r(m)=c{O)

C(O) - - - - - - - - - - -

--~--_----:._----

r (II)

C(II)

C{co)::O

## Transition phenomena Very often, in practice, the serni-variogram stops

increasing beyond a certain distance and becomes more or less stable
around a limit value ")'(00) called a "sill" value, which is simply the a priori
variance of the RF
(11.13)
")'(00)= Var{Z(x)} = C(O).

In such cases, the a priori variance exists as does the covariance. Such
variograms, which are characterized by a sill value and a range, are called
"transition" models, and correspond to a RF which is not only intrinsic but
also second-order stationary, d. Fig. II.4.

## Zone of influence In a transition phenomenon, any data value z(x) will be

correlated with any other value falling within a radius a of x. This correlation and, hence, the influence of one value on the other will decrease as
the distance between the two points increases. Thus, the range corresponds
to the intuitive idea of a zone of influence of a RV: beyond the distance
Ih! = a, the RV's Z (x) and Z (x + h) are no longer correlated.
Anisotropies There is no reason to expect that the mineralization will
exhibit the same behaviour in every direction, i.e., that the mineralization

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38

MINING GEOSTATISTICS

will be isotropic. In the three-dimensional space, x represents the coordinates (xu, xv, x w ) and h represents a vector of modulus Ih I and direction
a. Thus, in condensed form, y(h) represents the set of semi-variograms
y(lh I, a) for each direction o'. By studying y(h) in various directions a, it
would be possible to determine any possible anisotropies, such as the
variability of the range a (a) with the direction a.
In the example of Fig. II.S, the semi-variogram for the vertical direction
reveals a short range a I corresponding to the mean vertical width of the.
mineralized lenses, while the semi-variogram for the horizontal direction
has a larger range a2 corresponding to the mean horizontal dimensions of
the same lenses. The directional graph of ranges a (0:) thus represents the
average morphology of the mineralized lenses.
Horizontal
range

.- - Vertical
COre

1!vertlcal
rcnqe

.I

- - - Horizontal

01

02

lnl

## Behaviour of the variogram near the origin

The continuity and regularity in space of the RF Z (x) and, thus, of the
Re V z (x) that it represents, are related to the behaviour of the variogram
near the origin. In order of decreasing regularity, four main types of
behaviour can be distinguished, see Fig. II.6 and section II.A.S, Case
Study 3.
(a) Parabolic: y(h) ~ A Ih 2 when h -? O. y(h) is twice differentiable at
the origin and the RF is itself differentiable in the mean square
sense. This type of behaviour is characteristic of a highly regular
spatial variability.
(b) Linear behaviour: y(h)~Alhl when h-?O. y(h) is no longer
differentiable at the origin" but remains continuous at h = 0, and,
thus, for all Ih I. The RF Z (x) is mean-square continuous! but no
longer differentiable.
1

t In fact, the right and left derivative, /,'(0+) and )"(O~) exist but have different signs.

+A

## RF is said to be mean-square continuous if

lim E{{Z(x +h)-Z(x)J

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=0,

when h ..... 0.

## II. THE THEORY OF REGIONALIZED VARIABLES

39

FIG. II.6. Behaviour near the origin of the variograrn. (a) Parabolic behaviour;
(b) linear behaviour; (c) nugget effect; (d) pure nugget effect.

(c) Discontinuity at the origin: y(h) does not tend towards zero when h
tends towards zero, although by definition 1'(0) = O. The RF is no
longer even mean-square continuous; the variability between two
values z(x) and z(x +h) taken at two very close points may be quite
high and increases as the size of the discontinuity at the origin of
y(h) increases. This local variability can be compared to the random
phenomenon of white noise known to physicists. As the distance h
increases, the variability often becomes more continuous and this is
reflected in the continuity of y(h) for h > O.
The discontinuity of the variogram at the origin is called a "nugget
effect" and is due both to measurement errors and to micro-variabilities of the mineralization; since the structure of these microvariabilities is not accessible at the scale at which the data are
available, they appear in the form of a white noise. A whole section
(lILA) will be devoted to the study of this nugget effect.
(d) Pure nugget effect. This is the limit case when y(h) appears solely as a
discontinuity at the origin:
y(O)= 0

and

8.

## In practice, such a variogram can be represen ted as a transrtion

phenomenon with a sill value Co and a range a = E, where 8 is very
small in relation to the distances of experimental observation. For all
experimental distances, small as they maybe, the two RV's Z (x) and
Z (x + h) are uncorrelated. The pure nugget effect thus corresponds
to the total absence of auto-correlation. It should be pointed out that
such a case is very rare in mining.

Behaviour at infinity
Using the property that - y is a conditional positive definite function, it
can be shown that the variogram necessarily increases more slowly at
infinity than does Ihf:
when

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Ih I-l> 00.

(II.14 )

40

## As a consequence, an experimental variogram which increases at least as

rapidly as Ih 2 for large distances h is incompatible with the intrinsic
hypothesis. Such an increase in the variograrn most often indicates the
presence of a trend or drift, t i.e., a non-stationary mathematical expectation:
1

E{Z(x)}

= m(x),

where m (x ) depends on x.
11.A.4.

Coregionalization

## A regionalized phenomenon can be represented by several inter-correlated

variables and, in certain cases, it may be useful to study them simultaneously.
In a lead-zinc deposit, for example, the two mineralizations are closely
related and it may be of interest to know if, on average, zones that are rich
in Pb are equally rich in Zn, In other polymetallic deposits, it would be
useful to be able to use the information provided by one variable to make
up for missing information on another variable.
The probabilistic approach to a coregionalization is similar to that of the
regionalization of a single variable.
The simultaneous regionalizations (or the coregionalization) of K ReV's
{Zl(X), . . . , ZK(X)} are interpreted as a particular realization of the set of K
inter-correlated RF's {Z 1(x), ... , ZK (x )}.
Under the hypothesis of second-order stationarity we define:
(i) for each RF Zk (x), the mathematical expectation
E{Zk(X)} = m; = constant,

Vx;

= Cek(h),

Vx;

(I1.l5)

## E{[Zdx + h)- Zk'(X)][Zk(X + h)- Zk(X)]} = 2Yn(h),

Vx.

(11.16)

Properties of cross-moments
1. When k' = k, the formulae (11.15) and (11.16) yield the definitions of the
covariance and variograrn, Ci; (h) and 2 'Ykk (h).
t The term "drift" is preferred in geostatistical applications as it avoids the subjective

## interpretation that is often associated with the word "trend".

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41

## 2. A cross-serni-variogram l'k'k (h) can take negative values, whereas a

direct serni-variogram is always positive. A negative value of the crossserni-variograrn indicates that a positive increase in one of the variables
(k') corresponds, on average, to a decrease in the other (k), This could be
explained, for example, by a phenomenon in which element k is substituted for element k'.
3. Under the second-order stationary hypothesis, the existence of the
cross-covariances entails that of the cross-variograms and these two groups
of structural tools are related by the expression
(11.17)

## 4. The cross-variograrn is symmetric in (k', k) and (h, -h) while this is

not necessarily so for the cross-covariance:
'Yk'k (h) = l'kk,(h ) and 'Yk'k (h) = l'n (- h),
}
(11.18)
Ckk,(h) = C n (-h) and C n (-h):;i: Ci-; (h).
A lag effect in one variable in relation to another (e.g., rich Pb grades

## lagging behind rich Zn grades in a given direction) is represented by a

dissymetric cross-covariance in (h, -h). Such a lag effect would not be
visible on the cross-variogram, and, where it is suspected, the crosscovariance should be used, Otherwise, for reason of consistency with the
study of the direct variograms, it is preferable to use cross-variograms,
5. Correlation coefficient. The point to point correlation coefficient
between two variables Zdx) and Z; (x) located at the same point x is
written
Pk'k =

JrCkk(O). Ck'k'(O)] =

PH'

(11.19)

## 6. Linear combination. It may be useful in certain studies to define a

finite linear combination of ReV's {zdx), k = 1 to K} such as
K

f(x,

AkZdx),

(11.20)

k = I

## with, for example,

the profit or loss associated with a unit quantity of the metal k;
the grade of metal k on a given production unit;
or
costs associated with the extraction of waste and mineral;
Zk (x ), thickness of overburden and mineralized thickness.

Ak,
{

## The ReV [ix, Ak ) can then be interpreted as a particular realization of the

RF F(x, A k ) = Lk AkZdx).

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MINING GEOSTATISTICS

42

If the coregionalization {Zl (x), ... , ZK (x)} verifies the intrinsic hypothesis, then the RF F(x, Ak ) is also intrinsic and has the following
stationary serni-variogram:

I I

AkAk''Ykk'(h),

k k'

## and this single direct semi-variograrn r(h) will be used in subsequent

operations. In particular, the economic function [(x, Ak) can be kriged
(estimated optimally).
ILA.5. Case studies

## 1. Infinite a priori variance (Witwatersrand gold province)

Under the hypothesis of second-order stationarity, both a covariance and,
thus, a finite a priori variance, Var {Z (x)} = C(O), are assumed. But there
are theoretical models of RF's and also physical phenomena that display an
unlimited capacity for dispersion and for which neither an a priori variance
nor a covariance can be defined.
Brownian motion is an example of such a phenomenon and is well
known to physicists. It can be described by the Wiener-Levy process,
which has an infinite variance and, thus, no covariance. However, the
increments of this process do have a finite variance:
Var{Z(x+h)-Z(x)}= 2'Y(h)=A

.Ihl.

The variogram of the Wiener-Levy process is, thus, defined and it can be
shown that it increases linearly with the distance Ih I.
The dispersion of gold grades in the Witwatersrand provinces of South
Africa provides a more practical mining example. South African authors
(in particular D. G. Krige, 1951, 1976) have shown that the experimental
variance (T2( V) of gold grades within a field of size V increases continuously with the dimensions of the field.
The essence of Krige's original results is shown in Fig. II.7. The variance
of the logarithms of the gold grades increases without interruption as a
function of the logarithm of the area of the field V over which the variance
is calculated. A complete range of field sizes was considered, beginning
with the cross-sectional area of analysed core samples (0,0075 fe) and
followed in turn by the horizontal cross-sectional area of a mining panel
8
(63 000 ft\ the mean horizontal surface of a mine (about 34 x 10 fe)
and, finally, the total surface area of the gold bearing region of the Rand
(about 34 x 10 9 fe).

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43

## The experimental curve of Fig. 11.7 can be approximated by a straight

line representing the uninterrupted logarithmic growth of the variance with
the increase in field size V:
(T2(V)=a log (Vjv),

fe

where v = 00075
is the area of cross-section of the analysed core
samples, and V is the horizontal surface area of the field. The a priori
variance of the logarithms of the gold grades should be the limit of (T2( V)
as the field size V tends towards infinity. Experimentally, such a limit does
not exist and, as a consequence, a second-order stationary model of RF
cannot be used. It then becomes necessary to use a model that is only
intrinsic, having a variogram but no covariance.
06
0-5
0-4
4>
u

<:

<:::

...
c

03

0
OJ

>

02

to

toO
0

'"

.2:

,2
<lJ

0-1

QJ

::
".

o,

"0
C
0

'l-

'"

0:

-I

Log area

2.

## A quasi-stationary type of mineralization is illustrated by a massive copper

deposit explored by vertical bore-holes. The copper grade decreases
regularly with the depth, which indicates the presence of a trend (nonstationarity) in the vertical direction.
The experimental variograrn of the copper grades calculated using all the
vertical drill cores is shown on Fig. 11.8. The characteristics of this variogram are:
(i) a nugget effect, the amplitude Co of the discontinuity at the origin
(ii) a transition phenomenon between the origin and a distance of about
100 ft with a sill value of 1 and a range of about 50 ft;

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44

MINING GEOSTATISTICS

## (iii) beyond 100 ft there is a sharp increase in the variogram values,

indicating the presence of a trend due to Cu grades decreasing with
depth.
Thus, over neighbourhoods of less than 100 ft in the vertical direction,
the mineralization can be considered as stationary and characterized by a
semi-variogram with a finite range, i.e., we accept an hypothesis of quasistationarity. More precisely, the theoretical model adopted is known as a
spherical scheme with a nugget effect and is represented by the equation
y(h) = C o + CII'5hl a -O.5h 3; a 3 ] ,

\thElO, 100]ft

with
a = 50 ft.

Co = 0-4,

## This model is shown on Fig. 11.8 as the broken-line curve.

3

r'--~------~------------'

,/

,/

/'
/'

'"

50

100

150

200

II (f t )

## 3. Behaviour near the origin (pluuiometry at Korhogo)

The example given here has the advantage of demonstrating the four
different types of behaviour of the variograrn near the origin cited in
section II.A.3 and Fig. 11.6.
This variography is taken from a prominent unpublished study carried
out by J. P. Delhomme (1976) on Korhogo (Ivory Coast) piezometric data.
The data z (x, t) at each position x give the piezo height (height of the top

W'INI.minebook.blogfa.com

45

## of the water table) as a function of time t. The measurements are carried

out daily. The piezometers are placed at intervals of about 500 ill, see Fig.
II.9.

Surface

Rain

1
Water table

..

500 m

D~

of discharge
FIG.

## At every location x of a piezometer, a profile of the piezo height is

available over the rainy season (t varying from August to November).
Figure n.10 shows four of these profiles among the most characteristic
ones. The following observations can be made.
At piezometer no. 3 The water table is very close to the surface and the
water level reacts each time it rains (the amount of water retained by the
ground is proportional to the thickness of the ground through which the
water passes before reaching the water table). The profile from piezo no. 3
is very erratic and is directly related to the daily rainfall. The variogram
along this profile is linear with a nugget effect (Brownian motion for small
lags t).
At piezometer no. 4 The water table is deeper and the absorption effect of
the ground begins to take effect. The corresponding variogram has a
smaller nugget effect and a transition phenomenon is becoming evident
(appearance of a sill).
At piezometer no. 33 The water table is deeper still and the regularizing
effect of the ground absorption is almost total. There are, however, a
certain number of discontinuities evident on the profile and these are due
to abrupt refilling of the table. The corresponding variogram has a small

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46

MINING GEOSTATISTICS

## nugget effect representing the discontinuous refilling of the table followed

by a parabolic shape which is characteristic of the high continuity of the
ReV z(t). More precisely, the RF Z(t) is mean-square differentiable
except during the periods of refilling.
At piezometer no. 18 The water table is even deeper and the effects of the
discontinuous refilling are no longer evident. The nugget effect has
completely disappeared and the variogram presents an almost perfect
parabolic behaviour near the origin, characteristic of the differentiability of
the profile z(t)at any time t.
0

3
4

~
..c

.;:

l'

3:

'0
s:

n..,

Piezo no. 33

10
II

12

13

Piezo no. 18

14

July

-;:'005

l
,-!

October

November

December

I
./

f-

.'

.'
.~t

1-.

10

15

10

.'

15

10

l-

.'
." .'

...... I

15

.1
I

Time (days)

## FIG. 11.10. Variography of the piezo heights at Korhogo.

VNNI.minebook.blogfa.com

i
I

18
-

.,fI

.,........

33

September

Olf-

....
....
....

August

10

15

## II. THE THEORY OF REGIONALIZED VARIABLES

47

4. Coregionalizations at Exotica
The experimental direct and cross-semi-variograms for Fe, soluble Cu and
total Cu, shown on Fig. I1.II, were calculated from the grades defined on
13-m vertical bench heights in the Exotica copper deposit (Chile) and
represent the horizontal coregionalization of Fe, Cu S, Cu T grades.

IfCu T-Cu T
Cu S - Cu T
CuS~CuS

"}' 05-

.~.~.

Fe-Fe

01-

o f----------;---

h (m)

## Note that the cross-semi-variogram for Cu T -Cu S is positive while that

of Fe-Cu S is negative. The negative Fe-Cu S correlation can be explained
by a superficial alteration phenomenon in which water liberates the initial
sulphuretted copper elements (chalcopyrite in particular), and the almost
insoluble Fe remains close to the surface while the soluble copper Cu S is
washed away. As a consequence, a positive increment in iron grades
[zdx + h)- zdx)] > 0 corresponds, on average, to a negative increment in

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48

MINING GEOSTATISTICS

negative.

## 11.8. ESTIMATION VARIANCE

11.8.1. The estimation error and its distribution

## Every estimation method involves an estimation error, arising from the

simple fact that the quantity to be estimated generally differs from its
estimator z ", thus implying an error of estimation z -z*.
When the mean grade z (x.) of a vertical bore-hole through the middle of
a block V is used to estimate the true mean grade zv(x;) of the block, the
error involved is r(xi) = zv(x;) - z (x,), ct. Fig. II.12. Just as the ReV z (x)
has been interpreted as a particular realization of the RF Z (x), so the error
r(xi) also appears as a particular realization of the RV R (Xi) =
Zv(x;)-Z(x;) at the point Xi.

Vi

)(i

Block
Borehole

,ll'(

xl

r ( xl

I...

mE

95 01.0

pro b 0 hI'
I Ity

.1

## FIG. Il.12. Distribution of the error.

Suppose now that the entire deposit is divided into blocks of equal size
V, and that each block is intersected by a vertical bore-hole passing
through its centre. If the deposit is an homogeneous mineralization, i.e., if
the RF Z (x) can be considered as stationary, then the error R (x ) is also
stationary and any two errors r(xi) and r(xj) can be considered as two
different realizations of the same stationary RF R(x)=Zv(x)-Z(x).
If the histogram of experimental errors is available in a control zone,
then it will be possible, because of the stationary assumption, to infer the
complete distribution function of R (x). Even if such a histogram is not
available, it will still be possible to calculate the stationary expectation
mE = E{R (x)} and variance (T~ = Var {R (x)} of the distribution function of
the error.

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## II. THE THEORY OF REGIONALIZED VARIABLES

49

The particular error r(x;) = zv(x;) - z (x;) involved when estimating the
block V (x;) remains unknown, but the mean and variance of the errors (or
the complete distribution function if it is known) will provide a measure of
the quality of the estimation.
The distribution of the estimation error

## As with all RV's, the error R (x) is characterized by its distribution

function:
F(u) = Prob {R (x)< u}.

If the RF R (x) is stationary, then this function does not depend on the
location x. Given the function F(u), it is possible to calculate the prob-

ability that the error will fall within any interval [a, bJ;
Prob {a

~R(x)<b}=F(b)-F(a),

'Ix.

## In practice, the only error distributions considered will be those with

finite moments, at least the first two moments. This amounts to the
assumption of second-order stationarity of the RF R (x) with
(i) a mathematical expectation,
E{R(x)}=rnE=Constant,

'Ix;

## (ii) a variance, called the "estimation variance",

Var{R(x)}=E{[R(x)]2}-m~=O"~ = constant,

'Ix.

The expectation mE characterizes the mean error, while the variance O"~ is
a dispersion index of the error, cf. Fig. 11.12. Thus, a good estimation
procedure must be such that it ensures
0) a mean error close to zero; this property of the estimator is known as

unbiasedness;
(ii) a dispersion of errors very concentrated around this mean zero
value, this second criterion being expressed by a small value of the
.,

2
estimatron vanance O"E.
At the estimation stage, the type of the distribution function of the errors
is unknown in most cases. But, since the two most important characteristics
of this function - its expectation and variance - can be calculated, we shall
refer to a standard two-parameter (mE and t:T~) function which will provide
an order of magnitude of the required confidence interval.

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50

MINING GEOSTATISTICS

## Confidence intervals for the normal distribution

Among all the two-parameter distribution functions, the one most often
used to characterize an error is the normal distribution. The main
justification for using this distribution is that it is the one most often
observed in practice, particularly in mining practice, d. section II.B.3, Case
Studies 1 and 2.
In mining applications, the error distribution functions are generally
symmetric with a slightly more pronounced mode and larger tails than 8normal distribution with the same expectation and variance, cf. Fig. 11.13.
Thus, in relation to the standard normal distribution, there are more small
errors (in the region of mE = 0) and more large errors (in the tails of the
distribution). However, the classic confidence interval [mE 2crEl contains
approximately 95% of the observed errors. The 95% normal confidence
interval is then a good criterion for judging the quality of a geostatistical
estimation, but it should always, as far as possible, be verified experimentally through test or control zones, cf. section II-B-3.
,-hI

i...

## FIG. 11.13. Normal standard for the error distribution.

, Gaussian standard;
- - - -, experimental distribution of errors.

## Remark Under the non-bias condition (mE = 0) the standard gaussian

95% confidence interval can be expressed as [z* 20"d for the true value z
estimated by z*. It sometimes happens that the value of the estimation
standard deviation erE is such that the lower bound of the 95% confidence
interval (z*2aEl is negative, e.g., with z*=O'5% Cu and O'E=l%, so
that Z*-20'E= -1,5%. A practical solution to this problem is to define
the relative standard deviation ad i", i.e., the ratio of the standard deviation to the experimental mean
1

i*=-

L zt

ni=\

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51

## of the n available estimators. The 95% confidence interval of a particular

true value z, estimated by zf will then be
z,

*J

z* Z i
[ Z *i 2 (O'E)

(11.21)

This formula will not ensure that the lower bound is always positive,
since it becomes negative when the relative standard deviation is greater
than 50%,
z* > 0'5, but it is sufficient for practical applications. The
case where erE! f* > O 5 corresponds to poor estimations and, in such cases,
it would be better not to use confidence limits at all.
Formula (H.2l) corresponds to a non-stationary standard deviation of
estimation, since it depends on each estimator z 1. It can be shown (d. G.
Matheron, 1974, and section II1.B.6, Remark 3 and
that this formula
corresponds to a conditioning of the RF Z (r ) to the estimator i*, the
distri bution of Z (r ) being lognormal.

O'el

51

## 11.8.2. Linear estimators

Consider the problem of estimating the unknown value Zv (e.g., the mean
grade Zv(x) of a block of size V centred on point x) from a set of n data
values {Z(Xi), i = 1 to n}.
The estimator Z* will be a function of the available data:
Z* = f{Z(x'), ... , Z(x n ) } ,
but not any function of this' data. It must be a function such that
(i) it satisfies the non-bias condition, E {Zv - Z*} = 0;
(ii) it is reasonably simple, so as to permit the calculation of the estimation variance,

O"i = E{[Zv -

Z*]2}

## E{Zt} + E{Z*2} - 2E{Z vZ *}.

For any function [, the calculation of the first-order moment E{Zv - Z*}
and the various second-order moments of the expression of o-} requires
that the n-variables distribution {Z(Xl), . . , ,Z(xn ) } be known. However,
since it is generally not possible to infer this distribution from a unique
realization of the RF Z (x), we are limited-to the class of linear estimators
(at least in linear geostatistics). It will then always be possible to calculate
the mean and variance of the error from the variogram or the covariance.
Research into non-linear geostatistics is being carried out and has shown
that restrictions imposed by linear estimators can be removed, d. disjunctivekriging and transfer functions, G. Matheron, (1975a, b, c) and
section VIII.B. Such non-linear geostatistics requires a much more

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52

MINING GEOSTATISTICS

## complete knowledge of the spatial structure of the RF Z (x) than that

provided by the variogram.
With the exception of Chapter VIII, non-linear methods are outside the
scope of this book and we shall limit all further discussions to estimators
Z* which are linear combinations of the n available data:

Z*

L"

(11.22)

AiZ(Xi)'

i=l

Estimation variance
Let Z (x) be a second-order stationary RF with expectation m, covariance
C(h) and semi-variograrn y(h), ct. definition formulae (11.1) to (11.4).
Consider first the simple case of the estimation of the arithmetic mean ZK
of the K unknown values {z (xd, k = 1 to K}:
ZK =

1 K
K k~l

Z(Xk)'

## The linear estimator z~ is the arithmetic mean of the n available data

values {z (Xi)' i = 1 to n}:

* =1;'
- L.

ZK

Z(Xi .

ni=l

## The value z'k is interpreted as a particular realization of the RV Z'k and

the unknown error is, thus, a particular realization of the RV ZK ~ Z'k.
Under the second-order stationary hypothesis, the non-bias is automatically verified, since
1
E{ZK} = K tE{Z(xk)}=m

E{Zk} =

L E{Z(Xi)} = m
i

## The estimation variance is written

with
1

E{ZK} =

-2

E {L L Z(Xk)Z(Xe}}.

Kkk'

2

L and the

## E{Z K} = -2L L E{Z (xdZ (Xk')} =

Kkk'

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expectation operator E:
1

-2

L L [C(Xk - xd+ m ].

Kkk'

53

Similarly,

:I<

## E{ZKZK} =-I I [C(Xk -x;)+m ].

Kn k i

the constant term m 2 is then eliminated from the expression of (T~, leaving

O"i:: =
=

## E{[ZK - Z'J( ]2}

1 1 2
K2 L L C(Xk - Xd+2 L LC(Xi - X j ) - - I I C(Xk -Xi)'
n

k k"

Kn

(II.23)

## We will denote by C(K), (n ) the mean value of the covariance C(h)

when one extremity of the vector h describes the set {Xk, k = 1 to K} and
the other extremity independently describes the set {Xi, i = 1 to n}:
1
C((K),(n=-K L
n kfdK)

C(xk-x,l

iE(n}

(}"~

= CK), (K+C((n),

(11.24)

(n-2CK), (n.

## Generalization to the continuous case

Suppose now that the K points Xk are located within a volume V centred on
point x and the n points Xi within another volume v centred on point x'.
As K and n tend towards infinity, the preceding arithmetic means ZK and
Z k tend towards the mean values in V and v of the point variable z (y), i.e.,
ZK

~Zy(x)=.2.- f
V

z(y) dy

and

Vex)

Zk~z,,(x')=~f
v

z(y)dy.

,,(x')

These mean values zv(x) and z,,(x ') are interpreted as particular realizations of two RV's Zy(x) and Z,,(x'):
Zv(X)=

VI

Z(y)dy

y(x)

and

Zv(X')=~f
v

Z(y)dy.

,,(x')

## (Recall here that the preceding unidimensional integrals in fact represent

double or triple integrals according to whether V and v are two-dimensional or three-dimensional, cr. notations in Index D.)
Under the hypothesis of second-order stationarity, the expectations of
Zv (x) and Z" (x') are equal to m, and the non-bias condition is satisfied.

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54

MINING GEOSTATlSTICS

For the estimation variance, the formulae (II.23) and (II.24) for the
discrete case give, for the continuous case of estimating the mean grade of
V by that of v,

uk = E{[Zv(x)- z..,(X')]2\
j

dyf

=v\f
V(x)

-~ r
Vv

JV(X)

C(y-yl)dy'+~f
v

dyf

C(y-y')dy'

vex')

## C(y - y') dy.

dyf

JV(x)

dx')

(II.25)

u(x')

If the domains V (x) and v (x") are three-dimensional, each of the integral
signs of formula (II.25) represents a triple integral and the variance o'~ is a
sum of sextuple integrals. To simplify notations, we shall denote by
C (V, v) the mean value of the covariance C (h) when one extremity of the
distance vector h describes the domain V(x) and the other extremity
independently describes the domain vex'). It is then written
o'~=C(V, V)+C(v, v)-2C(V, v).

(II.26)

When the covariance C(h) exists, the serni-variogram y(h) also exists
and these two structural functions are related by formula (II A ):
C(h)= C(O)-y(h).

Formulae (II.23) to (I1.26) can then be rewritten in terms of semivariograms. Since the constant term C(O) disappears, it is enough to
replace C or C by -1' or -if and the estimation variance becomes

IO"~

## 2y(V, v)-y(V, V)-y(v, v),

(II.27)

where y( V, v), for instance, represents the mean value of y(h) when one
extremity of the vector h describes the domain V(x) and the other extremity independently describes the domain v (z '),
It can be shown that this formula (11.27) remains valid even when the
covariance C(h) does not exist, provided of course that the semi-variogram
is defined (the RF Z (x) is then only intrinsic).
Remark 1 The estimation variance of V by v is sometimes referred to as
the variance of extending the grade of v to V or simply the extension
variance of v to V and is then denoted by O'~(v, V).
Remark 2 The two formulae (II.26) and (rI.2?) are completely general
whatever the domains v and V. In particular, the domains need not

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55

## necessarily be connex (single continuous regions); the domain V to be

estimated may be, for example, two distinct blocks V = VI + V 2 and the
information domain v may consist of several samples v = L 51, some of
which may be situated within the domain V.

Remark 3 The variogram 2y(h) can itself be interpreted as the elementary estimation variance of a variable Z (x) by another variable Z (x + h) at
a distance h from Z (x):
E{[Z(x + h)- Z (x )]2} = 2y(h),

## cf. definition formula (11.6).

Remark 4 Formulae (11.26) and (11.27) express four essential and intuitive facts which condition all estimations. The quality of the estimation of
V by v is a function of the following.

## V to be estimated: term -Y(V, V)

appearing in formula (11.27). As the serni-variogram y(h) is an
increasing function of h, YC V, V) increases with the size of V.
Hence, if v and the interdistance (v, V) remain fixed, it is easier to
estimate the mean grade of a large block V than the grade of an
unknown point (V = x). Finally, if the size V is fixed, the term
YC V, V), and, consequently, the estimation variance cr~, depend
also on the geometry of V.
(ii) The distances between the domain V to be estimated and the
support v of the estimator: term +y(V, v) appearing in formula
(11.27). As the distance between V and v increases, so does the
value of HV, v) and the value of the estimation variance a~.
(iii) The geometry of the estimating domain v: term -y(v, v). As the
dimension of the domain v increases, the value of y(v, v) also
increases and, consequently, a~ will decrease. But, for fixed
volumes V, v and fixed interdistance (v, V), the estimation variance
will also depend on the configuration of the information v. The
estimation of the same block V on Fig. 11.14 by the two distinct
samples v, and V2 will be better than that provided by the two
closely spaced samples v ~ and v; ; indeed, y( v /, + v~, V /1 + v 2) will be
less than ii(VI + V2, VI + V2). This intuitive notion of the importance

## FIG 11.14. Influence of the information configuration.

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MINING GEOSTATISTICS

56

## of the sample configuration, which is formalized in geostansucs

through the term y( v, v), is ignored by the more usual estimation
methods such as weighting by inverse distances or their squares.
(iv) The structural function, semi-variogram or covariance. The quality
of the estimation should depend on the structural characteristics
(anisotropies, degree of regularity) of the regionalized phenomenon
under study. The serni-variogram l' may depend on the direction a
of the vector h, i.e., the serni-variogram will be a function of distance and direction, I'(lh I, a). In horizontally stratified mineralizations, for example, the variation in grade values will be much more
continuous in an horizontal direction than they will be vertically,
and, by taking the anisotropic semi-variogram into account, more
weight would be given to sample VI in estimating block V in the
same stratum than to sample V2 in a different stratum, ct. Fig. 11.15.
This fundamental geological notion of the spatial continuity of a
mineralization as a function of direction is quite often overlooked or
simply ignored in the application of many estimation methods such
as that of weighting by inverse distances.

---------

0.

~\lhI __

--

VI

1111

S, r al if ica 1ion

----

------

\OV2

------

## The estimation of a mean value by a weighted average

The general formulae (11.26) and (11.27) given above also cover the particular case of estimating the mean value Z of a block V by a linear
combination Z* of n available data values Z (Xi) taken at the points {Xi,
i = 1 to n}:
n

Zv=-lr
Z(y)dy
V JV(x)

zt = ;=1
L A;Z(x;).

and

## The non-bias condition entails that E {Zv -

E{zt} =

I
i

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A,E{Z(x,)} = m

zt} = 0, i.e.,

L Ai = E{Zv} =
i

m.

57

J1

L
j=

Ai = 1.

O'~ = E{[Zv -

z t. f}

## =2LA i -Y (Xj, V)-y(V, V)-IIAiAiY(Xi-Xj),

i i i

(II.28)

where Y(Xi, V) denotes the mean value of y(h) when one extremity of the
distance vector h is fixed at point Xi and the other extremity independently
describes volume V.

## Remark 5 Formula (11,28) expresses the estimation variance (T~ as a

linear function of the n weights Ai. The estimation procedure known as
kriging (d. Chapter V) determines the optimal set of weights {Ai, i = 1 to
n}, i.e., the weights Ai which minimize the variance O'~ subject to the
non-bias condition L Ai = 1. Hence, kriging appears as the best linear
unbiased estimator (in short a "BLUE" estimator).

## Remark 6 Formula (11.28) is completely general, whatever the geometry

of the block V and the choice of the weights Ai, subject of course to the
condition L Ai = 1. Thus, the formula can be used to calculate the estimation variance of any linear weighted average type of estimator, such as
inverse distance weighting or weighting by polygons of influence. This is
sufficient justification for the calculation of the serni-variogram (i.e., a
geostatistical approach), even if for reasons of cost or speed we are limited
to one of the usual types of estimator such as polygons of influence.
11.8.3. Case studies

## 1. Block kriging at Chuquicamata after I. Ugarte (1972)

Chuquicamata is an open-pit porphyry-copper deposit in Chile where
short-term planning is based on data from blast-holes. The mean copper
grade of each block V(2D m x 20 m x 13 m) is estimated by kriging from
the blast-hole data available from the nine neighbouring blocks which have
already been mined out, as shown on Fig. II.16, i.e., each block V is
assigned a weighted average of the blast-hole values, the weights being
chosen to ensure the minimum estimation variance. The kriged estimator
of block V will be written z t, Next, the block V is put into production and

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58

MINING GEOSTATISTICS

blast-hole samples are taken from it, an average of six being taken from
each block. Experimental observation has shown that the arithmetic mean
of these six holes, Z v, can be taken as the true grade of block V.
--~-~----------------~~~~-~

c
B

## Available infor motion

over nine blocks

Block V to be eslimated

FIG.

## A test zone of homogeneous mineralization which comprised 397 blocks

gave the histogram of the observed errors (Zv - zt), shown on Fig. 11.17_
The arithmetic mean of the 397 errors was +0-01 % Cu and, given that
the mean copper grade over the test zone was 2% Cu, it is obvious that the
non-bias condition has been met.

% Cu

## FIG. II.l? Histogram of kriging errors. - - , Histogram, m = 0-01, (J"~ = 0-273;

---, Gaussian distribution, m =0, (J"~ =0273.

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## II. THE THEORY OF REGIONALIZED VARIABLES

59

The experimental variance of the 397 errors was 0273 (% CU)2 while
the predicted theoretical estimation variance calculated from formula
(11.28) was 0261 (% CU)2. Therefore, the estimation variance was predicted with the very acceptable relative precision of 4%: (0' 273 -

0261)/0-273

004.

A normal distribution with zero mean and a variance equal to that of the
experimentally observed errors (u~ = 0,273) is also shown on Fig. 11.17. It
fits reasonably well the experimental distribution of errors with the following remarks.

## (i) The normal distribution underestimates the proportion of both small

errors (between O'5% Cu) and large errors (greater than
15% eu).
(ii) The standard normal interval (2UE) correctly estimates the
experimental 95% confidence interval: 96% of all the observed
errors fall within 20'E = 1 04 % Cu. If the predicted theoretical
variance, O'~ = 0261 had been used, the result would have made
little difference: 2UE = 102 % Cu.

## 2. Cartography at Mea after 1. Deraisme (1976, unpublished)

The nickel silicate deposit of Mea (New Caledonia) was sampled by vertical
bore-holes on a pseudo-regular square grid of 80 m, d. Fig. 11.18. One of
the problems in the exploration of this deposit was to map the hanging wall
and foot wall of the mineralized bed. Let Z (x) be the height of the hanging
wall at the horizontal coordinate point x. At each point Xi where a drill
intersected the bed, the height Z(Xi) of the hanging wall was measured. In
order to map the hanging wall, the unknown value z (xo) of the height at

Xj
Z(Xi)

,,/
x

Xo

Xi
0

'0

Foot
wall

"

(J

FJ

4 Qm

FIG.

## II.IB. Cartography of the hanging wall at Mea. 0,

the estimated grid.

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Xj

bore-holes; x,

Xo

node of

60

MINING GEOSTATISTICS

each node of a regular square grid of 40-m side was estimated. The
estimator used was the BLUE Z*(xo) provided by kriging.
Isobath contours of the hanging wall were plotted, using the estimated
40-m grid pattern. Similarly, the foot-wall depth and the mineralized
thickness (hanging-wall height minus foot-wall depth) were plotted.
The kriging estimator z*(xo) used was a linear combination of the nine
bore-hole data closest to X():
9

z*(xo)=

Ajz(xj).

j = 1

## Kriging provides the system of weights which minimizes the estimation

variance:
O'~ (xo) = E{[Z (xo)- z*(xo)f}.

Since the configuration of the available data was not strictly regular, the
estimation variance depends on the position of each point x(} to be estimated.
In order to verify the procedure, each one of the known values z(x;) was
estimated by kriging, i.e., by the optimal linear combination of the nine
closest data values {Xj, j = 1 to 9, and j oj:;
Using the theoretical estimation variance (T~ (x,) and the known experimental error z (x;) - z*(x;), the
deviation ratio e (x.) = [z (Xi)- z*(x; )1/ O'E(X;) was calculated for each data
point Xj. For the theoretical estimation variance (T~ (Xi) to be an effective
estimator of the true estimation variance, the variance of the n values of
e(x;) should be approximately equal to 1.
Figure 11.19 shows the histogram of 167 e(x;) values along with a normal
distribution with the same variance and a zero mean. The arithmetic mean
of the 167 experimental errors [z (Xi) - z *(x;)] is 015 m and, given that the
mean hanging-wall height is 840 ill and the mean bed thickness is 20 m, the
non-bias can be accepted as having been verified.
The variance of the deviation ratio is O'863, which is reasonably close to
1, and, thus, the theoretical estimation variance a~ (x ) provided by formula
(11.15) is an effective predictor of the experimental variance of the errors.
As in the preceding Chuquicarnata case, the normal distribution underestimates both the proportion of small errors (between O' 5 m) and the
proportion of large errors (greater than 2 m). Again, the usual normal
interval of 20'E correctly estimates the 95% experimental confidence
interval: 92% of the deviation ratios fall within this interval (2aE =

n.

186m).
Remark The verification of the estimation method by constructing the
histogram of the experimental deviation ratios, e (x ) = [z (x ) - z*(x)1/ aE(x)

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61

## is sufficiently simple to be carried out systematically, provided that the

support of the elements to be estimated is the same as that of the data used
in the estimation. This is often the case in cartography.

0,1

\
\
\
\
\

\
\

005

\
\

-3

-2

-I

## FIG. 11.19. Histogram of the experimental deviation ratio. - - , Histogram, m =

0-01, (J"~ = 0863; Gaussian distribution, m = 0, (J"~ = 0863.

## II.C. DISPERSION VARIANCE

ILC.1. Definition of the dispersion variance

## In selection problems, it is of little use to know the mean grade of a

working stope if we don't have some measure of the dispersion (or variability) of the grades of production size units within the stope.
Let V be a production stope centred on point x and divided into N
equally sized production units v (Xi) centred on points Xi:
N

V=

2:
i=1

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Vi

=Nv.

62

MINING GEOSTATISTICS

Let Z(y) be the ReV grade at pointy. The mean grade of each uni t v (Xi)
centred on point Xi is

1
Zy(x)=V

f z(y)dy
IN
=- I
N

V(x)

;~1

zv(x;).

## To each of the N positions Xi of the units v (Xi) inside stope V corresponds

their mean value zv (x) can be characterized by the mean square deviation:
2

1
(x)=N

(II.29)

[Zv(X)-ZveXi)] .

i=1

## The dispersion can also be represented by the histogram of the N values

Zv(Xi), i.e., by the frequency of occurence of each value z as shown on Fig.
n.20. In practice, however, at the estimation stage the true grades z,,(x.) of
the production units v are not known and neither is the true grade zv(x) of
the stope V; thus, the histogram of Fig. n.20 is not available.
,...-

<I>
u

,...-

.,

0::

~S2(X}

'"

Vi

1--,...-

In

## FIG. 11.20. Dispersion of units v grades within stope V.

The problem is then to estimate, if not the entire histogram, at least its
two main characteristics, mean zv(x) and variance S2(X).
Probabilistic interpretation

## When the point ReV z (y) is interpreted as a particular realization of a RF

Z (y), the unit grades z v (x;), as well as the stope grade z v (x ), appear as

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## II. THE THEORY OF REGIONALIZED VARIABLES

63

realizations of RV's Zv(x;) and Zv(x); the preceding mean square deviation, s2(x), also appears as a particular realization of a RV S2(X) defined at
point x, i.e., corresponding to the stope V centred on point x:
1 2

S (X)=-L[Zv(x)-Zv(Xi)] .
N;

(11.30)

Variance of dispersion

## Under the hypothesis of stationarity of the point RF Z (y), the stationary

expectation of this RV S\x) is, by definition, the dispersion variance of the
units v within V. The following notation is used:
1
2} .
D 2(vjV)=E{S 2(x)}=E {N~[Zv(x)-Zv(x;)]

(11.31)

Remark It must be stressed that the RV S\x) and its particular realization S2(X) depend on the particular position x of stope Vex), while, under
the hypothesis of stationarity, D 2 (v/ V) no longer depends on the position
x but only on the geometries of v and V and the covariance C(h), as will
be demonstrated.
For each one of K stopes, identical in shape and size, {V(xd, k = 1 to
K}, there is an experimental mean square deviation S2(Xk) calculated from
(II.29). As K ~ co, the arithmetic mean of these K experimental deviations
tends towards the stationary expectation of S2(X), i.e., the variance of
dispersion D\v/ V).
If we are only interested in one particular stope V(x), then D\v/ V) will
be an estimator of the experimental mean square deviation S2(X), just as
the expectation of a RV is an estimator of a particular realization of this
RV.
Example Suppose that a stope V(xd is divided into four equal units
{vex,), i = 1 to 4} and that grades Zv(Xi) are assigned to each unit according
to the result of casting an unbiased six-sided die.
The distribution resulting from casting the die is uniform with a mean
equal to (1 + 2 + 3 +4 + 5 + 6)/6 = 35 and a variance (J"2 = 2917.
Consider three stopes {V(Xk), k = 1 to 3} having the same mean grade
Zv = 35 and consisting of the following three series of four units:
k = 1,

k =2,
k

= 3,

= 6, 3, 2, 3,
z, = 6, 5,1,2,

Zv

= 35,

zv

= 35,

= 225;
S2 = 4.25',

z.;

zv = 35,

s = 275.

Z",

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6, 2, 2, 4,

64

MINING GEOSTATISTICS

## The variance of dispersion

the definition formula (11.31):

## (vl V) of these units v within V is given by

which is equivalent to
2

1
4

22

i~l

## Thus, the three experimental deviations, S2 = 225, 425, 275 fluctuate

around their theoretical expectation n 2 (vl V) = E{S2(X)} = 2917.

## Generalization of the definition of the dispersion variance

The definition (11.31) corresponds to a domain V divided into an exact
number of units v.
When v is very small in relation to the field of dispersion V, v V, every
unit v centred on a point y inside V can be considered as being wholly
within V. This means that the border effect (units v overlapping the border
of V) can be ignored, d. Fig. H.21. The mean square deviation S2(X) is
then an integral over the field of dispersion V:

1
S 2 (x)=V

[Zv(X)-Z1J(Y)] 2 dy.

V(x)

## Under the stationarity hypothesis, the dispersion variance of units v within

V is defined as the stationary expectation of S2(X):

v V.

FIG.

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(II.32)

## II. THE THEORY OF REGIONALIZED VARIABLES

65

It can be shown that the order of the integral sign and the expectation

## symbol can be reversed to give

D2 (v/ V ) = 2V

1
= V

E{[Zv(x)-Z,,(y)]2}dy
Vex)

f O"E'

(II.33)
(V(x), v(y)) dy,

u V.

Vex)

The variance of dispersion D 2 (vl V) thus appears as the mean value over
V of the estimation variance of grade Zv(x) by grade Z,,(y) of a unit v
located inside V.
(In practical applications, care must always be taken to distinguish
2
between the dispersion variance, denoted by D , and the estimation variance, denoted by aI.)

Remark For v = 0, the unit v becomes a point and is said to have a point
support. In this case, the restriction v V is verified and formula (11.33)
gives the variance of dispersion of a point variable within V, which will be
denoted by
1
D 2 (O/V)=V

E{[Zv(x)-Z(y)] 2}dy

(I1.34 )

v(x)

with
Zv(x)=

VI

Z(y)dy.

v{x)

## Experimental dispersion variance

Consider again the particular stope of Fig. 11.20, V consisting of N units v.
Now; a limited number n of the N unit grades (n ~N) may be known and
the mean square deviation of these n values can be calculated according to
formula (II.29):

with

If n is not too small in relation to N and if the n known values z,,(xd are
distributed approximately uniformly over the field V, then s~ can be

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66

MINING GEOSTATISTICS

## considered as an estimator of the dispersion characteristic S2(X) given by

formula (II.29) and we say that s~ is the experimental dispersion variance
of v within V.
II.C.2. Calculation of the dispersion variance

## Let Z(x) be a stationary point RF with expectation m, covariance C(h)

and semi-variogram )'(h).
2(vj
Expression (II.33) gives the dispersion variance D
V) in terms of the
mean value of the estimation variance over the field V:
1
D 2 (vjV)= V

2
CTdV(x),
v(ydy,

v V.

Vex)

## The general formula (11.26) expresses the estimation variance as

CT~(V(x), v(y)= C(V(x), V(x+C(v(y), v(y-2C(V(x), v(y.

As the covariance C(h) is stationary, the first two terms of this last
expression do not depend on the positions x and y of the volumes V and v
but only on their respective geometries; thus, we can write
C(V(X), V(x= C(V, V),

't/x,

and
C(v(y), v(y= C(v, v),

"rIy,

and, as a consequence, these two terms remain invariant when being taken
on average over the field V(x):
l
V

V(x)

~
V

C(V(x), v(ydy

if v V.

V(x)

## Because of the condition v V, the border effect can be neglected and

the displacement of v(y) within V amounts to displacing a point y within
V. When the field Vex) consists of an exact number of units u(y), the
condition v <t: V is no longer required, as the displacement of y within V(x)
can be limited to the discrete number of centres of units v, ct. formula
(II.3l).

## V 2 (vj V ) = C(v, v)-C(V, V).

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(11.35)

67

## The dispersion variance D 2(v/ V) appears as the difference of the two

mean val ues C over the two volumes v and V.
This formula can be rewritten in terms of serni-variogram by replacing C
by -1':

## D (v/ V) = ii( V, V)- ii(v, v).

(11.36)

It can be shown that this last expression remains valid even if the covariance C(h) does not exist, provided that the serni-variogram y(h) is defined
(the RF Z (x) is then only in trinsic).

Remark 1 Krige's relation. The additivity property of dispersion variances can be established as a consequence of the linearity of expressions
(II.35) and (II.36). This relation was found experimentally by D. G. Krige
using data from the gold deposits of Witwatersrand and is thus called
"Krige's relation":

D \ vi G) = D \ v/ V) + D \ VI G),

if v eVe G.

(11.37)

The dispersion of the unit v within the deposit G is equal to the sum of
the dispersion of v in the stope V and the dispersion of these stopes V
within the deposit G.
Note that D2(vIG)~D2(V/G)if v < V.
The dispersion variance decreases as the support increases; it is well
known that there is much more variability among the grades of core
samples (v of several kilograms) than there is among the mean grades of
blocks (V of several thousand tons).

## Remark 2 A priori variance. Consider a very large field of dispersion

(V ~ (0). Under the hypotheses of stationarity and ergodicity, the mean
value Z over this field tends towards the stationary expectation of the
point RF Z (x):
Zv(x)~E{Z(x)}=

when V . . .H )).

## Thus, the dispersion variance D (O/ 00 ) of the point variable in a very

large field V is given, according to formula (11.34), by
D 2(O/oo)= lim
V-a:>

.If
V

E{[Z(y)-m]2}dy.

## If it exists, the a priori variance of the point RF Z (y ) is precisely equal to

the term under the integral sign, i.e.,

Var {Z(y)}

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## = E{fZ(y)- mf} = C(O)

MINING GEOSTATISTICS

68

(ct. relation (11.3)), and, thus, the preceding limit exists and is written

2(O/oo)=

lim D

2(0/V)=Var{Z(y)}=

CeO).

(11.38)

v--,>c<)

When the RF Z (y) has a stationary and finite a priori variance, this
variance is the limit of the dispersion variance of the point variable Z(y) in
a very large field.
Under the hypothesis that this a priori variance exists and, thus, the
covariance C(h) exists, expressions (11.35) and (11.36) give
D

2(0/00)=

## D\O/oo) = y(oo, (0) - 1'(0) = Hco, co) = C(O),

since 1'(0) = O. Thus, for the finite a priori variance RF, we have
C(oo, (0) = C(oo) = 0

(11.39)

and
ii(oo, (0) = '}'(oo) = C(O) = Var {Z (y )}.
However, there are RF's which have an unlimited capacity of dispersion,
at least over an entire deposit or a metallogenic province. In such cases,
neither the a priori variance nor the covariance are defined and the dis2(0/
persion variance D
V) increases indefinitely with V, d. Fig. II. 7.
II.C.3. Case studies

## 1. Grade dispersions at Chuquicamata after I. Ugarte (1972)

This example is taken from the study of the Chuquicamata (Chile) copper
deposit already cited in section II.B.3. This porphyry-copper deposit is
mined by open-pit and the mining unit is a block V of dimensions 20 m x
20 m x 13 m, d. Fig. II.16. On average, each block V contains six blast
holes and experience has shown that the arithmetic mean Zv of the grades
of these six blast-holes can be taken as the mean copper grade of a block.
On a particular mined out bench of area 410 m x 160 m, there are
approximately 1000 blast-holes which are used to provide the mean grades
Zv of 160 blocks V. Using this set of data the following calculations were
(i) The histogram of the dispersion of the blast hole grades over the
field (bench) B, d. Fig. 11.22. The horizontal cross-section of the
blast-holes can be considered as zero when compared to the dimensions of the field B or of the block V and, thus, this histogram
corresponds to a realization of the dispersion law Fom of the

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69

## FIG. 11.22. Dispersion of blast-hole grades in the bench. 13-m blast-hole, m =

212% Cu, 52 = 0939 (% CU)2.

## quasipoint grades in the field B. The theoretical dispersion variance

of this law is D\O/ B).
A lognormal distribution with the same mean (2,12% Cu) and
same variance (S2 = 0939 (% CU)2) has been fitted. The fit is not
perfect, especially at the extremities of the curve (the quality of this
fit can be better appreciated by drawing the experimental and fitted
cumulative distributions on logarithmic probability paper).
The theoretical value D\O/ B) calculated with the semi-vario2(O/
gram through formula (11.36) is D
B) = 0'92, which is close
enough to the experimental dispersion variance S2 = 0939.
(ii) The histogram of dispersion of the block grades Zv in the field B, cf.
Fig. 11.23. This histogram is a realization of the dispersion law F Y1B
of the block grades in B.
2
The theoretical dispersion variance D ( V/ B) calculated from
2(V/B)=O'37,
formula (11.36) is D
which is close enough to the
experimental dispersion S2 = 0346 (% CU)2.
The adjustment of this histogram to a lognormal distribution with
the same mean (216% Cu) and same variance 8 2 is quite satisfactory.
(iii) The histogram of dispersion of the 2 V -block grades in the field B,
ef. Fig. 11.24. These blocks 2 V (40 m x 20 m x 13 m) were obtained
by grouping the blocks V in pairs in the N-S direction.
2(2
The calculated theoretical dispersion variance is D
VIB) =
0287, which is close enough to the experimental dispersion variance S2 = 0269 (% Cui.

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70

MINING GEOSTATlSTICS

~
~

0-1

\
\

I
I
I
J
J
I
'I

001

02

--

## FIG, II.23. Dispersion of V-block grades in the bench. Block V,20mx20rn x

13 rn: rn =2'16, S2 = 0346% Cu:t.

## The adjustment of this histogram to a lognormal distribution with

the same mean (217% Cu) and same variance S2 is not satisfactory
at all.
(The difference between the experimental means of the blastholes and those of the blocks V and 2 V is explained by the fact that
the grid of blast-holes is not regular and that the field B is not
divided into an exact number of blocks V and 2 V).

## As a conclusion concerning the type of the dispersion laws, it can be seen

that, if all three histograms of Figs 11.22 to 11.24 present the same asymmetry, only the histogram of the dispersion of the V -block can be correctly
fitted to a two-parameter (m and S2) lognormal distribution.
2. Grades dispersion in the Nimba Range
The iron deposits of the Nimba range (Liberia and Guinea) are of itabirite
type concentrated in iron oxides by leaching of the silica.
Figure IL25(a) and (b) gives the histograms of the dispersion of the
grades in Fe and Si0 2 impurity measured in core samples of constant
length 5 m. The field of dispersion is that of a particularly enriched surface
ore essentially made up of hematite.
The extreme homogeneity of this are is demonstrated by a clearly
unimodal Fe histogram of inverse-lognormal type. If Z (x) is the Fe grade

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71

\
~
a;

O~

>,

oc;

<1>

:>

...

<:T

\
\

-,
'\

001

-,

## FIG. II.24. Dispersion of 2 V -block grades in the bench. 2 V block, 40 m x 20 m x

13 m: m = 2170/" CU,

S2 =

()269

Clio

CU)2.

## expressed as a percentage (%), the variable log [70 - Z (x)] is distributed

according to a normal law. The value 70% corresponds exactly to the
maximum iron grade of a sample composed entirely of hematite Fe203.
The adjustment of an inverse-lognormal distribution with the same mean
(666% Fe) and same variance (52 = 23 (% Fe)2) is satisfactory.
The histogram of the Si0 2 grades is of straightforward lognormal type
and can be adjusted to a lognormal distribution with the same mean
(0,58% Si0 2) and same variance (S2 = 020 (% Si0 2)2). The adjustment is
satisfactory at least for Si0 2 grades greater than the mean O' 58.
However, the histograms are not always unimodal. Figure 11.26 shows
the histogram of the Fe grades over a much larger field, covering not only
the surface hematite are but also the deeper and poorer are (little altered
itabirite). Note that this global histogram is multimodal with a first mode of
67% Fe corresponding to the hematite are and several other modes around
50% Fe corresponding to the various poorer ores.
Figure II.27 shows the histogram of the dispersion of Fe grades within
the poor ores. This histogram appears as a mere development of the (30 to
60% Fe) part of the preceding global histogram.
Note again the extreme homogeneity (52 = 23 (% Fe)2) of the hematite
ore in relation to the variability of the grades of the deeper, poorer ores
(52 = 50.1 %).

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(a)

I I

II
II
)
~
~

II
I

0 I

>.

...

<::
::l

<:r
Ol.l

0-01

65

70

(b)

03

2
%

sic,

## FIG. 11.25 (a). Dispersion of Fe grades (hematite ore). m = 666% Fe, S2 = 2 3

(% Fef. (b) Dispersion of Si0 2 grades (hematite ore). m =0.58% Fe, s2=020
(% Si0 2 ?

VNNI.minebook.blogfa.com

~
s;
o-,

'"0'"
~

0-1

:::J

70

50

%.

Fe

s~

= 84.9% 2.

~
~

>-

..

0-1

1,I

....

:::J

0-

u,

I
I
I
I
~

I
30

I
50

"/0

70
Fe

## FIG.IL27. Dispersion of Fe grades (deep poor ores). m:= 505% Fe,

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S2::;:

50-1%2.

74

## 3. Grade distribution at Haartebeestfontein after D. G. Krige (1976) and 1.

M. Rendu (1976)

The field of dispersion B is a section (3000 m x 900 m) of the Haartebeestfontein gold mine in South Africa. Each unit v (75 m x 75 m) contains
between zero and four samples (two on average) and the arithmetic mean
of the samples, Z"" in each unit v is said to represent the cluster of samples
inside the unit v.
10000

;/

/"X
2000
'I

.,

....-:::;~~

./lC

~JP

log (Z,+o)

\
x?",
1.~~;7x

1000

~~>-

::>
C

>

I
200

x~x -.

-r:>
>:

x/

109

r,

..,.,.L.x~ xL
/
/x

100

/
10

20

Curnulc tive

FIG.

90

30
f req uancy (,,; x

99 995

## Figure II.28 shows the cumulative histogram of the dispersion of the

grades Z; in the field B in normal logarithmic coordinates. Note that for
the low grades Z; there is a significant deviation compared with the line
corresponding to the lognormal model.
The two-parameter lognormal model can be improved by introducing a
third parameter a, so that log (Z, + a), rather than log Z; is assumed to be
distributed normally. With a = 150 g em/ton, a lognormal model can
be fitted remarkably well to the experimental distribution of the grades
(Z, +a).

## II.C.4. Type of distribution functions

It would not be possible within the bounds of this book to present all the
following remarks, drawn from practical experience, can be made.

VNIW.minebook.blogfa.com

## II. THE THEORY OF REGIONALIZED VARIABLES

75

(i) When the dimensions of support v are small compared with those of
the field of dispersion V (e.g., v representing a core sample and Va
mining unit or a stope), then (a) the histogram of the grades of
elements of weak concentration (Au, Ag, precious metals, U, Cu,
Ni, impurities in weak concentration such as Si0 2 , Ah03, Fe, ... )
are asymmetric with a tail towards the higher grades (direct
lognormal type); (b) the histogram of the grades of macro-elements
of strong concentration (Fe in deposit, Mn in deposit, phosphate
P 20s, ... ) are asymmetric with a tail towards the lower grades
(inverse lognormal type).
(ii) As the support v increases in size (e.g., v representing a mining unit)
a blending effect is produced, which tends to slowly tone down the
previous asymmetries. Lognormality is not preserved when the
support v becomes too large.
(iii) A cleary multimodal histogram reveals, in general, t the existence of
heterogeneous mineralizations (the presence of several types of ore
or the influence of poorly mineralized areas on the fringes of the
deposit). The interpretation of such a histogram must be made with
due regard to statistics (e.g., fluctuation due to an insufficient
number of samples), geometry (e.g., position of the field of dispersion in the deposit) and geology (e.g., mineralogy, faults, directions of enrichment of the are).
(iv) The preceding remarks concern the grades of elements. For
geometric-type regionalized variables, such as mineralized or overburden thickness, or mining type ReV such as percentages of ore
recovery or granulometric ratios, the histograms are of a more
symmetrical type, as shown on Fig. 1I.29(a) and (b).
Figure 1I.29(a) shows the histogram of the mineralized Ni-Iaterites
thickness of the Prony are body (New Caledonia). The histogram was
constructed from 176 values measured on the 176 vertical bore-holes of a
regular square grid of lOO-m side.
Figure I1.29(b) shows the histogram of the percentage of are (mesh
>6 mm) recovered on mining units at Moanda (Gabon) manganese
deposit. This histogram was constructed from the results of mining 1400
unit blocks corresponding to an area of about 42 ha.

t But the presence of several modes on a histogram does not imply that each of these modes
necessarily corresponds to one type of mineral that can be distinguished at the mining stage.
For example, a banded hematite quartzite ore surveyed by drill cores (of a metre or so in
length) would show a multimodal histogram of the type shown on Fig. II.27 and still be
considered as an homogeneous are at the scale of mining (on blocks of 10 or 20 m in
dimension).

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76

MINING GEOSTATISTICS

## Of course, non-symmetrical histograms of thicknesses can be found just

as well as symmetrical histograms of grades. The previous remarks are not
to be taken as laws. As an example, Fig. 11.30 shows a histogram of
exponential type corresponding to the vertical thicknesses of bauxite lenses
within a zone of the Delphi deposit (Greece). Only the positive drill-holes
that actually meet a mineralized lens (i.e., with a non-zero mineralized
thickness) were retained for this histogram. Indeed 60% of the drill-holes

f---

l-

( 0)

(b)

f-

i:? 4

20 -

f-

<:
."

g.

f---

f-

."

10 f-

f-

mfm
10

124

~ rhh,n
20

30

40

4f-

2 f-n

rrf

51-5

Thickness (rn)

57-2

64-7

Mn (%)

~/

20

Va

'"
&
u
<:

<l>

10

'"
LI..

Negative
drillholes

I 2

5
Thickness {rn)

FIG.

## II,30. Dispersion of bauxite thicknesses.

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77

## were negative and would have appeared on a global histogram as a strong

atom at the origin (zero thickness).

11.0. REGULARIZATION

## 11.0.1. Regularization and estimation variance

Very rarely, in practice, will point data z(x) be available. Most often, the
available data z; (x) will be defined on a certain support v (x) centred on a
point x, v being, for example, a core sample or, more generally, the volume
of a sample. The grade zu(x) of this sample is the mean value of the point

z,,(x)=!
v

z(y)dy.

,,(x)

## The mean value z,,(x) is said to be the regularization of the point

variable z (y) over the vol ume v (x).
Let the point-regionalized variable z (y) be a particular realization of a
second-order stationary RF Z(y), with expectation m and covariance C(h)
or variogram 21'(h). The regularization of the point RF Z(y) over the
volume vex) is, likewise, a RF denoted by Z,,(x) and written as

Z.,(x)=!
v

Z(y)dy.

v(xl

## It can be shown that the regularized RF Z,,(x) of a second-order stationary

point RF is also second-order stationary, with
(i) an expectation identical to the point expectation m - indeed,

E{Z.,ex)}=!

v "

E{Z(y)} dy

=! f
v

u~)

m. dy

=m,

Vx;

## (ii) a variogram 21" (h) defined, according to formula (1104), as

21',,(h) = E{[Z,,(x

## + h)- Z,,(x )]2}.

The problem is then to derive this regularized variogram 21'v (h) from the
point variogram 2y(h). To do so, an easy way is to consider the expression
(HAD) of the regularized variogram as the variance of the estimation of the
mean grade Z" (x) by the mean grade Z; (x + h) separated by the vector h,
cf. section ILB.2, Remark 3. This estimation variance is then given by the
general formula (11.27):
21'v(h)= 2y(v(x), vex + h ) - y(v(x), v(x ) - y(v(x + h), vex + h).

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78

MINING GEOSTATISTICS

Since the point serni-variogram "1(h) is stationary, the last two terms of
the previous expression are equal and, thus,

(H.4l)

## denoting the support v translated from v by the vector h, y(v, Vk)

represents classically the mean value of the point semi-variogram y(u)
when one of the extremities of the vector u describes the support v and the
other extremity independently describes the translated support VkVk

Remark 1 For distances h which are very large in comparison with the
dimension of support u, the mean value y(u, Uh) is approximately equal to
the value y(h) of the point semi-variograrn and we obtain the very useful
practical approximation
y,,(h) ~ y(h)- Yev, v)

for h v.

(11.42)

## At large distances, h v, the regularized serni-variogram is simply derived

from the point serni-variograrn by subtracting a constant term Hv, u)
related to the dimensions and geometry of the support v of the regularization, ct. Fig. II.3l.

p,,", ,,"
C(O)

---- -----

--~-------I---=

~g\Jlon~_~.~__S.~_I_~_

/// -:
L -/

,--

FIG.

.....- .....-

/v(h)

...-

...-/

!y(v,v)

...-

,/

---

----------

## Remark 2 Regularized covariance and a priori variance. It can be shown

that if the point a priori variance Var {Z(y)} = C(O) exists, so do the
regularized a priori variance C" (0) and the regularized covariance C" (h),
which are written in terms of mean values C of the point covariance C(h):

(11.43 )

(II.44 )

## Hence, if the point serni-variogram y(h) is of a transition type with a sill

value equal to the point a priori variance, "1(00) = C(O) = Var{Z (y )}, cf.

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79

## formula (11.39), the regularized semi-variogram rv(h) is also of a transition I

type with a sill value equal to the regularized a priori variance:

(lIAS)
cf. Fig. II .31.

## This type of regularization, very frequent in practice, corresponds to the

construction of the variogram of the mean grades of core samples along the
length of a bore-hole, d. Fig. 11.32. It is assumed that all the core samples
have the same length l and the same cross-sectional area s.
s

x+h

~----+--.

--t-.

## Let Z (x) be a point RF which is either second-order stationary or at

least intrinsic with a semi-variogram y(h).
The RF regularized on the support v = s x I of the core sample is written

Zv(x)=!
v

Z(y)dy,

v(x)

where the sign Iv(x) dy represents, in fact, a triple integral over the volume
v. When the diameter of the core is smallt compared to length I, the
regularization effect of the cross-sectional area s of the core sample can be
t This approximation will not be valid for samples taken from sampling pits, for example,

because the diameter of the pits will be of the order of the distance between the samples (0' 5
to 1 m).

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80

MINING GEOSTATISTICS

neglected. The mean value over the length I of the core sample can then be
written in terms of a single integral in the direction of the length of the core
as
Zv(X)=Z/(X)=_,l

Z(y)dy.

(11.46)

I(x)

Regularized semi-uariogram

When the cross-sectional area s of the core sample is negligible, two core
samples can be considered as two aligned segments I and I" of the same
length I and separated by a distance h, ct. Fig. II.32.
According to formula (H.4l), the regularized semi-variogram can then
be written
(11.47)

## This regularization by cores assimilated to a segment of lengthI will be

illustrated on the four theoretical point variogram models in current use
(these models are precisely defined in section III.B.2).
This part is not essential for a complete understanding of the problem of
regularization, and the reader may go directly to the next section, ILD.3,
"Grading over a constant thickness", which provides another example of
regularization.
1. The linear model
y(h) = r

with r = Ih I.

If u denotes the coordinate along the length of the bore-hole, the first term
y(l, lh) of the expression (11.47) for the regularized semi-variogram is

written

## (Recall here that vector h is parallel to the direction of the bore-hole

length.)
According to whether or not r = ]hl is less than I, i.e., whether or not the
two segments land lh overlap (d. Fig. 11.33), we have the following two
expressions.

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r

r+1

I
-----... u

81
r+1

t,

u'

u
r~1

FIG.

0)

Ihl~/:
2

1 y (l, lh)=

du

## 11.33, Calculation of the term Hi, h,).

f+I(U'-U)dU+ fdU[i

(u-u')du'

f+1 (ul-U)du}

1

,2

## y(l,lh)=31 2(31-r)+3 forr=lhl~1

and, when h

0, y(l, I) = 1/3,

Oi) Ihl~/:

## and, thus, y(l, lh) = r for r = Ih I ~ l.

Applying formula (IlA?), the regularized serni-variogram is

'Yl(h) = !;;2(3/-r),
r

-1/3,

## Vr= IhlE [0,

III

(H.4S)

Vr ~ I,

d. Fig. 11.34.
For distances Ih I~ I, the behaviour at the orrgin of the regularized
semi-variogram is parabolic. The cores of length I overlap each other and
the regularizing effect is strong enough to change the behaviour of the model
at the origin from linear to parabolic.
For distances Ihl;?= I, which, in practice, are the only distances observable,
the linear behaviour of the point model is preserved and the regularized
model differs from the point model by a constant value
'Yl(h) = y(h) - y(l, I) = r -l/3,

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r = Ih I?: l.

,,"",U4i

CtJ J 0.-.. ~ ~

82

MINING GEOSTATISTICS

(Note that for the linear point model the approximation of formula (II.42)
becomes a strict equality.)
If the regularized curve for Ih I ~ I is projected towards h = 0, the
ordinate axis is intercepted at -1/3 and this negative value is called a
"pseudo-negative nugget effect" due to regularization.

2l f - - - - - - - r - - - - - -

'Y

L.

.:

. i //

.....

3~

## 2. The logarithmic model

y (h) =

log,

with, =

Ih I.

Note that for h = 0, log Ih 1 = -00 :;t- 0, so that this logarithmic model cannot
be used for point samples, only models Yo (h), regularized on non-point
supports, are defined.
For distances observed in practice, Ih I~ I, the serni-variogram
regularized by cores is written as

(r+/)2
yJCh) =

2z2

log (, + I) +

(,_l)2

2z2

log (r -I)

(11.49)

,2
- -}2 log r - log I,

'Vr ";3/.

In practice, this exact formula is rather difficult to use and the approximation of formula (11.42) is preferred:
ydh)=y(h)-y(l, l)=log

r 3

/+2'

with

## y(l, I)=log l-l

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Vr ";3/,

(II.50)

83

## The relative error involved in this approximation is 8% for r = I and 01 %

orr=51.
3. The spherical model
3 r 1,2
"2 a -2 a 3 ,

yCh)=

\ir =

1 = sill value,

## Ihl E [0, al,

\ir;:. a = range.

This model reaches a sill value, y(co) = 1, for a distance equal to the range
a, cf. Fig. IlLS.
The expressions for the semi-variograrn YICh) regularized by cores are
long and awkward. They can be found along with those for the exponential
model in Ch. Huijbregts (1971 b).
In practice we can make use of the following charts, given in section

ILEA:
(i) chart no. 21, which gives the values of the first experimentally
accessible points of Yi(h) and also of the sill value ydoo) as functions
of all;
(ii) charts no. 22(a) and 22(b), which give the values of YI(h) for
different values of all as a function of Ih Ill.
Furthermore, for the practical distances Ih I~ I, the regularized semivariogram can be derived from the point model by the approximate
formula (I1.42):

Yi(h)

0=

## yCh) - f(l, I),

for IhI;:'l,

the mean value f(l, I) for l ~ a (the length is not greater than the range),
being given by the expression
3

l
1
y(l, 1)= 2a - 20a3'

\il~a.

(I1.51)

Given the conditions, Ih I~ I and a ';3 31, which cover almost all practical
cases, this approximation is excellent.

## 4. The exponential model

yCh) = 1- exp (-rl a),

\ir = Ihl.

## This model reaches its sill value, y(co) = 1, asymptotically. In practice, a

practical range a I = 3a is taken, for which
yea') = 095

d. Fig. III.5.

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0=

sill 1,

MINING GEOSTATISTICS

84

## The semi-variograrn regularized by cores of length 1 takes the form

2
2
2
2
ar+ a- e -'/0.(2 -e -f/a) - a e (,-f)la
-2a (--I/O.
e
- 1) + Vr:%,; Z,
Z2

[2

YI(h) =

z2

Z2'

2)(1 -e -,/0.)

(11.52)

Vr';3l.

## In practice, the regularized serni-variogram can be deduced from the

point model by the approximate formula (HA2):
I'I(h) = y(h) - y(l, l)

for lh I;:;.Z,

## the mean value y(l, Z) being given by the expression

2

a [ 2 e -110. +~-2.
21
y(l, l)= 1-7

(11.53)

## Given the conditions, Ih 1;:;.1 and a

3a ;:;. 3/, which cover almost all practical cases, this approximation is excellent.
1=

Regularized range For a transition point model, i.e., for a point model
with a sill and a (practical or actual) range a, two point variables Z (x) and
Z(x + h) are uncorrelated when the distance h separating them is greater
than the range (lh I> a.)
As a consequence, the two variables regularized by cores of length I,
Z, (x) and Z, (x + h) are uncorrelated when the distance between each point
of l(x) and each point of lex + h) is greater than the range a, cf. Fig. II.35;
for which it is necessary and sufficient that lh I> a + 1.
The regularized range is, thus, (a + 1) and this is the distance at which the
regularized model )'f(h) reaches (actually or practically) its sill y/(oo). Note
that the previous practical approximation, 'Y1(h) = y(h) - y(l, l) amounts to
assigning to the regularized model the same range as that of the point

x+1I

0+1

## FIG. 11.35. Regularized range.

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## II. THE THEORY OF REGIONALIZED VARIABLES

85

model. In fact, for the two transition models in current use (the spherical
and the exponential), when IhI> a and a ~ 31, ydh) can be approximated
with excellent precision by its sill value y/{co).
11.0.3. Grading over a constant thickness

## This second type of regularization is also very common in practice and

corresponds to taking the mean over a constant thickness I of a horizontal
bench or of a vein of more or less constant thickness, cf. Fig. 11.36. Let
(xu, xv, x w ) be the three coordinates defining the position x of the point RF
Z (x) in the three-dimensional space. The point RF is assumed to be
second-order stationary or at least intrinsic and to have the semi-variogram
y(h u , hv , h w ) .
w

---f--- f
BenCh
__J

OeD

(x+h)

Horizontal plane

## If X w and h w represent the coordinates in the vertical direction, then the

mean value over the constant thickness I of a horizontal bench is given by

111 Z(X

Zo(X u , xv)=i

u,

xv, xw)dx w

(II.54 )

## We thus define a new RF, Zdx u , xv), which is a regularization of the

point RF Z (xu, xv, x w ) and is studied in the horizontal two-dimensional
plane (x"' xv) or, more generally, in the plane which is orthogonal to the
direction X w of regularization. ZG(x u , xv) is called the "grading of the point
RF over the constant thickness I".
(There is some difficulty here with terminology. The original French word
"mantee" has been translated as "grading" in the sense of a regularization
over an intervaL)

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86

## Example If Z(X u , xu, x",) is the point grade in a horizontal stratiform

formation of quasi-constant thickness l, then Zo(x u , xu) is the meant grade
of a vertical bore-hole positioned at the point of horizontal coordinates
(xu, xv) on the topographical surface.
Remark Even though the support of the regularization (I) may be the
same, care should always be taken to distinguish between grading over a
constant thickness l and regularization by core samples of length t along
the length of a bore-hole, d. Figs 1I.36 and 11.32. In the case of grading,
the two variables Zo(x u, xv) and ZG(x u + h.; Xu +~ are defined on two
segments I and lh which are parallel and separate(f by a distance vector
h = (h", h u ) orthogonal to the direction of the length l, while, in the case of
regularization by cores, the two segments I and lh are aligned and
separated by a distance vector h parallel to the direction of length I.

## The serni-variogram of the graded RF over the constant thickness I is

written in the standard form of formula (II .41):

(11.55)
where x = (xu, xv) and h = (h", h v) denote the coordinates in the plane
orthogonal to the direction of the grading thickness, d. Fig. 11.36.
In calculating the mean value y(l, Ih), three directions in space are
involved and the calculation is simple only when the point model
vtb: h.; h w ) is isotropic, i.e., if 'Y(h)= 'Y(r) with
r=(hl=J(h;+h~+h:)

## If the point model is isotropic in the three-dimensional space (h u , h.; h",),

the graded serni-variogram 'Yo(h u , h u ) is also isotropic in the two-dimensional space (h", h v ) and is written in terms of simple integrals, ct. Fig.
11.37, as

2 ('
yo(h)= ,2. J (l- Y)y(J(r2. + y2)) dy
o

-,2 Jf'
2

(l- y)y(y) dy

(II.56)

## with r = jhl = h~ + h~.

2

t The regularization effect of the cross-sectional area s of the core can be neglected in relation
to the thickness f of the formation when integrating over the support (v = s x f) of the
intersection of the bore-hole with the formation.

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87

## Application to point models in current use

This regularization by grading will be illustrated for the four point models
in current use. These point models will be supposed to be isotropic in the
three-dimensional space.
I

Mk::----------~

r
M'

## FIG. II.37. Calculation of the term HI, Ih ) .

This part is not essential and the reader may go directly to the next
section, II.DA.
1. The linear model
y(h)=r

withr=1hl.

## The exact isotropic expression for the serni-variogram graded over a

constant thickness I can be found by expanding formula (II. 56):
2

1+ J(r +, /2) 2 r
J 2 2
[
YG(h)=3 (r +[ )+,log
r
+3" [2(r- (r +l -3
1J

(I1.57)

## with y(l, [) = 113.

In practice, this exact expression will be used only for small distances
Ihl < l. For Ihl;;:: l, the standard approximation given in (HA2) is used
henceforth:
ydh) = y(h) - ii(l, I) = r -lI3,

V r = Ihi;;:: l.

(I1.58)

## The relative error corresponding to this approximation is 10% for Ih I= I,

1 % for Ihl = 31 and less than 01 % for Ih 1= 101.
Remark For distances h ";3 I, the model rdh) graded over the length I is
practically indistinguishable from the model Yl (h) regularized by cores of
length I, d. formulae (II.58) and (lIAB). On the other hand, for distances
Ihi;;;:; I, the effect of regularization by cores of length l is greater than that of

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88

MINING GEOSTATISTICS

## regularization by grading over I and we have l'/(h)::';; 'YG(h). Such distances

Ihl::.;; I are scarcely considered on the core-regularized serni-variograrn
'Y/(h).

## 2. The logarithmic model

y(h)=logr

withr=lhl.

(Recall that this logarithmic model should only be used when regularized
over non-point supports.)
The exact isotropic expression of the semi-variograrn graded over I is
found by expanding formula (I1.56):

28
log cos 8
.
'YG(h)=--+
2
-log Sin 8
tan ()
tan ()

(II.59)

## with () = arc tan 1/rand y(l, I) = log 1- 3/2.

This formula gives rise to the following limited expansions:
for' ~ I,
for,

I,

r r2
r 3,2 1 r4
l'G(h)=7f/+[ilog,-zP-12 [4+"';

1 [2 1 14
2+ log [+12 r2 ~ 60 ,4 + ....
3

YG(h) =

## In practice, for distances

the standard expression

(II.60)

(II.61 )

## The relative error involved in this approximation is 6% for r = I, 1% for

r = 2/, 01 % for r = 51.

3,4. The spherical and exponential models The analytical expressions for
these graded semi-variograms are long and awkward to handle and for this
reason the following charts are used:
(i) chart no. 1, which gives the graded serni-variogram of the point
spherical model (with sill 1 and range a) as a function of Ih a and
for different values of 1/ a;
(ii) chart no. 11, which gives the graded serni-variogram of the point
exponential model (with sill 1 and parameter a and, thus, practical
range a'=3a) as a function of Ihl/a and for different values of ll a.

1/

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## II. THE THEORY OF REGIONALIZED VARIABLES

89

The graded sill is defined as follows. If the sill of the point model is 1,
then, according to formula (HA2), the sill of the model graded over [ is
1'0(00) = 1- y(l, f),

the mean values .y(l, I) being given by formulae (11.51) and (II.53).
The model graded over l and the model regularized by cores of length [
thus have the same sill.
The graded range may also be defined. When the distance Ihl becomes
greater than the (actual or practical) range, every point of [ is independent
of every point of L; and, as a consequence, the range of the graded model is
equal to the range of the point model, whatever the grading thickness [
may be.
Recall that the range of the model regularized by cores of length I is,
strictly, equal to (a + I), where a is the range of the point model.
In practice, for both the spherical and the exponential models we have
the following.
(i) For distances
(IIA2):

## lh] ;3[, we will use the standard approximation of

yo(h) = 1'(h) - y(l, I).

(11.62)

## The relative error involved in this approximation never exceeds 10%

in the most unfavourable cases (e.g., Ihl = I and If a = 0,1) and is less
than 1 % in the majority of cases encountered in practice (!h! > 3I
whatever the ratio If a may be).
(ii) For distances !h] < l, we can either (a) use the exact formulae of the
graded models given in J. Serra, (1967 a), Vol. 2, pp. 446 and 461) or
(b) if a I (in practice if the actual or practical range is greater than
3[), we can assimilate the point models to their tangents at the origin
(linear model of slope w); the graded models can then be deduced
from formulae (II.57) or (11.58) corresponding to the linear model.

## II.D.4. Practical rule for regularization

The two types of regularization considered in the two previous sections are
particularly useful, for they correspond to the two main data configurations
encountered in practice: core samples aligned along a bore-hole and data
defined over the constant thickness of a mining bench.
Although these two types of regularization are different in their principle, they lead to a common approximate formula giving the regularized
semi -variogram
[ 'Y/(h)

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= y(h) -

y(l, f),

vlhl ~ I. 1

(11.63)

90

MINING GEOSTATISTICS

## The regularized model is deduced from the point model by subtracting a

constant term y(l, I) which depends only on the length I of the support of
the regularization.

## Remark 1 For the regularization by cores of length I of transition-type

point model with a (practical or actual) range a, the additional condition
a 31 should also be satisfied before applying the approximation (11.63).
In practice, this additional condition is often met, since the length I of the
data support should, generally, be much smaller than the range a of the
sought after structure.
Remark 2 Positive definite-type functions for regularized model. As it is an
approximation, the expression (11.63) for the regularized model is not
strictly a conditional positive definite function; hence, we are not sure that
all variances calculated from this approximate expression of 1'1 (h) will be
strictly positive or zero, cf, the condition (ILl 0) in section II.A.3.
Strictly speaking, if the point semi-variogram 'Y(h) is of a certain modelspherical for instance - the regularized serni-variograrn 'Y1(h) is not of the
same model, However, for practical applications, it is an acceptable
approximation to consider that the model is the same for the regularized
'Yt(h) - i.e., spherical - with, of course, parameters (regularized range and
sill) different from those of the point model y(h). This model for 'YI(h)
would then be a conditional positive definite function and ensures that all
variances calculated from it are non-negative.
Thus, if the point serni-variogram 'Y(h) is a spherical model with range a
and sill C, the regularized serni-variogram Yt(h) can be assimilated (for
distances Ihl ~ l) to a spherical model with
(i) a regularized sill, C I = C - y(l, I);
(ii) a regularized range at equal to (a + I) if the regularization is done by
cores of length I, and equal to a if the regularization is a grading over
the constant thickness l.

## Remark 3 Deconvolution (passing from 'Yo to 'Y). In practice, since the

available data are most often defined on a non-point support v, it is often
required to work out a theoretical point model y(h) which is coherent with
the observed regularized semi-variograrn 'Y~ (h). Deducing a point model
'Y(h) from a regularized model 'Yo (h) amounts to a "deregularization" or a
"deconvolution" of the model 'Yo (h). In practice, this deconvolution is
made through the following identification procedures.

0) A point model
curve

## 'Yt (h) experimentally available.

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## II. THE THEORY OF REGIONALIZED VARIABLES

91

(ii) The regularized theoretical expression 'Y... (h) (or its approximation
given by formula (11.63 of this point model y(h) is then calculated
and compared with the experimental curve y~ (h). The various
parameters (ranges, sills, ratios of anisotropies, etc.) of the point
model are then adjusted in such a way as to bring 'Yv (h) in line with
y~ (h).

(iii) Once the point model y(h) is determined, the theoretical expression
'Y.,{h) for the regularization over a possible second support VI can be
deduced and checked with the corresponding experimental curve
'Y~' (h).
The calculation of the regularized theoretical expressions 'Yo (h)
and 'Yv{h) is simple only when the point model consists of the
established models which are in current use, and for which formulae
and graphs are available, e.g., if the point semi-variogram y(h) is a
spherical model.
11.0.5. Case studies

## The iron (banded hematite quartzite) deposit of Tazadit (Mauritania) was

sampled by vertical bore-holes. The particular bore-hole considered for
this example was 177 m (L) long and was cut into core samples of constant
length 1 = 3 m.
Let Z (x) be the point RF representing the iron grade regionalization,
Z/(x) the RF regularized by cores of length 1= 3 m and Z4/(X) the RF
regularized by cores of length 41 = 12 m (obtained by grouping the available core samples in fours).
Figure II,38 gives the two experimental serni-variograms of the two
variables Z/(x) and Z4dx) (the circles. correspond to the experimental
points of 'YJ and the crosses x to those of Y41).
These semi-variograms reach a sill value for a range a of approximately
40 m. The sill C/ is approximately 133 (% Fef If a spherical model with
range a =' 40 m and sill C is assumed for the theoretical point model y(h),
then the theoretical value of the sill C, is written, according to formula
(11.51), as

with 1 =;: 3 m, a = 40 m and C, =;: 133. Thus, the sill of the point spherical
model is C= 138 (% Fel.

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92

MINING GEOSTATISTICS

In Fig. 11.38, the theoretical point spherical model y(h) with range
a = 40 m and sill C = 138 is shown as a dashed line. The theoretical
regularized model YI deduced from the previous point model by the practical approximation (II.63),
ydh) = y(h) - 1'(1, I),

Yh

a> 3/ = 12 m,

I,

## is shown as a first solid line. The regularized model

same approximation is shown as a second solid line.

200

'Y41

deduced by the

f-

150 f-

12

24

36
h

FIG.

48

60

72

(m)

points: e, 1'1; x, 1'41. L = 177 m, ~ = 492% Fe, S = 120 (% Fef

## The theoretical value of the sill C 4 1 is found from formula (I1.51) to be

41 (41)3
C 4 1 = 1'41(00)= C [ 1 - - +--3

2a

20a

= 117'5,

## and correctly fits the experimentally observed sill.

Thus, the practical rule (II.63) for the regularization by cores of length 1
provides models 'Yl and Y41 which are very good fits to the corresponding
experimental semi-variograms.

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93

## Variance of dispersion within L The experimental variance of dispersion

of the grades of the 59 cores of length I along the total length L of the
bore-hole was found to be S2 = 120 (% Fe)2 and the mean grade m =
492% Fe.
2
The corresponding theoretical dispersion variance
(l j L) can be calculated from formula (11.36) either (i) by using the previous point spherical
model y(h) and formula (H.9S),

## or (ii) by assimilating the core length I to a quasi-point support and by

considering a point spherical model with the same range a = 40 m and a sill
C, = 133,

111-9% 2.

52

120.

## 2. Regularized and graded variograms at Mounana after G. Matheron

(1962)
The massive Mounana (Gabon) uranium ore body was sampled by:
(i) vertical bore-holes cut into cores of length 1= 05 m and analysed for
(ii) several horizontal mine workings along which vertical channel
samples of length 21 = 1 ill were taken at regular horizontal intervals
of 1 m, d. Fig. II.39.

## As the histogram of the dispersion of the uranium grades in the deposit is

clearly lognormal, the ReV studied is the logarithm of the grade which is
then normally distributed.
Figure 11.39 also shows, in semi-logarithmic coordinates:
(i) the experimental serni-variograrn 'Yt (h) regularized by cores of
length I and computed from the vertical information (bore-holes);
(ii) the experimental semi-variograrn 1't (h) graded over the length 2l
and computed from the horizontal information (channel samples).

## As the experimental points of 1'7 (h) are reasonably well aligned, a

logarithmic model regularized by cores of length I = O'5 was fitted according to the approximation (II.50):

'Y/(h)= C[ log

~+~J

## h being a vertical distance, with C = 027.

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for h

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MINING GEOSTATISTICS

94

## If the mineralization is isotropic in the three-dimensional space, it should

be possible to fit the experimental horizontal semi-variograrn 'Y~ (h) to a

approximation as

2[.

## This graded model is given by the same

But the experimental curve 'Y~ (h) is clearly above the theoretical
straight line 'YG(h) derived from the hypothesis of isotropy, and we can,
thus, conclude that the phenomenon is anisotropic: the horizontal variability is greater than the vertical variability.
(Note that this study dealt only with small distances, h being less than
10 m in Fig. II.39.)

1 1-~

Vertical

2/

(l

Ho riz ont o l

10

05

7 8 9 10

20

2h (m)

## FrG. II.39. Core-regularized and graded semi-variograms.

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95

## The main geostatistical operations (estimation and dispersion variances,

regularization, kriging) make continuous use of mean values j of the point
semi-variograrn y(h) such as
j(v,

Vi) =

f f,,'

vv"

dx

## y(x - x') dx '.

(II.65)

y(v, Vi) denotes the mean value of the function y(h) when one extremity of
the vector h describes the domain v and the other extremity independently
describes the domain Vi.
It is recalled that if the domain v is three-dimensional, then the notation
Iv dx in fact represents the triple integral Ht d.r, dx, dx w , (xu, xv, x w ) being
the three coordinates of the point x.
Thus, in the three-dimensional space, the mean value f(v, Vi) between
the two volumes v and ' involves a sextuple integral. The advantage of
avoiding a direct analytical computation of these sextuple integrals is
obvious.
Two solutions are possible. The first one is to calculate the mean values j
numerically using a computer. The second one is to break the analytical
resolution of the multiple integrals into successive stages, some of them
being calculated in advance once for all, These stages or intermediate
integrations correspond to the definition of auxiliary functions which can
be presented either as charts or in their exact analytical form.

## This solution is completely general whatever the domains of integration v

and Vi, and is often much more rapid than might be supposed at first,
providing that there is no unnecessary generation of discrete points to
represent the volumes v and Vi.
Centred, regular grids of points (Xj, i = 1 to n) and (x; j = 1 to n ') are
placed over the domains v and Vi. The multiple integral Hv, v') is then
approximated by the double discrete sum
1 " n'
y(v, V')=-, I I Y(Xi -xf),
(II.66)
nn i= 1 j=l
ct. Fig. HAU, which represents a parallelepiped-shaped domain v and a
domain Vi consisting of a section of drill core.

Remark 1 The numerical integration formula (II.66) is the simplest discrete approximation method applying the same weight (1/ n) to each of the

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MINING GEOSTATISTICS

96

## n discrete points Xj which form a regular partition of the domain v. There

are, of course, other methods of discrete approximation (the methods of
Newton, Gauss, etc., d. R W. Hammings, (1971 which sometimes provide more precise numerical estimation of the multiple integral y.

r:

(Xi

-j(;)

x;

,
v

Vi.

## In practice, in mining geostatistics, taking account of

(i) the rapidity of calculation provided by the various micro-computers

now available,
(ii) the small number n of discrete points required (see remark 3
hereafter),
(iii) the simplicity and generality of the method of centred regular discrete approximation with uniform weighting,
the numerical expression of formula (11.66) will be used, provided that care
is taken to avoid any "zero effect" which may be produced when the two
domains v and Vi are not disjoint.

Remark 2 The zero effect. Whatever the adopted structural model, yeO) =
o by definition.
As a consequence, the two centred regular grids should be chosen so that
none of the discrete approximation points (Xj, i = 1 to n) and (x;, j = 1 to
n') coincide. Each time two points coincide (x, = xi) in the numerical
estimation of y, a greater importance is attached to the zero value yeO) = o.
This zero effect will cause the mean value of the semi-variogram to be
systematically underestimated and, correlatively, the mean value C of the

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97

## corresponding covariance will be systematically overestimated as C(h) =

C(O)-y(h).

Note that the zero effect can only be produced when the two domains v
and Vi are not disjoint, i.e., if there is a non-empty intersection of the two
domains within which points may coincide. It is particularly dangerous
when one of the two domains contains the other, e.g., when a mean value of
the type y(v, v) is to be estimated.
For such mean values Y(v, v), the zero effect can be avoided either by
slightly shifting each point of the second grid (xi, j = 1 to n ') independently
with respect to the corresponding point of the first grid (Xi, i = 1 to n), or by
using a more sophisticated weighting method for the calculation of the
multiple integral y(v, v) such as the Cauchy-Gauss method proposed in
remark 5.

## Remark 3 Required density of discrete approximation. The main rule is

that the density of discrete approximation must be such that all the
theoretical-order relations that are satisfied by the true mean values y must
also be satisfied by the numerical estimators y*. Thus, with v> Vi, it
follows that y(v, v)~y(v', v') and the two numerical estimators must
satisfy the inequality y*(v, v)~y*(v', v'), otherwise the dispersion vari2
ance D (V v) of v' within v will be negative.
The required density of discrete approximation can be found iteratively
by progressively increasing the numbers nand n r until any further increase
produces no significant improvement.
Experience has shown that in order to obtain an often unnecessary
accuracy, too high densities were considered, leading to a substantial
increase in computing time.
The densities of discrete approximation required for kriging have been
calculated as to ensure a loss of less than 1% of the kriging variances
(estimation variances minimized by the kriging procedure). In practice,
these values have proved to be largely sufficient for most of the other
purposes of the y calculation.
In practical applications of mining geostatistics, the following maximum
val ues are used:
I

## 10 points for a one-dimensional domain,

6 x 6 points for a two-dimensional domain,

}
(II.67)

## 4 x 4 x 4 points for a three-dimensional domain.

These maximum values are given only as indications. In many cases,
common sense will allow the number of points to be considerably reduced.
For example, if two domains v and v' are separated by a distance which is

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## MINING GEOST ATISTICS

98

greater than the largest range of the model y(h), then y(v, Vi) = sill value,
Again, if the dimensions of the two domains v and Vi, separated by a
distance h, are very small with respect to the range of the transition model
y(h), then y(v, v')= y(h), Now, if only the dimensions of v' are small with
respect to this range, then v' can be approximated by its centre of gravity
(n ' = 1) and v can be approximated by n discrete points in the usual
manner. Still again, if the domain v is a parallelepiped very much flattened
in the vertical direction, and if the model y(h) is isotropic, then v can be
approximated by its median horizontal plane - which is then approximated
by discrete points over two dimensions (n ~ 6 x 6 points); on the other
hand, if there is a strong vertical variability represented by an anisotropic
model y(h), it is advisable to approximate the vertical dimension of the
parallelepiped v also.

## Remark 4 Mean values y of linear combinations. As the operation of

taking the mean value -rev, Vi) is linear in y, more precisely, if y(h) =
Lk CkYk(h), then f(v, Vi) = Lk Ckiidv, Vi), each of the terms "Yk(V, Vi) can be
treated separately, Some of these terms can be estimated numerically,
while others can be determined directly either from charts or auxiliary
functions.
Remark 5 Cauchy-Gauss method for calculating -rev, v) and y(v, Vh). As
the following developments (rather tedious) are simply exercises in
numerical calculus and do not result in any new geostatistical concepts, the
reader may go directly to the next section, "Computer subroutines", which
will give, among others, the subroutine corresponding to this CauchyGauss method.
Let v be any domain in one, two or three dimensions. The mean value
j(v, v) can be written in the form of a weighted mean in which the
weighting function, (1/ v )k (x), is proportional to the indicator function
k (x) of the domain v:
1 f. dx
Y(V,V)=2
v
l)

f y(x-x')dx'=21 1+
V

l)

with

k(x)= {0

if x E v
if
I not

and

00

k(x)dx

-00

00

y(x-x')k(x')dx'

-00

1+
~
1

1+

00

k(x)dx=1.

-00

## By using the function K = k * k, auto-convolution of the indicator

function k, the preceding multiple integral can be reduced to
1
y(v, v) = v 2

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(00

too

y(u)K (u) du

(11.68)

## II. THE THEORY OF REGIONALIZED VARIABLES

with

1:

99

00

K(u)=k

*k=

k(u +x')k(x')dx'.

## Formula (II.68) is merely the Cauchy algorithm for reducing a multiple

integral of order 2n to a multiple integral of order n. This algorithm applies
equally well to the calculation of the mean values Hv, vd between two
equal domains v and Vh translated from each other by the vector h. The
formula then becomes
y(v, v,,) = 21

f. dx f
v

V

Vh

+00

## y(h + u)K(u) duo

-co

(11.69)

The Cauchy algorithm for reducing the order of the integral is of interest
only if the function K(u) is known a priori. Now, for rectangular domains v
with sides parallel to the system of axes, the corresponding function K (u),
also called the "geometric co variagram of v", can be readily determined.
(a) In one dimension, v is a segment of length l and its geometric
covariograrn is a function of one variable u:

K (u ) = { I -I u I ~f Iu I'!% t,
o
If not.
Consider now the symmetric function of y(h):
f(h

## Then, for aligned segments of identical length l translated one from

the other by the distance h, d. Fig. HAl, the Cauchy algorithm
(H.69) leads to the following expression for the mean value y(l, l,,):
y(l,1,,)=2

fa

(l-u)f(h+lu)du.

(H.70)

(II.70).

-----

I
h

...

## FIG. 11.41. Two aligned segments

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MINING GEOSTATISTICS

100

## (b) In two dimensions, v is a rectangle S with sides II and 12 parallel to

the two-coordinate axes (UI, uz) and its geometric covariogram is a
function of the two variables U I and Uz:
K(
U j,

2
0
if t
I no.

and

IU21~/2'

1 ",=+1

~2=+1

## r(h i + Ul, h z + uz) ="4 "1~'-1 "2~-'1 y(h 1 + FIUj, h z + zuz).

Then, for two coplanar rectangles of same dimensions (I I, [2) and
translated from each other by a vector h of coordinates ih, h 2 ) in
the directions of the sides (II, I z ) of the rectangles, cf. Fig. 11.42,
Cauchy's algorithm (11.69) leads to the following expression for the
mean value y(S, Sh):

t1
i

y(S, Sh)=4

(II. 71)

## The mean value y(S, S) is obtained by setting the vector h to zero in

formula (11.71).

L2

'I
--------------~--+-------~-----------,5

,
I
I

~I

I~"~-~--~---~-

## (c) In three dimensions, v is a rectangular parallelepiped with sides

II, 12 , l, parallel to the three coordinate axes (Ul, u. Uj) and its
geometric covariograrn is a function of the three variables Uj, U2, Uj:
K(UI, U2, U3)=

## (I I -Iu r!)(lz -luzi)(l3 -I U31)

if Iud ~ Ii,

if not.

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## II. THE THEORY OF REGIONALIZED VARIABLES

101

Consider now the symmetric function of the three-dimensional semivariogram y th, hz, h 3 ) :
1 ~1=+1
r(h, + u" h: +uz, h 3 + U3) = - L

I I

8~1=-1~2~3

## Then, for two rectangular parallelepipeds with the same dimensions

(II, 12 , f.:,,) and translated from each other by the vector h of coordinates (h 1, h. h s) in the directions of the sides (l biz, i 3 ) of the
parallelepipeds, cf. Fig. 11.43, Cauchy's algorithm leads to the following expression for the mean value i'(v, Vh):
,

y(v'Vh)=8

fa

11
,

(1-ud(1- U z)(1 - u 3)

(11.72)

## x f(h I + flU " h + Izuz, h + 13u3) dUl duz dU3.

The mean value nv, v) is obtained by setting the vector h to zero in
formula (11.72).
11
~

V
I, (
/1 2

hi

h3

## Thus, the calculation of a mean value f(v, v) over rectangular domains v

amounts to the integration over the domains (0, 1) of the nucleus 2(1- u),
4(1- ud(l - uz) and 8(1- u ,)(1- uz)(l - U3) in one, two and three dimensions, respectively.

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MINING GEOSTATISTICS

102

## This integration can be carried out by choosing a regular centred grid of

points within the intervals (0, 1) and considering uniform weights (1/ n if n
points are used). The Gauss numerical integration method provides much
better precision for the same number n of points. For example, four
abscissae points (Xi, i = 1 to 4) were chosen in the interval (0, 1) with the
corresponding four Gauss weights (A i, i = 1 to 4). The three integrals of
formulae (11.70) to (11.72) are then estimated by the following discrete
sums.
(a) In one dimension, the domain being a segment of length I:
4

y(l,l h ) =

(II.73)

Aif(h+lxi)'

i~l

## (b) In two dimensions, the domain being a rectangle S of sides (11, 12 )

parallel to the coordinate (h 1, h2 ) directions of vector h :
4

YeS, Sh) =

L L
i= 1 j= 1

AiA i f( h l

+ 11 X h h: + 12 xi)'

(II.74)

## (c) In three dimensions, the domain being a rectangular parallelepiped v

of sides (iI, l, 13 ) parallel to the coordinate (h 1, h, h 3) directions of
vector h:
4

y(v,

Vh)

I L I

i=1 ;=1

AiAiAd"(h 1 + 11 X i, h 2 + f 2 xj, h 3 + 13 x k ) .

(11.75)

k~l

The four abscissae Xi, Xi, Xk of the discrete approximation and the corresponding weights Ai, Ai' Ak have been calculated by A. Marechal (1976,
internal report) and are given in Table 11.1.
TABLE

points)

Xi

Ai

00571
02710

02768
04069

05836
02597

086025
006235

,=1

Ai = 0999 94 = 1

## Thus) in practice, the Cauchy-Gauss procedure reduces the numerical

calculation of the mean values Hv, Vh) and y(v, v) to that of 4 P values of
the symmetric serni-variogram function f(u), where p is the dimension of
the domain v considered (in practice, p = 1,2 or 3 dimensions).

## II. THE THEORY OF REGIONALIZED VARIABLES

103

For a spherical or exponential model y(h) and for practical values of the
range, the relative precision is always better than 1 %.

## Com puter su brouti nes

The following subroutines calculate the mean value Yev, Vi) between any
two rectangular domains v and Vi, whatever their respective positions in
relation to the system of axes on which the model y(h) is defined.
The first subroutine GBAR is completely general and considers a discrete approximation of each of the domains v and Vi by a centred regular
grid of n points.
The second subroutine F is more particular but much less time-consuming, and considers the Cauchy-Gauss discrete approximation to calculate
the mean value f(v, v) within a single domain v.

FUNCTION GBAR(---)
v and Vi can be any domain, provided that they are of rectangular form.
The dimensions of the two domains do not have to be the same, e.g., v can
be a three-dimensional parallelepiped (a mining block) while Vi is a segment (core samples along a bore-hole), cf. Fig. 11.44.
Each of the two domains is approximated by a centred regular grid of n
points, n taking the values given in (11.67).
When the two domains v and Vi have a non-empty intersection, and
particularly when one contains the other, the index INTER = 1 allows

w
101MI:3
( v)

-.h._-----CCAA 1
01

IDIM 2: I

CA2

## FIG. 11.44. Coding the domains v and v' in GBAR.

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MINING GEOSTATISTICS

104

avoiding the zero effect: each point of the discrete approximation of one
domain is shifted randomly from its corresponding node on the regular
centred grid.
The three-dimensional point semi-variogram y(h) is read as a subroutine FUNCTION GAM(HX, HY, HZ). The three coordinates
(HX, HY, HZ) of the vector h are relative to the reference system of axes.
(In section IILB.5, an example of FUNCTION GAM corresponding to a
very general structural model y(h) is given.)
Relative to the reference system of axes, each domain v or v' is defined
by one apex and one, two or three side vectors according to whether the
domain is one-, two- or three-dimensional. Thus, in the example of Fig.
11.44,
(i) the rectangular parallelepiped v is defined by the coordinates of one
of its apex 01 and the coordinates of the three side vectors CAl,
CBl and cci,
(ii) the segment v'is defined by the coordinates of one of its extremities
02 and the coordinates of the vector CA2.
FU-c T J C~j
lCC2,(C2,

:..

ICI~2.INTE:RJ

* ~

C
C
C
C
C

## COMPLTES tHE AVERAGE ~AllE CF GfJ~~fJ

eETWEEN TwC CO~Al~S ~ITH A~Y CRIENTATI(~ ~ITH
RESPECT Te CC~QDI~ATES SYSTE~ EIT~EP CC~Al~ ~IGhT
BE A SEG~E~T,KECTANGLE OR PARAlLELEPIPED CO~Ar~S
/oIIGHT BE rF A CIFFE;<ENT Cr",u,510N

C
C
C
C
C

C
C

C
C

C
C
C
C

~ARA"'ETERS
'vECTCI<S OEFfl\HC CC"AIt\ 1
VECTORS HAVE CC~/oIC~ ORIGIN LCCATED
ON CNE OF#THE APEXES OF T~E CC~AII\
COll31
(CORDINATeS CF THIS REFEREI\CE A~EX
IDJ~1
NuM8ER OF CI~ENSIO~S CF DOMAI~ 1
=1 SEGMENT
VEC1(~ (AI
=2 RECTANGLE
VECT(RS CAl,CEl
=3 PARALLELEPIPED VECTC~S CAl,CEl,CCl
UNUSEC VECTORS SHOUlC BE PUT Te C.
CA2U!tCE2(3),CC2(3},C02{31,HIM2
CEFHITIC/\ CF
CC"'AIr.. 2
CA1(3),CBIU),CClI3l

C SUBRCLTII\ES (ALLEe
C

C
C
C

## CAMIHX,rY,HZI SE~I-\~RIOGRA~ Fl~CTIC~

RANCOM(KJ RAND(~ NL~eERS CEI\ER~T(R (O,IJ
CPTIC/\S

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C

I"'TEP~l

C'
C

105

## CCRRECTION OF I~TEGRtTION ERRCP E)

GENERATiC~ CF A RANC(~ STRATIF{EC GRID
C,., SECC,.,C DC~AI'"

c .................................
c
CI ME,... S ION CA 1 (3 i ,C B1 (3 J ,C ( 1 (31 ,C A;: (3) ,C e 2 ( 3 )
1CC2(3) ,COU3) ,C02(3)
DIM~~SIO~ Ll{~),l2(3),l3(~}
CIME~SIO~

DATA

Xl(3),X2(64,3J,Alt3)

ll/1~,6,4/,l2/1,6,4/,l3/1,l,41

GB:O _
J\!-:llCICIMl}
p..,Z:l2 lID IMl)

N3=l3IICIMU
"'1-:ll(IDIM21
,.. 2 :L 2 ( I elM 2 )
M3:-L 3 (IO 1M2 J
I JK:O

IF(INTER.EC.lJGO TO 20C

c
c

fRICOING CF SECGNe

CO ICC

cc

C(~AI'"

12=1,~1

ICC J.2=1,~2
CO IOC K2=1,1oI;
IJK=IJK+l
CO llC 1(=1,3
X2 ( I J K, leI: C02 ( I C I ... ( I 2 - 0 ~ ) *c A.2 ( J ell ~ 1 ...
11 J 2- 0 5 j C8 2 ( I C) I foI' 2+ (K 2 - C ~ ) *c C2 ( rc : / 3

11C CO"'TI~UE
lac CCNTI1\UE
GC Te 30e
200 CO 2Ie I2=I,~1
co 2Ie J2=l,frl2
co z i c K2:l,tJo3
IJK=IJr<+l

en

220

220

Il=lr3

Al(Ill=-RA~CCfrl(K)

1(=1,3
1[J:C02(ICJ+( I2+Al (1) J*CA2( rc )/"'1+
1 ( J 2+ At (2) 1 -c 82 ( 1 C) / ~ 2 +( K ~ ito U ::= ) ) *c(2 ( 1 C) 1M3
CO 230
X2( IJK,

## 2.30 CONT HUE

210 CONT H,UE.

CCMPUTE GEAR

c
300 CC 1 ll=l,Nl

CO 1 Jl=1,N2
DC 1 Kl:;I,N3
DC 10 1(=1,3
XIt r c r =CO l( 1 C )+ ( I 1-0 .5 ) *c A I I rc i It'd'"

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MINING GEOSTATISTICS

106

lCJI-O ..5)*CBU

lCJ/N2+(Kl-C.~J*CCl(ICJ/f\~

10 CCNTlf\UE
DC 2

lJ=l,IJt<.

## HX=X 1 (lJ-X2 ( IJ,

HY=X 1 (~) -X2 ( I J,

1)
;2)

Hl=Xl(3)-X2( IJ,3)
GB=GB+GA~(hXtrYtHZ)

2 CONTIt>.UE
1 C(NT HUE
GHAR=GB/(Nl*N2*N3)/IJK
RETUFlf\
END

FUNCTION F(---)
This subroutine calculates the mean value y( v, v) over a rectangular
domain of any dimension (segment, rectangle or parallelepiped). The
subroutine considers the Cauchy-Gauss discrete approximation method
and the corresponding formulae (II.73) to (11.75).
w

FIG.

cc

## 11.45. Coding the domain v in FUNCTION F.

The three-dimensionnal point semi-variogram y(h) is read as a subroutine FUNCTION GAM(HX, HY, HZ), the three coordinates
(HX, HY, HZ) of the vector h are relative to the reference system of
axes.
Relative to this reference system of axes, the domain v is defined by one
apex considered as being the origin 0 of the coordinates and by one, two or
three side vectors. Thus, on the example of Fig. 11.45, the rectangular
parallelepiped v is defined by the coordinates of its three side vectors CA,
CB and cc.

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## II. THE THEORY OF REGIONALIZED VARIABLES

C a
~ ~ _
COMPLTES THE AVERAGE ~AllE OF GA~~A
C
C

~TTHIN

SEG~ENT,RECTANGLE

~ ~

OF FARALLELEPIPEO

C rAVI~G AN~ CRIENTATICN WITr PE~PECT Te CCCRDIC ~ATES SYSTEM COHAIN IS CEFI~E[ BY ~ECl(FS AlGCC RITH~ IS (AU(HY-GAUSS:POINTS FC~J,"EIGtT~ ftI41
C
C PARA~ETEPS
C
C

C
C
C
C
C

CA(3J,Ce(3),CCt~)

vECTO~S

[EFINI~G

CC~AIN

## vECTORS HAVE CC~~(~ CRIGl~ Le(ATEe

eN G~E OF THE APEXES OF TrE CC~Al~
IDI~
NUMeE~ OF DI~ENSIC~S
(F DC~Al~
SEGMENT vECTOR CA
=1
~ECTANGlE VECTCRS CA,CB
=2
=3
PARALLELEPIPED VECTCFS CA,CB,CC

C
C SUBROUTI~E CALLEe
GA~(HX,~y,rZI
SE~I-~~RIOGRA~
Fl~CTI[N
C
C e
C

DIMENSION CAI3},CB(3),CC(~J,P(4J,~(4)
DATA P/.C571,.27ef,.583t,.~tC251
DATA R/.2710,.406S,.25~6~,.0623~1

F=O.
GC Te

c
c
c

(1,2,3J,IDI~

1 CO 10 1= 1,4

HX=P ( I)*CAlll
liV=P(11*CA(2)
Hl=P(IJ*CAl3J
10

F=F+.tIJ.GAM(~X,r~,Hl)
RETUR~

REel H~GL E

C
C

2 Co 2C 1=1,4
CO 20 J=I,-4

G=O.
J 1=-1
J2=-1

00 21 IN=1.2
J l=-J 1
CO 21 1"'=1.2
J2=-J i:
HX=Jl~P(IJ.CA(1)+J2*P(J).(E(lJ

~Y~Jl.P(IJCA(2)+J2*P(J).CE(~1

Hl=JJ*PII)*CAt3J+J2*PCJJ*Ce(31

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107

MINING GEOSTATISTICS

108

21

G=G+G~M(HX,Hy,Hlj

F=F+wII)*WIJ)*G/4.
20 CONTINUE
~ETURf\

PAEUllElEPIPEC

C
C

:3 00 30 1=1,4

CO 30 J=l,it
CO 30 K=l,4
G-=O.

Jl=-1
J2:1:-1
J3=-1
CO 31 IN=I,2
Jl=- J 1
00 3 1 I ~:= 1 , 2
J2=- J 2
00 31 IL:tl,2
J3=- J ~
HX=Jl.PII).CA(1)+J2*PIJ).(E(1)tJ~.f{~}.CC{11

HY;Jl*P(1)*CAIZ)+J2*PIJ}'CEI2)+J3*FCK).CC(2)
31

~l=Jl*P(IJ*CA(3}+J2*P(J)'(E(3)+J~*FIK)*CC(3)
G=G+f~~(~X,hy,HlJ

F=F+W([)*W(J)*wIK)*G/8.
30 CONT HUE
HTUR ...

END

## An auxiliary function is a pre-calculated mean value .y(v, v) corresponding

to particularly simple geometries of v and VI, which are frequently found in
practice. There are four basic auxiliary functions, denoted by a, x, F and
H, defined for rectangular shaped domains in one or two dimensions.
L
A

L.

y(h) = y(r)

with r

= Ih I.

## Now, in one dimension, let AB be a segment of length L as in Fig. 11.46.

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## II. THE THEORY OF REGIONALIZED VARIABLES

109

The auxiliary function X(L) is defined as the mean value of y(h) when
the extremity of the vector h is fixed at the point A and the other extremity
describes the segment AB of length L:

1
X(L)= y(A, AB)= L I, y(u) duo

(11.76)

The auxiliary function X(L) thus represents a simple integral of the semivariogram y(h).
The auxiliary function P(L) is defined as the mean value of y(h) when
the two extremities of the vector h describe, independently of each other,
the segment AB of length L:
1
F(L)=y(AB,AB)=2
L

lL du lL y(u-u')du'.
0

(II. 77)

## The auxiliary function F(L) thus represents a double integral of y(h). By

means of the Cauchy algorithm of formula (II.70), this double integral can
be reduced to the following simple integrals:

fL (L-u)y(u)du=22

2
F(L)=2
L

iL

uX(u)du.

(11.78)

11.47.
L
A--'~B

1\

v
--L._---'

~--

I
FIG.

## The two-variable auxiliary function aiL; I) is defined as the mean value

of y(h) when one extremity of the vector h describes a side of length I and
the other extremity independently describes the opposite side at a distance
L:
a(L; 1)= y(AC, BD).

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MINING GEOSTATISTICS

110

It can be shown that this double integral of the isotropic function y(r) can
be reduced to the following simple integral:

2
a(L; I) = HAC, BD) = /2

## (1- u)y( J {L 2 + u 2 du.

(11.79)

Note that the auxiliary function a (L; l) is not symmetric in (L; I):
ai]; L) = y(AB, CD) y(AC, BD).

## The two-variable auxiliary function x(L; 1) is defined as the mean value

of y(h) when one extremity of the vector h describes a side of length I and
the other extremity independently describes the entire rectangle ABCD:

X(L; I)

## The auxiliary function x(L; I) thus

isotropic function y(r), and appears
function a(u; 1)= y(AC, UV) when
rectangle ABCD, as in Fig. I1.47. The
the"following simple integral:

## represents a triple integral of the

as the mean value of the auxiliary
the side UV describes the entire
triple integral can then be reduced to

IlL

## X(L; l)= y(AC, ABCD) = -

L ()

a(u; l)du.

(II.80)

Note that the auxiliary function X(L; I) is not symmetric in (L; I):
x(l; L) = y(AB, ABCD) oj; HAC, ABCD).

## The two-variable auxiliary function F(L; I) is defined as the mean value

of y(h) when both extremities of the vector h describe, independently of
each other, the entire rectangle ABeD:
F(L; l)

= y(ABCD, ABCD)

(11.81)

## This auxiliary function is symmetrical in (L; I). It represents a quadruple

integral of the isotropic function y(r). The Cauchy algorithm of formula
(II.70) can be applied to the auxiliary functions a(L; 1) and x(L; l) to
reduce this quadruple in tegral to the following simple integrals:
2
F(L; 1)="2
L

iL

(L - u)a{u; I) du

= 22 fL
L

UX{u; I) duo

(II.82)

Note the similarity between these formulae and those of (II. 78) corresponding to one dimension. In fact, the two-dimensional function Fil; 1)
is nothing other than a one-dimensional function peL) with the mean value
over I having been taken beforehand.
The two-variable auxiliary function H (L; I) is defined as the mean value
of y{h) when one extremity of the vector h is fixed at anyone of the

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111

## corners of the rectangle and the other extremity independently describes

the entire rectangle ABeD:
H(L; 1)= yeA, ABeD).

(II.83)

It can be shown that H (L; I) is also equal to the mean value of ')I (h) when
one extremity of the vector h describes one side of length I and the other
extremity independently describes the adjacent side of length L:
H(L; I) = y(AC, AB).
(11.84)
The auxiliary function H (L; I) is symmetric in (L; I) and represents a
double integral of the isotropic function ')I(r). It can be shown that H(L; I)
can be deduced from the auxiliary functions x(L; I) and F(L; I) by simple
and double differentiation, respectively:

1a

H(L; 1)= 21

a/

1
a
X(L; 1)= 41L al

22

(11.85)

## Remarks The two-variable auxiliary functions can be reduced to the

corresponding one-variable auxiliary function by contracting one of the
sides of the rectangle ABCD to zero, e.g., I ~ 0, as in Fig. 11.48. The
two-variable auxiliary functions then become
a(L; 0)= y(A,B)= y(L)
but
0'(0; 1)= HAC, AC)=F(l);
x(L; 0) = yeA, AB) = x(L)

but

and

and

A-

__ B

I
C
FIG.

(11.86)

I
D

## lIAS. Rectangle ABeD compressed (l

= 0).

The auxiliary function F(L; l) corresponds to the highest order (4) multiple
integral of y(h), all other auxiliary functions can be deduced from F(L; I)
by differentiation or by setting a parameter to zero:

1 i
a
2
a(L'l)=- - L
F (L ' 1)=-Lx(L' I)'
2
,
2 aL
'aL
"
1 a 2P(L;
x(L; l) = 2L aL L
I)
2

and

1
a 2 2
H(L; 1)= 41L al aL I L F(L; I);

## F(L) = F(L; 0).

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## X(L) = x(L; 0);

(11.87)

112

MINING GEOSTATISTICS

## In three dimensions, the three-variable auxiliary functions a, X, F and H

can be defined for a rectangular parallelepiped (11, /2. [3). In practice, these
three-variable functions are too awkward to use and the three-dimensional
case is restricted to the two-variable functions defined on a parallelepiped
P = ABCDEFGH with a square base (L x p) as on Fig. 11.49. The following auxiliary functions can then be defined:

(i) the mean value of y(r) between the two square faces (12) of P,

## a(L; e)= y(ACEG, BDFH);

(ii) the mean value of y(r) between one square face and P,

(11.88a)

## x(L; f) = y(ACEG, P);

(iii) the mean value of y(r) within P,
F(L; [2) = yep, P);

(II.88b)
(II.88~)

(iv) the mean value of y(r) between two adjacent faces of P which is
equal to the mean value between a side [ and P,

## H(L; 12 ) = y(ACEG, ABEF) = y(AE, P).

L

I(

E - - - -

(II.88d)

- - - - -

-L.F-..

~:""""G-----_..::..o H
......

## Remark By contracting one of the sides of the parallelepiped P to zero

(L ~ 0) these three-dimensional functions can be reduced to the two-

## dimensional auxiliary functions defined on the square [2.

Similarly, by contracting two of the three sides of P to zero (l ~ 0), these
three-dimensional functions can be reduced to the corresponding onedimensional auxiliary functions defined on the segment L.
The following relations are obtained:

e = 0)= y(L)

and

## a(O; 12)=F(I; I);

x(L; 12 = 0) = x(L)

and

## X(O; 12 ) = F(1; I);

F(L; [2 = 0) = F(L)

and

## F(O; f)=F(l; I);

= 0) = X(L)

and

H (0; [2) =

a(L;

H(L;

[2

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xtl: I).

(II.89)

113

## Mathematical expressions and charts

Expressions or charts of these auxiliary functions are given for each of the
four-point isotropic models -y(r) which are in current use.
1. The linear model

withr=lhl.

-y(h)=r

In one dimension,
and

X(L)=L/2

F(L)=L/3.

(II.90)

In two dimensions,

a(L; l)=-J(L +[
3

2
2
2
2 L2
L
l-r J(L 2 + [2)
)+- ~/2 (L-J(L +1 +-log
;

[L

1L
1L
I+J(L + f )
+ J(L 2 +[ 2 ) (1 - -1 -L-) +--log-----'-----.:....
X( L , [ ) = ~6 12
4 6 f
3 l
L
1 z2
L
+-12 L log

+ J(L 2 + [2)
[

;
(II.91)

2
3
IlL
1 -/2) +1 (L
F(L'/)=J(L 2+[2)(---2 - +z3-)
,
5151
15L 2
15 z2 L 2
1L

+ 6-/ log
1

H(L;

I+J(L + /

L
[2

2
l)=3 J(L + [2)+ 6L log

L+../(L 2 + Z2 )
+ (; L log
Z
;
1 l2

L+J(L 2+ f ) L 2

+ 6l

log

1+J(L 2 + z2 )

## Note the particular case corresponding to the squares, L

a (l; I) = 10765/,

x(l; 1)=O'7351l,

F(l; l)=O'5213[,

l:

## 2. The logarithmic model

-y(h) = log r

with r =

Ih I.

In one dimension,
X(L) = log L -1

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and

F(L) = log L

3/2.

(11.92)

MINING GEOSTATISTICS

114

## In two dimensions, with (} = arc tan (II L),

3
2() log cos ()
a(L; [)= log {-logsin 0--+--+
2
;
2 tan e tan (J
11
()
log cos ()
X(L; [) = log I-log sin 0 -6+ tan () + 3 tan 2 (J

0)
tan
23'

+ (7T_

(J.

(11.93)

## F(L; I) = log I-log sm

25 tan ' ()
12 +-6-log sin ()

(J -

2
+- - - () tan
6 tan () 3 2

2B
0+--.

tan

## These exact formulae are rather tedious to use. In practice, it is better to

use their limited expansions as functions of LII or l] L depending on
whether L is less than or greater than I. For L < I, we have

3
L L2
L 3 L 2 1 L4
a(L'/)=
log
1--+7T-+-Iog
- 12
- -[4+ ... .,
,
2
1
1-2(2

1T L 1 L 2 L 11 L 2 1 L 4
x(L; 1)= log 1-"2+2/+3 [2 log i-18 [2- 60 14+' .. ;
3

3 1T L

1L

25 L

1 L

(II.94)

+ ....
2 3 -+I 6 -log
[2
[72
P- 180

## For L> l, we have

1
1 1
a (L; i) = log L + 12 L 2 - 60 L 4 +,

.. ;

7T I 1 [2
1 14
-- -+ - +
.
X( L ', I) = log L -1 + -6 L 12 L 2 180 L 4 . .

(II. 95)

3 1T 1 1 [2
I 25 [2
1 [4
F(L; 1)=IOgL-"2+'"j L +"6 L z log L -72 L 2-180 L 4 + ....

Remark For values r = Ihl < 1 the function log r has negative values.
Consequently. for small values of L and I, the auxiliary functions a, x, F
can take negative values. Every estimation or dispersion variance defined
on non-point supports and calculated with these auxiliary functions is.
nevertheless, always positive. since every regularization of the logarithmic
model is a conditional positive definite model.

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## II. THE THEORY OF REGIONALIZED VARIABLES

115

Linear equivalents

written as

IS

## F(l) = log I ~ 3/2.

The auxiliary function F(a; b) for a rectangle with sides (a, b) and a > b,
is given by the first three terms of the limited expansion (II.95):
F(a; b)=log a

3
2

3
2

7Th
3 a

b
a

## (by considering that tTI3 = 1).

These three terms are also the terms resulting from a limited expansion
of the function log (a + b) with a> b, i.e.,

b)] -2=loga-2+~+""
3
3 b

3
[a( 1+-;;F(a;b)=log(a+b)-2=log

Thus, the two-dimensional auxiliary function F(a; b), for the rectangle
with sides a and b, is approximately eq ual to the one-dimensional auxiliary
function F{l) for the segment I = a + b :

with [ = a + b.

(11.96)

## The segment of length I = a + b is said to be the" linear equivalent" of the

rectangles with sides a and b.
More generally, when v is any domain in one, two or three dimensions
and not necessarily rectangular, the linear equivalent of v is the segment of
length I such that

r-

3/2.

(11.97)

## For the logarithmic model, the approximate linear equivalent of a

rectangular parallelepiped of sides (a ~ b ~ c) is l = a + b + O'7 c.
Application Let v and V(v c V) be two domains with linear equivalents I
and L, respectively. The variance of dispersion of v in V is given, according
to formula (II.36), by

D\v/ V) = y(V, V)- y(v, v) = (log L - 3/2)- (log 1- 3/2) = log (L/ I).
If the two domains v and V are geometrically similar, e.g., if they are

## homothetic parallelepipeds, then

V/v = (L/I)3

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and

## D\vl V) = ~ log (VI v).

MINING GEOSTATISTICS

116

l'(h)=

3 r 1 r3
2 a -2 a 3

for r =

1 = sill

for r

a (range)

## In one dimension, the auxiliary functions become

3L

1L

4a-'87'

VLE[O,aJ,

3 a
1--8 L'

VL~a;

X(L)=

"2
F(L)=

1 L
a - 20 a 3 ,

1L

(11.98)

VL E [0, a],
2

3 a 1 a
1---+-~
4 L 5 L2,

VL~a.

In two dimensions, the exact expressions for the auxiliary functions x(L; I);
Ftl.; l) and H(L; I) can be found in J. Serra (l967 a) and D. Guibal (1973a).
In practice, the following charts, which are graduated in terms of LIa and
II a, are used.
Chart no. 1 gives the function l'G(L) = a(L; /)- a (0; 1) = a(L; 1)- F(l).
This function is simply the serni-variogram graded over the constant
thickness l, ct. section 11.0.3, formula (II.55). As F(l) is given by
formula (11.98), this chart also provides the values of the function
a(L; I).
Chart
Chart
Chart
Chart
Chart

no.
no.
no.
no.
no.

2 gives
3 gives
4 gives
5 gives
6 gives

the
the
the
the
the

## function X(L; I).

function H(L; I).
function F(L; I).
function F(L; 12 ) .
2
difference a (L; /2) - a (0,1 ) = a (L; 12) - P(l; l)

## 4. The exponential model

l'(h) = 1 - exp (-r/ a)

with r =

Ih I.

## In one dimension, the auxiliary function becomes

(II.99)

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## II. THE THEORY OF REG IONALIZED VARIABLES

117

In two dimensions, the exact expressions for the auxiliary functions can be
found in J. Serra (1967) and D. Guibal (l973a). In practice, the following
charts, graduated in terms of L/ a and II a, are used.
Chart
Chart
Chart
Chart
Chart
Chart

no.
no.
no.
no.
no.
no.

11 gives
12 gives
13 gives
14 gives
15 gives
16 gives

## the function YG(L)=a(L; l)-a(O; l)=a(L; l)-F(l).

the function X(L; I).
the function H(L; I).
the function F(L; I).
the function F(L; [2).
the difference a(L; P)-a(O; p)= a(L; l2)-F(l; l).

5. The r B model
y(h)=r

withr=lhland

OE]O,2f.

## The function rO is conditional positive definite only when the parameter 0

is contained in the interval ]0,2[, the limits a and 2 being precluded.
Although this model is not currently used in practice, it has some
theoretical importance and, therefore, the following one-dimensional
auxiliary function can be of interest:

X(L)= [/(8+1),

(II. roo)

## II.E.3. Using the auxiliary functions

In mining applications, the geometry of the data supports and of the blocks
to be estimated is very often rectangular, and, thus, the previously defined
and pre-calculated auxiliary functions can be used when calculating the
mean values y.
By way of example, several geostatistical operations will be given, with
results that can be read from charts.
More charts for the spherical and exponential models

## All developments given hereafter correspond to an isotropic point model

y(h), i.e., such that y(h)= 1'(r) with r = Ihl.
The following elementary estimation variances are given in Charts no. 7
and 17.
(i) The estimation variance of a segment AB of length 1 by a central
sample O. This variance can be written, according to the general
formula (II.27), as

## ITt = 2y(O, AB)-y(AB, AB)- HO, 0)

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118

with

## ii(O, AB) = ii(O, OA) = y(O, OB) = x(l/2);

because of symmetry,
HAB, AB) = F(l)

and

ii(O, 0) = 1'(0) = 0

## and the extension variance becomes

O"t = 2xUI2)-F(l)
(ii) The estimation variance of a segment AB of length I by two samples
located at its extremities A and B. Letting if;' = {A + B} designate the
set of the two sample points, this variance is expressed as

with
'Y(g', AB) =

## yeA, AB) = y(B, AB) = xU),

because of symmetry;

## HAB, AB) = F(l),

Hit, g') = y(A, g') = t[y(A, A)+ y(A, B)] = ~y(l)
and, thus,

uL = 2x(l) ~ F(l) A

y(/)/2,
B

I
--------t

## (iii) The estimation variance of a square ABCD of side I by a central

sample 0, d. Fig. 11.50, is written as

## uL = 2HO, ABCD)- 1'(ABCD, ABCD)- 1'(0,0)

with
HO, ABeD) =

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yeo, OIBJ) =

H (l12; liZ),

## II. THE THEORY OF REGIONALIZED VARIABLES

because of symmetry;

and, thus,

and

HO, 0) = 1'(0) =

119

## O't : : ; 2H(lj2; tns-r, I).

(iv) The estimation variance of the square ABCD of side l by the set 'iC
of its four corners, d. Fig. 11.50, is written as

## O't = 2y('jg', ABCD)- y(ABeD, ABCD)- y(g, fg)

with
-y(g', ABCD)= yeA, ABeD)= H(l; l),

because of symmetry;

y(g, 'iC) = yeA, 'iC) = ~[y(A, A)+ yeA, B)+ yeA, C)+ yeA, D)]
=

and, thus,

!-yU) + ~'Y(lv'2)

## Charts no. 8 and 18 give the estimation variance of a rectangle ABCD of

sides I and L by its median IJ of length l, ct. Fig. II.51:
cr~5 = 2y(IJ, ABCD)-y(ABCD, ABCD)-y(IJ, 11)
=

A

C~D
L

## FIG. 11.51. Estimation of a rectangle by its median.

Charts no. 9 and 19 give the estimation variance of a rectangular parallelepiped P with a square base (L x 12 ) by its median square, d. Fig. II.52:
cr~6 = 2y(IJEF, P)-y(P, P)-y(IJEF, IJEF)
=

## 2x(Lj2; [2)-F(L; e)-F(/; I).

As the functions F(L; 12 ) and F(l; I) are given elsewhere (Charts no. 4, 5
and 14, 15), these charts no. 9 and 19 can be used to calculate the
2
three-dimensional auxiliary functions X (L; 1 ) .

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120

MINING GEOSTATISTICS

## Charts no. 10 and 20 give the variance of estimation of a parallelepiped

P with a square base (L x [2) by its axial drill core 11 of length L, d. Fig.
II.53:

(it = 2i(U, P) -

## ru.. [2)_ F(l).

L
FIG. 11.52. Estimation of a parallelepiped by its median square.

As the functions Fil; [2) and P(L) are given elsewhere (Charts no. 5 and
15 and formulae (11.98) and (11.99)), these Charts no. 10 and 20 can be
used to calculate the three-dimensional auxiliary function H(L; [2).
L

I
I

-\
J

\
\
\

FIG.

## Configurations for the estimation of a rectangle

This exercise is intended to familiarize the reader with the use of the
various auxiliary functions.
Let P be a rectangular panel with sides a and b, and located within an
isotropic regionalization represented by the point model y(r), cf. Fig. 11.54.

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121

## A practical analogy would be a parallelepipedic panel P with surface a x b

and height c, located within an isotropic horizontal regionalization
represented by a model graded over the constant height c.
b

I
I
I

I
[

K - - - - - - - T - - - - - - - La
I
I

I
I

J
FIG.

## Consider the estimation of the unknown mean grade of panel P by each

of the following four estimators:
(i) mean grade 2 1 of the perimeter ABCD of panel P (Z, could be
obtained by continuous horizontal channel sampling along the
length of the four side drives of panel P);
(ii) mean grade Z2 of the two parallel sides AB and CD;
(iii) mean grade 2 3 of the two medians IJ and KL;
(iv) mean grade 2 4 of the single median IJ.
The estimation variances corresponding to these four data configurations
will be calculated and a numerical example will be provided with a linear
model y (r) = r and a sq uare panel (a = b).

## O"t = 2y(ABCD, P)- yep, P)- y(ABCD, ABCD)

with yep, P) = F(a; b).
-(ABeD P)= -(AC+AB P)= ay(AC, P)+by(AB, P)

,Y

a+b

ax(b; a)+bx(a; b)
a+b

## y(ABCD, ABeD) = i'(AC+ AB, ABeD)

'#NNI.minebook.blogfa.com

a+b

122

Since

## -(AC ABeD)= a'HAC, AC)+Zby(AC, AB)+ay(AC, BD)

'Y

2(a+b)

aF(a)+2bH(a; b)+aa(b; a)
2(a + b)

=-....:..-..;~-----=-....;,.........:-....._....;,.......=--...:.

we finally obtain
2

UE
I

( b)
= Zax(b;a)+2bx{a;h)
-F a"
a+b
'
a 2P(a)+4ahH{a; b)+b 2P(b)+a 2a(b; a)+b 2a(a; b)
2(a +b)2

Considering formula (11.91)of the linear model and a square panel (a = b),
we get the following values for the auxiliary functions:
F(a)=a/3,

a(a;a)=1'077a,

Thus, (]'~l

= 0'0479a

and

xCa;a)=O'652a,

H(a; a)=0765a.

## ut = 2y(AB +CD, P)-y(P, P)~"Y(AB+CD,AB +CD)

with y(AB + CD, P) = y(AB, P) = x(a; b).

## yeAB + CD, AB +CD)= y(AB, AB +CD)

=~[y(AB, AB)+y(AB, CD)]

=~[F(b)+a{a; b)]

## and, thus, we have

u~z = 2x(a; b)- F(a; b)-!{F(b)+a(a; b)].

2a.

## Rectangle evaluated by its two medians II and KL

Similar calculations

2 ~
O"E 3 ~

bFa,
a+

## a 2 Fea)+ 2abH(a/2; b/2)+ b 2 F(b)

(a + b)2

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## II. THE THEORY OF REGIONALIZED VARIABLES

123

Considering the values x(aI2; a)= O'454a and H(aI2; aI2)=O383a, the
numerical application gives ui 3 = Q0288a, for a total sampled
length 2a.
Rectangle evaluated by a single median IJ
can be found:

ut

## and, for the numerical application,

= O056a for a total sampled
length a.
Thus, to sum up, the following inequalities hold:
U~3

Q0288a

## with the respective total sampled lengths 2a, 4a, a, 2a.

The centred configurations (medians) are obviously better than the
closed configurations (sides).
This case study shows how geostatistics, by means of the structural
characteristic y(h), is able to balance the quality (the median information is
better located) and the quantity of information (the perimeter has the
largest total sampled length but does not correspond to the smallest estimation variance).

II.E.4. Charts

All the charts given hereafter correspond to the two following spherical
and exponential isotropic models with a unit sill value. All the distances
considered by the charts are relative distances ria (divided by the
parameter a of the model).
Spherical model:
1 r3
y(r)= 2-;-2 a 3 '
{
1 = sill,
3 r

## VrE [0, a],

V r ;3 a (actual range).

Exponential model:
y(r)= l-exp (-ria),
with a sill equal to 1 and a practical range a'

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3a.

MINING GEOSTATISTICS

124

Chart no.

## Object of the chart

Spherical
model

Exponential
model

'YG(L)=a(L; 1)-a(O; I)

11

2
3

12

5
6

14
15
16

7
8
9
10

17
18
19
20

Auxiliary functions
X(L; I)
H(L; I)
F(L; I)
F(L; (2)
u.. f)-a(O; 12 )

## Regularized semi-variograrn over cores of length (

First point 1'1 (l) and sill 1'1 (co)
21
Serni-variograrn 'Y{(h)
22(a), (b)

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13

## II. THE THEORY OF REGIONALIZED VARIABLES

090 -----;..----i--

,-, ,-'--------t--

~-

____ ~_+---L

060

125

,
I
j--i---+-

-----j---+----l

------->---i-I------:.-

0,70

,
,

"T------i--+--r---,-!----'---+! ,,-

0_60

...J

'"
>...

---T-------+----.-.V.,

0,50_ ..._+-.-, ..

V. h.~

OAO

0,30

020

0.8

0,9

## CHART NO.1. Spherical model. Grading over I: I'dL).

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1.0

MINING GEOSTATISTICS

126

10
9
8
7
6

--------t ---::---~
J""'.....
" t~s~~:."~F'" "1\."" """ "" "~0!J o.!J!J~
r-,
<,
" " .:\. ~
_p.
.
.
-I 1"--,' . !"\.. '" """
0.\$l~ .~~
-----....,
.>:i"
:""'
"
""'
"
",,\"
<,
'\
r--.' :
'Po "-j"\
"
5
1'..

"I\. '\

'"

'j.,.

I 0'!J !J...
'PO'

0!J

I\.

4
3

1:)

-....

r---

'I'.

- ~ 1'-" 1\ 1\

0.,

0,

--I---- 'I'- ~ ~
o. '., ~
~ I--- I'--- 1'-.. [\ 1\
~
r'\
0:>, \

-=-I---I----I'---~

_~~r---...!'..f'.

-,

I
O,g
0'8

-......,,,

0-4

0'3

02

_,

\
~ -~
o \" \

~~. o~ .~ -'v--\
\

-,

~" 0'
_" \ \
0

a I '1,"'0\
0-1

~"
G'"" ~\

I"

'

~ "~ ~- ~.

~~.

1\

r"o \
\

r\

1\f\ r\ [\

l\l\
[\1\"
r, 1\ 1'\ l'~~
i\ \
\

1\

i\ \ \ \

1\

1\

"

"\ \
1\

1\

\"

\\\ \ \\ \ \1\

1\

f\~'

<, -, "~o""" 1\ \

'\ \
\ \ \ 1\ \ \
,,0"'0" \

r-, '"

r-,

r=====::::: <,

05

00

:~~I-"':'>'"

-......... ,............

---r-

0-7
06

1\

1\
\

1\

02

0-3 04

OG 08

45678910

i/o

CHART NO.2.

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## II. THE THEORY OF REGIONALIZED VARIABLES

127

10 r-------..,.--==-----.------,-.o::::-r--..:--r""'r"O<l'""""ll-.....--....-.........,..,...................,...,......................~.......__ .............._______,

9 I----=o::;;i:::::;:=-8
+--~~-....-:-____+_:::......I,._----=~P-Irt~~~~----"!.~~>J....."~~~~
7 ;----k=---_+__

61---"""""4:::.::::
5 ~==_.jk-~""_<d-.p"..J-Nx.:~~-~~~~~~

j----_~.:::- -~-.,-----f"--+~""-<

2!---=:t::::------''''''''''--.;;;;::-

0::.

....

<,

0'9

oa
07
06
0-5

0-4
03
02

02

03

04

06

08

L/o
CHART No.3. Spherical model. Function

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H(L; l).

6 7 8 910

MINING GEOSTATISTICS

128

~
.....

0-9
08
07
0-6

05
04
0-3
0-2

06

08

Llo
CHART NO.4. Spherical model. Function F(L; I).

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6 7 8 910

I:l
-....

129

,
0'9
06
0-7
0-6

05
0-4
0-3

02

0-3

0-6 08

04

Lla
CHART No.5.

## Spherical model. Function F(L; f).

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5678910

090

~.

i
i

----t-

s-

oso

060

---i "

.L.

f
I

+-

I-

;-

II

070

- ... _-f

-;

--

'"

r5 0-50
1:1
I

j---_._-

--t
N

..; 040

-t- ...

f---

1:1

,
(;

\'
'"'

030

__ -v} -

-t--,.
'I/o

020

010" "

11:

I/o

140

,'10,160
~- ~/O: 180
1/0'2,00
., I/o' 2 50 '1'-"""'----' ---- .,,['300!
= 4'00_-

_,n

L'IU-OO

01

02

03

04

05

06

07

OIS

' 09

Llo

f:

_1..___

131

a: E~

-, 18 -- 10
.............
t

-I

0-4

06 08

~,

03

02

01
009
008
007
006

005
/

004
003

002

0-2

03

I/o

## CHART NO.7. Spherical model. Various extension variances.

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6 7 B 910

MINING GEOSTATISTICS

132
Liz LIZ

10 r-------...,...~-rr__r-..,_____,_,~..,..,,,.....,..._r____,_______..-------,...,...-7T-_,___,________,~.....,_____,__,
9
8
7

6
5

4
:3

<,

09
0'8
0-7
06
05
04
0'3

02

0-6

0,8

Uo
CHART NO.8.

VNNI.minebook.blogfa.com

crL.

7 8910

## II. THE THEORY OF REGIONALIZED VARIABLES

r:====r==r=~o=r==l+===:L~~r==C~*:r:l

## ~ ---~~-=::~f~f-~~- c__ -+;

10

133

---+------;.++----+--+-------r-

----:io-""-

6~-----:---+

51-----

4
3 1--------.;----._..--

21-----+--+--f'------+------+-

'"

<,

0-9
0-8
0-7
0-6
05

0-4
03
0-2

06 0'8

L!a
CHART NO,

## 9. Spherical model. Extension variance CT~".

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6 7 8 910

MINING GEOSTATISTICS

134

Ia r - - - -........-------,,..-----rT"-..,..-...-----,----y,-----r--,-__.,~r____.,_-___._-_r___,______,_____,......,..___r_l
9
8
7
6 I------------cf-

5
4
3

<,

-..j

0-9
0-8

07
0-6
0-5
0-4

0-3
0-2

(/a
CHART NO.

## 10. Spherical model. Extension variance (T~").

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6 7 B 910

....-

1. .
+.

090

~,

SI-

080

';n

070

f--

_~r~M

060

---'

::.'" 0 50

040-

,
(;

030
110

020

h_

/101000

~.
~

'"

11:

700

110 02000 -,

10

0-5

15

2-5

,~

35

L/a

## CHART NO, 11. Exponential model. Grading over

I: ydL).

45

MINING GEOSTATISTICS

136

I~ c===::r:::::=::F3;;;J==r::r:S:r=:S=3:S::::I~Qr=D-:-1
-- ,------i'
----8 r=====;;===1===r:="

~ ~=~=~r===
5

~===-~+:-~~-.--J

4r-----....=-4:::::::.:=-=~

...----.--...-t.. -....
r------'---__

0:,
<,

....

09
08
07
06
0-5
004

-+- ..

03

---

02

0-2

0-3

04

06

0-8

L 10
CHART NO.

## 12. Exponential model. Function X(L; I).

VNNI.minebook.blogfa.com

6 7 8 910

## II. THE THEORY OF REGIONALIZED VARIABLES

137

10 r-----.--~~====:'

9
8 F"---'=c;;;..;.:=:.....f--"-"::=;;,;.e:.;....-.~~..:.-

~ ~=--:----:-=:::-~~-~
5 F='--"-~=-=l~-.:--.--:..~~

3--

II-----~
09r---~~

08t--

07r--_~

r---_

06 -05

04

o 3 ~. . .

o I 01oQ~---'----'~~-'----'--;~~:-'::"------;~.J.J...7-::"-'7::-,-:"-------'_.L..~LL..Il......l...l-~-.-L....L.L-~-.L....L~L-.....J
2

CHART NO.

"3

## 13. Exponential model. Function H(L; I).

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6 7 8 9 10

138

MINING GEOSTATISTICS

I~ C:==:I:=::======Eb:====:=:;;]C~=s[~h:5IS:CS:O;~
:~--_._._---i...-._-------T---_._----':----_ ....j,~".,_.

8 ---------------

~ r::==r=+=::=;=-=:"-+=4::~
5F==:~~~~~
....----.i.__~.~~m ~

2 ---

09
08
07

06
0-5

04
03

0-2

0-1

01

0-4

06

0-8

Ua
CHART NO.

## 14. Exponential model. Function F(L; I).

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7 8 9 Ie

139

8
7
6

I
0-2

03

0-4

06

08

Lla

## CHART NO. 15. Exponential model. Function F(L;

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[Z).

6 7 8910

I
I

---+-----;
i'
_, ~i . -----t,,------,
---+-------- - ----

090

__

_,

,
e

I-

080

f---= ",
1 -. t
-~_
t--"-r -. .
--_.... -.+--_._M'

~I

l--J-~
'

070

l J0

~-=- ~

,-----1---+---'-

-.,.0,10

_ .-

__'_" _"

~,-

-----t- "-~

~--+----------+-

II 0

060

'

"b--+

~ ..

._----~-+--

o
'" 050

'----i------

'"

'-

-:::J

040

l:l

030

(;

\'
'"'
~

020 ----

11:

05

10

15

-----'~_

2-0

i
25

30

35

40

45

Lta
CHART NO. 16. Exponential model. Grading over

n.

50

## II. THE THEORY OF REGIONALIZED VARIABLES

141

--

1--'
{
I 0

0-9
08

,--------r--:-...,--,.----r--r--r--,-,-r-----,----r---,----r~,..._I"1___rl
..
.._ .. ._.._ .....1...-.
---i
i- I ; -1 - --- -------~-"'~""" -------,--,,,,,,,--.
-'I, "'''-ri ----------,'
~

## -J, ~~~ ~t ;_ :~_:;' ~ _,. ~.~

--------------i-------+----+---"+ ,
t
t

0,7

-:--1;; . .

- __

....

...

__._

',j;;

0'6

-----1----

05

--i-- .

0-4 -

0'"

......... "'-r

03

_-t

-----

02

"It

'

E~ ~t"'-

...--1-........
,
:

~.

1"I

## -r---.----- +------._-- ..-r----t----+---+---1

01
009
008 007
a 06 f------------ ------ ------------ --i-------'/---

---.--

-f-'!

005

04

~--

,L..

--

~i!----+----

-----.-

--i-----~--_i_-~---~-"----------------------------------

---- --;--------

t------!

0- 03

J._
/

a2
CHART NO.

03

04

06

08

I
I/o

6 7 8 9 10

## 17. Exponential model. Various extension variances.

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142

MINING GEOSTATISTICS

10 ,-----------;--~-_.,___.~~__,__._______"_____,______,....__.____r_____,_______._______r_______r..,.....__,..,.__....,.......,,,......____.__r.>""l

9 . . -------S
7
6

5
4

<,

09
08 .
07
0-6

05
004

03

0-2

02

0-3

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004

6 7 8 9 II

## II. THE THEORY OF REGIONALIZED VARIABLES

,..,

1"11

143

.-1

i/2 i/2

10r-------r---,--....,...-,..---,--..,.....,.....,.......,.---."--x--~...,....-........,........,-.,...,,........_r__co

~ f----~~_~---+_--+-+----+----!--++-!------7''---,,L-_____7'''II______~~__!r_+_7'q._+___:J,.~
61-------f--51-------+---+
4 r------+-----t--_+___

31-----+---+--+
21-------f--------1f--

0-8 f-----.-"f--------Jl'-------+-+-+---++-+-+-+

0-6 r----+---+--,'--t--------.+-++---+-:fi---If--i---if--r

o 5 f----------t'--------+-+---I--- .f--f------A-----1I'------hI--+-----tf-t-.f--+--t

o 2 1-----+-------11-1'---......<-,

02

0-3 04

06

0'8

Llo

## CHART NO. 19. Exponential model. Extension variance (J"~6.

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6 7 8 910

MINING GEOSTATISTICS

144

8
7

6
5 ---4

2 ---------

1-

08
0-7
O6 1-------,1-----

05
004

02

06

08

l/a
CHART NO.

## 20. Exponential model. Extension variance a-~.,.

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6 7 8 9 10

I
~.

s-

00

090

080

.
__ :
: I '

" ',' .

I-

';n

070

f---------+--

;.

060

o 50 f------------+040

~---

-- -.....

....-----;------.-.-t- "...

MM"'''

030

.+_~_+c_.+_

020
I

(;

t-

---I

~.
~

'"

11:

01

02

0-4

06 08 10

5 6

8 10

20

30 40

60 80 100

0/1

CHART NO. 21. Spherical model. Regularized semi-variogram. First point, y/(l);

I
~.

s-

07

oil: 1- 3

I-

06 I

m __

---+--

~-:-~

0-9

0-511-----

S
~

04r

I'

031

5=1

___

___
0-21

\\'

';/",/:;>/

JB'C;;,/J,..r

:;;;0>"

--=

05

J
J-

o 1I

R I ::

_I

0-4

03

02

2-0

25

'"'

1,
1:

05

CHART NO.

10

15
hit

## 22(a). Spherical model. Regularized serni-variogram I'I(h).

I
01

-: 7 / / v / /
....--

09

~.

so

"/ /

/'

08

./

v /

I-

./

0-7

0-6

';n

--c::
~

0/1 =2-0

0-4

./

18
H3

-:

,31//// / / /

r!}///IJ j/ -:
If,IV/li' / V V
II//;W /20/V
W//J'// /
/

-"'"

.//

/"
..-/

/"

,/

./"

1-4/

"oil'"
0-5

//

/ / 1/ /

~/ I / / /
J.7 / / I / / /
, A / I I I / /v /
/

./

./

"/

/'

15/

0-3

0-2

~.
~

'"

1
1:

01

~~V
V
2

10

II

h/t

CHART NO

12

13

.---

III

## Structu ral Analysis

SUMMARY
The structural analysis of a regionalized phenomenon consists in constructing a
model of a variogram which characterizes, in an operational way, the main features
of the regionalization. This modelling requires good physical knowledge of the
phenomenon under study as well as good "craft" in the practice of fitting geostatistical models.
Section lILA introduces the concept of nested structures to model the succession
of various scales of variabilities of a natural phenomenon; e.g., from the petrographic scale to the macro-scale of the distribution of deposits within a metalliferous province. The micro-variabilities, i.e., the variabilities of which scale is
overpassed by the available grid of information, are represented on the variogram
by an apparent discontinuity at the origin called the "nugget effect". '
Section III.B gives the various models of regionalization and coregionalization in
current use and shows how they should be combined to provide an operational
representation of structural features, such as nested structures, anisotropies, quasistationarity. Then, the study of the statistical inference of a moment of order two
gives an order of magnitude of the reliability of an experimental variograrn or
covariance.
Section III.C shows how a structural analysis should be guided in practice. A
critical review of the available data, along with elementary statistics of these data,
should be made before proceeding to the effective geostatistical analysis. Three
main steps can be distinguished in this analysis, successively: the construction of
experimental variograms, their interpretation and the construction of a synthetic
structural model.

## III.A. NESTED STRUCTURES AND THE NUGGET EFFECT

III.A.1. Nested structures
Consider two point or quasi-point (very small support) grades Z(x) and
Z (x + h) in a deposit separated by a distance vector h. The variability
148

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149

## between Z(x) and Z(x +h), which is characterized by the variogram

E{[Z(x + h)- Z(x )}2}, is due to many causes, which appear over a range of
different scales, for example:
(i) at the level of the support (h "'" 0), there is a variability due to

## measurements, i.e., fluctuation in the rate of recovery of the core

sample, sampling errors;
(ii) at the petrographic level (e.g., Ih 1< 1 em) a second variability
appears due to the transition from one mineralogical element to
another:
(iii) at the level of strata or mineralized lenses (e.g., [h 1< 100 m), a third
variability may be due to the alternation of strata or of lenses with
waste material;
(iv) at the level of a metalliferous province (e.g., Ihl < 100 km), a fourth
variability may appear, due to the distribution of the deposits
related to the orogenesis of the province;
etc.
All these sources or structures of variability, and possibly many more,
come into play simultaneously and for all distances h. They are called
"nested structures".
Scale and support

0/ observation

## this would require an enormous amount of data of quasi-point support

covering the entire field of variability from 1 u.m to 100 km.
At the scale of observation, when counting elements on a thin section
(where the information has a quasi-point support), it is not possible to
distinguish between the first variability due to errors of measure (e.g.,
counting errors) and the second due to petrographical variability.
At the scale of evaluation of a mining face by channel samples, the
support of the channel samples integrates the first two variabilities into one
single undifferentiated variability that will be defined later as a "nugget
effect". However, it is possible at this scale to distinguish the third variability, due to lenticular beds in the same stratum.
At the scale of the evaluation of the entire deposit by drill cores spaced
at 50- to 100-m intervals, the first three variabilities are indistinguishable.
However, it is possible to distinguish the fourth variability due, for example, to the alternation of mineralized strata, or trends in grade values at the
borders of the deposit or at lower depths.
Just as in a set of nested tables, each table covers all the smaller tables, so
each scale of observation integrates the variabilities of all the smaller
scales.

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MINING GEOSTATISTICS

150

## As far as the second-order moments of the RF Z (x) are concerned, these

nested structures can be conveniently represented as the sum of a number
of variograms (or covariances), each one characterizing the variability at a
particular scale:
(IlL 1)

## For example, 'Yo(h) may be a transition model (spherical or exponential)

which very rapidly reaches its sill value Co for distances h that are only
slightly larger than the data support. This model thus combines all the
micro-variabilities (e.g., measurement errors and petrographic differentiations). 'YI(h) may be another transition model with a larger range (e.g.,
a1 = 10 m) characterizing the lenticular beds and 1'2(h) may be a third
transition model with a range (az = 200 m) representing the alternation of
strata or the extent of homogeneous mineralized zones.
At smaller distances (h < 30 m), the observed total variability depends
on 'Yo(h) + 11 (h), d. Fig. III. 1, while for large distances it will depend on all
the Yi(h).
r

1--F-~

--

- -- ------

Yo+Jj +Y2

70+ 71

--

71

--/

/
V
I

Co

Yo

--

-=='------

72

-- --

-----~-~

(10 m)

(200 m)

## The previous representation (III. 1) of nested structures is not unique. It is

currently used in mining geostatistics because it is convenient in order to fit
several experimental variograms with different levels (sills) of variability
and, above all, because the expression (III.1) is linear in 'Y.

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151

## As a matter of fact, each of the main geostatistical operators is linear in

y(semi-variogram) or in C (covariance when it exists), d. formulae (11.27)
for the estimation variance, (11.36) for the dispersion variance, (11.41) for
regularization. Thus, using the previous linear representation (III. 1) for the
point semi-variogram y(h), the regularized semi-variogram on a support v
is the sum of the regularization on v of each of the component structures:
yv(h) = I Yi,l)(h) = I ['Yi (v, Vh) - Yi(V, v)].
i

## More generally, if '#; is anyone of the previous operators, then

?5'(y)=I '#;(1';)

ify(h)=IYi(h).

(111.2)

The operator 'l acts identically on each of the components Yi of the nested
model y.
The same holds true for the covariance; the operator 'itt operates identically on each of the components C, of the nested model C:
g(C) =

gee)

if C(h) =

L C(h).

(111.3)

## 1II.A.2. Genesis of a nugget effect

If we study the spatial variability of the gold grades of core samples, two
core samples that are very close together may have grades Z (x) and
Z (x + h) which differ considerably when one of them contains a nugget and
the other one does not.
This nugget effect appears on the semi-variograrn as a component yo(h)
which rapidly increases to reach a sill Co almost as soon as h is greater than
zero (more precisely, when h is greater than the range ao equal to the
dimensions of the gold nugget), d. Fig. 111.1.
To this micro-structure Yo, of very small range, a structure 1'1(h) of
larger range at can be added, characterizing, for example, the regionalization of mineralized veins or of a concentration of nuggets. The total
variability can then be characterized by the nested model y(h) =
1'o(h)+YI(h).

## At the level of sampling a gold deposit by drill cores, the distances of

observation h > 0 are generally larger than the first range ao and, thus,
yo(h) = Co

and

y(h)= C o + Yl(h),

## with 1'(0) = 0, by definition.

Thus, at the level of observation (e.g., h = 20 m ao), the residual microstructure yo(h) appears on the total variability as an apparent discontinuity

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152

MINING GEOSTATISTICS

at the origin of amplitude Co, which is called the" nugget constant", d. Fig.
liLt.

Definition
The term "nugget effect" can be generalized from its specific meaning in
gold-bearing deposits. It is used to characterize the residual influence of all
variabilities which have ranges (ao) much smaller than the available distances of observation (h ao). A nugget effect will appear on the variogram (or covariance) as an apparent discontinuity at the origin.
It must be stressed that the definition of a nugget effect is
closely related to the scale of observation. The same structure 'YI (h) with
range al = 10 m and sill C 1 would be quite evident on a sampling grid of 3
to 5 m, but would appear as no more than a nugget effect on a grid of 30 to
50 m, cf. Fig. Il I.L Experimentally, on such a large grid the two
components Co and C 1 of the nugget constant would be indistinguishable
unless data are available to put into evidence the range a I.
For this reason, it is often advisable when a sampling programme has
been carried out on a systematic grid (of, for example, sides b = 100 m) to
take a few other samples on a smaller grid (for example, b/IO = 10 m). In
this way, data are available at two different scales of observation, which
makes it possible to study the nested structures and interpret any possible
nugget effect.
Scale effect

## When the observed semi-variogram appears as a single nugget effect

(completely flat serni-variograrn), it is said to represent a pure nugget
effect, ct. Fig. IIL2 and Case Study 1 in section IILA.5.
y ,----------------,
y(h)=Co=C(Ol

c(Ql

- ---------

C( h)= 0

o .. - -------------'
h

## FIG. lIL2. Pure nugget effect.

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153

## The corresponding model yo(h) is a transition-type model (spherical, for

instance) with a range e which is very small compared with the smallest
distance Ih I of observation;
yo(h) = {O

Co

for h

G,}

'tlhl> e.

(IlIA)

## The corresponding covariance is written according to formula (1104);

Co(h) = Co(O) - yo(h).

In the course of this book, a pure nugget effect will often be represented
by the notation with square brackets: yo(h)=[Co(O)-Co(h)], so as to
distinguish it from a structure 'Y1 (h) with a greater range.

## Absence of spatial correlation

The corresponding covariance model Co(h) for a pure nugget is, thus, of
transition type, its value at the origin CoCO) being equal to the sill Co of the
semi-variograrn model, according to formula (IL13), cf. Fig. 111.2,

= Co
for h = O,}
Co - yo(h) = 0, \:Ilhl > e.

Co(h)= {CoCO)

(IlLS)

## The pure nugget effect thus corresponds to a total absence of spatial

correlation between two variables Z (x) and Z (x + h), at least for all
available distances (Ihl>e), and is equivalent to the well-known
phenomenon of "white noise" in physics. In so far as geostatistics is only
concerned with second-order moments, a pure nugget effect will be interpreted as spatial independence.
In order to establish the standard statistical results for independent
variables, it is sufficient to apply the various geostatistical formulae to a
pure nugget-effect model.
In practice, a pure nugget effect at all scales with a function y(h)
amounting to a single discontinuity at the origin is exceptional.
In fact, if the pure nugget effect at all scales is associated with stationarity, it represents an incredible degree of homogeneity of the
mineralization. It means, for instance, that no local differentiation is
possible between mining panels and, thus, no selection is possible. There
are no rich zones nor any poor zones (a zone being defined on any scale).
At every point in the deposit the best estimator is the mean grade of the
deposit.

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154

MINING GEOSTATISHCS

## IIf.A.3. The nugget effect and geostatistical operations

Let C (h) be the covariance, defined on a strict point support, characterizing the variability of a sequence of nested structures:
k

C(h)= Co(h)+

C(h).

i= 1

## The nugget effect can be distinguished as a transition covariance Ca(h)

with a very small range ao = E:

Co(h) = 0

## when Ihl > ao.

It was shown previously that the main geostatistical operators are linear
functions of the covariance C. If 'l: is one of these operators, then, from
(III.3), we have
k

g'(C)=

~(Co)+

~(C;).

i= 1

## In this section we shall be concerned with the effect of these operators on

a pure nugget effect, i.e., with the term fg(Co). In terms of the semivariogram, with yo(h) = Co(O)- eo(h), we shall be concerned with the
operation g'(Yo).

Mean values

## Co and Yo of a pure nugget effect

Mean values of the type Co(v, v) or Co(v, V) are calculated for the four
possible relative configurations of the two supports v and V. The following
results are obtained.
(i) v

A
Co(v, v)=-

A
v

Co(O)~-.

and

yo(v, v)=

and

## )lo(v, V) = Co(O)- V'

(III.6a)

(ii) v c V, V contains v:

Co(v, V) = V

(III.6b)

## (iii) v V = 0, v and V are disjoint and the distance between them is

much greater than the range ao:

Co(V, V) = 0

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and

Yo(v, V) = CoCO).

(III.6c)

IS5

## (iv) v n V:F 0, v and V have a non-zero intersection, the measure of

which has dimensions much greater than the range ao:

Co(v, V)=A

Mea:; n

yo(v, V)=

and

Co(O)~AMea~; n V.
(III.6d)

In all four cases, we have v, V ao, i.e., the range ao of the pure nuggeteffect model is very small compared with the dimensions of the supports v
and V~ also, A = f Co(h) dh (recalling that, in the three-dimensional space,
h is a vector and this integral in fact represents a triple integral over the
whole space).

Proof:

Co(v, V)=

v~

v.

1L

Co(x - y)dy.

dx

## If the distance between every point x

than the range ao, then, strictly,

## fv Co(x - y) dy = L(00 eo(h) dh

A,

'VXEV,

oo

and, thus,

f
v

1
CoCv,
V) =-

A
-dx
v V

=A
-.

If not, i.e., if there exist points XoE v such that the sphere Bao(xo), centred
on Xo and with radius ao, is not entirely contained within V, then, for these
points xo, fv Co(x - y) dy ~ A. But, as regards the second integration, Iv dx,
the influence of these points Xo is negligible (since ao is very small
compared with the dimensions of v). Thus, the previous expression holds
true: Co(v, V) = AI v.
Having established the mean values Co of a pure nugget effect, we can
now proceed to the study of its effect on the three main geostatistical
operators.
(i) Regularization on a support v with e ao If "Yo(h) is the point semivarlogram, then the serni-variogram regularized on a support v is written,
according to (lIA1). as
'Yov(h) = 'Yo(v, Vh)- -ro(v, v),
where v" is the translation of the support v by the vector h.

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MINING GEOSTATISTICS

156

is expressed as

o
YOlJ(h)=

if h =0,

[1-

A
v -

Meas v

## n VhJ If. Ihl < v, t.e..,

. if v n Vh ,t 0 ,

A
v

(I1I.7)

if Ihl > v,

with all the dimensions of observation much greater than the range ao, i.e.,

v, Meas v

n V,

and:

h ao

## Generally, the supports v and Vh do not overlap, so that the only

discontinuity at the origin is of amplitude A/v, and the a priori variance of
the regularized RF Z, (x) is given by expression (11.45) as

Vv ao.
In practice, point data are seldom available and the nugget effect will be
revealed by the regularized nugget constant Cov = A/ 0, when 0 is the
smallest support of information (e.g., piece of core). For all other supports
V, the regularized nugget constant Cov is given by the inverse proportionality relation:
v

COy

= Cov V'

with v, V ao

(III. 8)

## (ii) Estimation variance of the mean grade Zv on the support V (mining

block) by the mean grade Z; on the support v According to formula
(II.27) and taking the expressions (111.6) of Yo into account, this variance is
written

if V contains v,

## (T~(V, V)= A(;-+ ~) =

C ou( 1+

~),

A(!+~-2 Meas v n
v

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V\

vV-;

if v and

V are disjoint,

if v and V have a
non-zero intersection,

(III. 9)

157

## where C o u is the nugget constant regularized on the support v and shown

on the regularized semi-variograrn You (h).
In the estimation of a block V, the measure v of the data support is often
negligible when compared with the measure V of the block. In such cases,
all terms in AI V in formulae (111.9) are negligible and we have
'r;f V

v ao,

(III. 10)

## which is the standard relationship found in the statistics of independent

variablest (or, more precisely, variables without auto-correlation, the
structures of which are, thus, a pure nugget effect). The variance of estimation of the mean grade Zv of a field V by the arithmetic mean

## of the grades of n samples v (x;) located at the n points {Xi, i

written, according to (111.10), as

VV no,

1 to n} is

u ao,

## since the measure V' of the data support is V' = nv.

This estimation variance is directly proportional to the a priori variance
C ou of the grades Z; (x) of the samples, and inversely proportional to the
number n of samples.
(iii) Dispersion variance of the mean grade of a volume v in a field
V. According to formula (11.36), and taking the expressions (111.6) of Yn
into account, this variance is written
(III.ll)

## In the presence of a pure nugget effect, the dispersion variance

2(v/oo)
are inversely proportional

## D (vl V) and also the a priori variance D

to the support v.

t Strictly speaking, the standard formula a.2 J n of the statistics of independent variables
corresponds to the estimation variance of the expectation E{Z(x)} of the RF Z(x}.

However, under the hypothesis of stationarity and ergodicity, the mean grade Zv~'
( I J v>S vZ (x) dx of a fixed domain V, with V (10' tends towards the expecta lion E {Z(x) } .

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158

## 1. Pure nugget effect at Adelaida

The scheelite (W0 4Ca) of the Adelaida (Spain) deposit is concentrated in
veinlets and small nodules which are distributed more or less homogeneously in a network of quartz veins.
Drilling was carried out in a direction perpendicular to the plane of the
greatest density of veins, The variable measured was the cumulative
thickness of the quartz veins intersected by a drill core, the support of the
variable being core samples of constant length I = 5 m. Thus, the variable
measured, ZJ(x), is the regularization of the point variable "density of
quartz" over the length of a core sample. The core samples can also be
grouped in pairs to provide the variable Z2r(X) regularized on core samples
of length 21 == 10 m.
Figure IIL3 shows the two regularized semi-variograms )'/(h) and )'21 (h),
as well as the means and experimental dispersion variances of the data used
in the calculations. Both these serni-variograms are flat up to around
h = 50 m (pure nugget effect), after which (h > 50 m) a slow increase
indicates the presence of a macro-structure with a range that cannot be
determined from the available data.
The respective nugget constants (sills of the flat semi-variograms)
1
adopted are 60 and 30 (em m- )2, which verify the rule (IlL8) of inverse
proportionality to the support of the regularization.
The experimental dispersion variances, sf = 668 and S~I =
35,3 (em m- 1)2, verify well enough the rule of inverse proportionality,

..

-----------,
h)

.----------~,".~-

r, (

60
>-..

30

~~.,

10

60

50
h

## FIG. 111.3. Pure nugget-effect serni-variograms. YI(h): 1= 5 m, m = 5 em m -1, a~

668 (em m~1)2, Y21(h): 2l = 10 m, m = 5 em m- 1, a31 = 35,3 (em m- 1)2,

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## III. STRUCTURAL ANALYSIS

159

These experimental variances are greater than the above nugget constants,
the difference corresponding to the additional variability and nesting effect
of the macro-structure observed at large distances (h > 50 m).
On the scale of observation, hE [5,50 m], which is also the scale on
which production is carried out, this pure nugget effect indicates that
selective mining is not possible in this deposit, unless the size of the
selection unit is of the order of the vein (5 to 40 em). In fact, the absence of
spatial correlation makes impossible any local differentiation by estimation: at any location in the deposit, the best local estimator of the cumulative quartz thickness is the global mean, rn = 5 em m -1 in the direction
studied (approximately vertical).
This example is one of the rare examples of deposits in which the main
variable exhibits no spatial auto-correlation at the scale on which the study
is made. There is, however, a macro-structure which appears at greater
distances.

## 2. Nested structure at Maz augues, after A. Marechal and J. P. Roullier

(1970)
The bauxite deposit at Mazaugues aval (France) consists of subhorizontal
stratiform lenses aligned principally in the North-South direction, d. Fig.
III.4la). These lenses are characterized by a regular hanging wall and a
very irregular foot wall which consists of a dolomite karst. The mean
dimensions of the lenses are between 200 m and 300 m in the N-S direction and 100 m and 200 m in the E-W direction.
The deposit was sampled by a network of vertical bore-holes with a
regular density; on average, each square 100 rn x 100 rn contains one borehole.
Figure III.4(b) shows the semi-variograms of the thickness of bauxite in
the two main directions N-S and E-W. These serni-variograms reveal
transition models, the ranges of which correspond to the observed mean
dimensions of the lenses. Thus, the structure of the variability of the
thickness observed on a hectometric scale is that of the lenticular beds.
If a spherical model of range aNS = 250 rn is fitted to the experimental
serni-variogram N-S in Fig. IIl.4(b), the nugget constant would be Co =
O' 25. On the hectometric scale of this observation, it is not possible to
specify the structures between 0 and 100 m to any greater degree of
precision.
During the course of mining, the bauxite thickness at small sampling
distances (2 and 10 m) was measured within one particular lens. Figure
III.4(c) shows the semi-variograrn calculated from these measurements. A

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MINING GEOSTATISTICS

160

## transition with a range of approximately 20 m can be noticed, which is

characteristic of the karstic structure of the foot wall, d. Fig. III.4(a).
Beyond this transition, the semi-variogram increases continuously in a way
remarkably similar to the linear behaviour at the origin of the previous
transition model with range aNS = 250 m, which characterizes the macrostructure of the lenticular beds.

100

15 to 20 m

(a )

(c)

( b)

,,-

-N- S

,,- ./

io

N-S

/
"/

I
I _

"

/.

05

I I
/

./
/

1/
II

Co

Co

100

500

0 10

50

100

h (m)

## lIlA. (a) Sampling a bauxite thickness. (b) Nested structures at Mazaugues On

a hectometric scale. (c) Nested structures at Mazaugues on a decarnetric scale.

FIG.

Thus, what is seen on the hectometric scale as a nugget effect is, in fact,
the micro-structure of the jagged outline of the karstic foot wall. The total
variability of the bauxite thickness, up to h = 500 m in the N-S direction,
can, thus, be represented as a nested model which is the sum of two
transition models (spherical, for example) with ranges an = 20 m and a I =

250m.
This nested succession of variabilities at different scales is a very
frequent phenomenon in mining geostatistics. The best example of it is the
nested structures of the ferriferous basin of Lorraine (France) studied by J.

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161

## Serra (1967a, b, 1968). Serra observed and decomposed an entire series of

structures from the petrographic scale (100 urn, the regionalization of
oolites and limonite) up to the hyperrnining scale (5 km, the regionalization
of deposits related to the stream currents at the time of the deposition of
the Lorraine basin).

## The four main operations of linear geostatistics (variances of estimation

and dispersion, regularization and kriging) involve only the structural
function of the RF (covariance or variogram). Thus, every geostatistical
study begins with the construction of a model designed to characterize the
spatial structure of the regionalized variable studied.

## 111.8.1. -Positive definite conditions

It will be recalled (d. section II.A.3) that the covariances C(h) of a
second-order stationary RF Z (x) must be a positive definite function. If
the RF is only intrinsic with a serni-variogram y(h), then -y(h) must be a
conditional positive definite function.

## Positive definiteness and dimensions of the space

It can be shown that the positive definite property of a function g(h) is
related to the dimensions of the space on which the distance h is defined.
Thus, a function g(h,,) which is positive definite in a one-dimensional space
may no longer be so in an n-dimensional space (n > 1); e.g., the function
g(J(h~ +h~ of the two parameters (h", h v ) may no longer be positive
definite in the two-dimensional space.
On the other hand, every function which is positive definite in an
n-dimensional space is also positive definite in every space of smaller
dimension, n' < n.
These properties also hold true for conditional positive definite
functions.
In mining practice, the only spaces involved will be those of dimension
less than or equal to three. Thus, all the theoretical models presented in the
following sections III.B.2 and IILB.3 are positive definite or conditional
positive definite in three dimensions and can, therefore, also be used in two
or one dimensions.

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162

Two properties
By means of the two following properties of positive definite functions, a
vast family of theoretical covariance or covariogram models can be defined
in terms of basic positive definite schemes.
(i) Every linear combination of covariances with positive coefficients is
a covariance. The same holds true for variograms:
C{h)=

i:

n1

L:

y(h)=

A7C(h),

(111.12)

Ail'i(h),

i= 1

## For the proof, it is sufficient to consider the RF Z (x) as the weighted

sum of n independent RF's Yi(x) with covariances C;(h). The
covariance of
n

L:

Z(x)=

AiYi(x)

.=1

## is then given by formula (III.12).

(ii) Any covariance product is a covariance:
n

C(h)=

TI c,.(h) =

(III. 13)

j=1

n

y(h)= C(O)-C(h)=

TI

C(O)-

;=1

TI

Ci(h).

(III.14)

i=1

## Again it is enough for the proof of consider the RF Z (x) as the

product of the independent RF's Yi(x) with covariances C{h). The
covariance of
Z (x ) =

Il" Y 1(x )
1=1

## In practice, we can take advantage of the linearity of the geostatistical

operators, d. formula (111.2), and use only the first property. The covariance and variogram models given by formula (111.12) can then be considered as representations of nested structures according to formula
(111.1).

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163

## Let Z(x) be an intrinsic RF with serni-variogram y(h). The two main

characteristics of a stationary variogram are:
(i) its behaviour at the origin, the three types of which are shown on Fig.
II.6 (parabolic, linear and nugget effect);
(ii) the presence or absence of a sill in the increase of y(h), i.e., l' (h) =
constant when Ih I> a.
Thus, the currently used theoretical models can be classified as:
Models with a sill (or transition models)
and a linear behaviour at the origin
(a) spherical model
(b) exponential model
and a parabolic behaviour at the origin
(c) Gaussian model
Models without a sill (the corresponding RF is then only intrinsic and
has neither covariance nor finite a priori variance)
(a) models in Ih 18, with 8 E ]0, 2[
.
(b) logarithmic model
Nugget effect It has been seen in section IILA.2 that an apparent
discontinuity at the origin of the semi-variogram, i.e., a nugget
constant Co, can be interpreted as a transition structure reaching its sill
value Co at a very small range compared with the available distances
of observation.
Remark For the moment, only isotropic models will be considered, i.e.,
RF's which have the same spatial variability in all directions of space. Thus,
in the three-dimensional space, the isotropic notation y(h)= y(r) denotes

y(.J(h~+h;+h~)),
where (h u , h", hw ) are the three coordinates of the vector h.
1. Models with a sill, or transition models

It will be recalled from formula (11.13) and Fig. 11.4 that the sill value of a
transition structure is the a priori variance of the RF Z (x) which is secondorder stationary and has a covariance C(h) = y(oc.) - y(h).
The models presented below have been normed to 1, i.e., they correspond to RF's with unit a priori variance:
Var {Z(x)}= C(O)= ')'(00)= 1.

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MINING GEOSTATISTICS

## To obtain a model with a sil! C(O) = C :- 1, it is enough to multiply the

given expressions for y(h) or C(h) by the constant value C.

## This is the most frequent type of behaviour

untered in mining practice (variogram of grades and accumulations)
it is most often accompanied by a nugget effect.

## (I) Spherical model:

y(r)= {

3 r 1 /'
2" a -"2 a 3 '

VrE[O,a],

1 = sill,

'rfr';3!;a.

(III.1S)

b) Exponential model:

## y(r)= 1-exp (-ria).

(III. 16)

Note that the spherical model effectively reaches its sill for a finite distance
r = a = range, while the exponential model reaches its sill only asymptotically, d. Fig. IlLS. However, because of experimental fluctuations of
the variogram, no distinction will be made in practice between an effective
and an asymptotic sill. For the exponential model, the practical range a
can be used with
I

a'

3a

O 95 = l.

## The difference between the spherical and exponential models is the

distance (abscissa) at which their tangents at the origin intersect the sill, d.
Fig. 111.5:
r = 2a13, two-third of the range for the spherical model;

r= a

a'/3, one third of the practical range for the exponential model.

Thus, the spherical model reaches its sill faster than the exponential model,
r(r)

....-----------------------------------,

I u0-F========*?~====3?~=::::::::~;;=:::::::::=======4
095I Ex ponenucl
I

I
I
I
I

IT

2/3

## FIG. III.5. Models with a sill.

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r/o

## III. STRUCTURAL ANALYSIS

165

Parabolic behaviour at the origin This very regular behaviour at the origin
is seldom found in mining practice. In the absence of nugget effect, it
corresponds to a RF Z (x), which is mean-square differentiable, and, thus,
to a ReV z(x), which is very continuous in its spatial variability. Thickness
variograms of continuous sedimentary beds sometimes display such a
parabolic behaviour at the origin, but are usually accompanied by a slight
nugget effect due to errors of measure.
(c) Gaussian model:
(Ill. 17)
The sill is reached asymptotically and a practical range can be considered
with a' = a.J3, for which yea') = 095 = 1, d. Fig. IlLS.
A parabolic drift effect (d. section ILA.3, "Behaviour at infinity" and
section II.A.5, Fig. 11.8) does not stabilize around a sill for large distances
and, thus, cannot be confused with the parabolic behaviour at the origin of
a Gaussian model.
However, for small distances (r < 2aj3), it makes little difference
whether a very regular local variation is interpreted as a drift effect or as a
stationary Gaussian structure.

## 2. Models without a sill

These models correspond to RF's Z (x) with an unlimited capacity for
spatial dispersion; neither their a priori variances nor their covariance can
be defined. The RF's Z(x) are only intrinsic, d. section II.A.S, Case Study
1.
(a) Models in -".
y(r) = r

with 8 E ]0,2[,

(JILI8)

## the limits 0 and 2 being precluded.

These models have a particular theoretical and pedagogical importance
(they show a whole range of behaviours at the origin when the parameter 8
varies and they are easy to integrate).
In practice, only the linear model is currently used, cf. Fig. 111.6:
y(r) = cor,

## with w the slope at the origin.

Remark 1 For small distances (h ~ 0), the linear model can be fitted to
any model that has a linear behaviour at the origin (e.g., spherical and
exponential models).

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166

MINING GEOSTATISTICS

## As () increases, the behaviour of 'Y(r) = / ' at the origm

becomes more regular and corresponds to a RF Z (x ) of more and more
regular spatial variability.

Remark 2

y( r) , - - - - - - - - - - - - - - - ,

## Experimentally, models in rO for (j E ]1, 2[ are often indistinguishable

from a parabolic drift effect, d. Fig. 111.6. The choice of interpretation, as a
drift (non-stationarity) or a stationary model in r" with () close to 2,
depends on whether or not it is desired to make a drift function m (x) =
E{Z(x)} obvious.
For (j ~ 2, the function (-rl'l) is no longer a conditional positive
definite function and, in particular, the increase of r" at infinity is no longer
2
slower than that of r , d. section II.A.3, formula (11.14). This strict limitation 8 < 2 prohibits, in particular, any blind fitting of a variogram by
least-square polynomials, even though some polynomials of degree higher
than two do satisfy the conditional positive definite property, cf. the
expression (111.15) of the spherical model.
Remark 3

## (b) Logarithmic model:

y(r) = log r.

(111.20)

Note that log r ~ -00 as h ~ 0, so that the logarithmic model cannot be used
to describe regionalizations on a strict point support. On the other hand,
this model, once regularized on a non-zero support v, d. section II.D,
formulae (11.49) and (11.59), can be used as the model of a regularized
serni-variogram 'Yv (h). In particular, the regularization of a point
logarithmic model is a function which is null at the origin: 'Yv(O) = O.

## The logarithmic model.

has been studied exhaustively by the initial workers in geostatistics (D. G.

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167

## Krige, 1951; G. Matheron, 1955, 1962; Ph. Formery and G. Matheron,

1963; A. Carlier, 1964). Indeed, up until around 1964-66 this mode! was
practically the only theoretical model used; this is explained by certain
analytical properties which make this model a very convenient one (cL
section ILE.2, "Linear equivalents"), and also by the fact that the first
geostatistical applications were carried out on deposits (gold on the Rand,
uranium) for which the variograrns of the characteristic variables had no
sill. Since that time, however, geostatistics has been applied to a great
number of other are bodies for which the variograrns show more or less
clear sills. J. Serra, using a very detailed study of the different scales of
variability of the mineralization of the ferrif'erous basin of Lorraine
(France), showed that a logarithmic model is the limit model of a nested
succession of transition models, the ranges of which increase geometrically,
d. J. Serra (l976b, 1968).
As the nested model has more parameters, it is more flexible and lends
itself more readily to a geological interpretation. Moreover, the particular
analytical properties of the logarithmic model are less attractive now that
computers are readily available and it is a relatively simple task to program
the evaluation of any integral y of any nested structure y(h).
This explains why the various sill models have increased in use since
around 1966 at the expense of the logarithmic model (except perhaps in
South Africa). In our opinion, the use of a logarithmic model or of various
nested spherical models is more or less a question of habit and does not
alter the result of geostatistical calculations. What is important is that the
chosen theoretical model y(h) should be a good fit to the experimental
semi-variogram within its limits of reliability (Ih 1< b).
In other words, the results of the geostatistical calculations prove to be
robust in relation to the choice of the model- provided that the parameters
of this model are correctly estimated (see the following example and, in
section III.C.6, the exercise on robustness).
Example Consider the regularization of the logarithmic model y(r) =
log r by core samples of length 1= 1. According to formula (11.50) the

## regularized model can be written as

'Yl(r) = log (r/I) + 3/2 = log r + 3/2,

Vr~I=1.

## On Fig. 111.7, a nested model has been fitted to this regularized

logarithmic model. The nested model consists of a nugget effect and two
spherical models Yl and Y2:
yt(r)= C O + YI(r ) + Y2(r ),

with

Ci,>

05; C 1 = 15, al

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= 3;

C 2 = 2, a2

"rEU,131]

= 13.

MINING GEOSTATISTICS

168

Within the interval r E [I, 13i] the relative error involved in fitting the
nested model "If (r) to the regularized model 1'1 is less than 3 %, 11'1"If 1/1'1 < O 03. In practice, this can be considered as a very satisfactory fit.

x
2

FIG.

7
rll

models (x).

I
9

I
II

13

yt

## Of course, the. sill value

(00) "= C o + C j + C 2 = 5, which is attained for
distances r ~ a2 = 13, is a pure artefact of the nested model. Similarly, the
finite character of the a priori variances of the type D2(vloo), calculated
using the nested model vt. is a pure artefact. However, when the use of the
regularized model
is strictly limited to the interval r E fl, 131], there is
no difference between calculations using )'1 and those made using
Now, if the logarithmic growth of an experimental semi-variogram
continues well beyond the interval [i, 13/], it would be better to use the
regularized logarithmic model 1'1 rather than a nested structure yt consisting of too many transition models.

yt

yt.

3. Hole-effect models
A semi-variograrn y(h) is said to display a hole effect when its growth is not
monotonic, The hole effect can appear on models with or without sills, d.
Fig. III.8(a) and (b).
The following hole-effect model, positive definite in the three-dimensional space, is given:
C(r)=sinrlr

and

y(r)=l-sinr/r.

## r being expressed in radians

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(III.21 )

## III. STRUCTURAL ANALYSIS

169

This model has a sill and parabolic behaviour at the origin, cf. Fig. III.8(a):
y(r)=r 2j6 when r~a.
Amplitude of the hole effect Let a be the relative amplitude of the hole
effect, i.e., the minimum value of the covariance divided by the sill value
C(O):

## For the model given in (IlL21), this relative amplitude

obtained for r., = 44934 = 37Tj2.
, . . - - - - - - - - - - - - - - - - - - - - - - - , r(r)

(o)

IS

a212,

r--------.....,
(b)

.:

C(r]

4".

FIG. III.S. Hole-effect model. (a) With sill; (b) without sill.

## It can be shown that this value a = O'212 is the maximum amplitude of

an isotropic hole effect in the three-dimensional space. If an experimental
hole effect with an amplitude greater than 0212 is observed in practice,
then either the hole effect is not significant and is nothing more than a
fluctuation of the experimental serni-variogram, or the hole effect is directional, i.e., it is not present in every direction of the three-dimensional
space.
If we desire to specify a particularly strong directional hole effect, then a
positive definite hole-effect model in only one dimension must be used,
e.g.,
y(h)= I-cos h

(I1I.22)

for one-dimensional h.
In this case, the relative amplitude a is equal to 1 and superior to 0-212.
The cosine model C(h) = cos h is not positive definite in three dimensions
and, thus, its use must be restricted to one particular direction.

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MINING GEOSTATISTICS

170

Note that the model (111.22) is periodic without any dampening of the
oscillations. In practice, however, such a hole-effect model is always associated, in a nested manner, with other models with or without sills and the
considerable oscillations of the cosine will, thus, be dampened, d. Case
Study 6 in section IV.B.
Interpretation A pseudo-periodic component of the regionalization may
appear on the experimental serni-variograms as a hole effect; for example,
the stationary sequence of two well-differentiated types of mineralization
in an ore body. If this sequence is not isotropic (and, in general, it is not),
then the hole effect will be observed only in certain directions, for example,
the vertical direction if the phenomenon is a pseudo-periodic sequence of
horizontal stratification, d. Fig. 1I1.9(a).
If the grades of Fig. II I. 9 are point grades, then the vertical semivariogram will exhibit a hole effect with an amplitude possibly greater than
0217 and the abscissae b, and b2 of the oscillations can be related to
the mean vertical dimensions of the waste and rich are strata. The
horizontal semi-variograms will differ according to whether they are
located within a rich or waste stratum, cf. Fig. III. 9(b).
z{ x)

~ - - - - - - - - - -,_._-~-,--

( b) I z( x)
I
;

Rich

"'tI'''i+f-I'tH-l/lI"ttf\t-+-tIf-'''i
I

40{.,\>~44>-,,: I
x

Tin'

~rI.-h-)----~-----------'
{a )

R,c"

__

. - - - - _ . ._----

Waste

ol~.
h

FIG. II 1.9. Verti cal hole effect. (a) Vertical dire ction ; (b) hori zontal direction.

## A hole effect may also be due to an artificial pseudo-periodicity of the

available data. Thus, when a first sampling programme on a square grid b is
completed by a second one on a grid b12, the heterogeneity of the drilling
material used for the two programmes would appear as a more or less
dampened serration on the experimental variograrns.
In practice, the natural variability observed in an ore body is, in general,
nested, i.e., a pseudo-periodic structure is most often associated with other

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171

## structures, the consequence of which is a dampening of the observed hole

effect. For estimation purposes, an experimental hole effect that is open to
a doubtful interpretation, or is not very marked, can simply be ignored, d.
section IV.B, Case Study 6.
III.B.3 Theoretical models of coregionalization

## Warning Even though the study of coregionalizations IS In every way

analogous to that of regionalizations, there is an additional difficulty due to
the number of indices involved. Thus, the reader who is not familiar with
the index notation of matrices is advised to go directly to the next section,
IILB.4. A thorough understanding of the study of regionalizations will
make the study of coregionalizations more understandable on a second

## Consider a set of K second-order stationary RF's {Zk(X), k = 1 to K}. The

object of this section is to establish cross-covariance models Cu,(h), ct.
section II.AA.
The positive definite condition

## Just as any function g(h) cannot be considered as the covariance or the

variogram of a second-order stationary RF, any set of functions {gkk'(h); k,
k' = 1 to K} cannot be considered as the matrix of covariances or variograms of a set of co regionalized RF's. The matrix of co variances {Ckk(h)}
must be positive definite so as to ensure that the variances of all finite linear
combinations of the basic RF's Zk (x) are positive.
In practice, to construct a matrix model of coregionalizations, we shall
first build a model of the spatial intercorrelation of the RF's Z; (x). The
matrix model of this coregionalization will then be, by definition, positive
definite.
The linear model (for coregionalizations)

## Consider n second-order stationary RF's {Y,. (x), i = 1 to n} with the

functions K;{h) as direct covariances, These n RF's are orthogonal, i.e.,
their cross-covariances are such that Kif (h) = 0, 'r/hand i -,i; j.
Consider now the K second-order stationary RF's {Z, (x), k = 1 to K}
defined by the K linear combinations of the type
Zk(X)=

"

akiY;(x),

1=1

## whatever the parameters akl: negative, zero or positive.

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(111.23)

MINING GEOSTATISTICS

172

## The set of K intercorrelated RF's thus generated has a positive definite

cross-covariance matrix {Ckk(h )}:
n

Ckk,(h)=

L L

ak,ak'iKjj(h)=

,=1,=1

akiak'jKj(h).

,=1

## In the definition expression (III.23) of the RF Z; (x), some basic RF's,

Y,,(x) and Y,.{x), with i ':;t: i. can have the same direct covariance, K,,(h) =
Kj(h), while remaining orthogonal, K'i(h) = O. An expression similar to
(III.23) can be written by grouping the RF's which have the same direct
covariances:
(III. 24 )

yr

being
with y~ and y~' having the same direct covariance K;(h); y~ and
orthogonal except when i = j and 1=[' simultaneously. The cross-covariance of Zdx) with Zdx) is then written as
n

Ckk'(h)=

lIj

[La~ja~.;]K,(h)=

,=11=]

,~l

b~k,Ki(h),

with
Vi = 1 to n.
This last expression gives the n matrices of coefficients [b~k'] and is nothing
more than the general expression of a positive semidefinite matrix.
The linear model for coregionalization is then a model where

(i) all the direct and cross-covariances arc derived from linear
combinations of n basic direct covariances {Ki (h), i = 1 to n}, i.e.,

Ckk(h)=

I"

,=1

b~k" K;{h)

with

b~k'

b~,k,

Vi;

(111.25)

(ii) for a fixed index i, the coefficient matrix [b~k'] is positive semidefinite.
Remark 1 The condition enforcing the coefficient matrices [b~k'] to be
positive semidefinite is a necessary and sufficient condition for a set of
functions {Ckk(h)}, defined as linear combinations of n basic covariances
K(h), to constitute a linear model of coregionalization. The condition is
only sufficient for this set of functions {Ckk(h)} to constitute a model of
coregionalization.
Remark 2 Remember that a matrix [bkk,] is positive semidefinite if all its
eigenvalues are real and non-negative.

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173

## In practice, there is no need to look for the eigenvalues of the matrix

b k k, = bk'k> it is positive semidefinite
when its determinant as well as its diagonal minors are strictly positive.
Thus, for a matrix of dimension K, we have the following K conditions:
b 1 1 b 12
blK

b 12

bn

~O,

~o.

... ,
bK

bK 2

(111.26)

bK K

## In mining practice, coregionalizations more than three variables (K > 3)

are seldom considered, mainly for difficulties of statistical inference of the
covariances matrix.
For K = 2, verification of the positive definiteness of the matrix rb~k']
amounts to checking that, Vi, the two following inequalities hold true:

bL >0;

## Ib~21 = Ib~11 ~J(b~lb;2)

(Schwarz's inequality).

## For K = 3, in addition to the two preceding inequalities, one must verify

that the determinant Ib ~k' I is positive or zero, Vi.
The linear model written in terms of semi-uariograms

## By replacing the covariances Ckk,(h) and Kj(h) by the corresponding

semi-variograrns 1"kk{h) and 'Yi (h), the definition formula (111.25) can be
rewritten as
l'kk,(h)

"

= L b~k"

with b ~k'

Yi (h)

i=1

b~'b

Vi,

(III.27)

## where, Vi fixed, the coefficient matrix [bL,] is positive definite.

Restrictions of the linear model
Restriction 1 As b~k' = bk'k, Vi, the linear model only provides symmetric
cross-co varia nces:
Ckdh)= Ck'k(h)

and

Ckk,(h) = Ckd-h).

## Now, there are non-symmetric cross-covariances, d. formula (11.18).

However, non-symmetric cross-covariances can be obtained from
formulae (III.23) and (III.24) by shifting some of the RF's Z (x ) by a
vectorial lag h k :
Zk(X + h k) =

Zdx) =

~ aki Y;(x),

ac-, Yj(x).
j

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(111.28)

174

MINING GEOSTATISTICS

## or, with the change of variable, h

h k + h',

This last cross-covariance Ckk,(h) is no longer symmetric in (h, -h) as, Vi,
Ki(h -hk)r=. Kj(-h -h k) unless the lag h k is zero.
Restriction 2 The coeffident matrices [b ~k'], \;/ i, must be positive definite.
This condition entails Schwarz's inequality:

Vi,

Vk rfk'.

(111.29)

## Consequently, b~k' > 0 entails b ~k . b~'k' > O.

Coming back to the definition (111.25) of the linear model, Schwarz's
inequality implies that every basic structure K;(h) appearing on the crosscovariance Ckk,(h) must also appear on the two direct covariances C kk (h)
and Ck'k,(h).
Thus, for example, if Ckk,(h) reveals two nested ranges a 1 and a2, these
two ranges must also appear on the direct covariances Ckk(h) and Ck'k,(h).
On the other hand, a structure may appear on the direct covariances
without being present on the cross-covariances, For this, it is sufficient that
b~k' = 0 when b~k > O.
Intrinsic coregionalization

## This particular example of the linear model corresponds to the case in

which all the basic RF's Yj(x) of formulae (III.23) and (111.24) have the
same direct covariance Ko(h) (or the same direct semi-variogram 'Yo(h )):
K,(h) = Ko(h),

\Ii = 1 to n.

## The model of intrinsic coregionalization is then

Ckk,(h) = bkk, . Ko(h), }
'Ykk(h) = bkk, . 'Yo(h),

(III.30)

## the matrix of coefficients [b kk'] being positive definite.

In practical terms, the intrinsic coregionalization model means that all
the direct or cross-covariances (or variograms) can be derived by affinity
(multiplication by the constant b kk ,) from a single basic model Ko(h) or

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175

## III. STRUCTURAL ANALYSIS

yo(h). This model thus corresponds to variables Zk(X), Zk'(X), the variabilities of which are proportionally related to a unique principal cause
characterized by the model yo(h). This occurs in mining practice in the
study of the two proportional variables, thickness t(x) and accumulation
a (x) = t(x) . z (x); z (x) being the mean grade of the drill core of length t(x).

Restriction of the intrinsic coregionalization model The intrinsic coregionalization model is particularly convenient for calculations, since studying the coregionalization amounts to the study of the single regionalization
with model l'o(h). On the other hand, this model is a very particular one
and has an important limitation in applications: it can be shown, d. section
V.A.4 "Cokriging", that if two variables Z; and Zk' are in intrinsic coregionalization and if these two variables have been sampled in an identical
way (e.g., each sample is analysed systematically for Z; and Zd, then there
is no gain in using the Zk' information in the estimation of the Zi:
However, if the variable Zk has been undersampled with respect to Zk',
then the estimation of Z; can be improved by using Zi. values.
111.8.4. Models of anisotropy
As x represents a point with coordinates (x"' Xl" x w ) in the three-dimensional space, the notation y(h) represents a function of the vector h of
rectangular coordinates (h", h v , h w ) or spherical coordinates (I h I, a, 'P ),
where Ih I is the modulus of the vector h and a and (() its longitude and
latitude.

Isotropy When the function y(lhl, a, q; depends only on the modulus Ihl
of the vector h, the phenomenon is said to be "isotropic". The variability of
the three-dimensional RF Z (x"' xv, x w ) , characterized by the serni-variogram y(ih I), is identical in all directions of the space and, thus,
(i) in spherical coordinates,

Va and

tp ;

## (ii) in rectangular coordinates,

Anisotropy The phenomenon is said to be "anisotropic" when its variability is not the same in every direction. The structural function
y(lh I), a, 'P) characterizing this spatial variability in mean square thus
depends on the direction parameters a and tp,

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176

MINING GEOSTATISTICS

## Interpretation The notion of mean variability in the above definition is

important. In a massive stockwerk-type ore body, rich, small, local veinlets
can be found in specific directions, but, on average, these veinlets will occur
over the entire mass in every possible direction of space and the
phenomenon will be isotropic on average.
In practice, a real anisotropy of the structural function 1'(h) corresponds
to the existence of preferential directions at the time of the genesis of the
studied phenomenon. These preferential directions are, in general, known
a priori and the variogram quantifies their respective variabilities. For
example,
(i) the vertical direction in a deposit formed by deltaic deposition;
(ii) the horizontal directions of the deposition currents in an alluvial
deposit;
(iii) the radial directions around a volcanic pipe in an intrusive deposit.

## Anisotropies and nested structures

We have already seen that the observed variability of a phenomenon is
most often due to many causes ranging over various scales. Consider a
nested model y(h) consisting of the sum of a micro-structure 1'1 (h) and a
macro-structure 1'2(h):

## There is no reason for these two-component structures to have the same

directions of anisotropy. Thus, the micro-structure 1'1, characterizing, for
example, the measure errors and the phenomena of diffusion and
concretion at very small distances, may be isotropic, while the macrostructure 1'2, characterizing, for example, the lenticular deposits of the
mineralization, may reveal directions of preferential alignment of these
lenses, d. Fig. II.5. In such an example, the structure 1'1 depends only on
the modulus Ih I of the vector h and it can be represented by anyone of the
isotropic models presented in the previous section, III.B.2. On the other
hand, the macro-structure 1'2 will require a model 1'2Clh I, a, If') which
depends not only on the modulus Ih I but also on the direction of the vector
h.
Representation

## All the theoretical models of regionalization presented in section 111.8.2

are isotropic, i.e., y(h) depends only on the modulus r = Ih I of the vector h.

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177

## Anisotropies will be represented by the method of reducing them to the

isotropic case either by a linear transformation of the rectangular coordinates (h u , h v , h w ) of the vector h in the case of geometric anisotropy, or
by representing separately each of the directional variabilities concerned in
the case of zonal anisotropy.
1. Geometric anisotropy

## A serni-variogram y(h u , h.; h w ) or a covariance C(h", h.; h w ) has a

geometric anisotropy when the anisotropy can be reduced to isotropy by a
mere linear transformation of the coordinates:
y(h u , h o, h w ) = yi( J(h ~2
anisotropic

+ h ~2 + h ~))

(111.31 )

isotropic

with
h~ = as-h; +a12hv +a13hw,

## h'; = a-ah; + Q2Zh" + aZ3hw,

h~

= as.h; +a32h,,+a33hw,

## [h']:= [AJ . [h],

where [A] = [ail I represents the matrix of transformation of the coordinates, and [h] and [hi] are the two column-matrices of the coordinates.
As a first simple example, suppose that we are only interested in two
particular directions, aJ and (Xl, in space. Let h , and h : be the distances
measured along these two directions, cf. Fig. III. 1O. The two serni-variograms Ya] (h \) and Ya2(h l ) calculated in these two directions have been

- ;;>"T""------=~-----1

o
FIG.

## III.10. Geometric anisotropy.

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Ihj

178

MINING GEOSTATISTICS

represented by two spherical models with the same sill 1 and different
ranges aOl l and a 0l 2 Thus, since 1''''1 (h)"= y"'2(h), the phenomenon represented is anisotropic: the variability depends on the directions Cl'j and a2.
In order to make the two curves Ycq and 1''''2 in Fig. 111.10 coincide, it is
sufficient to multiply h, by the ratio of affinity QOI7./a"" or, conversely, to
multiply the distances h by the ratio a",) a",7.' Indeed, consider the two
spherical models

and

## The following relation is then derived:

i.e.,

a
h',1 -h
I ~.
a",
Subject to the change of coordinates h', = h1a",)(l"'Il the variability in the
two directions a 1 and a2 is characterized by a single spherical model with a
range a"'2'
More generally, if n directional semi-variograms {y",Jh i ) , i = 1 to n} can
be represented by n transition models of the same type (spherical, for
example) with the same sill, and the n ranges of which form an ellipticalshaped directional graph (in two dimensions), or ellipsoidal (in three
dimensions), geometric anisotropy is present.
An example is given on Fig. 111.11, which shows four semi-variograms
for four horizontal directions ai, Cl'2, Cl'3, Cl'4. Spherical models with identical
sills and ranges of a",l' a"'2' a"'3' a"'4 have been fitted to these semi-variograms. The directional graph of the ranges, i.e., the variation of the ranges
a"" as a function of the direction ai, is also shown on Fig. 111.11. There are
three possible cases.
(i) The graph can be approximated to a circle of radius a, i .e., a = a,
for all horizontal directions ai, and the phenomenon can thus be
considered as isotropic and characterized by a spherical model of
range a.
Q ,

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179

## (ii) The graph can be approximated by an ellipse, i.e., by a shape which

is a linear transform of a circle. By applying this linear transformation to the coordinates of vector h, the isotropic case is produced (circular graph). The phenomenon is a geometric anisotropy.
(iii) The graph cannot be fitted to a second-degree curve and the second
type of anisotropy must be considered, i.e., zonal anisotropy in
certain directions, CK4, for example, on Fig. 111.11.

"'I
{i) lsotropy

(Ii) GeomelrlC

FIG.

orus otropy

## IILll. Directional graphs of the ranges.

If, instead of transition structures of range a"" as in Fig. III. 11, the directional semi-variograms are of the linear type, 'Y",; (hi) = W",,h'i, then the
directional graphs of the inverses of the slopes at the origin WO: will be
considered. An hypothesis of isotropy, geometric anisotropy or zonal
anisotropy will then be adopted according to whether or not this directional graph can be fitted to a circle or an ellipse.
j

## Consider a geometric anisotropy in two dimensions (e.g., the horizontal

space). The directional graph of the ranges a", is elliptical, cf. Fig. III. 12 .
Let (x .., xv) be the initial rectangular coordinates of a point, ({! the angle
made by the major axis of the ellipse with the coordinate axis Ox .., and
A > 1 the ratio of anisotropy of the ellipse. The three following steps will
transform this ellipse into a circle and, thus, reduce the anisotropy to
isotropy.
(1) The first step is to rotate the coordinate axes by the angle q; so that
they become parallel to the main axes of the ellipse. The new

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MINING GEOSTATISTICS

180

## coordinates (Yl' yz) resulting from the rotation can be written

matrix notation as

[;J = [R~I[::]

In

with

tp,

Xv

Y1
FIG.

## 111.12. Correction of geometric anisotropy.

(ii) The second step is to transform the ellipse into a circle with a radius
equal to the major range of the ellipse. This is achieved by multiplying the coordinate Yz by the ratio of anisotropy A > 1. The new
coordinates (y~, yD are then deduced from the coordinates
Y2)
by the matrix expression

(Yl'

[Yy;~ J = 1Al[Yl]
Y2

with

1 ~J.
/\

[1]-=[0
/\ -

(iii) This last step is not necessary for a mere correction of geometric
anisotrophy.
The initial orientation of the coordinate system can be restored by
rotation through the angle -cp, and the final transformed coordinates (x~, x~) are given by

The final transformed coordinates (x~, x~) can then be derived from
the initial coordinates (xu, Xl!) by the transformation matrix [AI
which is the product of the three matrices [R-..,][AJ[R~], i.e.,
(III.32)
with

[Al=[;

~J

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and

a = cos
. 2
b =sm
c

= (l -

181

\.")

'1'+/\ s1O-'P,
2

## 'I' +A cos 'P,

A) sin

({!

cos tp,

If h is any vector in the two-dimensional space with initial coordinates li.; h,J, then, to obtain the value of the anisotropic
semi-variograrn y(h) = y(hr.;, h,J, we first calculate the transformed
coordinates (h ;,j, h ~) from

[hh;J= fAJ[hr.;]
hI;
L

## and then we substitute these coordinates in the isotropic model

y'(lhll), which has a range equal to the major range of the directional
ellipse, i.e., according to formula (III .31),

anisotropic

isotropic

## Remark 1 In practice, in two dimensions, three or four directions (Xi must

be studied in order to establish the directional graph of ranges and
subsequently decide whether a geometric anisotropy is present.
Remark 2 Optimum sampling grid. The minor axis of the ellipse (OY2 on
Fig. III.12) is the direction in which the phenomenon varies fastest, and the
major axis the direction in which variability is slowest. Thus, the optimum
sampling grid for this phenomenon is a rectangular one, aligned in the
directions of these two main axes (OYl and OY2) with sides in the ratio A,
the smallest side corresponding to the direction 0Y2 of fastest variability.

Remark 3

## It is possible to determine the matrix rAJ for the correction of

a geometric anisotropy in three dimensions. However, reality seldom conforms to such a particular model as an ellipsoid of geometric anisotropy. In
practice, a zonal anisotropy will most often be used to distinguish certain
directions (e.g., the vertical direction from the horizontal directions).

2. Zonal anisotropy
The model of zonal anisotropy is the one most currently used in practice,
since any observed anisotropy which cannot be reduced by a simple linear
transformation of coordinates will call for this model.
Let y(h) be a nested model characterizing a variability in the threedimensional space y(h) = 'i.m(h), where h is a vector with coordinates (hr.;,

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MINING GEOSTATISTICS

182

hv , h w ) . Each of the components Yi(h) of this nested model can be anisotropic in h, i.e., y;(h) is a function of the three coordinates (h u , h-; h w )
rather than of just the modulus Ihl. Moreover, the anisotropy of YiCh) may
be completely different to that of YiCh). Thus, the structure Yl (h) may have
a geometric anisotropy, while Y2(h) is a function of the vertical distance h-:
only:

## A third structure may be isotropic in three dimensions: Y3(h) = Y3(lh I).

The model of zonal anisotropy can thus be defined as a nested structure
in which each component structure may have its own anisotropy.
An obvious anisotropy of the structural function y(h) will most often
correspond to a genetic anisotropy known beforehand, so that any preferential direction is well known and, thus, can be differentiated when
modelling the three-dimensional structural function y(h).

""'-~~~_~~~~~

~.~

__

~~----l~

## On Fig. 111.13, the semi-variogram y(h w ) constructed in the vertical

direction reveals a strong variability due, for instance, to surface weathering or enrichment with depth. The mean semi-variogram, isotropic for the
horizontal directions y(J(h ~ + h ~)) reveals a weaker variability due, for
example, to lenticular beds in the horizontal strata.
On Fig. 111.13, the difference 'Y(hw)-y(J(h~+h~)) is approximately
linear, at least over the smaller distances, so that a three-dimensional
nested model y(h) can be adopted, y(h) being the sum of an isotropic
transition model 1'1 (ih I) and a linear model Y2(h w ) :
y(h", h-; hw ) = Yl(lhl)+ Y2(h w ) .

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183

## The model ')11(Ih I) is fitted to the mean horizontal serni-variograrn and

the model Y2(h w ) to the difference y(h w ) - y(J(h ~ + h; ).

Remark. The model of zonal anisotropy, built from the nest of structures
having their own anisotropy, is flexible enough to be used as a model for
almost any type of experimental anisotropy.
However, the flexibility of this model must not be misused. When
a specific direction is differentiated (e.g., the vertical direction h.; of
the previous example), this direction must correspond to a preferential
direction of the mineralization. The known geology should always be
used as a guide when fitting a mathematical model y(h) to a natural variability.

## Ill. B.S. General structural model

Now that the different models of anisotropy have been studied, the following general expression of a structural model y(h) can be proposed, 1'(h)
being made up of the nested sum of N isotropic structures {"i(lhil), i = 1 to
N}:
N

y(h) =

(111.33)

1'i(lh i l)

i=1

The particular anisotropy (geometric or zonal) of each component structure 1'i is characterized by the linear transformation matrix [A;] which
changes the vector h into the vector hi. More precisely, let (h u , h.; h w ) be
the initial coordinates of the vector h; the coordinates (h i u , hi,v, hi,w) of the
transformed vector hi are deduced from the initial coordinates by the
linear relations

[hd
Example

## Consider the following three-dimensional model:

(111.34)
with Yo(lh I) = Co when Ih I> I; characterizing an
1'1 (!h I)
characterizing
a
first
isotropic
1'z(J(h~ .... + htv + htw)), characterizing a second
regard to the coordinates of the transformed
For example, this second structure may depend

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## isotropic nugget effect;

structure;
)l2(lh zl) =
structure isotropic with
vector [h z ] = [A 2 } . [h].
only on the horizontal

184

MINING GEOSTATISTIC\$

directions (hu , hv ) with a geometric anisotropy. The previous linear transformation is then written

h3 w

a. h., + a. hv + a. h;

O.

## The last component structure, Y3(h w ) , depends only on the vertical

direction (h,,,). This structure, Y3(h w ) , can be considered as isotropic with
regard to the coordinates (h3 , u , h3 ,,, , h3 , w) of the transformed vector [h 3 ] =
fA 3 ] [h]:

The general structural model of formula (111.33) is particularly convenient since it is made up of the sum of N isotropic structures Yi(!hjl)
which, in the course of a particular geostatistical operation, will be treated
separately one by one, ct. section lILA. I, "The linearity of geostatistical
operators".
All the case studies and practical exercises given in this book make use of
this general structural model.

Computer subroutine

## The subroutine FUNCTION GAM(HX, HY, HZ) gives the values of a

three-dimensional model y(h) of the general type (111.33). The component
structures 'Yi are transition models, either spherical or exponential. A
nugget effect is represented by a transition model with a zero range.
Example The parameters of the previous model (IIL34) given as an
example would be: (HX, HY, HZ) = initial coordinates of the vector h;
NST = 4 = component structures, with respective sills C(NST) and ranges
A(NST). For the nugget effect, Yo(lh I), we have A(I) = O.
The matrices of linear transformations of the coordinates are

CAX(4,3)=

I
I

all
O

0
0

al2
0

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-;.

## for the first coordinate of the first component Yo

for the first coordinate of the third component Y2

## III. STRUCTURAL ANALYSIS

CAY(4, 3)=

1
0

1
1

a2 1

a 22

0
0

FUNCTION

1
and

185

1
0
CAZ(4, 3)=
0

1
0
0

a a

GA~(rX,rY,HZJ

c
C
SE~I-VARICGRA~ FU~CTIC~
C
C DEFINED AS TrE SUM OF EXPJ~E~TIAL CR SFrfRICAL
C ~E~TED STRUCTURES TrES= SIRLCILRES ARE ISCTRCPIC

C
C

## AFTER L!NfAP TOA~SFOPMATIO~ ( f INITIAL CC(RDINATES

hX.Hy,Hl ~E~ CCORDINATES ARE [X,DY,Ol

C CCMMCf\S

IGAfJ"Aj

JF

t\ST

NU~~ER

(NSfl

A.NST)

SILLS
RA"IGES <0 .. EXPCP\E!\ liAl

## =0. NLGGEI EFFECT

C
C

>0. SPHERlcn

CAX{I\iST,~}

C/. Y ( ~J ST ,

CAI(NST,~)

ELE~ENIA~Y STRU(TL~ES

LINEAR TRA~SF(~~ATI[~
tJATRICES CF
:3 COGROIf\ATU

......................................................

## CIMEt\SION /l(ll,C(ll,CA)l(!),CAY(ll ,Ct.l(l)

COMMC~/GA~~A/~Sr,c.A,cAX,C~Y,CAI

GAM'"

o.

H=SQRT(HX*rX+rY*~Y+rlZ*rZ)
IF(H.LT.l.E-03IRETUR~

CU 1 IS=l,NST
I JS= IS
IJSl=IJS+\JST

[JS2=IJSl+NST
CX=HX*CAX( IJS)+HY*CAX( IJSll+Hl*CA}l( IJS2J
ov =H X*C II V( I J S I +H V*C A'(( 1 J S 1 ) +HZ C A V ( I J S 2 )
Cl=HX*CAZ( IJSI+HV*CAZ( IJS 1 )+Hl*CAZ (1 .. 52)
H=\$QRT (C)(*Cx'+-CY*CY+DZ*CZ)
IF!A{lS) 110.12,11
Gt\M=G/lt-l+(IS~*(I.-ExP( .. JtO{I5HI
GC Tel

10

11 IF(H.GE.A(ISllGO r c 12
G /J ~= Gll ~ + c( 1 S ) * I 1 '; "" H I A ( IS)1 C. 5* r *H'1' 1--/ ( A( IS) * A( IS) *A ( IS
GO TO 1
12 G" M.::; G A,.. + C( I S J
1 Cf'NT I f'.UE

RETLJRI\
END

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J) I

186

MINING GEOSTATISTICS

## 111.8.6. Proportional effect and quasi-stationarity

Let x and x' be two points with coordinates (x"' xv, x w ) and (x~, x~. x ~).
Without any hypothesis of stationarity,
(i) the expectation of the RF Z(x) depends on the position x, i.e., on the
three coordinates (x". xv. x w ) ,
E{Z(x)} = m(x". x"' x w ) ;

(ii) the semi-variogram y(x, x') or the covariance C(x, x') depends on
the two locations x and x' of the R V's Z (x) and Z (x '), i.e., on the six
coordinates (x"' x"' x.,) and (x;" x ;" x~),
~E{[Z(x) - Z(x')f} = y(x, x').

## It was indicated in section II.A.2 that an hypothesis of quasi-stationarity

is generally sufficient for geostatistical applications. This hypothesis
amounts to assuming that:

## (i) the expectation is quasi-constant over limited neighbourhoods and,

thus, m(x)=m(x')=m(xo) when the two points x and x' are inside
the neighbourhood V(xo) centred on the point Xo, cf. Fig. III.14;
(ii) inside such a neighborhood V(xo), the structural functions y or C
depend only on the vector of the separating distance h = x - z ' and
not on the two locations x and x'; however, this structural function
depends on the particular neighbourhood V(xo), i.e., on the point xo,
y(x, x') = y(x - x', xo) = y(h, xo).

## VX, x' E V(xo).

The construction of a model of quasi-stationarity thus amounts to building a model of the structural function y(h, xo) which depends on the
argument xo.

## FIG. 111.14. Neighbourhood of quasi-stationarity.

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## III. STRUCTURAL ANALYSIS

187

Proportional' effec t

## In practice of mining applications, it has been observed that experimental

serni-variograms y(h, xo), -y(h, xo), ... , defined on different neighbourhoods V{xo), V(xo), . . . , can be made to coincide by dividing each one of
them by a function of the corresponding experimental mean m *(xo),
m*(xo), of the data available in each neighbourhood, d. Fig. 111.15:
y(h, xu)/ fUm * (xo)] = y(h, xo )/f[m *(xo)].

This amounts to assume that there exists a stationary model yo(h) independent of the neighbourhood location Xu and such that:
y(h, xu) = f[m*(xo)]. yo(h).

(11.35)

The two quasi-stationary models y(h, xo) and y(h, xb) are said to differ from
each other by a proportional effect.
r
___- - - - - - - - - - y(h,xo)

---------~

___- - - - - - - - - - y ( h , x O )

## The experimental mean m *(xo) appears as an estimator of the expectation

E{Z (x)} = m (xu), Vx E V(xo), which is quasi-constant over the neighbourhood V(xo).
The proportional effect function f is determined by studying the proportional relationships between the various experimental semi-variograrns
y(h, xo), y(h, xo), .... Of course there must be enough data available on
each neighbourhood V(xo) to form a correct estimator m*(xo) of the
expectation E{Z(x)}, x E V(xo). Thus neighbourhoods that are too small
compared to the data grid cannot be considered. Examples of the fitting of a
proportional effect are given in the various case-studies of Chapter IV.

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MINING GEOSTATISTICS

188

## Remark 1 Direct and inverse proportional effect. The proportional effect

is said to be direct if the experimental semi-variogram increases with the
corresponding experimental mean, i.e., when the function f[m *(xo)} in
formula (III.35) increases with m *(xo).
Such a direct proportional effect can be expected when the studied ReV
z (x) has a lognormal-type histogram, the mode M being less than the
expectation m. Examples are the regionalizations of Cu, U, Au grades
or, more generally, of grades of mineralizations of weak concentration,
d. Fig. 1I1.16(a).

r
r(h,xo)

r(h,xol

m*(xol>m*(x~l

m*(xom*(xo)

r(h,xo)

r(h,xo)

(c)

FIG. 111.16.

## (a) Direct proportional effect. (b) Inverse proportional effect.

(c) Proportional effect on the nugget constant only.

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189

## The proportional effect is said to be inverse when the sills of the

experimental serni-variograms decrease as the experimental mean
increases.
Such an inverse proportional effect can be expected when the studied
Re V z (x) has an inverse lognormal-type histogram, the mode M being
greater than the expectation m. For example, the grades of Fe-hematite or
tricalcite phosphate or, more generally, the grades of mineralizations of
strong concentration, d. Fig. III.16(b).

## Remark 2 Proportional effect and geostatistical operations. As the main

geostatistical operations are linear in y (or in C-covariance), d. formula
(III.2), a proportional effect present on the structural function yore will
appear identically in each of these operations.
More precisely, if 't is anyone of the three main geostatistical operations (i.e.
estimation and dispersion variances and regularization), and if the proportional effect model of (111.35) is considered, then:

and:
g[ y(h, xu)] = f[m*(xo)] . g[ yo{h)].

(III.36)

It is enough to work on the stationary model yo(h) and multiply the result
\E[ ')Io(h)] by the corresponding proportional effect factor f[ m * (x 0)].

## Remark 3 Proportional effect in m*2. Very often, in mining applications,

the proportional effect observed is a function of m *2, i.e., the function
f[ m *(xo)] is proportional to the square of the experimental mean; thus,
(i) for a direct proportional effect,

## f[m *(xo)] = [m*(XO)]2;

(ii) for an inverse proportional effect,

f[m*(xo))

= [A -m*(xo)]2,

## where A is the maximum value that m*(x) can take.

With a direct effect in m*2, the relative semi-variograrn, i.e., the local
serni-variogram divided by the square of the local mean, appears as a
stationary function independent of the localization Xo:

## VXo and x~.

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190

MINING GEOSTATISTICS

## Consequently, the various relative variances of estimation and dispersion

also appear as stationary and can be caiculated directly from the stationary
model1'o(h), i.e. according to formula (111.36):

## The 95 % confidence interval attached to the corresponding estimate is then:

Note the similarity of this expression with the one quoted in (11.21), m *(xo)
being replaced by the estimate zf itself.
Remark 4 Proportional effect and nested structures. When nested structures are present, l' = LiYi, the proportional effect may be different for each
of the component structures, i.e., the function f;[m*(xo)] may be different
for each of the models 1'i.
Thus, on Fig. III. 16(c), the direct proportional effect affects only the
nugget constant Co:

## '+Ix and h >0,

the structure )'1 (h) being stationary (independent of the location x).
A good practical way to reveal a proportional effect is to draw the
relative serni-variogram (when a direct effect is suspected) on transparencies: the visual comparison thus shows whether or not the proportional effect affects the micro-structures (e.g., nugget effect) and the
macro-structures equally.
Remark 5 Genesis of a proportional effect. The proportional effect is an
experimental observation and can be interpreted as follows.

## (i) As an effect of the non -stationarity of the RF Z (x) which is only

locally stationary or quasi-stationary. This is the interpretation given
up to this point.
(ii) As a conditioning effect to the local experimental means m *(x), the
RF Z (x) being strictly second-order stationary.
In fact, the local serni-variogram y(h, xo) calculated over the neighbourhood V (xo) is relative to the RF Z (x) conditioned by the set of
available experimental data on V(xo) and not to the RF itself. This local
conditioning effect on V(xo) can be represented by a new RF

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191

## on V(xo). If y(h) is the stationary model of Z (x), the semi-variograrn

y(h, xo) of the conditioned RF is no longer stationary (it depends on xo).
Knowing the spatial law of the RF Z(x) and the type of estimator m*(xo)
considered, it is possible to calculate the theoretical expression of
the conditioned model y(h, xo); the calculations can be found in G.
Matheron (1974). Here, we shall limit ourselves to the following principal
results.
(i) When the RF Z (x) has a Gaussian spatial law and when it has been
conditioned by the best possible linear estimator m*(xo) (i.e., by the
BLUE kriging) of the mean grade Zv(xo) on V(xo), then the conditioning does not alter the stationary character of the variogram and,
thus,

## 1"(h, xo) = 1"(h),

VXu.

This does not hold true for any other definition of the conditioning
variable m *(xu).
(ii) When the RF Z(x) has a lognormal spatial law (i.e., Y(x)= log Z(x)
has a Gaussian spatial law) and if it has been conditioned by the
kriging BLUE estimator m;:"(xo) of the mean grade Yv(xo) of Y(x)
on V(xo), then the conditioning is expressed on the local semivariogram 1"z (h, xo) of the conditioned RF {Z (x)/ m 't-(xo)} as a pro2
portional effect in m t (xo):

where mv(xo)=E{Z(x)/m't-(xo)} is the expectation of the conditioned RF. In practice, this expectation will be estimated by the
experimental mean m~(xo) of the data Z(x) available on V(xo).
r(h) is a stationary reference semi-variograrn (independent of xo)
and is related to the stationary semi-variogram 1"(h) of the nonconditioned RF Z (x).
From this brief survey of the possible genesis of a proportional effect, it
is enough to remember that the experimental observation of such an effect
does not necessarily indicate a non-stationary phenomenon.

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MINING GEOSTATISTICS

## In practice, all that is known of the structural function y(h), or C(h), is an

estimator y*(h) derived from the data Z (x.), Since the expressions for the
various geostatistical operations have be.en constructed using the function
y{h), it is useful to study the possibility of estimating this function y(h)
from a finite network of experimental data {z (x,)}.
Let Z (x) be a RF assumed to be stationary on domains of limited extent
V. On one such domain V the experimental data {z(x;)} are available, from
which the following experimental variogram can be calculated:

(111.39)
where N' is the number of pairs of data values on V separated by the
distance vector h.
If the domain V was perfectly known, i.e., if values of the realization
z (x) were known for every possible location x in V, the following variogram, called the" local variogram" on V, could be calculated:
2Y{Vl(h)

= ~,

t.

[z(x

+ h)-Z(X)]2 dx,

(111.40)

where V' represents the intersection of the domain V and its translation by
the vector h ; in other words, x E V' = V V h is equivalent to x, x + h E V.
The experimental variograrn 2y*(h) and the local variogram 2y(VJCh)
are both relative to the particular realization z (x) of the RF Z (x); these
two variograrns thus appear as random variables, the expectation of which
is precisely the theoretical variogram of the RF Z (x):

2y(h) = E{[Z (x

(111.41)

## In practice, only the experimental variogram 2y*(h) is available; this

gives rise to the first deviation [Y(vlh) - y*(h)], which is the difference
between the experimental value and the local value, due to the fact that
only a finite number N ' of data pairs, [z(xi+h),z(x;)] separated by a
distance h, is available. This deviation is characterized by the variance
E {[Y( VJ(h) - y* (h) ]2} called the" variance of esti rnalion" of the local semivariogram.
However, even if the domain V, and, thus, the local semi-variogram
"Y(v)(h) were perfectly known, there would still be a second deviation
between this local expression and the theoretical serni-variogram y(h).
This deviation is called a "fluctuation" and is characterized by the variance

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## III. STRUCTURAL ANALYSIS

193

E {[ y(h)- y(v)(h )]2} called the "fluctuation variance" of the local semivariograrn.
In fact, since the ReV z (x) is the only available reality (the RF Z (x)
being a mere probabilistic model), only the local semi-variograrn y(v)(h)
and its estimator y*(h) have any physical meaning; consequently, only the

## estimation variance of the local semi-variogram is of practical interest. The

fluctuation variance merely gives an order of magnitude of the deviation
which can be tolerated between the local semi-variogram y(v)(h) and the
fitted model y(h) which is chosen to be as simple as possible.

## Order of magnitude of the variances of estimation and fluctuation

The estimation variance E{[ y(v)(h) - y*(h )]2} depends on the number N'
of experimental data pairs available on V, while the fluctuation variance
EH l'(h)- y{V)(h )]2} depends on the dimensions of the domain V.
These two variances involve moments of order 4 of the RF Z (x).
Therefore, it is necessary (at least) to calculate moments of order 4 to
characterize the quality of the inference of a moment of order 2; this
apparently paradoxical fact is a characteristic of all statistical tests.
The calculations involved in these moments of order 4 are tedious and of
no real interest in a book devoted to practical mining geostatistics. The
study can be found in G. Matheron, (1965, Chap. XIII), and a summary of
the principal results follows.
These results are for an intrinsic RF Z (x) with a Gaussian spatial law
s
and a theoretical serni-variogram y(h)= Ihl . The domain V is, in the
one-dimensional space, a segment of length L (e.g., the length of a drill
core). The following results are obtained.
(i) For the estimation variance,
2

E{[Y(L)(h)-y*(h)] }=4y(h).

2(O/L)

N'

'

where D\O/ L) is the dispersion variance of the point-support variable Z(x) in L. N' is the number of experimental data pairs [z(x; +
h), z(x;)] used in calculating the experimental semi-variograrn y*(h).
(ii) For the fluctuation variance,
L
for h <3

prohibitive value
with A = 4/3 for the linear model y(h) = Ih I.

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for h >2

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MINING GEOSTATISTICS

For the linear model and for h > L/2, the relative fluctuation variance
E{[y(h)-Y(Llh)]2}/fy(h)]2 is about 2 (i.e., a prohibitive value of 200%).
Note that even if the estimation variance vanishes (very large number N'
of pairs), the fluctuation variance allows the choice of any possible
behaviour of the theoretical model y(h) for distance Ihl > L/2 greater than
the distance of reliability which is half the dimension of the field L considered.
Note again that this considerable value of the fluctuation variance prohibits the construction of goodness-of-fit tests for the fitting of a theoretical
model y(h) to an experimental semi-variograrn y*(h): such tests would
almost never invalidate the fit.
Practical rule
Although there is no reason for the RF Z (x) representing the ReV z (x) to
follow a Gaussian spatial law , the preceding expressions for the estimation
and fluctuation variances provide the following orders of magnitude:

and

(111.42)

## An experimental serni-variogram should only be considered for small

distances (Ih 1< L/2) in relation to the dimension (L) of the field on which it
has been computed, and provided that the numbers N' of available data
pairs are not too small.
This practical rule holds true for estimators of:
(i) cross-variograms or non-centred cross-covariances;
(ii) non-centred covariance, the stationary expectation m
being supposed known, i.e., for the estimator

C*(h)+m

= E{Z(x)}

N'

## Bias of the covariance estimator

The preceding estimator [C*(h) + m 2] of the non-centred covariance
[C(h)+m 2 ] is without bias, since
E{C*(h)+ m

}=

## E{Z(x +h). Z(x)}= C(h)+m 2

On the other hand, to get an estimator of the centred covariance C (h), the
expectation m must also be estimated from the same set of data {z (x;)}, and

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## III. STRUCTURAL ANALYSIS

195

a bias becomes obvious. Indeed, if N data are available on the quasistationary field V, the unbiased estimator of this expectation is
1

m" = ~
N

with

Z(Xi)

E{m*} = m.

j=l

## A possible estimator of the centred covariance is then written as

1 N'
1
C*(h) = N' i~l [Z(Xi + h). z(xdl - N2 ~ ~ z(Xj)z(x{)

(111.43)

and, thus,

## = C(h)- N2~ ~ C(Xi J

j').

The estimator C*(h) of formula (111.43) then shows a bias which is nothing
more than the mean value of the theoretical covariance C(h) when the two
extremities of the vector h independently describe the set of the N available data:
1

## E{C(h)-C*(h)}= N 2 j~l j"~1 C(Xj-x/).

(III.44)

This bias can be assimilated to the mean value C( V, V) when the N data
are uniformly distributed on the field V. When the covariance model C(h)
is of transition type, i.e., C(h) = 0 when the distance Ih I is greater than a
range a, then the bias C(V, V) vanishes if the dimensions of the field V are
much greater than the assumed range a of the covariance model.
This bias is linked to the simultaneous estimation, from the same set of
data, of the expectation m and the non-centred covariance E{Z(x +
h) . Z (x)}. This difficulty does not appear in the estimation of the uariogram,
since the expression
E{[Z(x + h) - Z(x )]2}

## Structural analysis, of mining phenomena in particular, is undoubtedly the

field of geostatistics which lends itself the least to totally computerized
methods. A structural analysis draws continuously on all the practical

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MINING GEOSTATISTICS

knowledge gained from the studied phenomena, and also involves a certain
amount of "craft" in the choice and use of the various geostatistical tools.
While skill can only be gained by experience, structural analysis has certain
general attributes. In particular, it should:
(i) be adapted to the proposed aim of the study. Structural analysis is
most often a prelude to an intended estimation and, as such, should
be limited to the purpose of the estimation. Thus, it is no use to
specify structures on a petrographic scale if it is intended to estimate
blocks on a hectometric scale.
(ii) take critical account of all qualitative information. The experimental
knowledge of a geologist may lead him to give all importance to a
data file; conversely, and more rarely, an erroneous geological
hypothesis may bias the use of such a file.

## Before beginning the geostatistical treatment of a data set, it is advisable

to become familiar with both the physical nature of the phenomenon under
study (geology, mining technology) and the available data. This preliminary phase is essential in any structural analysis, and also for a correct
formulation of the study; very often, it even defines the framework within
which geostatistical methods will be applied.
The next step is the "variography" phase, i.e., the construction, analysis
and fitting of the variogram models. The purpose of the basic geostatistical
tool- the variogram - is to condense the main structural featurest of the
regionalized phenomenon into an operational form. Variography makes as
much use of the previously introduced geostatistical concepts as it does of
the gained (qualitative or not) knowledge of the physical nature of the
phenomenon (metallogeny, structural geology, tectonics, mineralogy,
mining and treatment of are, etc.).

## Before making any geostatistical calculations, it is advisable to carry out a

critical assessment of the available data. Errors at all levels can be corrected and, very often, such an assessment can provide simple explanations
for results which may seem, at first sight, rather odd.
t Most readers have seen feasibility studies in which excellent studies of structural geology or

geomorphology are presented, only to be completely ignored in the estimation phase when
methods such as polygons of influence are used. If estimation is divorced from geology, one
might rightly wonder about the purpose of the geological study. Obviously, geological
information must playa part in the estimation and the variograrn is used in geostatistics to
interpret the structural information to be used in the estimation procedures.

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197
IS

given

## Study of the reconnaissance plan

What are the criteria upon which the plan is based? Has it been biased in
favour of a priori geological hypotheses or financial or technical constraints? Is it suited for the purpose at hand? A reconnaissance plan
designed for a tectonic survey may prove to be unsuitable for grade
estimation. The sampling material should also be suitable, e.g., type of
drilling equipment, diameter of cores, channel samples or core samples?
What type of sampling grid is used? e.g., regular grid, pseudo-regular
grid with constant sample density, preferential grid. Have the different
facies encountered and the variables of interest been sampled in a systematic and homogeneous manner?
Causes of systematic error

The various geostatistical tools are constructed from the available data. A
systematic error in these data will be reflected at every stage of the
geostatistical analysis. Geostatistics does not create data, it is a method for
treating information; in other words, geostatistics complies with the in situ
reality only as long as the data are an accurate reflection of this reality.
Thus, it is very important, especially at the data review stage, to detect all
possible causes of systematic error.

## Adherence to the sampling plan

Does the actual layout of data correspond to that of the plan? Geometric
errors may have been made, or the drilling platform may have been
displaced for accessibility reasons. Such displacements, even though small,
may cause considerable biases in estimation, e.g., if preferentially rich or
poor zones are the ones accessible to the drilling platform.
Has the deviation of bore-holes been controlled? Quite often, some
types of samples will tend to follow preferentially rich pockets of soft are
(e.g., bauxite).
What are the causes of breaks in drill cores? More or less random
accidental causes are not so important as definite physical causes, e.g.,
when a boulder of hard waste is encountered.
Have the different facies been sampled in an homogeneous manner?
Samplers do not always take the same care in sampling waste as they do
ore, but mining may recover part of this waste.

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MINING GEOSTATISTICS

## Representativeness of the data

What is the recovery rate of the core samples and how much does it vary?
Could the various methods of sampling, quartering and analysing have
introduced any systematic error? Is it possible to evaluate the variance of
measurement errors (account can be taken of this variance in the process of
cokriging, d. section V.A.5, Case study 3)? A fairly large variance of the
measurement errors can be tolerated, provided that these errors have a
zero mean (non-bias). However, this measurement error variance must not
be such as to mask a natural variability of interest.

## Homogeneity of the data

Has the representativeness of the data been constant through time and
space?

Through time An original sampling campaign may have been carried out
by different companies with different objectives and different equipment.
In space The same are body may have been sampled in different places
by very different methods, dragline samples on the surface, drill cores in
depth, samples taken along a drive, etc. Topography, rock type and
equipment used may vary from place to place and can lead to different
recovery rates and to biases (loss of fines in blasting holes for example).
Has the sampling of the various variables (thicknesses, grades, densities)
been homogeneous? In polymetallic deposits, certain trace metals,
although economically important, are not always as well sampled as other
metals (e.g., Ag and Au in Pb deposits). Some metal grades are measured
from other variables by regression, deconvolution (e.g, U), construction of
a metal "balance sheet", etc.

## Compiling the data

How and by whom have the data - from drill log sheet to magnetic tape been compiled? Aberrant grades can sometimes be traced to breaks in the
sequence of data compilation, e.g., changes in groups collecting samples or
transferring data from one type of computing device to another,
What type of error tests are made when the data file is being completed?
Is it possible to retrieve rejected data at a later stage?
On what basis have the geological maps and cross-sections been drawn?
how much of the interpretation is ascribable to the geologist or to the
draughtsman?
.

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199

## In practice, the following must be specified to define a regionalized variable.

(i) Significance, e.g., thickness (in metres) of a given geological horizon
or the volumetric grade of uranium obtained by deconvolution of
well logs.
(ii) Support, i.e., the volume on which the variable is defined, as well as
the method used to obtain the samples; e.g., the support may be a
piece of drill core of diameter d and constant length I obtained by a
specific type of drilling, or the support may be assimilated to a point
at which the thickness of a bed is measured, or again it may be a
grab sample of variable volume and approximately 1 kg of material.
(iii) Field of extension, i.e., the domain in which the spatial distribution
of the variable is to be studied. This extension field may cover an
entire mining concession or only a zone of primary mineralization,
the zone of secondary mineralization being studied separately.
A geostatistician should always participate in the definition of the ReV
that he intends to study, i.e., he should at least be consulted at the time that
a systematic sampling campaign is being planned. In fact, all potential users
of the information to be extracted from the sampling campaign should be
consulted to ensure that the information is relevant to the intended studies.
Such users would include the mining engineer and metallurgist as well as
the geologist. For example, a detailed lithological study in which only the
rich facies are analysed is not suitable for estimation in the frame of
non-selective mining.
In choosing and defining the ReV for the purpose of estimating a
deposit, the following rules should be observed.

1. Additivity of the variables The ReV must be such that all linear
combinations of its values retain the same meaning. If f(x) is a grade value,
:LA!(x;) and especially the arithmetic mean (l/nYLf(x;) must have the
Thus, a volumetric grade (in kg m", for example) defined on a constant
defined on variable supports is no longer an additive ReV and, in fact, the
mean grade of two different supports is not the arithmetic mean of their
grades. In such a case, the additive ReV to be considered should be the
accumulation (or quantity of metal in kg), i.e., the product of the volumetric grade (kg m:') and the support volume (m'). However, when studying the spatial variability of this accumulation, it would not be possible to

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MINING GEOSTATISTICS

distinguish between the variability due to the grade and that due to the
variable supports.
If the preceding volumetric grade z (x) is associated with a granulometric
fraction r(x) and both analyses are defined on variable supports v (x)
centred on point x, then the additive ReV accumulation or quantity of
metal is defined as
a (x) = r(x) . z (x) . v (x).

## Other examples of non-additive variables are permeability, direction of

a vector a (x) with origin at point x, indices of all sorts (hardness, colour),
ratios of grades, certain metallurgical recoveries, etc. The particular case of
the non-additive variable Y (x) = log Z (x), logarithm of a grade, for
example, is discussed later.

2. Adequateness of the variables chosen for the purpose of the study The
chosen ReV's should be such as to encompass the main characteristics of
the problem at hand. They should also lead to operational solutions
without any superfluous precision or calculation.
Consider the example of a thin subhorizontal sedimentary bed with
obvious foot and hanging walls. Rich and poor zones will be selected by
mining on a horizontal plane, but no vertical selection is to be considered
(the entire bed thickness will be recovered). Since there is no vertical
selection, it is of no use to study the vertical regionalization of the grades,
at least for the strict purpose of ore reserve estimations. The chosen
additive ReV's for this estimation are:
(i) the thickness l(x) of the bed measured along the bore- hole located at
the point x of the horizontal two-dimensional space;
(ii) the vertical accumulation a(x)= t(x)xz(x), defined as the product
of the thickness and the corresponding mean grade z (x) sampled on
the mineralized section (of length t(x of the borehole.

## Actually, the thickness variable is used in the estimation of the volumes

of recoverable ore, while the accumulation variable is used in the estimation of the quantity of recoverable metaL The recovered mean grade within
a zone is given by the ratio of the quantity of metal to the recovered
tonnage in this zone.
In this example, it can be seen that the following are true.
(i) It is of no use to consider the three-dimensional regionalization of
(ii) It is of no use to cut the cores into small lengths to represent the
various facies met along a vertical. Moreover, to do so would risk

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201

## biasing the estimation if some facies assumed to be waste were not

analysed (in fact, all facies within the bed will be mined and their
grades, however small, should be considered).
(iii) The information will be focused on the recognition of. the horizontal
variabilities since there will be a horizontal selection.

## 3. Homogeneity of the variable within its extension field The significance

and support of a ReV must be constant, or vary very little in its field of
definition. In particular, this homogeneity allows the observed variability
between two data z (x ) and z(x + h) to be ascribed to the single regionalization of z (x).
In a porphyry-copper deposit, for instance, grades from the oxidized
zone should not be mixed with those from the sulphide zone. Similarly,
data from two drilling campaigns carried out with different drilling equipment should be considered separately, at least in the first step.
It is useful to recall here that the concept of homogeneity and its
probabilistic counterpart, the concept of stationarity, are relative to the
scale of observation: the same are considered extremely homogeneous
during mining and industrial treatment may appear on a petrographic scale
as consisting of a number of very different minerals.

## In some cases, the spatial variability of an additive ReV Z(x) is so great

that it is convenient to study it through a transformed variable; for example, the logarithm Y (x) = log Z (x). A gold grade in a nugget deposit can
vary from 1 to 1000 or more from one sample to another; it is then more
convenient to study the regionaJization of the grade logarithm. This new
Re V is less variable, but is no longer additive.
The structural function yy(h) = E{[ Y(x + h)- Y(X)]2} of the transformed variable Y(x) is calculated and a linear estimator Y*(xo)=
:L" Y(x a ) = Le. log Z(x a ) is built. It would then be erroneous to think that a
mere exponentiation of Y*(xo) provides a good estimator of

## Z (xo) = e Y(x o).

First, the non-bias condition may not be met, as E {Y*(xo)}
not entail that
E{e v*(x,,)} = E{e y(x)};

= E {Y(x)} does

then, the required estimation variance E{[Z(xo)-e v*(x o )j2} is not readily
derived from the variance
E{Y(xo)- Y*(XO)]2}
calculated from the structural function of the transformed variable Y(x).

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202

## As a matter of fact, the estimator e Y"(xol is no longer linear with respect

to the data Z(x"J; therefore, its study requires not only the second-order
moment covariance or variogram of Z(x), but also the knowledge of its
entire distribution function. When Z (x) is lognormal, Y (x) = log Z (x)
being normally distributed, it will be shown that a correct estimator Z*(xo)
can be derived from the linear combination Y*(xo) = La AaY(x",). This
technique, called "lognormal kriging" will be presented in Chapter VIII,
"Introduction to Non-linear Geostatistics". Until then we shall limit
ourselves to the study of additive regionalized variables.
always be applied with flexibility. Obviously, recovery rates from drill cores
cannot be perfectly constant. Similarly, a drill core cut into segments of
very variable length (non-constant support), as unsuitable as it may be,
represents a considerable investment and it is advisable to make the best of
what is available, e.g., to create an homogeneous set from this heterogeneous information that will allow a geostatistical treatment of the data.
Reconstitution of core samples of constant length
On a zone considered as homogeneous for the purpose of the study, one of
the first requirements of the geostatistical approach is that the available
data be defined on a constant support. It is recalled from section II.D,
"Regularization", that the same spatial variability shows up differently
according to the size and geometry of the support on which it has been
sampled.
If the cutting and analysis of the core samples on this zone has been
carried out heterogeneously, then it is advisable to restore the homogeneity, which, in practice, means defining the data on core sections of
constant length. Two cases can be distinguished.
1. The actual core samples are not of constant length but have all been
analysed, c]. Fig. II!.17

c~

C2

C1

I~(------"I~I
I

/,

12

I I~

I
I

He t erog eneous
cutting

~I~_I
I
I
I

I
I

Reconstitution of
homogeneous cu r tinq

## FIG. III. 17 . Reconstitution of core samples of constant length.

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203

## The grades of the core samples of constant length l are obtained by

weighted averages of the grades of the actual core samples. Thus, on the
drill core of Fig. 111.17, the grade z, of the core sample l is made up of the
three actual core sample grades z), 22, Z3, weighted by the respective lengths /1, l2, l) of their intersections with the reconstituted sample
l, i.e.,
(III.45)
Smoothing error In such a reconstitution, it is implicitly assumed that the
grade is constant within each actual core sample. This causes a smoothing
effect as the reconstituted grades give a smoothed idea of the actual
variability of the phenomenon. This smoothing becomes even more
important as:
(i) the grades become more variable within each actual core sample
length, i.e., as the micro-variabilities (nugget effect) become
stronger;
(ii) the reconstituted constant length I becomes smaller with respect to
the mean length of the actual core samples;
(iii) the distance h considered for the serni-variograrn becomes smaller.
In fact, two contiguous supports I may be constructed from parts of
the same actual core samples. This smoothing at small distance h
may mask a micro-structure, bias the evaluation of a nugget effect
at the origin of the variogram.

## The theoretical calculation of this smoothing effect would require the

knowledge of the distribution of the two variables, length of the actual core
sample and its grade, and quickly becomes tedious. However, the following
practical rules can be stated.

## 0) The dimension I of the reconstituted constant support must not be

smaller than the mean dimension of the actual supports; if it is, then
the reconstitution amounts in fact to an artificial creation of
information. Very often, technical considerations largely determine
the choice of the support, i.e., the length I is equal to a submultiple
of the bench height in an open-pit operation.
(ii) Use caution when interpreting the behaviour at the origin of
an experimental variogram derived from reconstituted data; an
excessively regular behaviour may simply be an artefact due to
smoothing.

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MINING GEOSTATISTICS

2. The actual sample lengths are not constant and are not all analysed
This case is more often associated with a preferential analysis (e.g., of rich
facies only) rather than with a random one (e.g., one sample on two
analysed).
Random analysis is more favourable, and the non-analysed samples can
be considered as missing data. The missing datum is sometimes in rich facies
and sometimes in poor facies and, thus, does not cause any bias in the
structural analysis.
Preferential analysis is much more critical, especially when the nonanalysed facies (generally the poor ones) may actually be recovered by
mining. To reconstruct an homogeneous and unbiased data set, it is possible to act in one Or more of the following ways.
(i) To simulate the missing analyses. If the missing analyses correspond
to intersections with pure waste, a zero grade is given to the intersection; if the waste is, in fact, poor are which will be recovered (all
or in part) during mining, it is advisable to simulate the missing
analyses from the knowledge of the characteristics of this poor are,
deduced, for example, from systematically analysed drill cores.
(ii) To limit the calculation of the variograms to within the rich analysed
facies, thus characterizing the variability of these rich facies only.
This supposes that, during the mining stage, it will be possible to
select perfectly those facies with a negligible amount of contamination from other facies.
(iii) To calculate an inter-facies variogram by assigning a global grade
+ 1 or 0 to the various facies, according to whether they are rich or
poor. The drill core is then represented by a series of values of the
"all or nothing" type with constant support and the sequences of
facies may be characterized by the ranges of the variogram.

## Before starting variography (i.e., the study of the spatial auto-correlations),

it is advisable to carry out an elementary statistical analysis of the available
data and to compare the results with the geological hypotheses,
It is not within the scope of this book to present any of the various
statistics or other techniques of data analysis; it is enough to say that such
techniques can, and in some cases must, complete the geostatistical
approach, Among the many works dealing with the statistical analysis of
geological data, some of the most recent are: G. S. Koch and R. F. Link
(1970-71), J, C. Davis (1973) and F. P. Agterberg (1974).

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205

## The statistical techniques of histogram and correlation plots arc

sufficiently simple to be carried out systematically before any variography
is attempted.
The histogram, or the experimental curve of the frequency of occurence
of the different values of the random variable, is a remarkably simple tool
Ly means of which the homogeneity of the spatial distribution can be
verified, extreme or suspect data can be identified and the type of proportional effect to be expected can be predicted (d. section III.B.6, Fig.
111.15). In addition, the histogram is an estimator of the stationary distribution function of the RF under study, as well as a fundamental tool for
the geostatistical techniques of are reserves calculation (d. Chapter VI)
and conditional simulations of deposits (d. Chapter VII).
Correlation diagrams are a means by which the homogeneity of a spatial
coregionalization can be verified. Mineralizations or a set of homogeneous
data can sometimes be distinguished by apparent groupings in correlation
diagrams, d. the following case study. In addition, the correlation diagrams
are essential tools for simulations to the extent that they reveal inequalities
of the type
a X

## + {3 x grade in metal B ~ fixed limit.

Such inequalities often characterize the mineral of interest and it is advisable to take them into account when simulating the deposit.

Case study
The mineralization of the precambrian iron deposits of West Africa
(Mauritania, Liberia, Guinea) is made up of itabirites concentrated in iron
oxides by departure of the silica and is essentially composed of two types of
ore:
(i) type cr, rich shallow hematite ore corresponding to a maximum
departure of the silica (67% Fe, and Si0 2 < 1%);
(ii) type {3, mixed poor ore (55 % Fe and SiD 2 > 5%) located at the
fringes of the deposits or deep at the contact with the clean itabirites.
Of course, when mining within each of these two main types of are,
several subdivisions will be considered according to the granulometry and
the degree of alteration (i.e., presence of goethite).
These two types Ct' and {3 can be clearly distinguished by

## (i) their statistical. distributions (histograms and correlation diagrams of

the three main components, Fe, Si0 2 and AhO;> of the ore);
(ii) the structure of their spatial variability (variograms);
(iii) their spatial localization within each deposit.

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206

MINING GEOSTATISTICS

Figure 111.18 shows the Fe-Si0 2 correlation diagram derived from the
core samples (of constant support) available on one of the Nimba range
deposits (Guinea). This diagram clearly reveals two sets corresponding to
I

I
t

40

I
I

"

2
3

30

I I
II

,
'"

:2: 2 I
I

20

1 I 2

J"

10

Fe ("Ie,)

## the two previously defined types of ore. This statistical discrimination is

confirmed by the following.
(i) The geographic localization of each core sample. The samples
classified on the diagram as type a correspond to the shallow ore
while the .B -type samples correspond to the deep ore.
(ii) The study of the Fe and Si0 2 histograms. The global histograms
derived from all the core samples available are multimodal, d. Fig.
11.26 for the Fe grades. The main mode (67% Fe) corresponds to
the a -type hematite ore which, when considered separately, shows
the unimodal lognormal-type histograms of Fig. 11.25(a) and (b).
The histograms of the ,8-type ore remain multimodal with strong
dispersion variances, d. Fig. II.27 for the Fe grade, which confirms
the mixed character of the .B -type ore.
(iii) The study of the Fe and Si0 2 variograms calculated on each of the
two distinguished types a and ,8, d. Fig. II1.19(a) and (b). For the
a-type Si0 2 , there is a quasi-absence of structure (pure nugget

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207

## effect); the silica appears as mere non-structured traces in the

homogeneous hematite are, whereas the ,8 -type Si02 semi-variogram shows a structure quite similar to that of fJ-type Fe; the
(since the ore is enriched by departure of the silica).
It is interesting to compare the diagram of Fig. IlL1S with those of Fig.
1I1.20(a), (b) and (c) corresponding to the Fe-SiOrAb03 grades of the
deposit of F'derick (Mauritania). In his study of the F'derick deposit, G.
Matheron (1962) also distinguished two main types of ore, called I and II,
the characteristics of which are very similar to those of the two types of ore
(a and fJ) distinguished in the Nimba range deposit.
Fe

(0 )

{ b)

30

20

'"!'...:::e
>-.,

Fe

10

OI~(O)
005
....---.-J1

10

FlO

..L.-

50

## 111.19. Experimental semi-variograrns (Nimba range). (a) a -type ore;

(b) 13 -type ore.

## IlI.C.4. Variogram calculation

Let h be a vector of modulus r = Ihl and direction a. If there are N' pairs of
data separated by the vector h, then the experimental serni-variograrn in
the direction a and for the distance h is expressed as
1 N'
(111.46)
y*(r,a)=2N' i~l [Z(Xi+h)-Z(x;)f
for a regionalization, and
1 N'
ytk' (r, a) = 2N' i~l [Zk(Xj + h)- zdx;)][zdxj + h) - Zk'(Xi)]

(III.47)
for a coregionalization.

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(c)
20

18
16

14
,.('
0

12

'"

VI

10

4
2

Fe (0/0)

><'
0

'"
'"
"!
0

7
6
5
4
3
2

(b)

56

58

62

66

Fe (%)
(c)

16
14
12
><' 10

0'"
Vl

:3

Al 20 3 (%)

FIG. W.20. (a) Fe-SiO z correlation diagram (F'derick). (b) Fe-Ah03 correlation
diagram (F'derick). (c) Ah03-Si02 (F'derick),

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209

## Although these expressions are unique, the methods used in constructing

variograms depend on the spatial configuration of the available data.
Various cases can be distinguished according to whether or not the data are
aligned and to whether or not they are regularly spaced along these
alignments.
1. Data aligned and regularly spaced

## This category covers most configurations resulting from the systematic

reconnaissance of a deposit. The preceding experimental expressions can
be applied for each direction of alignment Q'.
Thus, the configuration represented by a rectilinear drill core in the
direction a, which has been cut into constant lengths l and analysed,
provides an estimator vj' (kl, a) of the semi-variograrn regularized by core
samples of length l in the direction Q' and for distances which are multiple
of the basic step size I, cf. Fig. III. 21 (a).

n Q
1
I
1

JI (k{,a)

1
I

?
b

0011

1
1
'1
I

1
I
1

0
1
1

I
1
1

JrI1
t

(0 )

FIG.

III.2L Data aligned and regularly spaced. (a) Rectilinear drill core;
(b) channel samples along a gallery.

## Vertical channel samples of the same height l and spaced at regular

intervals b in the same direction a along a horizontal drift also fall into this
category and provide an estimator y~ (kb, a) of the semi-variogram,
graded over the constant thickness I, in the direction a and for distances
which are multiples of the basic step size b, ct. Fig. 111.21 (b).

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210

## 2. Aligned but irregularly spaced data

To construct the experimental serni-variogram in the direction of alignment o, the data are grouped into distance classes. Every data pair which is
separated by a distance [r e(r)] is used to estimate the value y(r).
In practice, it is easier to consider a constant tolerance dr) whatever the
distance r, i.e., e (r) = constant, Yr. Otherwise, e (r) should be smaller for
smaller distances and larger for greater distances.
This grouping of data pairs into distance classes causes a smoothing of
the experimental semi-variograrn y*(r) relative to the underlying
theoretical serni-variogram y(r), cf. Fig. III.22. If the N' available data
pairs separated by a distance r, E [r e(r)] are used instead of those
separated by the strict distance r, then it is not y(r) that is being estimated
but rather a linear combination of the y(r;); more precisely, the theoretical
mean value

The significance of the effect of this smoothing over the interval [r e(r)]
decreases as the tolerance e (r) becomes smaller with respect to the range
of the theoretical model y(r) to be estimated.
y

(r) , . - - - - - - - - - - - - - - - - - - - - - - ,

---+---r---

Range

## In practice, the following points should be heeded.

(i) Take the pseudo-periodicities of the data locations into account

## when defining the tolerances e (r) of the classes.

(ii) Ensure that the section of interest of the variogram (e.g., the

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## III. STRUCTURAL ANALYSIS

211

increase to the sill value) has been calculated using at least three or
four classes; thus, the tolerance e (r) should not be too large.
(iii) Ensure that each distance class [r s(r)] contains enough pairs so
that the corresponding estimator of the variogram is reliable; thus,
the tolerance s(r) should not be too small.
(iv) Detect any risk of bias due to preferential location of data.

3. Non-aligned data
This category can be reduced to one of the former two.
(a) By defining approximately rectilinear pathways passing through the
available data locations. Each of these approximate alignments is
then treated separately with a possible grouping into distance
classes, cf. Fig. 1I1.23(a). The disadvantage of this method is that all
the available data are not used, and it is difficult to program.
(b) By grouping the data into angle classes followed by distance classes,
d. Fig. III.23(b). To construct the variogram in the direction a, each
data value z (xo) is associated with every other value located within
the arc defined by [a da ]. Within this angle class, the data can be
grouped into distance classes [r 8 (r).

(0 )

(b)

FIG. 111.23. Non-aligned data. (a) pathways; (b) grouping into angle classes.

## In the three-dimensional space, the arc (or angle tolerance) is defined by

a solid angle [a do, (tJ dcp]. In practice, however, it is seldom necessary
to consider the data field in three dimensions, as it is usually possible to
discern directions of alignments (bore-holes) or planes of alignment
(transverse cross-sectional planes or horizontal levels). Such a threedimensional data network can be regarded as the sum of several data sets
in one or two dimensions which can later be grouped together to provide
the information required to construct a three-dimensional variogram
model.

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212

MINING GEOSTATISTICS

## This grouping into angle classes provides an experimental variograrn

without distinction between the directional variabilities within the arc
[a da] and, thus, a smoothing effect will be produced when the theoretical
semi-variogram y(x, a) varies in the interval [a da l.
In practice, when the existence of a privileged direction of anisotropy ao
is suspected (and such a direction is generally preferentially sampled), then
an arc [ao dao] with a small tolerance is defined so as to estimate
accurately the variograrn in this direction. Alternatively, when such a
direction is not suspected, it is then enough to consider four directions in
the two-dimensional space, each with a tolerance angle of 7T/4, at least as
a preliminary step. If an anisotropy becomes obvious, the next step would
be to reduce the tolerance of the arcs.

## Information associated with each experimental oariogram

The following information must be associated with each of the elementary
experimental semi-variograms y*(r, a) given by formula (111.46).
(i) The number of pairs N'(r, a) of data values used in the construction
of the value y*(r, a). This number of pairs acts as a weight when
grouping various elementary experimental semi-variograms into a
mean semi-variograrn. Moreover, it is an indicator of the possible
estimation error, d. section III.B.?
(ii) The dimension of the zone V covering the data used in the calculation of the semi-variogram y*(r, a). When these data are all
aligned in the direction a (e.g., the data from a rectilinear borehole), then the maximum extension L of this alignment defines the
significant section of the experimental semi-variogram as r ~ L/2,
d. section III.B.7, "Practical rule", formula (III.42).
(iii) The arithmetic mean i and the experimental dispersion variance S2
of the N data [z (x.), i = 1 to N} used in the semi-variogram calculation, i.e.,
i

=N

~ z(x;)

and

=N

~ [z(x;)- z] .

## The arithmetic mean i is an estimator of the mean value m(xo) of the

variable z (x) studied in the zone V (xo) centred on Xu, and can be used to fit
a proportional effect between two experimental variograms calculated in
two zones V(xo) and V(x~) with different mean characteristics, d. section
III.B.6, formula (III.35). When the N data of constant support v are
approximately uniformly distributed over the zone V, then the experimental dispersion variance s 2 is an estimator of the theoretical dispersion
2
variance D (v/ V), cf. section Il.Cvl .

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213

## In the same way, an experimental cross-serni-variogram 1'%k' (r, a) given

by formula (IlIA 7) is associated with
(i) The number of pairs N'(r, a) of available data;
(ii) The dimensions and features of the zone V containing the data;
(iii) The means, Zk and Zk" the dispersion variances s~ and s~, of the data
used and the correlation coefficients PH'.

## Grouping into mean experimental uariograms

Consider the two following elementary experimental variograms:

1't and 1'~ are calculated from two different zones A and B, or from
core samples for one and channel samples with the same support for the
other, or, again, 1'X and 1'~ correspond to two different directions Q'A and
as
Suppose that, after examining the two experimental curves, it is
concluded that they are not significantly different, and it is required to
group them together to form a single mean experimental variagram
21':t+s (r). This mean variogram is calculated from all pairs of data values
separated by a distance r, whether they come from A or B, which amounts
to weighting each of the elementary variograms 21't (r) and 21'~ (r) with
the respective number of pairs used in their construction:

## More generally, the grouping of K elementary semi-variograms {yt, k

1 to K} into a mean semi-variograrn is expressed as

INk(r)yk(r)
k

1'*(r)=-----

N~(r)

(I1I.48 )

## The procedure for grouping elementary covariances or cross-variograms

is analogous: a mean weighted with the number of data pairs.

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214

MINING GEOSTATISTICS

## Each mean variogram obtained by combining elementary experimental

variograms is associated with the following information.
(i) The total number of data pairs, Nr(r) = Lk N'; (r).
(ii) The dimensions and features of the zone V containing the data. For
example, the mean variagram may be calculated over the zone of
primary mineralization from one set of essentially vertical drill cores
and another smaller set inclined at 45.
(iii) The arithmetic mean i and the experimental dispersion variance 52,
calculated from the set of the N data used. Care should be taken to
consider these N data only, since in practice, there may be data
within the zone that are not used in the calculation of any elementary variogram, and therefore play no part in the construction of the
mean variogram.
Note that the experimental dispersion variance 52, calculated from all the N
data used in the field V, will only be significant as an estimator of the
theoretical dispersion variance D 2 (v/ V) of the support v in V if these N
data are distributed uniformly over the field V, and it is not always so in
practice. For example, let V be a three-dimensional zone proven by
vertical drill-holes. The N data from the vertical core samples of constant
length I are thus not uniformly distributed through the volume V, since the
spatial distribution of these data will favour the vertical interdistances. On
the other hand, the data are uniformly distributed in the one-dimensional
field represented by the extension of a vertical drill core of mean length L,
and, thus, a significant experimental dispersion variance can be defined on [
(but not on V).
More precisely, consider K such vertical drill-holes of slightly variable
length {L k , k = 1 to K}. An experimental dispersion variance s~ can be
calculated along each elementary drill core k. This dispersion variance is an
estimator of the theoretical dispersion variance D\l/ Ld of the support I
of the core sample in the one-dimensional field of length Li, As the lengths
Li; of the drill cores are variable, a mean length [= (1/ K) Lk L k can be
defined and the experimental dispersion variance on L can be calculated
from the weighting formula

LNkS~
S2([/ [) = \ : N

estimator of D

(l/ L),

(111.49)

## where s~ is the experimental dispersion variance of the N, data aligned on

drill core no. k of length L k = N k I.

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215

## In the same way, if the three-dimensional zone V is proven by regular

sampling of selected horizontal levels, an experimental dispersion variance
can be defined over a mean horizontal surface S.

## A rule for combining the uariograms

The rule is to group homogeneous information only.
The definition of the ReV (meaning and support) must be reasonably
constant from one elementary variogram to another. Thus a variogram
calculated from the grades of diamond drill core samples should not be
grouped. without caution, with a variogram calculated from the grades of
blast-hole cuttings; a variogram for sulphurous copper should not be
grouped with a variogram for oxidized copper.
The representativeness of the elementary variograms must be approximately constant. Thus, for large distances r, it is sometimes advisable to
distinguish between variograms calculated from long drill-holes and those
calculated from short drill-holes, as the fluctuation error for the latter
samples can be very high. d. the practical rule (111.42) for assessing the
reliability of an experimental variogram.
When grouping variograms together, care should be taken not to favour
a zone of the deposit which is better sampled and perhaps preferentially
rich or with a regular variability. Such a zone will contain an increased
number of samples that will cause the characteristics of this zone to he
predominant in a mean variogram supposed to be representative of the
whole deposit.

## Exercise. Construction of a variogram

The data set used in this exercise is sufficiently reduced to allow the various
directional variograrns to be calculated by hand or with the help of a pocket
calculator. The example is very simple and is designed to prepare the way
for the programming of variograrn calculations. If only such a small amount
of data were available in practice, the experimental fluctuations on each
directional variogram would be so great as to render these variogram
curves useless.
The data are located at the corners of a square grid a, d. Fig. 111.24. The
directions to be studied are the two main directions at and a:2 and the two
diagonal directions a3 and a4. Note that the basic step size in the diagonal
directions is a.J2, while it is a in the main directions. Table IlL 1 gives the
number of pairs of data used and the corresponding values of the experimental semi-variograrn for each of the four directions and for the first three
multiples of the basic step sizes. Isotropy is verified and the mean isotropic

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216

MINING GEOSTATISTICS

semi-variograrn is calculated by combining the four directional semivariograrns, d. Table III.2 and Fig. 111.25. A linear model with no nugget
effect can be fitted to the mean serni-variogram:

Vria

)I(r)=4.1 ria,

[0, 3)2].

al

I
I
I
I
I
I
I

I
1
I
I
I
I
I

35-35-35-

a21

I
I
I
I

I
I
I
I
I
I
I I I
I I I
-

37-35-37---35-

I
I
I

37-40-42-

37-

I
I

-41-

--

--

FIG.

I
I
I
I
I
I
I

---35-33~

I
I
I
I
I

I
I
I I
I I
I I
I I
I I

I
I

-41

~37-37-39-39-41

-34-36-41-

-35-

I
I
I
I
I

-33-

--

---

~-

---

--

--30-

-34

---

-42-33~

I
I

--39-

--

-31

Direction
0'1
0'2
0'3
0'4

Step 1
N'(I)
y(1)

24
22
19
18

TABLE

41
425

5
65

Step 2
A (2)
N'(2)
20
18
16
14

84
82
119
113

Step 3
N ' (3 )
y(3)
18
15
10
8

121
109
173
154

## III. 2. Isotropic mean semi -variogram

N'
y(h)

a,J2

2a

2a,J2

3a

3a,J2

46
41

37
57

38
83

30
116

33
116

18
163

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## III. STRUCTURAL ANALYSIS

217

Computer subroutines

## Practical programs for the construction of experimental variograms are,

above all, programs for the creation of data files and the classification of
data. This section will be limited to the presentation of three basic subroutines which can be adapted to any particular data file.
/

/
/'

/
/

15

/'

/
/

'*~

10

5
/
/
/

/
/
/

r/a
FIG. Il1.2S. Fitting a linear model to the mean isotropic serni-variograrn.

These three subroutines correspond to the three types of data configuration most often encountered in practice in mining applications.
1. One-dimensional configuration of aligned and regularly spaced data,
d. Fig. Ill. 21; e.g., a rectilinear drill core cut into samples of constant
length t.
2. Two-dimensional configuration of data at the corners of a regular and
rectangular grid, d. Fig. JIL26; e.g., the sampling of a subhorizontal
sedimentary deposit by vertical drills analysed over the entire
mineralized thickness, the drills being placed horizontally on a
regular, rectangular grid.
3. Two-dimensional configuration of data irregularly distributed over a
plane, cf. Fig. JIL27; e.g., the sampling of the previous sedimentary
deposit by vertical drills placed anywhere on the horizontal plane (but
not preferentially on rich or poor zones).

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218

MINING GEOSTATISTICS

Remark The very general configuration corresponding to a threedimensional mass sampled by n drills in all directions can be regarded as
either none-dimensional configurations along the length of the n drills or
K two-dimensional configurations on each of K horizontal benches, the
data on each bench consisting of the intersections of the drill cores with the
bench.

## 1. Aligned and regularly spaced data (subroutine GAMA 1)

The alignment of the ND data is input to subroutine GAMAl in the form
of a vector VR(ND); some of these data may be missing or are to be
eliminated. The subroutine calculates the experimental semi-variogram
along with the experimental mean and dispersion variance. By mean of
another program, this serni-variogram may then be combined with others
according to the weighting formula (III.48).
SUBRGLTI~E

GA~Al(VRt~D.K~A)tPAS,NC,GtltV,N,ISJ

c ..................................
c
SEMJ-VARICGR~~ AlC~G ~ ll~E
c REGULAR GRIC OF POI~TS M~~ t~~E ~ISSl~( CAlA
c
c ..... PARAt" ETERS
c VP(~C)
CtT~ ARRAY

NO

K~AX
p~S

C
C

C
C
C

## NLMBER CF PCINTS CF THE LI~E

Nu~aER cF CC~PUT~TI(~ LAGS
OF LAG
NU~8ER OF COUPlESIl1G
SE~I-V~RIC{RA~
VALLES/LAG
A~ERAGE
I
V~RIANCE I OF DAT~ )lEST
NUMBER
I
~~Xl~U~
LENGT~

NC(KMAXI
G(KMAX)
U
V
N

C CPH C ~s

C
C

IS.NE.l

RESULTS ARE

PP(~TEC

8Y

SLBRC~lI~E

C CCp.l.MC ~S

C
C
C

IGUT
TESI

## LINE P~INTER UNIT ~lMBE~

INFERreR BCUNCAR~ ( f EXISTI~G CAIA
IF V~.LE.IEST MISSING OR ElI~INATEC CATLJ'"

C
C
OIME~SIO~

CC~MC~

VR(lJ,~C(l),G(l)

lCUT,TEST

INITIALIZE

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~U4i ~j J ~ ~-rU

## III. STRUcrURAL ANALYSIS

CO 1 tK~l,KMA)t
t-C (110=0
1 G e J K ) =0.

IFeVRCNC)-TEST)12,12,1:
12 "'=0
tJ=o.
\1=0.
GC

rc

11

13 U=VR no
V:VR o..r.*VR(NC)

11

"'=1

ccnr H.UE
COMP~TE

SEMj-VARIOGRA~

A,.,C

## ,., C1=='" C-1

co

1=I.NOl

'vRl= \lJH l )

IFIVRl.LE.TEST)GC TO 2
NaN+l

U=U+VRI
~~,,+vF11JR.1

I1~I + 1
J~~MI~O(I+KMA~tNCI
CO 21 J~Il,JM

I(:oJ-t

tF(VPtJI.lE.TESTJGO TO 21
t\C.K).1:~C(K)+l

'w~R'Rl-~R(JJ

G(K)=GtK)+0.5*VRR*VRR
21 CONT H,UE

c
c
c

2 CONT INUS
RESLLTS
IF(~.eQ.O)GO

yotV-l*U/N)/N

TC 3

U=U/N
co 30 IK-l,KM.AX
30 GlIKj-G(IKl/M.AXotl,NC(IKII
3 CONT HUE
IFtlS.EQ.1IGO TO 5

## PRINT RESULTS If IS.NE.l

C
~~lle(ICUT,20CO)

RPITE(OUT,2001JU.V.N
DC 41 1<.=1,K/o(,6X
C-K*P,aS

41
4

~RITE(IOLT,2002)K,O''''C,G
caNTI~UE

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STA11~lI(S

219

MINING GEOSTATISTICS

220
C

2000

FORMAT(1~l,53Xt'SE~1-VA~I((R~M'/Ir

1~3Xt'****

GRIC 1 CI~EN~I(~)')
" AVERAGE = ttFIC.5,cX,'v~RIA~CE = t
I.cll.5,' NUM8ER OF DATA = 'tl5/1H ,'LAG CIST'
3,'ANCE I NC
VAQIOGR~~
I')
2002 FOR~ATtl""" tlXd2,2XtF8.?,~)I,' I ',I4,2),Ell.5,
12X,' I ')
21(REGLl~R

2001

****.* '.lC>,

FORMAT(l~

5 RETURI\
ff\ 0

## 2. Data on a regular rectangular grid on a plane (subroutine GAMA2)

2 - - - - - - NCO columns

I
I
I
I

0
0

I
Nli

lines

## FIG, III.26. Data configuration considered by GAMA2.

The data for subroutine GAMA2 (NLI lines and NCO columns) is input in
the form of a column vector VR(NLI*NCO). The subroutine calculates the
four experimental serni-variograms corresponding to the four main directions (the two directions of the grid and the two diagonal directions). The
subroutine also supplies the experimental mean and dispersion variance
corresponding to all the data available on the plane.
SLBRClTI~E

GA~A2(VR,l\lI,~C(,K~AX,I\(,(,L,V,~,IS)

C
SfMI-VARICGR~~ IN T~( CI~f~~IC~S
C
C REGULAR GRIC CF POI~TS M~Y ~~vE ~ISSI~f CATA
C TrIS IS A SA~PLE SUAROUTINE ~rC~I~G Cr~PlTATIC~
C fOR FOUR tIRECTlOl\S HQwEVER !vECTORS r r Jf\C JC tJAY
C ~E DEFINED FeR OTHER CIRECTI(~~
6

C
C

...... PARA~fTEf1S
vP(~LI*N(OI CATA ARRAY (SlCRE[

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COLL~N.I~fl

221

NLI
NCO

K~AX

C
C
C
C
C
C

10(4)
JD(4)

C
C

## NUMBER OF LI~ES CF [ATA GR!C

NUMBER CF CCL~M~S
~~xIMU~ NUMBER CF C(~PUTATI(N
LAGS
DIRECTION INOICAT(~ ALONG tJ~ES
OIRECTION INOICATCR ALO~G C(ll~~S
~UMBER Of COUPlE~/lAG/GIRE(TIC~
SEMI-VARIOGRAM vAllES/lAG/CIRECTICN
~vE~AGf
I
vtRIA~CE I OF DATA >TEST
~UM8ER
I

~C(KMAX*4)
G(K~AX*4)

U
V
~

C (PTtCl\S

## RESULTS ARE PPII\IEC

IS.NE.1

C
C CCMMCl\S
leUT
LINE PRINTEq
C

C
C
C

TEST

LNIT "l~BER
INfERIOR BCUNCAR~ CF EXISTr"G CA Ttl
IF VP.Lf.TEST MISSING OR EllMINATEC CATU,..

C

C
C

C
C
C
C
C
C

## INDICATOPS rO,JC SHOk ThE REUTIVE ll~E AI\D

COLUMN PCSITICNS OF CL05EST I\EIGHeCLP Te PCI"T
I,J IN TrE CONSICEREC DIRECTI(N,EXA~PlE:
OIRECTIC"ll N-S
(//COlU"'f\~)
Ie=1 JD=C
2 ~-E
(1IlI~ES)
IC:C JC=1
3 NW-SE
(CIAGC~~l
1)
IC~l
JD=l
:3 NE-SW (OIAGCf\AL 2)
I[}~-l JD=l
CTrER DIRECTIONS ~A~ eE DEFI~EC
e

~_~

CIMEt--SION

CATA

VRll),NC(I),(I"lO(4),JC(~J

leUT,TEST

COMMC~

IAII

'I

INITIALIZE

c
I(~M"'K"AX*4

u=o.

11=0.
Nc::O

CC 1 IK=l,KM~
"'((IK)=O
1 G(lKJ"O.

cr

10

1""1,-4

IOU 1=1

c
C

10 JO(J '=1
10(21=C
10(4)=-1
JC(!J"'O

CCMPUTE

SfMI-IJARIC(qAM,NE~

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PCINT

222

MINING GEOSTATISTICS
CO 2 l=l,NlI
CO 2 J=l,NCO
[J=IHLI*(J-l)

ST~l1STICS

C
C
vlH=\lR( IJ)

IrIVRl.lF.TESrtGO TO 2

N=N+l
V=V+'-'Rl*'vRl
l;=U+\lRl
3 U1NTlf\UE

C
C

NEk C(RECT!CN

CO 30 KC"lI:l,"
J3:JCCKDJ

13'-'= I C{ K C)
I 1.. I
J 1 ~J
(

/liE, LAG

C
C
CO 31 j(:l,KMAX

11'-11+13
IF(Il.LT.l.OR.Il.GT.NLOG( TO ;C
Jl=JliJ3
IF(Jl.LT.l.CP.Jl.GT.NCCIGC TO 3C
JJ1= 11+MI*CJl-l)

IK="..KMAX*CI(C-IJ
IF(VRUJl).LE.TESTIGC

31

rc 11

f\C(IK)=NC( {KIf!
Yf.lR=VR(lJl)-YRl
GCIK)=GC IK)+O.S*VPR*vRR
CONTtr...UE

30 CONTIf\UE

2 CONT II\UE

flESUI.TS

lfIN.EC.O.GO TO
Y-IV-U*U/NI/N

U-U/M
DO 40 II(-l.KMfIl

.0 GIIKJ-GIIKJ/.AXOC1,NCCJKIJ

c
e

,. CONTINUE

PRJNT If IS.HE.1

IFCIS.EC.l.GO TO ,
WIUT E CI OUT. 2000)
WRITE.(OUT.2001)U.Vt~

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## III. STRUCTURAL ANALYSIS

223

WRJTEfIOUT,2002JIIA,IO,ICt,4)
IKMaK'AX*3
WRITECIOUT,2003)IIA,IO1,~)
DO ~5 K-t, KMAX

IK2-K+IK"
~S

WRITf(IOUT,2004JK,INCIIK)"tIK),IK=K,IK2,K~A})

153X, ',10)

2001

## 2'(REGUlAR GRID 2 DIMENSIO.~)I)

FORMATI1~ , I JVERAGE ',F1C.5,6X,'\JRIA~CE :
I,Ell.5, NUMBER OF DATA ',15)

## 2002 FORMAT'l~ ,6X,'I',6(Al,2J,'Dl~ECTICN ',{Z,3X,'I'))

2003 FORMATCIH " LAG I ',6(Al,'NC 1/2 V~~10GRAMI')
2 004 FORM AT U H ,IX, 13, 2X, I I E( 14 t 2 X.sr i, ~ "1 X, I

.5 flETURN

END

Oistonce
tolerance--y

- -....

)l.

## The ND data values are input to subroutine GAMA3 in the form of a

vector VR(ND). The location of each data value is defined by its rectangular coordinates X(ND) and Y(ND). The NDI directions considered in
the subroutine are defined by their angles ALP(NDI). The angle tolerance
(2DA) and distance tolerance (DP) are unique for each of the NOI directions.

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MINING GEOSTATISTICS

224

The subroutine supplies the NDI experimental directional semi-variograms as well as the experimental mean and dispersion variance corresponding to all the ND data available on the plane.

SUBROUTINE GAMA3(VR,X,~,~C,K~AX,PAS,CP,NOI,
lAlP,O_,NC,G,O.U,V,N, IS)

C
SEMI-V_RIOGR'M IN lwe [IME~SIO~S
C
C IRREGULAR GRID.TMERE MAY BE ~1~SlNG DA1A
C CALCULATION BY CLASS OF ANGLE 'hD DIST.~(E

C
C
C

C
C
C
C
C
C

C
C
C
C
C

C
C
C
C
C

C
C
C
C

PARA ..ETEIliS
VRCHOI
O'TA ARRAY
XCNDI,YCNOJ
X ANO Y (VORDIN'TES CF FClhTS
NO
NUMBER OF POINTS
KMAX
H'XIHU~ HUMBER CF CCfIIPUTATICN LAGS
PAS
LENGTH Of BASIC LAG
Op
WIDTH OF DISTANCE CLASS.IF CP~O.
OP=PAS/2. IS TAKEN
HDI
NUMBER OF DIRECTIO~5
ALPeNOl} ANGLES DEfINING Dl~ECTIONS ehlTH
RESPECT TO X AXIS IN OEGREE~1
OA
WIDTH OF ANGLE CLASS.IF CA=C.
OA=45. DEGREES IS TAKEN
NC(~HAX*NDl)
NUHfER Of CO~PlES/LAG/OIPE(TION
GCK~AX*NDl)
VARIOGRAH VALUES/lAG/CI~EC110N
OCKMAX*NDI) AVERAGE DISTANCE/lAG/DIRECTION
U
AVERAGE I
V
VARIANCE I OF DATA )TEST
N
NUMBER
OPTIONS

## IS.NE.I RESULTS ARE

PER EAC~ 5

PRINTEO:O~E

P~GE

CF

PR[~TCUT

DtRECTICN~

C
C COMHO~S
LINE PRINTER LNIT NlMBER
C lOUT
INFERleR 8CUNnA~V OF EXISTI~G DATA
C TeST
IF VR.LE.TEST MISSING OR ELIMINATED CATUM
C
C
C CAP4ClTY:
C 10 DIRECTIONS:SANCIO),CANCICI
C
C

OIMENSION VRCll,XClJ,Y(1)
DIMENSION AlP(ll.CANtlC),S'NClOJ
DIMENSION Gtl),Otll,NCClJ
COHMON I(Ul,1EST
DATA tAl' 'I

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## III. STRUCTURAL ANALYSIS

c
c

225

INITIALIZE
PI=3.1"1~92t5

Tel: CF
IFITOL.LE.O)TOl=PAS/2.
DALPH_=D,6

IFCOA.lE.O)0,6LPHA~5.

CO 1 KO=l,NOl
_LPHA:PIAlPC~O)/180.

CANIKCJ=COSIAlPHA)
1 SANIKC)=SINIALPHA)
THETA=PJ*OAtPHA/180.
COAzCtlSITHETA)
KMM#KflAX

DO 10 IKzl,Kf!4f11
NCClK)=O
OI'K I :z0.
10 G( IK }aO ..
IFIV~INCI-TEST.12,12,1~

12 N-O

u~o.

v-=o.
GO TO 11
13 H.-I

c
c
c

U-VRCfl,CJ
V=VAChO}.VR(NCJ
11 CONT INUE
COMPUTE

seM'-VARtCGP_~,NEW

P~tNl

N01-NO-l

00 2 lal.NCl
VIUaYAl1 J

IFeVR1.LE.TESTIGO TO 2
NaN.}

lJ::aU+ VR 1
Y-V+VRl*VRl
nal +1

NEW LAG

00 21 J- n,NC

IFIVRIJ).LE.TESTJGO TO 21
DX-X J'-JCC J)

Oy,.y (J ,-YI

I)

H=SQATCOX*OX+OY*OY)
IFIH.LT.I.E-Q3JGO TO 25
K=INTIH/PAS+O.5)+1

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MINING GEOSTATISTICS

226

"lAeSCH-(~-l)PAS)

IFtK.GT.KMAX.CR.Hl.GT.TCL1EO TO 21

c
c
c

NEW OIRECTICN
CO 22

KO~l,NO I

COSDzCOX*CANCKO)+OY*SANCKCII/H
IFfABSfCOSOJ.GE.COAIGO TC ;3
22 CONT INUE
GO TC 21

23

IK.K.K~AX*fKC-l)

NCU";)zNC( IK)+l
OfIK)=OlIK)+ti
VRR=VFI CJ I-VR 1
GfIKI-GfIKI+O.S*VFlP*VRP
GO TO 21
25 WRITECIOUT,20001 I,xt U .... Ol,J,XCJ},'t(JI
21 CONT I~UE

c
c
c

2 CONTINUE

AESUlTS

IFlN.EQ.01GO TO 3
V.. (V-l*U/NI/N
U=U/N

CO 30 IK-=1.KJ4M
O(IK)~O(tK)/MAXOCl,NC(tKII

30

CONTJ~UE

## PRlhT RESULTS IF IS.NE.!

C
C

IF(IS.EQ.l)GO TO 5
It<-CI-IHS+l

IMP=

IO~~FlOATlNOII/FlOAT(I~P)+C.9999

00 42

1f~al,

IMP

WRITE(IOUT,20011IM
WRITECIOUT,Z002)U.V,N
101=1+10 .... ( 114-1)

I02;MINOINC].IOM.IM)
WRITElIOUT,200310AlPHA,P.S,TOl,
l(

lA, ] C, I Os J 0 1, 10 2 )

WRITECIOUT,2004)(AlPIID),J[I01,tD~1

WRITECIOUT,2005)(IA,IO=I[l,ID2)
IKO: KP'AX*( ]01-1)
lKM=]KO+KMAX*CIC2-101)
00 43 K: 1 , K/'til A)(

43

IK1=K+IKO
IK2= K+ IKf'
WR]TEIIOUT,2006)K,CNCCJKJ,C(IK).GCIK),
11K=IKl,IK2,K"~X)

tt2

CONTlf\.Ue

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227

41 CONT II\UE
C

OEFJ~EC

P(JNT [AIU~,

## X=',F9.4,' ~=',F9.4,' DATlJftI',I4,' X=',

2F9.4,' Y=',F9.4}
2001 FORHAT(1~1,53X,tSEHI-VARICGRAH ',3EX,' PAGE:'
1,I2/54X,' ',lCX,(IRREGlLAR GRIO',
2' 2DIMENSION5}'/1
ZOOl FORMAT(l~ " ~~ERAGE = ',FIC.5,6X,'\~PIANCE = t
i,Ell.5, NUMBER OF DATA = ',15
2003 FORMAT(l~ ,'OIRECTION TOLErANCE = ,F4.0,
l' DEGREES ,LAG = '.Fl.2,' ,OISTANCE lCLERANCE'
114,'

## 2,' = ',F7.2/1H ,5x,'1',SCAl,5X,' CIlHCTlGN',IZ

3,6X,'I'j
2004 FORM.T(lH ,5x.'I',5()X,F1.1,'DEGREES',6X,"')
2005 FORMATIIH " LAG 1',5(Al, 'NC DISTANCE "
1'1/2 ~ARIO I '1)
2006 FORMATI1H ,lX 3,' 1'.5(I?,tX,F1.2,1),
lEl1.5,'I'))
C

5 RETURN

END

## IlI.C.5. Variogram analysis

The object of variogram analysis is to detect the major structural characteristics of the regionalized phenomenon under study by analyzing the
various experimental variograms, The structural information gathered
from such an analysis must be continually compared with the known
physical characteristics of the phenomenon (geology, tectonics,
mineralogy, sampling procedures, etc.). Under no circumstances should a
variographical study take the place of a good geological survey of the
phenomenon; on the contrary, the variography should always be guided by
geology. The variography completes and enriches the geological knowledge of the phenomenon, since the variogram quantifies the structural
information for use in estimation procedures.
For instance, geological studies may indicate a direction of preferred
variability of the mineralization due, for example, to alteration or diffusion.
A study of the anisotropies of the various directional variograms will
permit the verification of such an hypothesis and, moreover, will allow a
quantitative evaluation of the amount of total variability due to the
particular phenomenon of diffusion or alteration (e.g., in section IV.F,
Case Study 13 - the case study of the radial anisotropy of EI Teniente
deposit). Conversely, a significant anisotropy displayed by the variograms

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228

MINING GEOSTATISTICS

## must be accounted for, either by the geology of the phenomenon or as a

structural artefact due to measurement (preferential data locations, biased
analyses, etc.),
Again, through geological studies the various scales of variability of the
phenomenon can be qualitatively evaluated, e.g., the petrographic scale of
the transitions from one element to another, the lithological scale of the
various facies, the tectonic scale of the transitions delimited by the main
faults, etc. Through the concept of nested structures, variography
quantifies the respective part played by each of these scales of variability
and evaluates their importance in relation to the various problems of
estimation, optimization of production, quality control, etc. For example,
what is the residual macroscopic effect of the micro-variabilities on the
this problem is provided by a study of the effect of regularization of these
micro-variabilities when changing from the data support (core sample of
several kilograms) to the support of a production size unit (several hundred
tonne block) d. section IV.F, Case-Study 15 - the case study of the grade
fluctuations in a nickel deposit.
Each of the case studies given in this book, and particularly those in
Chapter IV, demonstrates the complementary nature of the quantitative
approach provided by variography and the more qualitative geological
approach.
Because of this essential interaction between the geostatistical approach
and a qualitative experimental knowledge of the studied phenomenon, it is
not possible to define a uni-versal model or plan for the analysis of variograms which would apply to all types of deposits and to all methods of
proving these deposits. At the most, several general remarks can be made
on the various data artefacts which may bias the structural picture given by
the variograrns.

## Causes of heterogeneity between variograms

Before ascribing the observed difference between two variograrns to a
physical heterogeneity of the phenomenon under study, it should be
ensured that this difference is not due to unrepresentative variograms or,
more generally, to an artefact of measurement or calculation. Causes of
heterogeneity due to measurement are numerous.
First, each of the two variograms may be calculated from an insufficient
number of pairs or a neighbourhood which is too small (cf. practical rule
(III.42)) to be representative. For example, if the available data consist of
vertical drill cores and horizontal channel samples, and it is intended to
detect anisotropy, then it is advisable, if possible, to compare the two mean

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## III. STRUCTURAL ANALYSIS

229

experimental variograms calculated from the two sets of data, rather than
to compare a single vertical drill core with a single horizontal channel
sampling profile.
Second, if the two experimental variograms are defined on different
supports (channel sample and core sample in the preceding example) it is
advisable to determine whether the difference in support is sufficient to
explain the observed difference between the variograrns.
Third, if the means of the data used to calculate the variograms are
different, then it is advisable to look for a possible proportional effect; for
example, by plotting and comparing the relative semi-variograms y* / m *2
if a direct proportional effect is suspected.
Fourth, the preliminary review of the data may immediately reveal the
causes of an observed difference between two variograms: a single aberrant
data value, when squared, is enough to completely change the behaviour of
the variogram; the two sets of data may not have been collected with the
same equipment, etc.
Variography does not create data, it simply presents it in the synthetic
and quantified form of the variogram, A significant heterogeneity displayed by a variogram usually has an obvious physical explanation; when
this is not the case, an artefact of measurement or calculation should be
suspected. If only a slight difference is observed between two variograms
and there is no obvious physical explanation, then it is best to ignore it - by
assuming isotropy, for instance - rather than developing a complicated
model for anisotropy that may turn out to be illusory. The rule in such a
case is to adopt the simplest model.
Pure nugget effect or experimental fluctuations?
Let yo(h) be a point transition model with a range ao which is very small
when compared with the dimensions of the data support: ao v.
In practice, when v> ao, the regularization on the support v covers the
entire variability 1'0 so that it no longer appears on the regularized semivariogram except for a nugget constant, ct. formula (II. 7):
1'ov(h) =

COl) =

constant,

Vlhl > v,

with the rule of inverse proportionality to the support: Cov = (v' / v )C ov'
When the variability consists only of micro-structures of the type 1'o(h),
i.e., structures with ranges much smaller than the supports of the data, the
experimental semi-variograrns will appear flat or, more exactly, fluctuating
slightly around their sill values, as in Fig. III.28(a).
Before assuming the hypothesis of a pure nugget effect, it should be
ensured that nugget-like behaviours (i.e., without apparent increase) of the

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MINING GEOSTATISTICS

230

experimental variograms are not due to a smoothing effect in the calculation of the variograms (e.g., due to reconstruction of core samples of
constant length, d. formula (III.45)) or a fluctuation effect (e.g., too few
data used for the construction of the variogram). Contrary to an apparent
nugget effect due to experimental fluctuations, a pure and stationary
nugget effect only shows slight fluctuations around the sill value, d. Fig.
I1I.28(a) and (b). A good way to verify a pure nugget effect is to test the
rule of inverse proportionality of the sills to the supports, d. in section
III.A.4, the case study of the Adelaida mine.

(a)

(b)

FIG. 111.28. Nugget-like variograms, (a) Actual pure nugget effect; (b) apparent
nugget effect.

## It should be stressed that under the hypothesis of stationarity a pure

nugget effect corresponds to an hypothesis of extreme homogeneity of the
underlying mineralization, cf. section III.A,2. In mining practice, an actual
pure nugget effect is extremely rare; on the other hand, insufficient or
inadequate data will quite often result in experimental variograrns having
the appearance of a pure nugget effect.
III.C.6. Fitting a model

As was the case for the analysis of the variograms, there is no unique
technique or method for fitting a theoretical model to a regionalization
identified by a set of experimental variograms. At the most, the following
Operational representation

The representation of a natural variability by a point or regularized variogram can never be unique. In practice, it is enough that this representation

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## III. STRUCTURAL ANALYSIS

231

reflects the various experimental features which have been found (experimental variograms, anisotropies, a priori geological knowledge, etc.). This
representation must also be operational, i.e., simple and adapted to the
purpose of the study. In particular, certain minor features revealed by
structural analysis may be omitted from the adopted representation.
Similarly, when two geological or mineralogical causes contribute in the
same way to the observed variability at some scale, it is useless to complicate the model by distinguishing between these two causes.
Support and regularization

The dimensions of the support v of the data are often very small with
respect to the distances involved in the intended estimations, and also with
respect to the dimensions of the units V to be estimated. In such cases, the
support v of the data can be approximated to a quasi-point support (v ::::: 0)
and the experimental regularized semi-variogram 1'~ (h) can be treated as
if it were an estimator of the point model y(h).
The techniques of regularization (passing from y to y,,) or, conversely, of
deconvolution (passing from y" to y) are required only when comparing
variograms defined on different supports (y" and y,,' with v oF- v'), d. in
section II.DA, remark 3.
These techniques should not be used to deduce a point model which is
illusively detailed, e.g., a sophisticated point model with a large number of
parameters. To be rigorous, it is not possible to reach a greater degree of
precision than that of the smallest support v of the data without introducing supplementary and unverifiable hypotheses. It is always better to avoid
such hypotheses and stick to the available data; for example, by assuming a
quasi-point support.
Sill and dispersion variance

## The choice of a theoretical model with or without sill is always made by

examining the experimental variograms, taking account of the fact that
these variograrns are subject to significant fluctuations at large distances.
In the same way, a sill is chosen by fitting a line through the experimental
fluctuations. The experimental dispersion variance
2
1
2
s =N~{z(Xj)-i]

can sometimes be used in fitting the sill, provided that it is in fact a good
2(vjoo),
estimator of this sill (i.e., of the a priori variance D
when v is the
support on which the serni-variogram is defined), ct. section Il.Cd and
section III.C.4 (the note about the experimental dispersion variance S2

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MINING GEOSTATISTICS

232

## associated with each mean variogram). In particular, it should be stressed

that the existence of a sill is not related to the existence of an experimental
dispersion variance S2; the experimental variance of N data can always be
calculated even when a sill does not exist (the corresponding RF being only
intri nsic).

## Behaviour at the origin

Most variables encountered in mining applications yield a semi-variograrn
with a linear behaviour at the origin and a nugget effect, d. Fig. 111.29. The
nugget constant Co is obtained by extrapolating the average linear
behaviour of the first experimental points to the ordinate axis.
y* (II) .------------~-

x x

xl
x

/'

/'

II

## FIG. III.29. Linear behaviour at the origin.

Nested models
It will be recalled from III.A.l that the nested structure y(h) = Lty,.(h) is a
model particularly well suited to applications because of the linearity of
geostatistical operators. Thus, it is not strictly necessary that the different
components 1", have a distinct physical interpretation. However, two
components Yi and l'i with markedly different sills and ranges will, in
practice, have a clear physical meaning.

Automatic fitting
As far as mining geostatistics is concerned in this book, any attempt at
blind automatic fitting of parameters to experimental variograms - such as
least squares methods - should be avoided.
It should be recalled that each estimator point y* (h) of an experimental
serni-variogram is subject to an error of estimation and fluctuation which is

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## III. STRUCTURAL ANALYSIS

233

particular to that point; this error varies according to the separation distance h, the zone or data set from which the varicgram is calculated, etc.
The best method for weighting each estimated point y*(h) with its
"coefficient of reliability" comes from a critical review of the data and from
practical experience.
Among other things, physical knowledge of the phenomenon (geology,
for example), common sense involved in approximations and adapting
available information to the ultimate purpose of the study are all as
essential to a structural analysis as the calculated values of the experimental variograms.
Robustness of the geostatistical results with respect to the chosen type of
model

Consider the two transition-type models in current use which have a linear
behaviour at the origin, i.e., the exponential and the spherical models.:
y(r)= 1-exp(-r/a);

and

3r

y(r) =

1 r3

2a - 2 a 3,
{

1,

Vr,;;;;.a,

Vr~a.

The object of the study is to demonstrate the robustness of the geostatistical results with respect to the choice of either one of these two
models, provided that a correct fit to the experimental semi-variograrn has
Consider a spatial regionalization characterized by a point isotropic
exponential model with a unit sill and a parameter a = 1:
y(r)= l-e- r

Suppose now that this theoretical model is not known, and that a nested
model y*(r) consisting of two spherical models is fitted to the experimental
serni-variogram (1- e -r) in the interval r E [0, 3], cf. Fig. III.30:

## the parameters of the two spherical component structures C 1 Yl and C 2 Y2

being
Ql = 1.2
ranges { Q2 = 3.6.

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MINING GEOSTATISTICS

234

For these two models, exponential y(r) and its fit y*(r), the values of the
following geostatistical operations in the one, two and three-dimensional
spaces are given in Table Il1.3:
(i) The dispersion variance of a point within a segment, square or cube
of side 1= 3a = 3, d. formulae (11.98) and (11.99) and Charts no. 4, 14 and
5,15;
(ii) The estimation variance of this segment and this square by a central
point, and of the cube by an axial core of length I, d. Charts no. 7, 17 and
10,20.
1-0 I--------------:==::;;:;;:;;;:;;,;;~~

...

0-5
7
'/
;t

'/

'/

If
0-1

rio
FIG. 111.30. Exponential model

TABLE

## (~-) approximated by two nested spherical

models (- - - -).

## 111.3. Robustness of the geostatistical variance with respect to the type of

model chosen

Dispersion
vanance
Exponential y
Fit
Relative deviation

,.*

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Estimation
variance

R1

R2

R3

R1

R2

R3

0544
0535
0017

073
072
0014

082
081
0012

042
0419
0002

057
0571
0002

018
0179
0006

## III. STRUCTURAL ANALYSIS

235

Note that the relative deviation between two results derived from the
two models is of the order of 2% for the dispersion variances and of several
per thousand for the estimation variances. The excellent concordance of
the results is to be expected since the fitting of y*(r) to y(r)on Fig. III.30 is
remarkable, at least within the working interval r E [0, 3].

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IV

## Case Studies of Structural

Analysis
SUMMARY
The 15 case studies given in this chapter were selected to cover the main structural
models of variograms in current use as well as the most critical problems which may
be encountered in the course of an actual structural analysis.
The first five examples of section IY.A show how to fit the various variogram
models currently used and introduced in section III.B.2.
Section IV.B gives a typical example of model fitting to a regionalization which
shows a hole effect and a proportional effect.
Section IV.C gives two case studies of coregionalization and their fit to the linear
model introduced in section I1LB.3.
The two examples in section IV.D present the two models of anisotropy introduced in section 1I1.B.4, i.e., the geometric and the zonal anisotropies.
While the presentation of each of the previous ten case studies is focused on a
limited number of theoretical concepts, the three case studies of section IV.E give
three examples of structural analyses in three dimensions, where most of the
previous concepts are involved simultaneously.
The two examples of section IV.F are particularly important in mining practice,
as they show how the two geostatistical tools - estimation and dispersion variances
- can be used in a simple but operational way to control grades during production.

## IV.A. FITTING MODELS

Case Study 1. Regionalization of oil grades, after P. A. Dowd (1977)

This study concerns a section of the Athabasca tar sands (Canada) that
were proven by 40 vertical diamond drills, the cores of which were cut into
approximately constant lengths, IE (1S to 22 tr]. The drill-holes themselves
236

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237

## had an approximately constant length L = 350 ft, thus providing a set of

some 7000 samples analysed for oil expressed in per cent saturated weight.
As there was little variation in the data support, it was not deemed
necessary to reorganize the data into strictly constant lengths. However, an
effective grouping into distance classes was made by approximating the
data to a point grade z (x) centred on the mid-point of the corresponding
core sample, the class interval being equal to the mean length, 1= 2 ft, of a
core sample. It will be recalled from section I1I.C.4 that the grouping into
distance classes causes a smoothing effect particularly significant for very
small distances, and, for this reason, the first point of the experimental
variogram, i.e., y*(2), has been rejected.
The mean vertical semi-variograrn obtained from the weighted average
of all the elementary vertical semi-variograms for each drill core is shown
on Fig. IV.I. The following information is relevant to this variogram.
(i) The first experimental point )1*(4) was estimated with 3363 pairs of

## data values, and all points up to h = 150 ft were estimated with at

least 2000 pairs each. Toward the limit of reliability (h = Lj2 =
175 It, one half of the mean length L of the drill cores), the number
of pairs was still greater than 700. Thus, the reliability of this mean
vertical serni-variogram can be considered as excellent.

/
(783)

10

100

50

150

hOI)

FIG.

## IV.1. Spherical model fitted to a regionalization of oil grades.

(ii) The arithmetic mean of the 7000 grade values used is m"

= 88%
saturated weight.
(iii) The mean experimental dispersion variance of a core sample of
length I within the total core length Lis, s 2(lj L) = 173 % 2. This

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238

MINING GEOSTATISTICS

value was obtained by weighting the elementary dispersion variances for each drill core according to formula (III.49). This mean
experimental variance is an estimator of the theoretical dispersion
2
variance D (l/ L) of the grade of a sample, 1= 2 ft, in the mean
length of a vertical drill-hole L = 350 ft.
A spherical model with a nugget effect was fitted to the experimental
semi-variograrn y(h)=Co+Ydh), Vh vertical s j Z, 175ft], with C o = 6 % 2
for the nugget constant, and '}It(h) a spherical model with a sill C t = 13% 2
and a range a 1 = 36 ft, cf. formula (III.15).
The fit was achieved by considering the following features of the
experimental serni-variogram.
(i) The tangent to the curve at the origin was approximated by the
average linear behaviour of the first three experimental points. This
average linear behaviour gives the value Co = 6% 2 when
extrapolated to h = O.
(ii) The total sill value, Co + Ci = 19% 2 was estimated as the value
around which the serni-variograrn becomes stable.
(iii) The value of h at which the preceding tangent intersects the sill was
taken as two-thirds of the value of the range at of the spherical
model, i.e., 2 al/3 = 24 ft, and, thus, at = 36 ft.
Verification One way to verify the chosen model is to compare the
theoretical dispersion variance D 2(l/ L), derived from the fitted model, to
the experimental dispersion variance s2(l/L) = 17,3%2.
This dispersion variance is a linear operator of the serni-variogram y, d.
formula (III.2), and is the sum of the variances D6(1/L) contributed by the
nugget effect and Di(l/ L) contributed by the structure Yl:
2

## According to formula (111.11), the nugget-effect contribution is written as

A A A( I)

( 2)

2
Do(l/L)=---=1 ~- = Co 1 - - = 596
1 L
I
L
350

and, from formula (11.36), the variance contributed by the" structure 'Yt is
given by
Di(l/ L) = iit (L, L)- ii1 (l, I),

where 'Y1 (h) is the underlying point model of the model 'Yl(h) regularized

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

239

by core samples of length 1. The range a ~ of the point model is thus equal
to a ~ = al -I = 34 ft, and the point sill value is given by formula (I1.51) as

[3] = O 97C

= C l 1- 2a~ - 20a~j
I [

1,

giving C t = 1339. Approximating the point model 1'1 (h) to a new spherical model with range a~ = 34 ft and sill value C; = 1339%2 and using
formulae (II.51) and (II.98), the following mean values y~ are obtained:

## fiU, 1)= Ci -C1 =039,

I

,[

'h(L , L)= C 1 1-

1 a ~2]
,
T
+s
""2
=O947C 1 = 1267.
4
3 ai

## Thus, the variance contributed by the structure Y1 is

Dt(l/L) = 12-67-0'39= 1228

## and the theoretical dispersion variance is

D 2(//L) = 596 + 1228 = 18,24,

which is close enough to the experimental value s2(//L) = 17'3, and this
can be taken as a verification of the chosen model.

Remark 1 As the support 1=2 ft of the data is small with respect to the
experimental range al = 36 ft, it could have been approximated by a point.
The fitted model, y(h)= Co+'Yl(h), would then be considered as a point
model and the experimental dispersion variance could be accepted as an
estimator of the theoretical dispersion variance D 2(O/ L) of a point in L.
This variance D 2(O / L) can then be calculated directly from the model
y(h), i.e.,

at]

al
+:S1 L: =18,02.
D 2(O/L)=y(L,L)=CO+C l [ 1- 3 T

This value of D 2(OI L) is very close to the value 1824 calculated assuming
a support of length 1= 2 ft. Note that the quasipoint support approximation
greatly simplifies the calculations (see the remark on support and
regularization in section III.C.6).

## Remark 2 Exponential models, like spherical models, also have a linear

behaviour at the origin and a sill. However, if such a model is adopted for
Yl(h), the practical range would be three times the abscissa (h = 24 ft) of
intersection of the tangent at the origin with the sill, cf. section IILB.2,

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MINING GEOSTATISTICS

240

## remark on the definition formulae (111.15) and (1II.16), i.e., al = 3 x 24 =

72 ft. Now, this value (72 ft) is too large to correctly represent the rapid
increase of the experimental serni-variograrn towards its sill.

## Case Study 2. Regionallzation of copper grades, after J. Damay and

M. Durocher (1970)

## The copper deposit of Los Bronces (Chile) is a breccia mass impregnated

with chalcopyrite and can be seen as a series of fault planes delimiting
zones of homogeneous grades, ct. Fig. IV.2(a).
The deposit has been sampled by vertical drill-holes and horizontal drills
from the galleries. The drill cores have been cut into constant 2-m lengths
and analysed for copper.

I
I

Co
01

20

a/

40

60

r( m)
(0 )

(b)

## FIG. IV.2. Regionalization of Cu grades (Los Bronces). (a) Breccia structure;

(b) fitted exponential model.

The experimental serni-variograrns for each direction verify the hypothesis of isotropy of the mineralization, and they can thus be grouped
together to provide a single isotropic semi-variogram, y(h) = y(r), with
r= Ihj.
This experimental isotropic serni-variogram constructed from 8000 m of
drill core or approximately 4000 samples is shown on Fig. IV.2(b). For a

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241

## distance up to r = 60 m, equal to the limit of reliability, an exponential

model with a nugget effect can be fitted remarkably well:
Yo(r)=CO+Yl(r),

VrE[2,60m]

## with nugget constant Co = 016 (% CU)2 and Yl(Il) an exponential model

with sill C j = 0,57%2 and practical range a; = 3al = 255 m , cr. formula
(111.16).
This model was fitted in a similar manner to that of the previous example, except that an exponential model was chosen instead of a spherical
model, to take account of the slower increase of the experimental curve
towards its sill.
The practical range a ~ = 25 m of the model Yl corresponds to the mean
distance, observed during mining, between the main fault planes defining
the zones of extreme homogeneity of the mineralization. The nugget effect
Co is due both the the measurement errors and to the micro-variability
introduced by a network of micro-faults through which the chalcopyrite has
been able to impregnate. In fact, during mining, it was seen that the
mineralized breccia consisted principally of two levels of heterogeneity, the
first one corresponding to the main fault planes with range a ~ = 25 m and
the second one corresponding to a micro-breccia with a 20 to 50 cm range.
For a survey on a support equal to the core sample length (2 m) this
micro-structure appears as a nugget effect.

## Case Study 3. Regionalization

of geopotential,

after

P. Delfiner

(1973bl
In meteorology, geopotential is the name given to the height Z(x) at a
point x of a given isobar surface (in the present case study this isobar
surface is 500 mbar). Taking the earth as a sphere, the distance h between
two points x and x + Il is a geodesic distance, i.e., a distance measured on
the great circle passing through these two points.
The experimental semi-variograrns of the 500-mbar geopotential for the
two main directions N-S (longitude) and E-W (latitude) are shown on Fig.
IV.3. These two directions are perpendicular and parallel to that of the
general circulation of winds, which is from West to East.
The N-S semi-variograrn shows a marked parabolic-type increase which
is characteristic of a drift effect, d. section ILA.3, formula (II. 14). The
E~W serni-variogram, however, shows a very regular behaviour at the
origin and attains a sill at a distance of about 3000 km. A Gaussian model
with a nugget effect has been fitted to this E-W semi-variogram:
y(h)= Co + Yl (h),

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## 'rill E] 0,4000 km],

MINING GEOSTATISTICS

242

## where h is the geodesic E-W distance, Co = 300 m2 is the nugget constant,

and 'rt(h) is a Gaussian model with sill C 1 := 25000 m2 and practical range
a; = at..)3 = 2500 km, ct. formula (III.17).

..a

E-W

2..-

'"E

2000

4000
Ii (km)

FIG.

## IV.3. Semi-variogram of geopotential.

The N-S drift can be seen directly on the data themselves; the N-S
profile of the mean values of the geopotential within zones of 5 of latitude
is shown on Fig. IVA. By contrast, the E-W profile of the mean values of
the geopotential within vertical bands of 10, shown on Fig. IV.S, appears
stationary.
(km)

50 52

Pole 90

5-4

0 '"-------

56

58

6,0

.....:...--J

EQualor

## FIG. IV.4. N-S profile of geopotential.

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243

## The non-stationary RF geopotential Z ex) can then be modelled as the

sum of a drift function m(x) and a stationary RF Y (x):
Z(x)= m(x)+ Vex),

where m(x) = E{Z (x)} is the meridian drift, i.e., m(x)= m (x + h), whatever
h in the E-W direction may be.
60...--------------------------,
5-8

f-

..

~6.....

I '.-.-.
\ .-.'
~ 5 -4 f-- ,_~,
E

52f---

- - .-.- \..

/--., /.-

-e"

/-

, .......\ ,.

- \..

\ _/

..

\/

."

l.onqltude Eosl

FIG.

## Vex) is a stationary RF with zero expectation, E{ Vex)} = 0, and with an

isotropic Gaussian model with nugget effect fitted to the experimental
semi-variogram in the E-W direction.
It is advisable to verify that this model is consistent with the two
experimental serni-variograms of Fig. IV.3. As these serni-variograms were
calculated from the quadratic deviations of the data z (x.), they are, in fact,
estimates of the moment !E{[Z(x + h)- Z(X)]2} which, according to the
above model, is
!E{fZ(x + h)--'-Z(X)]2} = ~E{[ Y(x + h)- Y(x )]2}+Mm(x + h)- m(x)]2
=

## It will be seen that if h is a vector in the E- W direction, the difference

[m(x+h)-m(x)] will be zero and the experimental E-W semi-variogram
is an estimator of the Gaussian model y(h) with nugget effect. Also, if h is
a vector in any direction except E-W (N-S, for example), the experimental

## semi-variogram will consist essentially of a marked parabolic increase due

to the drift term -Hm(x + h) - m(x)f
Case Study 4. Topographic regionalization. after J. P. Chiles and P.
Delfiner (1975).

The first step in the mapping of the Noiretable region of France by kriging
was to construct the semi-variogram of the variable Z (x), defined as the
altitude at point x.

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MINING GEOSTATISTICS

244

The directional experimental semi-variograms showed no clear anisotropy over small distances (h < 300 m), and so they were grouped together
to provide a mean isotropic semi-variogram, as shown on Fig. IV.6.
A model without sill was fitted to this semi-variogram: y(r) = C/', with
C=07, and f) = 14<2, cf. formula (III.lS).

/'

/'
/

/
/

/'

10

'/'"

><

/'

.,.-

,/

/
,/

200

100

(m)

Matheron (1962)

## The bauxite deposit at Mehengui (Guyana) consists of a horizontal bed

with a vertical thickness varying from 5 to 10 m. The deposit was sampled
by vertical drills on a 50-m-square grid and the drill cores were analysed
for Ah03 over all the mineralized thickness, cf. Fig. IV.7. On two 50mx
50 m squares, samples were taken on a 5-m grid in order to study small
range structures. Horizontally, the mineralized section of the deposit can
be approximated to a rectangle 1125 m x 670 m. Vertically, the mean
thickness was found to be t = 725 m.
The support on which the Ah03 grade analyses were made is the
mineralized thickness t(x) at point x. As this support is not constant, the
ReV grade z(x) defined over t(x) is not an additive variable, d. section
II.e2. For this reason, the additive variable a (x) - accumulation at point x
- was defined as the weighting of the grade z(x) by the vertical dimension

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245

## I(X) of the support, i.e., a (x) = I(X). z (x ), and is expressed in percent

Ah03-metres.
Using the data on the 50-m grid, the semi-variograrns for the N-S and
E-W directions were calculated for distances h in multiples of 50 m.

50m

1=725

(al

FIG. IV.7. Survey of Mehengui deposit. (a) Vertical section; (b) horizontal section.

## Semi-variograms were also calculated in the diagonal directions NW-SE

and NE-SW for distances h in multiples of 50../2 = 707 m. Finally,
these same four directional semi-variograrns were calculated from the data
on the 5-m grid for distances h e (5, 50 m]. No significant directional
anisotropy was evident on the serni-variograms and, thus, they were
grouped together to form a single mean semi-variograrn representing the
horizontal regionalization of the accumulation a (x), cf. Fig. IV.8(a) and (b)
on a logarithmic and arithmetic scale, respectively.
A logarithmic model was fitted to the semi-variogram drawn on the
logarithmic scale, and, on that scale, this model appears as a straight line.
Matheron's preliminary fit (broken lines on Fig. IV-8(a) and (b)) corresponds to the model
y(r) = 288 log r - 069,

Vr";::;5 m.

## The semi-variogram drawn on the arithmetic scale gives a greater visual

importance to the larger distances (r> 350 m) and a sill model seems more
appropriate in this case. A nested model was fitted, consisting of a nugget
effect and two spherical models:
'r/r > 0,

with

C o=32,

C 1 = 7,3,

and ')11 and /'2 being two spherical models with unit sill values and respective ranges al = 50 m and a2 = 350 m.
Although the two chosen models are very different, the first one
having no sill while the second one has a sill equal to Co + C 1 + C 2 = 16, it

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246

MINING GEOSTATISTICS

is not possible to determine visually which is the best fit, at least within the
interval of reliability of the experimental semi-variogram (r < 500 m).
The theoretical calculations using both models lead to completely
similar results, d. the following evaluation of the theoretical dispersion
variance.

(a)

10
(b)

200

300

400

r(m)

## FIG. IV.8. Horizontal regionalization of accumulations (Mehengui). (a)

Logarithmic scale.
, Experimental variogram; - ' -, two nested spherical
models; -----, logarithmic model. (b) Arithmetic scale. - ' -, Two nested
spherical models, y(h)=CO+C!'Yl+C2'Y2' C o=3'2, C j = 7 ' 3 , C 2",,55, aj=50,
az = 350; - - - - -, logarithmic model y(r) "" 288 log r - 069.

This case study shows that the important thing in fitting variograms is not
the type of models, but rather-the agreement of the model with the various
experimental curves available, d. section III.C.6 (the paragraph on
robustness). Note the importance of the graph scale in plotting the variograms.

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247

## The experimental dispersion variance of the accumulations a(x) within the

deposit is s 2(01 D) = 15 3 (% m This variance was calculated from the
data on the regular square 50-m grid (uniform distribution within D) and
can thus be considered as an estimator of the theoretical dispersion
D 2(OI D) within the deposit approximated by a rectangle D =
1125 m x 670 m. The support of the horizontally regionalized variable a (x)
is the horizontal section of a core and is assumed to be a point.
The experimental value s2(0ID)= 153 will now be compared with the
two theoretical values derived from the two previous models.
From the general formula (11.36), the theoretical dispersion variance is
written as

## D\OI D) = y(DjD)- ,.,(0) = y(D, D).

For the logarithmic model, the mean value "l(D, D) in the rectangle D is
given by formula (H.97), which considers the linear equivalent (l = 1125 +
670 = 1791 m) of the rectangle D;

## .y(D, D)= 2'88[log 1- 3/2] -0-69 = 166.

For the nested sill model, 1'(r) = C o + C1'YI(r)+ C 21'2(r), the mean values
11(D, D) and Y2(D, D) of the spherical models
and
are found from

"'1

"'2

## chart no. 4 in section ILEA:

y(D, D)= Co+ C 1 xO999+ C 2 x 0918 = 155.

## The latter theoretical value of 15 5 agrees with the experimental value

153. The theoretical value 166 derived from the logarithmic model seems
a little large, due to the fact that this model assumes a continuous increase
of the theoretical serni-variogram for large distances r e [500,1125 m]
which are beyond the limit of reliability of the experimental semi-variogram. In this case, a slight preference should be given to the second model.

## This study concerns the regionalization of uranium grades along a vertical

cross-section of a sedimentary uranium deposit in Niger (Central Africa).
Twenty-five vertical drill-holes were available, each of which was cut into
core samples of 20 em length and chemically analysed for U. The mineralized length L, is defined for each drill-hole i. The experimental mean rn i of
uranium grades and the corresponding dispersion variance sf have been

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248

MINING GEOSTATISTICS

## calculated r along each of these lengths L; This experimental variance s~ is

an estimator of the dispersion variance D 2 (l j L i ) of a core sample I in the
vertical segment of length L;.
Figures IV.9 and IV.I0 show the experimental vertical semi-variograrns
for four holes S4, S14, SlO and S16. The semi-variograms are clearly different
from each other both in shape and variance: semi-variograms S4 and SlO
show a marked hole effect between 1 m and 3 m; the increase of S14 is very
rapid while that of S16 stabilizes quickly. A proportional effect is also
evident: the variability increases with the experimental mean m t.
r - - - - - - - - - - - - - - - - - - - . - - - . ' ...' ...... ---.. - - - - - - - - - - - - - - - - - - ,

/~

I
I

"

I
(

I
4

FIG.

## The experimental mean semi-variogram obtained by grouping all 2S

elementary serni-variograrns is shown on Fig. IV.11. This mean semivariogram corresponds to a mean mineralized length . = 2\ L L, = 13 m
and is calculated from more than 1500 samples with a mean grade m * =
22%"0 U. The hole effect is no longer evident. Two nested spherical models
with a nugget effect have been fitted to it, i.e.,
y(h)

Co + C l l' l (h ) + C 2 Y2(h ),

Vh vertical E [0,2 to 4 m]

with Co=3'5 (';(0 U)2, as nugget constant, and e l Yl and C 2 1' 2 being two
spherical models with sills C j = 37, C 2 = 11 (%'o U)2 and ranges a I = 1 m
and az = 3 m.
The two ranges correspond roughly to the abscissae of the maximum and
minimum of the hole effect observed on the serni-variograms S4 and SIO.
The two ranges can be interpreted as the mean thickness of the rich seams
(a 1 = 1 m) and the mean distance between two such seams (az = 3 m) which
tFor proprietary reasons, all the mean grades quoted in this example have been multiplied by
a factor.

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S'4

L '22m,286%"

249

10

::J
0

::<:

)...

SI6

L'IT4m,I16%"

I~

[l-__-----'---

L
2

4
3

h (rn )

## FIG. IV.IO. Semi-variograms S14, SIO, S16.

,-----~~~-----~~----~-~-~--__.

/
I

h (m)

## FIG. IV.Il. Mean experimental serni-variogram and its fit.

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MINING GEOSTATISTIC5

250

can also be seen on the vertical profile of the grades of drill-hole S4 on Fig.
IV.12.
Fitting a proportional effect

## The preceding nested spherical model is only representative of the mean

vertical structure over the zone sampled by the 25 drill-holes with mean
grade m * = 22%0 U. In order to represent local values on a zone with a
- 22, a proportional effect should be added to the model.

mt

%"U

o
I

10

~-------------

10

- -

-- -

------

g
~

~
E

15
L4'11-6m

s:
a.

<II

~
20

~
~

------- ----

FIG.

- - -

--~---

---

## To do this, the experimental dispersion variance sf of the grades in each

mineralized length L; were considered, and, in order to make a significant
comparison, the 12 drill-holes with the longest mineralized lengths were
selected, L; E [11 to 33 m]. As these mineralized lengths are larger, the

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

251

variance s; is calculated with a larger number of samples (five per metre) and
can be considered as an estimator of the a priori variance D 2 (I/ oo ) of the
grades of core samples of support 1= 20 ern. A scatter diagram of cxperimental variance s1 is shown on Fig. IV.13. A parabola has been fitted to the
scatter diagram:

'im*E [13 to 3% U]

15

~ 10

t\l

V,

.:
,

. r..
I

FIG.

## IV.13. Fitting the proportional effect (experimental variance

experimental mean m;).

sl

versus

## The mean experimental variance S2 = 83 calculated by weighting the 25

and
the
vertical
model
'Y(h) =
elementary
variances
C o + C1'Yt(h) + C 2'Y2(h ) corresponds to the overall mean value m*=22.
So, for a local vertical serni-variogram in a zone with constant mean grade
m, the following quasi-stationary model applies:

s;

## with h vertical E [0 20 to 4 m] and m E [1' 3 to 3% U]. However, note

that this model may not be adapted for a precise local estimation around
drill 54.

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MINING GEOSTATISTICS

252

## Fitting a hole-effect model

By way of example, an attempt has been made to fit the best possible
model with a hole effect to the experimental serni-variogram of core 54.
There is no practical purpose in fitting this model, as this hole effect has
only a local significance and disappears on the mean structure of Fig.
IV, 11, which is the only one used in the geostatistical calculations, It does,
however, serve as a good exercise in fitting such a hole-effect model.
The first step is to estimate the amplitude of the hole effect (ct. section
III,B,2, subsection 3, "Hole effect models"),
a = Imin {C (h )} I/ C (O) =

6'8-48
=0417,
48

## The maximum value of the experimental serni-variogram on Fig. IV.9 is

68 and, thus, (6'8 - s~) is the minimum absolute value of the corresponding covariance. The experimental variance s~ = 48 is an estimator of the a
priori variance C(O).
As the value of 0417 is larger than the maximum amplitude of an
isotropic hole effect in three dimensions, CX ma x = 0217; the hole effect
observed on the semi-variograrn S4 will not be reproduced in the nonvertical directions of the deposit. And, indeed, in such a subhorizontal
sedimentary deposit, the pseudo-periodicity of the alternance of rich
mineralized seams and waste has no reason to be isotropic,
As this hole effect of amplitude a = 0417 is limited to the vertical
direction, it can be modelled by positive-definite-type models in only one
dimension, in particular the cosine model, y(h) = 1 - cos h, for which the
hole-effect amplitude is a = 1, d. formula (III. 22), The following nested
model has been fitted and is shown on Fig. IV.9:
y(h) = 1'1(h) + 'Y2(h),

Vh vertical e [02 to 4 m]

with
'Yt(h) = A[l-exp (-f.Lh/ A). cos J.Lh],
A

= 2,26(% U)2,

A = 10 m,

Y2(h)={Bh,
Ba,

J.L = 25;

VhE[O,a],
\:fh??:- a,

with
B = 9(%'" U)2 m-- I ,

a =0'28m,

The model ')It(h) is constructed from formula (III. 13 ) for the covariance
function exp(- uh] A). cos J.Lh, which is the product of an exponential
covariance and the cosine covariance.

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253

## The model ]'2(h) is a transition model with range a and is conditional

positive definite only in the one-dimensional space.
Note that the sill of this model is A + Ba = 4'78(%'o U)2 and the amplitude of its hole effect is

645 ~ 478
478

0349.

IV.C. COREGIONALIZATIONS
Case Study 7. Coregionalization of uranium radiometry, after M.
Guarascio (1975)

## The pitchblende mineralization of the Novazza deposit (Italy) is contained

within a tuff bed dipping 20 to 30, see Fig. IV.14.
The deposit has been sampled by bore-holes of various directions drilled
from the galleries, The samples have been analysed, either chemically in

o
I

50m
I

Hanging tuff

EZJ

as'"

B:SM
'4":

FIG.

.'

Mineralized tuff
Foot tuff
Sandstone

## IV.14. Vertical cross-section of the Novazza deposit.

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254

## U 3 0 8 grade on cores of variable length, or by radiometric measurements.

The purpose of the study is to provide a local and global estimation of the
deposit through "cokriging", i.e., considering simultaneously both types of
information available, the chemical analyses and the radiometric
coregion alization.
E-W

N-S

Vertica 1

15

10

20

h(m)

## The analysed grades of core samples of constant length I = 1 m were

reconstituted according to formula (111.45), and experimental serni-variograms along each of the reconstituted drill-holes were calculated. These
elementary serni-variograms were then grouped into three mean semivariograms which are characteristic of the three directions N-S, E-W and
vertical (the angle tolerance around each direction being da = 15). The
chemical-grade semi-variograms are shown on Fig. IV.iS. They indicate
very different levels of variability (the ordinate axis of Fig. IV.iS is graduated in logarithmic scale). However, a direct proportional effect can be
observed. Such an effect is classical in low-grade uranium deposits and is
corroborated by the scatter diagram (s;,
between the experimental
mean m1 and the experimental dispersion variance sf calculated on each
chemically analysed drill core: a parabolic relation (sf, mt) is observed.
Thus, a direct proportional effect in m1, d. formula (III.37) has been
considered, and the three semi-variograms of Fig. IV.i5 have been divided

mn

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255

## by the squares of their respective mean. These three relative semi-vario-

grams are shown on Fig. IV.16, the coordinate axis being graduated now in
arithmetic scale. A remarkable isotropy between the three directions
studied is observed and a last grouping provides the mean isotropic relative
semi-variogram of chemical grades, 'Y~(h)/m:2, ct. Fig. IV.17.
E-W
Verti col

N-S

15

10

20

II (rn)

--2
.-'

//
v

.>:

.-

--- -- -----

/"
,/

.-.-

,/

'/

10

15

20

II (m )

- ' - , 'YR

## The radiometric measures reconstituted on the same support (cores of

constant length 1= 1 m) provide experimental serni-variograms with structural features quite similar to those of the grade semi-variograms. A direct
proportional effect in m~ has been considered and the various directional

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256

MINING GEOSTATISTICS

## serni-variograms have been grouped into a unique mean isotropic

relative semi-variograrn of radiometric measurements, y~ (h)/ m ~2, d.
Fig. IV.17.
This same figure, Fig. IV.17, shows the mean cross serni-variogram,
Yi-R/mi. mt isotropic and relative to the product of the two experimental means of the grades (m;) and radiometries (m~) which have been used
to calculate it.
On Fig. IV.17 it should be noted that the cross-variability Yi-R/
m~
2
is always greater than the direct variabilities
and yV ~2, which
would entail that the experimental correlation coefficient p*(h) between
the two increment variables [Zg{x+h)-Zlx)] and [ZR(x+h)-ZR(X)] is
superior to 1. Indeed, the theoretical correlation coefficient is written

yi/ m:

m; .

(h) _
E{[Zg(x + h) - Zg(x )][ZR(X + h) - ZR(X)]}
P
- J(E{[Zg(x + h)- Zg(x )f}. E{[ZR(X + h)- ZR(X)]2})

Thus, the three experimental curves of Fig. IV.I? are not consistent. One
possible explanation is that the experimental cross-serni-variogram
has not been calculated from the same data as the direct semi-variograms
(in practice it is difficult to do otherwise, unless we consider only the few
points where the two measurements - grade and radiometry - exist simuland
taneously). Inconsistent and excessively small values of the means
m ~ can then explain the excessive variability of the experimental relative
cross-serni-variogram Y:-RI
~ as observed on Fig. IV.I?

Y;-R

m;

m: .m

## The Broken Hill lead-zinc ore body in Australia is "currently" considered

as an exhalative sedimentary deposit with a secondary metamorphism.
Three types of ore can be clearly distinguished in th; studied zone, according to their gangue minerals:
(i) rich ore in a quartz gangue;

## (ii) medium grade ore in a cllicite gangue;

(iii) poor ore in a rhodonite gangue,
A schematic vertical cross-section of the mineralized horizon with its three
gangues is shown on Fig. IV.18(a). The horizontal cross-section on a
particular mining level exhibiting the various mineralized surfaces is shown
on Fig. IV.18(b). Note, at this level, that the principal ore is the one
associated with the calcite gangue.

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

257

Mining is carried out using a bore and pillar method in an E-W direction;
pillars are 19 ft wide and stopes are 33 ft wide, cf. Fig. IV.19. On each
mining level, both faces of each pillar were sampled continuously by 10-it
chip samples, providing a set of data on a regular grid measuring 10 it in
the E-W direction and alternately 33 it and 19 it in the N-S direction.
Each chip sample is analysed for Pb, Zn and Ag.

100lj
1
30 m

(0 )

FIG.

(b)

## IV.18. Cross-sections showing the three types of ore. (a) Vertical;

(b) horizontal.

~===- -

-f--

-=-'=::::::;-";)133 ft
"=:::J 19 ft

Chip

FIG.

samPlin(~~~~===~=-'LJ 33 It

## IV.19. Sampling of a mining level.

For the E-W and N-S directions, the experimental direct semi-variograms for lead, zinc and silver grades are shown on Fig. IV.20(a), (b), (c), as
well as the experimental means and dispersion variances of the data used in
the calculation. The experimental variances calculated from formula
(III.49) are estimators of the dispersion variances D 2 (11L) of the grades of
a chip sample of length l within the mean length L of the drives ([' = 400 it
in the E-W direction).

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258

MINING GEOSTATISTICS

## The following points can be stressed about these experimental variograms.

(i) There is a particularly clear structural isotropy for small distances
(h < 150 ft), confirmed by the extreme homogeneity of the mineral-

## ization within a given gangue.

100
(a)

E-W

m:::;13-2Ofc,
5 2 *' = 4 9 .5
N

:0-

~-

n,

---

_. _.

:::"':"_...,..:==--~::=oo:::::'.

~
a;

lo...

10

40
(b)
N

""

N
~

">...

20 10

---

N-S

30 ,...
~'..-

..-

... - - - - - - - - - - - - - - - - ~.

E-W

m 10.3 %

..../

5 2"-:::;

209
I

5
(c)
4

m=372 oz/ton
5

"'
.....

*= 2 -6 2

___

---- - ---.-=-=-~ _.

<:
0

.........

----~

"" ....
N-S

~-W. _ . _

lo...

10

100

lid

200
(ft.)

## FIG. IV.20. Direct semi-variograms, (a) Pb grades regionalization; (b) Zn grades

(ii) There are obvious sills which are well approximated by the experimental dispersion variances S2 and which, in all cases, are reached at
distances of about [hi = al = 60 ft.
(iii) There are obvious nugget effects.

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

259

(iv) The variabilities begin to increase above the sills for distances lhl
greater than 150 ft. Although located in a zone close to the limit of
reliability of the experimental semi-variograrns (L/2 = 200 ft), this
increase may be due to the non-stationarity at large distances
caused by the presence of three different gangues: for distances Ihl
greater than 150 ft, when a point x is in one gangue, the point x + h
will probably be in a different gangue causing the square deviation
[z(x + h) - z (x)] to become particularly large.
silver-zinc are shown on Fig. IV.21(a), (b) and (c). According to the
preceding hypothesis of isotropy, only the E-W direction has been considered. There are no nugget effects on these cross-semi-variograms, and
there is a transition structure with a range approximating that found for the
direct serni-variograms (al = 60 ft).

Fitting a model
The simultaneous presence of the same transition structure with range
al =60 ft on all direct and cross-serni-variograms suggests that the linear
model of coregionalization (111.27) would provide a good fit. This model is
given by
Vh horizontal E [0, 150 tt].

The indices (k, k' = 1 to 3) represent, respectively, the grade in lead, zinc
and silver. The index (i = a and 1) represents the two basic structures; a
transition model yo(h) with a very small range or nugget effect, and a
transition model Yl (h) with range a I = 60 ft, respectively.
As the nugget effect is not present on the cross-semi-variograms, the
coefficients b~k" Vk - k' are all zero.
A spherical model with a unit sill and a range al = 60 ft has already been
fitted to the basic structure Yl (h), and Table IV.1 gives the two matrices of
coefficients [b~k'] for i = 0 (nugget effect) and for i = 1 (Yl structure). The
corresponding fits obtained using this model are shown by the dotted lines
on Figs IV.20 and IV.21.
T ABLE IV.1. Coefficient matrices of the linear model

i~ ~ ~n

gg]

Ag

1.1

i=O

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39
14
[ 5_5

14.5
15
3.8
i=l

5 ]
3.8
1.8

260

MINING GEOSTATISTICS

Remark 1 The coefficients b~k' verify the constraints of the linear model,
i.e., both matrices of Table IV.l are positive definite, d. conditions
(III. 26).
30

r---~~----~~---------.

(a)

20
----~_:-::_=-=:_~_::""':--

- - - - --

10

(b)

I
5

(c)

10

100

200

r(f!)

FIG.

## IV.21. Cross-serni-variograrns. (a) Pb-Zn coregionalization; (b) Ag-Pb

coregionalization; (c) Ag-Zn coregionalization.

Remark 2 Note the quality of the fit achieved for small distances (h <
range al = 60 ft). This coregionalization model is thus ideally suited to the
geostatistical procedure of co-estimation of one grade (e.g., Ag) by the
three data sets (Pb, Zn and Ag), d. section V.A.5, Case Study 2.
This co-estimation will be carried out within an horizontal neighbourhood
of approximately 60 ft around the unit to be estimated.

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261

## Remark 3 One possible way of interpreting the chosen linear model of

coregionalization is to consider the three RF's Zdx) as issuing from two
principal causes of variability represented by the basic RF's Y Ok (x) and
Y1(x), d. the genesis of the linear model in formula (111.23):
Vk = 1 to 3.

The RF's YOk (x) represent the micro-variabilities of the grades for
distances (h < 10 ft) which are not accessible from the available data; these
micro-variabilities vary from one metal to the other and appear as different
nugget constants on the direct serni-variograms. The unique RF Y1(x)
represents a second variability common to all three metals. When the
hypothesis of a sedimentary origin of the deposit is accepted, this common
variability can be explained in terms of horizontal beds of deposition
micro-basins, the mean horizontal dimension of which is around 60 ft, ct.
Fig. IV.22.

60 It
FIG.

## It is interesting to note that the study of the vertical regionalizations of

these three grades, from one mining level to the other, produced direct
serni-variograms with the following features.
(i) Nugget constants strictly equal to those found for the horizontal
structures: the micro-variabilities Yodx) would then be isotropic in

## the three-dimensional space.

(ii) A common transition-type structure with a range a ~ = 30 ft
representing the mean vertical dimension of the micro-basins of
deposition, d. Fig. IV.22. A proportional effect is added to this
geometric anisotropy of ratio a 1/ a I = O 5, due to the fact that the
three types of gangue are not encountered in the same proportions in
each level.

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MINING GEOSTATISTICS

262

IV.D. ANISOTROPIES
Case Study 9. Geometric anisotropy, after J. Serra (1967a)

## As presented here, the study concerns the regionalization of iron grades on

a hectometric scale (from 100 m to 2 km) in the ferriferous basin of
Lorraine (France). The data come from two mines, Hayange and Bois
d'Avril which, although geographically distinct, mine the same sedimentary iron seam known as "red seam".
This red seam is sub horizontal, very regular, and has a fairly constant
thickness of about 3 m. The horizontal regionatization of the iron grades of
this seam has been sampled along the length of mining galleries by
regularly-spaced small drills (90, 100 and 120 m according to direction and
mine). Data is thus available from both mines on a regular grid in the E-W
and N-S directions of the galleries. These galleries are very long, from 2 to
3 kID, and, consequently, the limit of reliability of the experimental variograms extends up to 1 km.
The E-W and N-S experimental semi-variograms for Hayange and Bois
d' Avril are shown on Figs IV.23 and IV. 24, respectively. The N-S semivariograms for both mines are compared on Fig. IV .25.
10..---------------------------.........,

/-

--

-_./ ./

/'

E-w

'"

1000

500
h(m)

## Continuity of the red seam

Although both mines are geographically distinct (6 km distant from each
other), their variograms reveal exactly the same structures on the
hectometric scale. This is quite normal, since both variograms are relative
to the same seam, which is found over a large portion of the Lorraine
ferriferous basin. In addition, the means of both sets of data used in the
calculations are almost identical (34% Fe).

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263

## Hole effect and geometric anisotropy

There is a marked hole effect on the E-W serni-variograms, with a maximum and a minimum variability occurring at around Ihl = 200 m and
Ihl = 350 m, respectively. This hole effect is also present on the N-S semivariograms with the same mean sill of variability (around 83 (%Fe)2) but
with the maximum and minimum variabilities occurring at around lh I =
700 m and lhl = 1000 m, respectively.

--

10 .---------------------------~
/

,...

--"E-W

"-

"-, ..... /

N-S

## .... .......... ......

................ >

I
I

01'

500

100

1000
h(m

## FIG. IV.24. Horizontal regionalization at Bois d' Avril (red seam).

10.-----------------------------

## Bois d' Avril

.-

.-

....-

OL---'------------I-100

500

----I..

------l

1000
h(m)

## FIG. IV.25. N-S direction variograms at Hayange and Bois d'Avril.

This right shift of the hole effect along the distance axis Ih I suggests a
model of geometric anisotropy between the E-W and N-S serni-variograms. When the E-W distances are multiplied by the anisotropy ratio
A = 700/200 = 3,5, the E-W and N-S maximum values of the hole effect
are made to coincide, cf. section IILBA, Fig. III. 10.
Genetic interpretation
It is generally accepted that the origin of the Lorraine ferriferous basin is
sedimentary with marine deltaic deposition of the upper toarcian, d.

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264

MINING GEOSTATISTICS

## J. Serra, (1968). Iron-rich sediments coming from inland were deposited in

the sea between the main current lines of the river and formed thin
horizontal pseudo-elliptical sedimentation units, ct. Fig. IV.26. Sandbanks
in a river bed provide another example of this type of sedimentation. Thus,
the transition structure with hole effect and horizontal geometric anisotropy as shown on the variograms, corresponds to the alternance of these
sedimentary units; the N-S direction corresponds to the main direction of
the ancient fresh-water currents and the mean :N-S and E-W dimensions of
these units are given by the abscissae (700 m for N-S, 200 m for E-W) of
the maximum variability observed on the variograms.
It should be mentioned that sedimentologists explain the vertical succession of the different mineralized seams in terms of transgressions and
regressions of the marine sedimentation: during certain epochs and at
certain places of the delta, sedimentation was stopped and then started
again at a later period. Thus, the red seam occurring in both mines
corresponds to the same sedimentation sequence (and, more precisely,
within the granulometric differentiation 80 to 300 J,Lm in the sequence).
It is interesting to note that there are similar oolitic iron deposits
presently in the course of being formed in the sea at the estuaries of the
Meuse (North Sea) and the Senegal (West African coast) rivers.

Marine sedimentation

ErOSion

+
N-S

Sea

700 m

## The grey seam at Saney

The Sancy mine is a third geographically distinct mine in which a different
seam (the grey seam) is mined. The same structural analysis was carried out
on the horizontal regionalization of the iron grades.
The experimental semi-variograms for the four directions E-W, N-S,
NW-SE and NE-SW are shown on Fig. IV.27(a) and (b). The length of
the galleries sampled at Sancy is much shorter than at Hayange and Bois

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

265

(a)

100

378

500

1000

(h)
s2

10

NE-SW

\-

100

260

500

1000 1060

h (rn)

( c)

s
FIG. IV.27. Horizontal regionalization in the grey seam (Sancy). (a) N-S and E-W
directions; (b) NW-SE and NE-SW directions; (c) ellipse of geometric anisotropy.

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266

MINING GEOSTATISTICS

d'Avril, and, thus, the experimental semi-variograrn from Sancy will not
reveal the hole effect quite so clearly. However; the beginning of the hole
effect is quite obvious on the NW-SE semi-variogram which shows the
most rapid variability.
The first step in establishing a model of the geometric anisotropy at
Sancy was to approximate the sill of the various serni-variograms by the
experimental dispersion variance, 52 = 115 (%Fe)2 of all the data considered in the calculations. Next, a line representing the average slope of
the first few experimental points of each semi-variogram was extrapolated
to this sill, providing an abscissa which was transferred to the directional
graph on Fig. IV.27(c) (e.g., 260 m for the NW-SE direction). An ellipse of
geometric anisotropy can be fitted to this directional graph and shows that,
at Sancy and in the grey seam, the sedimentary units seem to be elongated
in the NE-SW direction and appreciably longer than those of the other
two mines, all facts which are corroborated by various lithological
observations.

## Case Study 10. Zonal anisotropy, after M. Guarascio and G. Raspa

(1974)

The Salafossa zinc deposit is located in the dolomitic Alps (Italy) and
consists of an ellipsoidal mass with a main axis in the N-S direction,
dipping between 0 and 30 W. The mineralization (blend and galena)
consists of dolomitic breccia cement. Three types of ore can be clearly
distinguished during mining, each one being related to the dimension of the
elements of the host breccia, d. Fig. IV.28(a) and (b):
(i) type lore in the heart of the deposit, associated with gravelly

## breccia (elements with dimensions of several centimetres);

(ii) type 2 are in the upper part of the deposit, associated with fractured
dolomitic rock (elements with dimensions of several decimetres);
(iii) type 3 ore at the base of the deposit in karstic cavities consisting of
dolomitic sand rich in sulphurous mineralized cement.
Thus, the two upper ores (types 1 and 2) are associated with altered
dolomite, within which stratification can still be distinguished, cf. Fig.
IV .28(b). The direction D perpendicular to this stratification is vertical in
the upper part of the deposit (type 2 ore) and then tends progressively
towards a slope of 30 West in the central part of the deposit (type lore).
No stratification can be distinguished in the type 3 are. The following
structural analysis by variography will identify these preferential structural
directions.

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

267

Data available
The data come from fans of drills set in an E-W vertical plane approximately perpendicular to the N-S direction of advance of the main drives,
ct. Fig. IV.28(b). The cuttings from the drills were analysed in 15-rn and
3-m lengths according to the zone. Because of this configuration, no data is
available in the N-S direction.

IE'J

Ty pe lore

Ed Type 3 ore

~ Type 2 ore
NJ\I.1215

Niv,1I81

-------------

200

400

~O

(a )

FIG. IV.28. The three types of ore at Salafossa mine. (a) Vertical N-S cross-section;

## (b) Vertical E-W cross-section and location of the N-S drives.

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MINING GEOSTATISTICS

## Regionalization of the zinc grades

The experimental semi-variograms for the zinc grades were calculated for
each ore type and for each of the four main directions on an E-W vertical
plane.
The semi-variograms for direction 1 and the two combined directions
3 +4 are shown on Fig. IV.29. They correspond to the upper type 2 are for
which the data support is 3 m. There is a marked anisotropy: the variability
is obviously greater in the vertical direction (l) than in the subhorizontal
directions (3 + 4), the horizontal direction corresponding to the mean
direction of stratification in the type 2 ore.
D

Direction 1

10
N

)--_ _~3

30

45

h(m)
FIG.

## IV.29. Regionalization in the type 2 are. - - - , Experimental;

model; support, l::::: 3 m.

## The combined semi-variograms for directions 1 + 4 and 2 + 3 are

shown on Fig. IV.30. They correspond to the central type 1 ore for which
the data support is 1 5 m. In this case, the main variability direction (l + 4)
tends towards directions D (30 W), perpendicular to the mean direction of
stratification in type lore.
The semi-variograms for directions 2 and 4 and the combined semivariogram for directions 4 + 2 + 1 are shown on Fig. IV.31. They correspond to the lower type 3 ore for which the data support is 1 5 m. In this
case, no significant directional anisotropy can be distinguished, which
confirms the absence of stratification in the type 3 ore (dolomitic sand
filling the karstic cavities).
0

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269

## Fitting models to the zonal anisotropy

The three-dimensional point support model adopted for each of the three
ore types consists of an isotropic structure with a nugget effect and an
anisotropic structure depending only on the direction D of zonality (D =
direction 1 for the type 2 are, and D = 30" W for the type 1 ore):
(IV. 1)

## where r = Ih I is the modulus of the three-dimensional vector h; bo is the

coordinate of the vector h along the direction D of zonality; {Co(O)Co(x)} represents an isotropic nugget effect which appears on the

30
":-

<::

~
0

>20
3

.....

-, ..... '

/'".

10

...... -.

-"..--' /

Direction 2 +3

_I
I
/

30

15
Ihl (rn )

(=:

15 m.

## regularization of support I as a nugget constant COL; C j 'Yl (r) is an isotropic

spherical model with sill value C 1 and range at; C 2 'Y2(h D ) is a spherical
model with sill C 2 and range a2, depending only on the distances ho in
direction D.
The values of the various parameters of this three-dimensional point
model for each of the three ore types are given on Fig. IV.32.

## Fitting the parameters

The parameters of the point model are derived from the experimental
semi-variograms regularized on the support l by the standard procedure of
deconvolution and identification, d. section II.D.4.

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MINING GEOSTATISTICS

270

## For type 2 ore and the combined direction 3 + 4 perpendicular to the

zonality direction D, Fig. IV.29 gives the following estimators of the
parameters, regularized on I == 3 ill, of the isotropic point spherical model:
COl

= 1,

= 37 (%

ell

Zn)2,

a1/

= 24 m.

20

....
15

\ Di rection 4
Direction 2+4+1

'"C

Direction 2

~ 10
e;

>-..

J5

Inl
FIG.

30

(rn)

## IV.31. Regionalization in the type 3 ore. Support, I"" 1 5 m.

'~3 m JC O ~ I (%Zn)2
01 = 21 m i C 1 ~ 4(%2n)2
02 = 36 m ; C2" 8{'%Zn)2
Zone E
Type 2

- 30 0 W

, " I 5

m ; Co = 47 (% Zn) 2

12 m i C 1 '" 12('%Zn)2
02= 24 m i C 2 " 20[%Zn) 2

01"

Zone W
Type 1

~
~

L = 1-5m; Co =3-3(%Zo)2
01= 9 m ; C 1 = 15(%Zn)2
C2 = 0 (%2n)2

Zone W
Type '3

## The parameters for the point model are then

al = all-3 = 21 m

and

Ct

C II / [1 - "Yl(l, I)]

= ell

x 1075 = 4.

## The regularized theoretical semi-variogram of this point model for a vector

h o perpendicular to the direction D of zonality is then written
Vh o;::.3 m,

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271

## ClI'YlI(h o ) being a spherical model of sill C li = 37 and range all = 24 m.

This regularized model 1'1 (k o) is shown on Fig. IV.29 and is a good fit to the
experimental semi-variograrn for direction 3 + 4.
Using the experimental difference [ y1(h D ) - ydh o)] between the
experimental semi-variogram 1't (h D ) in the zonality direction (D = 1 on
Fig. IV.29) and the preceding isotropic theoretical semi-variogram 'Y1(h o),
the regularized parameters can be fitted to the anisotropic component of
the point model (formula (IV. 1)), giving
a21

= 39 m

and

21

= 77 (% Znf

## The point model parameters are then

and

C 2 = C 2 1/ [I - Y2(l,I)]
=

e 2 1 x 104 = 8.

written as

v; 3 m,

Direction 1

10

/.

/
/

/
/
/

Direc lion 3 t 4

/
/
/

/
/

/'

30

15

45

h (m)

## FIG. IV.33. Regionaiization of support 2l::::: 6 m in type 2 ore.

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272

## MINING GEOST ATISTICS

C 21Y21(h D ) being a spherical model with sill Cu and range a21. This
regularized model 'YI (h D ) is shown on Fig. IV.29 and is a good fit to the
experimental semi-variogram for direction 1.

## Verification The expressions for the theoretical semi-variograms

regularized on the double support 2/ = 6 m, Y21(h), were calculated from
the previous point model (formula (IV.1. These theoretical curves were
then compared with the corresponding experimental serni-variograms
(obtained from the data of support 2l) on Fig. IV.33, for the type 2 ore.
The verification is positive.
IV.E. VARIOUS STRUCTURAL ANALYSES
Case Study 11. Structural analysis of El Teniente Cu deposit after A.
Journel and R. Segovia (1974)

The El Teniente copper mine in Chile has a daily production of 65000 tons
of crude ore assaying from 1 5 to 1 7% total copper and ranks among the
largest copper mines in the world.
Two mineralizations can essentially be distinguished:
(i) an intensive so-called "secondary" mineralization with high grades
of around 2% in the upper levels within a very fractured andesite
favourable to the flow of solutions.
(ii) a so-called "primary" mineralization with lower grades of about 1 %
in the lower levels within a much clearer andesite.

## On a given horizontal level, the mineralization shows a radial drift

around a vertical volcanic chimney: the copper grades decrease as the
distance from the chimney and its aureole of fracturation increases, cf. Fig.
IV.34.
The underground mining is done by standard block-caving. The secondary are, the degree of fracturation of which allows excellent recovery, is
being exhausted, and, thus, the present study concentrates on primary are
only.

Available data
Two sampling campaigns are available:
(i) systematic horizontal channel sampling along the drives and crosscuts. The channel samples have been cut into 10-ft lengths and
analysed.

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

273

(ii) drills are collared in each drive and developed in fans. The directions
are variable but the cores have been analysed in constant 10-ft
lengths.
The channel sample and core sample supports can be mixed with good
approximation and statistical studies have shown that there is no significant
bias between the two data sets.

Granodiorite

## S econdory min era Ii lOt i on

~~~

----------_
..............
Primary mi n arol ilotion

Andesite

FIG,

IV.34,

E~W

## Object of the study

Each mining block (60 m x 60 m x 180 m) is evaluated by the channel
samples of the drives above and below the block and by the drill cores
through the mass of the block. The object is to define the optimal proportion of each type of sample and the optimal configuration of the drills.

## Regionalization of the channel sample grades

Six levels of drives and cross-cuts were considered within the primary
mineralization. The N-S direction of the cross-cuts is distinguished from
the E-W direction of the drives.
By way of example, the E-W and N-S serni-variograms of the 10-t
channel samples on level T4 are shown on Fig. IV.35. The variability of the
grades in the E-W direction is obviously greater than that in the N-S
direction; in fact, at the T4 level studied here, the E~W direction is a radial
direction with regard to the chimney and therefore takes the previously
mentioned drift of the grades into account. A proportional effect is also
present in the micro-structures: the experimental nugget constant c;

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274

MINING GEOSTATISTICS

increases with the mean m" of the data from which the serni-variogram was
calculated.
All the data of the channel samples from six levels were combined to
provide the two mean serni-variograms for the N-S and E-W directions
05

---

[-w

## FIG, IV.35. Channel-sample serni-variograms (at level T4). E-W: N= 1619, m =

1,66, ct =0240. N-S: N=207, m = I-51, C6 =0190.

shown on Fig. IV.36. A considerable number of data were used: 3360 for
the E-W direction and 2730 for the N-5 direction. Again, the radial drift
effect on the E-W semi-variograrn and the proportional effect on the

-04

[- w

N-S

15

30

60

90

Ih!(m)

## FIG. IV.36_ Channel-sample rnean-variograms (at six levels). E-W: N = 3360,

m = 153, ct = 0-200. N-S: 2730, m = I-51, ct = 0-128.

~_~_~'#!)NW.f!linebook.blogf
. .com

275

## nugget constants are observed. On the N-S curve, there is a change of

slope around 10 m and the curve stabilizes around 100 m, which suggests
the fitting of two nested structures with lO-m and tOO-m ranges, respectively.
Regionalization of the drill core grades

As the inclinations of the drill cores are very variable, they have been
grouped into angle classes, d. section IILC.4 Fig. III.23(b). Five classes,
corresponding to the following five mean directions, were considered: at;
vertical direction; (Y2, N-S horizontal; a3, E-W inclined at ~45; a4, E-W
inclined at +45"; as, E-W horizontal.
The mean serni-variograms for each angle class are shown on Fig.
IV.37(a) to (e). The following features can be discerned:
0) a very significant radial drift effect in the horizontal E-W direction,
and a less significant one in the E-W directions inclined at +45 and
-45;
(ii) a change in slope around to m;
(iii) the curves for directions al and a2, which are unaffected by the
radial drift, stabilize around 100 m;
(iv) a proportional effect on the nugget constant values.

## Fitting a three-dimensional model

If all the preceding semi-variograrns not affected by the E-W drift are

superimposed, they will coincide except for the nugget constant, This
means that the two nested structures with IO-m and 100-m ranges are
isotropic. First of all, these two structures were fitted to the best-known
experimental serni-variogram, i.e., that for the channel samples in the N-S
direction, as shown on Fig. IV.36. The adopted model is
V r = Ihi> 10ft,

(IV.2)

## where C o=0-128, C1Yl(r) is a spherical model with sill C l =0-045 and

range at = 9 m, C 2 y z(r) is a second spherical model with sill Cz = 0-065 (%
CU)2 and az = 105 m.
To determine the experimental nugget constant (2~ for each of the
semi-variograms, a transparency of the preceding model was moved over
each of the serni-variograms along the ordinate axis until the best fit was
achieved. Table IV.2 shows the increase of this experimental nugget
constant cri as a function of the experimental mean m" of the data used in
the corresponding serni-variogram calculation (direct proportional effect).

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MINING GEOSTATlSTICS

276

0-2,.-------------(0 )

01

_--_

- - - - . -..

01

..__ ._----.-----~---------'

~ _...

(b)

... .-

-----------

--~------~--

/'

02

I
------",.,-,.,

..---

(c)

--~

.........................

--

01
./

N
OJ

e~

0'2 ( d)

>-..

--01

,.-----------------'~""--------

(e)

03

02

--

01

15

30

60

Inl

90

(rn)

## FIG. IV.37. Core sample directional semi-variograms, (a) Vertical: N = 694, m =

095, C6 = 0043. (b) N-S horizontal: N = 543, m = 065, C6 = 0004. (c) E-W
minus 45: N = 1588, m = 094, C6 = 0035. (d) E-W plus 45: N = 1331, m =
1-12, C6 =0-063. (e) E-W horizontal: N= 1199, m=O'
C6 =0030.

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I
~.

s-

I-

TABLE

';n

## IV.2. Proportional effect observed on the nugget constant

Direction

Drills
A

Horizontal
Vertical
N-S

m*

,
(;

~.
~

'"

11:

C6'
Fitted parabola

C0

095
0043
0-037

065
0004
0004

- 4)

Channel samples

--.,

,,'-

E-W

+45
E-W

Horizontal
E-W

T4
N-S

T4
E-W

094
0-035
0-035

112
0-063
0-064

088
0-030
0-026

151
019
0184

1-66
0-24
0-242

-0

--

--,

All levels
N-S
E-W
131
0128
0118

153
0-20
0-19

278

MINING GEOSTATISTICS

These experimental points (CS, m*) are shown graphically on Fig. IV.38;
the parabola Co = 018 [m - 0'5]2 is a remarkably good fit.
To take account of the radial drift in the E-W directions, it is necessary
(formula (IV.2)). The deviations from the isotropic model of the E-W

020

0-10

./a

001

.::'-~

----l

---"--

---'

1-5

10
m(%Cu)

## FIG IV.38. Parabola of proportional effect. . ,

CS,

experimental; ~-, Co =

## Q18[m -osf, model.

horizontal semi-variograms are shown on Fig. IV.39. A linear serni-variogram can be fitted to these deviations, at least for small distances (h u <
60m):

## The drift effect, only slightly evident in the 45 E-W directions, is no

longer visible in the N-S and vertical directions. This calls for a threedimensional model which cancels out the drift in all directions except the
E-W. An ellipsoidal model with an anisotropy ratio of a 14/a", = 20 was
adopted for the slope C 3 , cf. Fig. IVAO. The slope C 3 is equal to the radius

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279

## vector p of the ellipsoid and is given in terms of spherical coordinates (p, a,

'P) as
C3 =

. 2

au SIn

'P

2 [ 2

cos cp +a u

SIn

cp

. 2

SIn

a + COS ffJ J

196x 10-8
= . 2
2
400 [sm
. 2 'P Sin
. 2
Sin 'P cos a +
a + COS2cp]
from which it follows that

01
( 10

C)2
U

~'1

## for the E-W horizontal direction, C 3 = 14 X 10- = au;

for the 45 E-W directions, 'P = Trj4, a = 0 and C 3 = Q. 07 au ~
for the N-S and vertical direction, C 3 = au = O'OSau = O.

0'15

~ 0-10
x

005

DOl

x_--..

-)(

~.

x~-

----------

.~-=''''';~'''------'-------_ _----L
3 9 15
30

L -_ _--------J

....L..

60

90

hu

FIG. IV.39. E-W horizontal deviations from the isotropic model. ., Core samples;
x, channel samples.
E

## FIG. IVAO. Ellipsoidal model for the radial

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E~W

drift.

280

MINING GEOSTATISTICS

## To sum up, the three-dimensional model adopted to represent the

regionalization of the Cu grades of 10-ft samples is:
y(h) = 018[m.- 0'5]2 + 0'045YI(r)+ 0065Y2(r)+ C 3(a, 'P )r,

## where m > O 5% Cu is the mean grade of the zone considered.

)'1 and )'2 are spherical models with ranges al = 9 m and a2 = 105 m, the
three-dimensional vector h being defined by its spherical coordinates
(r, a, cp). This model is valid for r = Ihi < 150m and for horizontal E-W
distances less than 60 m.
This model proves to be an excellent fit to all the available experimental
figures, cf. the dashed curves on Figs. IV.35 to IV.3?
'Case Study 12. Structural analysis of Exotica Cu deposit, after Ch.
Huijbregts and R. Segovia (1973)

## The mineralization of the Exotica copper deposit in Chile is generally

supposed to be of detritic sedimentary origin, consisting of material coming
from the neighbouring massive Chuquicarnata porphyry-copper deposit.
The deposit consists of a large stratiform slab with an average plunge of
10. The total surface area is approximately 1 krn x 2 km, with a thickness
varying between 50 and 100 m.

Available data
The results from three types of sampling campaigns are available.
(i) The entire deposit was sampled by vertical diamond drilling (DDH)
on a pseudo-regular 100 m x 100 ill horizontal grid. The diamond
drill-holes were cut into constant lengths 1= 13 m and analysed for
copper. The 13-m core sample length corresponds to the height of
mining benches.
(ii) The primary production zones were sampled by rotary drilling
(RDH) on a regular 50 m x 50 m horizontal grid. These holes were
also cut into 13-m lengths.
(iii) The production benches were sampled by the same rotary drills used
for blast-holes. The drill cuts coming from these blast-holes (BH)
were approximately 14 m long. The blast-holes are located on an
irregular horizontal grid with constant density and a mean 8-m
distance between two BH. At the time this study was carried out, the
are production had just begun and the information from the BH was
very limited.
All samples were analysed for total copper (Cu T) and soluble copper
(Cu S).

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

281

The three types of samples (DDH, RDH and BH) were studied
separately. The vertical regionalization can only be studied through the
DDH, which are the only sampled long enough. On the horizontal plane
(by grading over the constant 13-m thickness of a horizontal bench), these
DDH provide data over large horizontal distances (h > 100 m.) The RDB
provide a comparable horizontal information, i.e., graded over 13 m, for
horizontal distances (h ~ 50 m). The BH provide horizontal information
graded over 14 m (14 = 13), but at smaller distances (h ~ 8 m).

## Horizontal variograms graded over 13 m

DDH samples. The horizontal semi-variograms of the Cu T and Cu S
were calculated bench by bench over 14 benches and in three directions
(the two directions of the grid and one diagonal direction). The number of
data pairs used in the calculations were too few for the bench variograms to
be significant. However, a strong proportional effect can be seen on the
nugget effect: the nugget constant increases with the mean grade of the
data from the bench considered.
For each of the three directions considered, the 14 bench serni-variograms were combined into a mean directional serni-variogram, d. Fig.
IV AI. As the three directional serni-variograms proved to be isotropic,
they were combined into a single mean isotropic serni-variograrn, d. Fig.
IV.41. For both Cu T and Cu S, the mean isotropic semi-variograrn shows
a macro-structure with a range a2 """ 200 m and a very strong nugget effect
engulfing all the structure with ranges less than the horizontal scale of
observation of the DDH (100 m).

-......::..

cu

/-~-~-~-..

10

/
/
/

/
/
/

--..:::.
Cu S

250

300
Ihl

FIG.

350

400

450

(rn)

## ry.41. Horizontal regionalization of DDH samples (graded over 13 m),

~~-,---, -----, Directional;
,Mean isotropic.

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282

MINING GEOSTATISTICS

The two other groups of samples - RDH and BH - were used to evaluate
these small-range structures. Care was taken to choose benches and zones
for which the Cu T and Cu S grades were similar to the corresponding
DDH grades (1,1 % Cu T and 08% Cu S). This precaution cancelled the
influence of the proportional effect.

## RDH and BH samples. The mean isotropic serni-variograms obtained

from the RDH samples (on a regular 50-m grid) and the BH samples
(grouped into classes because of the irregular grid) are shown in Fig. VI.42.
There is a remarkable agreement between the structures revealed by the
three types of samples. The long-distance structures revealed on the RDH
and DDH semi-variograms are almost exact extensions of the small-distance (h < 50 m) structures revealed by the BH samples. Conversely, the
BH semi-variograrn explains part of the strong nugget effect observed on
the RDH and DDH serni-variograrns.

-- ~:-..;:--~--

10

/"/"~-

Cu T

/"

/"

,
,,

/"
/"

Cu S
-~-----

-,

-,

0-5

50

150

200
Ihl

250
(rn

300

350

FIG. IV,42. Horizontal regionalization of the three types of samples (DDH, RDH,

BH).

-~-,

........ , BH.

## Horizontal model fitted

The following model of the horizontal, isotropic, point support regionalization has been considered:

## y(h)=[Co(0) - C o(r )] + C t "Y t( r ) + C 2 Y2(r ),

'V horizontal vector h with
(IV.3)
modulus r E [0,400 m].

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

283

{Co(O)- Co(r)} represents the nugget effect which appears on the grading
over the I = 13~m support as the nugget constant COl, ct. formula (lII.7).
Cj)'1 (r) and C 2 Y2(r ) are two spherical models with respective sills C 1 and
C 2 and ranges a 1 = 45 m a2 = 200 m.
The theoretical model of the semi-variograrn graded over the constant
13-m thickness is obtained from the point model by applying the practical
rules of section II.D.4, and is given by
yo(h)= C O/+CllYI(r)+C21l'2( r ),
V horizontal vector with modulus rE
[13,400 m],
(IVA)

## where the graded sills are written as

Cll = C 1[l- Yl(l, l)]

and

C 2 1 = C 2 [1 - Y2(/, I)].

## The graded parameters Co/, C II, C 2 1 are fitted to the experimental

graded semi-variograms, By way of example, the fitted parameters for total
copper are
COl

= 0,22,

Cll

= 0,10,

C2 l

= 064 (% ce T)2

and, thus, the parameters for the point model (IV.3) are

C j = Cllxl16=O12

and

C 2=C2 I Xl03=066(%CuTl.

The graded model (IVA) thus considered is shown on Fig. IVA2 and is a
good fit to the experimental curves.

## Vertical uariograms regularized over 13 m

Only the DDH have a vertical extension long enough for the construction
of vertical variograms regularized over core samples of 13 m length. The
mean vertical serni-variogram calculated from the set of DDH samples is
shown on Fig. IV.43, along with the preceding 13~m graded model. There
is a marked anisotropy between the vertical and the horizontal directions:
the vertical variability is clearly greater than the horizontal variability.

## A three-dimensional model can be fitted to the 13-m regularized vertical

variability C 3 IY3(h w ) which depends only on the vertical coordinate h.; of
the vector h = (h", h-; hw ) . The following point support three-dimensional
model is then obtained:
Vr =

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Ihl.

(IV.5)

MINING GEOSTATISTICS

284

## This three-dimensional model is the sum of an isotropic component and

an anisotropic, specifically vertical component C 3Y3(h w ) . For every
horizontal vector h (such that h.; = 0), this three-dimensional model
reduces to the preceding horizontal point model (IV.3).
1-5,-----------------------------------,
E x per rme nt cl
semi- vor ioqrorn

,/ ">---

.....

/ <.

.>

c><>
10

./

".

./

<,

......

The 0ret i ca I
vertical model

-....-.

"

I
I

..... .....

.:

/
/

I I

1/

;1
;1
;'

05

~ Theoretical

horizontal model

13

26

s;
FIG_

65

52

39

78

91

(m)

CuT.

## For a vertical vector h; (Stich that h;

model is obtained:

h.,

## and the theoretical model obtained from regularization by core samples of

length l = 13 m is, thus,
(IV.6)

## Regularization over t = 13 m reduces the isotropic component to the

preceding theoretical I3-m graded model YG(h w ) . An estimation of the
vertical anisotropic component C 31'Y3(h w ) regularized on l, is then given by
the experimental difference ['Yt (h w ) - 'Ydh w ) ] between the experimental

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285

## vertical semi-variograrn and the theoretical 13~m graded semi-variogram,

d. Fig. IV.43.
A spherical model, C3 1Y 3( h w ) , with range a31 = 65 m and sill C 31 = 061
(% Cu T)2, without nugget effect, was fitted to this experimental difference
[1'1 (h w ) - yc(h w )]. The underlying point model C 3 Y3(h w ) is, thus, spherical
without nugget effect, with range
a3 = a3/- j =

65 -13

52 m,

## cf. section II.D.2, and sill

The regularized model (IV.6) thus considered is shown on Fig. IV.43 and is
a good fit to the experimental vertical serni-variogram.
completely similar, with the same vertical anisotropy, the same values for
the nested ranges at, a2, a3 and the same nugget constant COl, but with
different values for the sills Cj, e 2 , C 3

Remark.

## The values of the previously fitted parameters COl, C j , C 2 , C 3

correspond to the well-determined mean grades (1,1 % Cu T and 08%
Cu S), which are equal to the mean grades of the deposit as determined by
the DDH. These parameter values thus define an average model valid for a
global study of the deposit. For local estimations on quasi-stationary zones
or benches, which have different mean grades, it is advisable to take the
proportional effect into account (this will not be presented here).

Case Study 13. Structural analysis of the Mea nickel deposit, after J.
Deraisme (1976)

The nickel silicate deposit of Mea (New Caledonia) consists of altered basic
rocks. The following three components can be schematically distinguished
on the vertical, cf. Fig. IV.44.
(i) A seam of yellow laterites with low, uneconomic nickel grades.
(ii) A seam of silicate ore consisting of various facies of altered basic
rocks (dunites, harzburgites). These facies are more or less nickel
rich. Unaltered inclusions of variable sizes are found within this
silicate ore and consist of hard blocks caned pure waste, that can be
distinguished both when drilling and mining (selective crushing and
selection at the face by the miner himself).
(iii) A karstic bed-rock consisting of peridotites.

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286

MINING GEOSTATISTICS

## The deposit was evaluated on a pseudo-regular, square (80 m x 80 m)

grid of vertical drill-holes. These holes were used to determine the hanging
and foot walls of the silicate seam. Kriging was used to map the isobaths of
the hanging wall and foot wall as well as the isopacks of the silicate
thickness, cf. section 1I.B.3, Case Study 2.

t.

,80m
I

I
I

____ Laterites

"

'-waste

## the Mea deposit.

The drill cores were cut into constant l-m lengths within the silicate
seam. The pure waste was removed from the softer, altered material with a
geologist's pick and the lengths of the two separated materials were
measured and analysed for density and grades (of dry are) d. Fig. IV.45.
L en g t hl I-Ii

...r~------"7'Altered

Density.
{

Of

Grodes,Nf , Co
Co and impurities

1m
Pure

:J-------~waste

L engt h, Ii
{ Density, a,

## Besides mapping of the silicate seam, the study consisted of a local

estimation on panels and a global estimation on the deposit of the four
quantities
tonnage of pure waste, Tw;
tonnage of altered ore, T A ;

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287

## tonnage of nickel metal, QA;

mean nickel grade, gA = Q AI T A
The three additive regionalized variables regularized on the I-m core
sample support that were used in the study, are
the accumulation of waste, zw(x;) = I, x d.:
the accumulation of ore, ZT(Xj) = (1 -Ii) x a; ;
the accumulation of metal, ZQ(Xi)= (l-I;)X Q; x g,;
where l, is the length of pure waste measured on each core sample, d,
is the density of this pure waste, a, is the density of the altered ore (of
length I-I,) and gi is the corresponding Ni grades of this altered ore, d.
Fig. IV.45.
The preceding tonnages are obtained by multiplying these three
regionalized variables by the volume in cubic metres.
The following regionalizations were studied, using the data available
from the vertical drill cores cut into / = J -m lengths:
(i) the vertical regionalizations through the semi-variograrns 'Y1(h)
regularized on I, d. Fig. IV.46(a);
(ii) the horizontal regionalizations through the semi-variograms 'YG(h)
graded over the constant thickness 3/ = 3 m of a horizontal mining
bench, ct. Fig. IV.46(b).
Ii
( horizontcl l

Core

If

sample _

(a)

FIG.

(b)

## IV.46. Available structural information. (a) Regularization over I; (b) grading

over 31.

Vertical regionalization
The I-m and 3-m regularized vertical serni-variograms, 'Y/(h) and 'Y31(h)
for each of the above three variables, were calculated from the 167 vertical
drill-holes and are shown on Fig. IV.47(a), (b), (c).

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288

MINING GEOSTATISTICS

## The semi-variograms of the waste and ore accumulations are remarkably

similar, and a transitive model with a range all = 4 m (regularized on
1=1 m) was fitted. The semi-variogram for the metal accumulation shows a
net increase after reaching the first range, which suggests a second nested
structure with a range around a21 = 15 m.

Support i

( o)

0-5

Support 31

--------

/
/"
/"

01

Support l

----

-----~-----

Support 3'

Support l

(cJ
2

Support 3t

--/

10

20

r (m)

## FIG. IV.47. Regularized vertical semi-variograms. (a) Waste accumulation Zw;

(b) Ore accumulation ZT; (c) metal accumulation ZQ.

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## IV. CASE STUDIES OF STRUCTURAL ANALYSIS

289

Horizontal regionalization
The three accumulation variables graded over a constant thickness of
31 = 3 m were defined by the intersections of the vertical drill-holes with
each of the 3 m thick horizontal mining benches, cf. Fig. IV .46(b). The
graded variables were defined by weighting the analyses of the I-m core
lengths by the intersection of these lengths with the bench, according to
formula (III.45).
Using these variables graded over 3/, the graded experimental semivariograms were calculated, bench by bench, and for various horizontal
directions. The bench variograms are not significant, due to the small
number of data pairs used in the calculations, and they were combined into
mean directional variograms for all benches. These mean directional
variograms show no significant anisotropy and, thus, were combined into a
single mean isotropic, horizontal variogram valid for all benches. These
three mean isotropic semi-variograrns, graded over 31, are shown on Fig.
IV.48(a), (b), (c).
On each of these experimental curves, there is a horizontal macrostructure with a range a3 = 180 m and a very prominent nugget effect with
an amplitude approximately equal to the sills of variability of the vertical 31
regularized experimental serni-variograms 'Y3/(h), d. Fig. IV.47.
Thus, a three-dimensional model can be established by adding a macrostructure with a range a, = 180 m to the structure revealed by the vertical
regionalization,

## Fitting a three-dimensional model

The theoretical, point support, three-dimensional model adopted for each
of the three accumulation variables considered, is of the type
(IV.?)

## where r = Ih I, [Co(O) - Co(r)J represents the nugget effect that appears on

the I and 31 support regularizations as nugget constants CO,l and C O,3l'
C 1'Yl(r), C 2'Y2(r ) and C 3'Y3(r ) are isotropic spherical models with respective
sills C l , C 2 , C 3 and ranges aj = 3 m, az = 14 m, a3 = 180 m.
This model was fitted in two stages.
(i) The parameters of the small-distance structures, i.e., the nugget
effect and the first two structures C j 'Yl and C 2 'Y2 were fitted using the
vertical regularized serni-variograms. The influence of the macrostructure C 3 'Y 3 can be neglected for these small distances (r ~ 25 m).
(ii) The parameters of the macro-structure C 3 1' 3 are estimated from the
horizontal graded serni-variograms. Over large distances (r;;?: 80 m),

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MINING GEOSTATISTICS

## the short-distance structures (nugget effect + C 1 'Yt + C 2'Y2) show up

as a single nugget constant equal to

Cb,31

## C O,3 1 + CUI + C2,31

can then be written
'YG(h) = Cb,31 + C 3.31'Y3(r ),

\:1,= Ihl~80m_

## The nugget constant C O,3 1 is obtained from the preceding study of

the short-distance structures, while the parameters C 3 . 3 1 and a3 are
estimated from the experimental semi-variogram graded over 3/.
0-5,-------------------------------.,
l a)
-~-~----------~----

-----

Suppor t 3/

02

01

0.--

---------_.~,,_._-----------------'

02r~lb)-~-,

"

--------------~

- --

Support 3/

01

o1'-

.,

(c )
2
Support 3t
I

Of

10

100

200

300

FIG. {VAS.

## Graded horizontal serni-variograms. (a) Waste accumulation Zw;

(b) ore accumulation ZT; (c) metal accumulation Zoo

Remark Strictly speaking, the theoretical model C 3.31'Y3(r ) of the macrostructure, estimated from only the horizontal graded semi-variograrns,
describes only large-distance (r ~ 80 m) horizontal regionalizations. This
horizontal macro-structure is then extended to three dimensions by adopting an hypothesis of structural isotropy to give the isotropic term C 3'Y3(r ) of
the model (IV. 7).
This hypothesis has little effect on the vertical study as the vertical
thickness of the Mea silicate seam never exceeds 35 m and for such small

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291

## distances the influence of the macro-structure C 3 Y 3 is very weak, if not

negligible.

To sum up, nested models of the type (IV. 7) have been adopted for the
three-dimensional point support regionalizations of the three accumulation
variables. The values of the sills, regularized on I and 3[, are given on Table
IV.3. The corresponding theoretical models, regularized by core samples
of lengths [ and 3/ and graded over 31, agree remarkably well with all the
available experimental curves.
TABLE

Waste accumulation
Ore accumulation
Metal accumulation

## IV.3. Fitted sill values

(a) Support I

CO. I

C Ll

C:u

C3 .1

L: C.I

024
012
063

022
0105
079

0
0
061

0184
007
070

0644
0295
273

(b) Support 3l

Waste accumulation
Ore accumulation
Metal accumulation

C O3 1

CUI

C2,J1

C 331

L CD J

008
0-04
021

0'145
0-07
052

0
0
057

0185
0-07
070

041
018
20

## Logarithmic model or nested spherical model

One drawback of the previous structural models is the large number of
estimated parameters, five for the accumulation of waste and are and seven
for the metal accumulation, namely the four sill values Co, C t , C2 , C 3 and
the three ranges al, a-, a3.
The roughly geometric progression of the three ranges a 1> a2, G3 suggests
a logarithmic structural model for the metal accumulation, d. section
III.B.2, the subsection entitled "Logarithmic model or nested models?"
The advantage of such a logarithmic model is that it depends on only two
parameters rather than the seven parameters of the nested spherical
model.
The logarithmic model with nugget effect is expressed as
y(h) = [Co(O) ~ Co(r)] + Clog r,

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MINING GEOSTATISTICS

292

## for the point support;

1'31(h) = C O,3 1 + C[ log

;1 +~l

## for the 31 support; cf. section ILD.2, regularization of the logarithmic

model.
For the two supports land 3/ considered here, these models become
r
1'1(h)= [CO) + 15C]+ Clog!,

'Vr=31;

r
1'31(h) = [CO,3i + C(l' 5-log 3)] + C log I'

'r/r';331;

thus
1'1(h)- 1'31(h) = CO,I - C O,3i + Clog 3
=

2eO,3 1 + Clog 3,

## Consequently, if the logarithmic model is adopted, the experimental

serni-variograms for metal accumulation plotted on logarithmic paper
would be approximated by straight lines with slope C. The two lines
corresponding to the two supports I and 31 should differ by a translation of
modulus 2CO ,3i + Clog 3.
On Fig. IV.49, the two parameters fitted to the experimental curve
regularized on 31 are C O3i = 021 and C = 036.
Over small distances (r:s;: 25 m), there is a very good agreement between
this logarithmic model and the experimental curves. On the other hand, the
logarithmic model slightly overestimates the variabilities at large distances

-- -

Support I

-- -

2
Su pporl 31
F------,~,--~~--+-----

10

rlt

## FIG. IV,49 Fitting a logarithmic model to the regionalization of metal accumulation.

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293

## (r ~ 80 m). The nested structure of spherical models (IV.?) is preferred in

practice, essentially for reasons of homogeneity with the models chosen for
the other two variables used, accumulations of waste and ore.

## IV.F. GRADE CONTROL DURING PRODUCTION

The two following case studies show how the characterization of a spatial
variability through the variograrn model allows the prediction and, thus,
the control of the production grades. These two case studies can also be
considered as examples of practical applications of the two geostatistical
variances: the estimation and the dispersion variances.
Case Study 14. Prediction of the grade of planned production from
several stopes

## Consider a deposit the daily production of which comes from N blocks

{B i , i "= 1 to N} of same size V. The true mean grade Oi of the block B; is
estimated without bias by the estimator 07 with estimation variance
=
E{[O; - 0[]2}.
For a day j, the total production is programmed from N mined blocks.
Each block B, provides PH tonnes with a true mean grade t;j estimated by
of block Bi; ct. Fig. IV.50. The elementary
the global estimator

U';,

ot

Block 8i'

SloCk 8;
True qr o de, 8,Estimated grade,!lt

Ta"noq~, Pri

Tonnage. p;,;

--"

tJ,*

~t

To to I tannoge,~-

Day

FIG.

{

T;

Ii*

## IV.50. Production planned from several blocks.

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t r-".j

294

## MINING GEOST ATISTICS

tonnages Pu are calculated so that the predicted mean grade T] of the total
production P, of the day j satisfies the sales or milling constraints. Thus, for
a day i. the following relations hold:
N
= 1 Pij;

P, = L

## total tonnage produced,

L pijeU Pi;
true mean grade actually recovered, T, =.L Pii . lii/ P;
Tj

The object of the study is to control the difference between the predicted
grade Tj and the true grade actually recovered. This is done by means of the
estimation variance:
CT~i = E{[ t,

Tj ]2}.

Suppose that the N blocks B, are within the same stationary regionalization characterized by the grade expectation m. The various grades O!, fh
tij thus have the same expectation m and the non-bias for the prediction of
E{7j - Tj} =

L PijE{l ij - ef}/ P, =

O.

## If the distance is sufficient between the N simultaneously mined blocks,

then the two errors, (tij - (1) and (ti'! - 8~), for i 01:- i', can be considered as
independent and, th us,

CTt = L pt E{[tjj -

eil 2 } / Pi.

## The elementary error can be written as

ti j

et =

(tif -

OJ)+ (8 i

--->

en

and, thus,

The fluctuation (tij - Oi) of the elementary grades Iii within the block B I can
be assumed to be independent of the estimation error (01 - On of this
block, and, thus,
E{(ll; - Oi)(ei

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(1)} =

o.

295

## If the exact location of the tonnage Pu in block B, were known, the

extension variance of the grade tii of this tonnage Pi! to the grade I)i of
block Bi; i.e., the term E{Uii - t9 i i}, could be calculated. In practice, it is
sufficient to consider the mean value of this term E{(tii - Oi )2} when the
tonnage Pu is taken anywhere within block B; i.e., the dispersion variance
of an average production unit Pi in block Bi:

1
2
2
V JE{(fij-Oi) }dx=D (PdB i ) ,
8,

## cf. formula (11.33).

If the block B, is mined in k days, then the average production unit Pi
can be expressed in terms of tonnage as
1
Pi = k

.L p.;
J=1

As for the term E{(Oi - Or )2}, it is, by definition, the estimation variance
of the mean grade of block B; If this block has been estimated by
kriging, then
is simply the kriging variance of the block given by the
kriging system of equations.
Finally, the estimation variance of the mean grade TI of the total production PI of day j is written

a;,

0";,

<rEi =

2[ 2 / )
2]
Py1:!;::-1 Pu D (Pi B, +a c , .

(IV.8)

This variance thus depends not only on the dispersion of the grades of the
production units Pi in each block B, but also on the quality of the estimators of the mean grades of these blocks B;
Remark 1 If the programming of the mining for a day j is based on the
estimators tt of the grades tij of each production unit rather than on the
estimators
of the blocks or stapes B i , then the previous formula reduces
to the sum of N independent elementary estimations:

ot

1;'

2
aE i =~ L,

Pi

2
2}
1*
PiiE{(tii
ij )

(IV.9)

;=1

This case corresponds, for example, to an open-pit deposit for which the
mining program is based on blast-hole analyses which provide the estimator t~ of each production unit.
Remark 2 However, it sometimes happens that it is not possible to
directly estimate the grades (ii' either because the exact location of the

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MINING GEOSTATISTICS

## corresponding production unit Pii is not known (e.g., extraction by block

caving of Pij tonnes from block B i ) , or because the necessary data is not
available. These two cases will be studied later in the numerical applications.
In such cases, it is obvious that the grade control has an absolute limit
imposed by the dispersion of the production grade units in the blocks, and
obtained by equating the estimation variances IT;; to zero in formula (IV.8)
(i.e., assuming that the true grades Oi of blocks B, are perfectly known).
The minimum value of the estimation variance a~j of the daily grades is,
thus,
(IV.I0)
If the daily production units Pu extracted from each block B, are more or
less equal, then

and, thus,
(IV. 11)

## Remark 3 If it is desired to improve the quality of the control of daily

production grades (i.e., to decrease on average the deviations between the
predicted grades rj and the actual recovered grades T i ), formula (IV.8)
must be used, either to decrease the estimation variances IT~i so that, with
the grades of the blocks B, being better estimated, the production planning
is on a much surer basis, or, preferably, to reduce the average dispersion
2
variance 0/ N)LD (pJ B i ) by increasing the number N of stapes - the
compensation effect of the estimation errors (t ii - ot) between the different
stapes is then increased.
Numerical application 1 Consider a massive copper deposit mined by
block-caving with blocks of constant dimensions 60 m x 60 m x 180 m (e.g.,
EI Teniente deposit in Chile). The estimation variance of such blocks by
the available holes is calculated to be (T~ = 0003 (% CU)2, and, with a
mean grade m = 1 % Cu, the mean relative standard deviation is o: e/m =
0-055 =55%.
The daily production of 65 000 tons of are is obtained from N = 8
blocks. This production is assumed to be equally distributed among the
eight blocks, providing an average daily mining unit in each block of

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297

## approximately 8125 tons, corresponding to a mean support p of dimensions 60mx60 mxO75 m.

2
The mean dispersion variance D (P/ B) is calculated from the threedimensional serni-variogram model of the grades. This results in
D 2 (P/ B) = 0026, i.e., a mean relative standard deviation of dispersion
D(P/B)/m =0-16= 16%.

formula (IV.8):
1

2
(TE

0-029

## i.e., a relative standard deviation CTE/m = 0-06 = 6% and, thus, when a

Gaussian standard is adopted, in 95% of cases the actual grades T, will
differ from the predicted grades Tj by less than 2CTE/m = 12%, i.e.,
T;ETj (1012).
If the grades of the blocks B, were perfectly known, i.e., with an almost
unlimited sampling budget, this variance could be brought to its lower limit
(IV. 11):

## which amounts to a maximum relative decrease of (363-325)/325 = 0116_

less than 12%. For a better daily grade control in this deposit, it is
more efficient to make mining flexible by increasing the number N of
blocks being simultaneously mined, than spending more money on extra
sampling.
Numerical application 2 Consider now a subhorizontal deposit mined by
open-pit with 5-m benches (e.g., a New Caledonian silicate nickel deposit).
constant thickness of the bench is characterized by an isotropic model with
a range a = 300 m, a sill C = 0- 3 (% Ni)2 and a nugget constant, defined on
the support I = 5 m of the grading, COl = 0- 2 (% Nii. The mean nickel
grade over the deposit is m = 2% Ni.
A direct proportional effect in m *2 is observed, which means that,
locally, in a production zone with an estimated mean grade m *, the relative
semi-variogram 'Ylm*2 is a spherical model with the parameters range
a = 300 m, sill C/ m 2 = 0-3/2 2 = 0-075, nugget constant C otlm *2 = 0-05.
The deposit has been evaluated by vertical drills on a regular square grid
of 60 m x 60 m. The workings on which each shovel is located can be

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MINING GEOSTATISTICS

## approximated to this grid unit, i.e., to a panel B of dimensions 60 m x

60 m x 5 m, cf. Fig. IV.51. A preliminary estimation of the mean grade ()i
of each panel B; is obtained by taking the mean grade Or of its central
drill-hole (classical but not optimal estimation procedure by polygons of
influence), the estimation variance (7"; is, thus, given by chart no. 7, and the
relative estimation variance is

(T;'/ ei 2

0056,

V the panel i.

82.,2'''10 Ni
x

'%
V//

"/.

%%

I'

;5

------ x

------x

1--------1

60

8 3,1 5"/" Ni
1

///

2700 t cloy -I

+ 2700

///

1 - --

+900tdoy-1

t doy-I

## It is desired to produce 6300 t of ore per day with a grade average of 2%

Ni from three shovels located in workings with different mean grades, d. Fig.
IV.51.
The following situation is typical of any given day i.

(i) Shovel no.1, which has a small capacity (900 t day -1) is located in
workings B 1 with a rich mean grade estimated to be of = 35 % Ni.
This shovel moves a unit of size Pt = 20 m x 5 m x 5 m each day.
(ii) Shovels no. 2 and 3 have thrice the capacity (2700 t day") of shovel
no.1. Shovel no.2 is located in a workings of average grade (O! =
2% Ni), while shovel no. 3 is in a poor grade site (ot = 15% Ni).
Each shovel moves a unit of size P2 = P3 = 60 m x 5 m x 5 m per day.
The predicted mean grade of the 6300 t day -1 so produced is
ok

T' =

6300

.
200 % Ni.

## The dispersion variances D 2 (pJ B i ) of the grades of units Pi in the

working sites B; are given by the standard procedure of the difference

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## IV_ CASE STUDIES OF STRUCTURAL ANALYSIS

299

between the two auxiliary functions P(Pi) and F(B;) defined on the two
rectangles Pi and B; cf. section I1.E.2, formula (11.82), e.g.,

(p d 8 1 ) =

"1(8[,

## F(60; 60)~ F(20; 5).

(For the calculations of the values D (pJ B i ) , note that as the supports Pi
and B, are considerably larger than the support of the 5 m-Iorig core
samples, the influence of the nugget effect disappears and the relative
spherical model with range a = 300 m and sill elm 2 = 0-075 is sufficient.)
Immediate calculation and chart reading provide the following values:
D?(PI/ Bd = 3-5 2 (0-0117 ~ 0-001 95) = 0119 (% Nif,
D 2 (pzl B2) ="2 2 (CHH17 ~ 0-0(338) = C).(l33 (% Ni)2,

## D\PJI B J) = 1 52 (0-0117 ~ 0-0(338) = C).(H9 ('Yo Nit

Thus, from formula (IV.S), the estimation variance of the daily grade
production is

(T~
I

## =_1_J900 2 (0-119+0'686)+2700 2 (0,0033+0,224)

6300
+ 27002 (0-019+ 0'126)] = 0-0903 (% Nif,

## i.e., a standard deviation (TE = 030% Ni.

Thus, adopting the Gaussian standard, the mean grade T, actually recovered each day will, in 95% of the cases, fluctuate in the interval
[2 0-6] % Ni around the predicted value
= 2 % Ni.
The mediocre result of this grade control is due to the rather hasty
estimation of the panel grade 0; by the grade of the central drill core
(polygon of influence method). Using the same data (drills on a regular
60 m x 60 m grid), simple kriging would, without any additional cost,
reduce the estimation variance
by half, and daily grades could then be
controlled with a variance of
= 0-0511 (% Ni)2, i.e., a standard deviation of (TE, = O' 23% Ni, which represents a noted improvement.
The lower limit of the variance (T~'i corresponding to the perfect knowledge of the mean grades ()i of the working sites is given by formula (IV.IO):
j

r:

a;;

at

## The interval of fluctuation is, thus, T, E [2 O-22j % Ni.

In practice, if the grade control is deemed insufficient in the interval
T, E [2 0-46]% Ni provided by kriging and the data on a 60 m x 60 m grid,
it is advisable to establish some sort of balance between the cost of
reorganizing the daily production plan and the cost of additional sampling.

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MINING GEOSTATISTICS

300

## Case Study 15. Dispersion of production grades

Consider again the New Caledonian type of nickel silicate deposit considered in the latter numerical applications of Case Study 14.
In the present example, only the two large capacity shovels are retained,
providing a weekly production of 5400 X 6 = 32 400 t, which can be
achieved in either of the two following manners, cf. Fig. IV.52.

1. The two shovels can advance in line on the same front. The weekly
production unit 5 can be approximated by a rectangle with horizontal
dimensions 360 m x 5 m. The height of the bench is 5 m, and the
average density of the dry ore is 18.
2. The two shovels can advance in parallel on two fronts, getting
progressively farther away from each other. The weekly production
unit 5 thus consists of two rectangles 51 and 52 with equal horizontal
dimensions 180 m x 5 m.
In the first case, the annual production (1,62 million tonnes over 50
weeks) can be approximated by a rectangle S with horizontal dimensions
720 m x 125 m; in the second case, it will be approximated by a rectangle S
with horizontal dimensions 500 m X 180 m.
Taking the weekly production as the homogenization unit, it is desired to
2(sj
calculate the dispersion variance D
S) of the weekly mean grades
actually obtained in one production year, using both types of shovel
thickness (5 m) of the benches is stationary with expectation m = 2% Ni
and characterized by an isotropic spherical model with a nugget effect,t
range a = 300 m and sill C = 03 (% Ni)2
The dispersion variance is given by the standard formula:

## D 2(sjS) = y(S, S)-y(s, s).

The first term y(S, S) is simply the auxiliary function F defined on the
rectangle S. Chart no. 4 gives
1'(S, S) = F(720; 125) =

ex 0754 =

0226

## for method 1,and

1'(S, S)=F(500; 180)=

ex 0700 = 0210.

for method 2
t There is no need to give the value of the nugget constant, as the nugget effect vanishes when
regularized on the previous supports sand S of the weekly and annual productions, d.

formula (IlLS).

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301

## Similarly, for the first method, the term ii(s, s) is given by

ii(s, s) = F(360; 5)=F(360) =

ex 0514 = 0-154.

The calculation of the term yes, s) for the second method is a little more
involved. As 5 is the union of the two rectangles 51, 52, it follows that

## Y(s" 5d = F(l80; 5) = F(l80) =

ex O'289 = 0087.

The term y(s}, 52) depends on the distance 2x between the two working
sites 51 and 52, cf. Fig. IV.52(b). The production year consists of 50 weeks
(i = 1 to 50) and, during week i, the distance between the two sites is
5(2i -1) metres. Approximating each working site by its median line of
180 m length, the term Y(Sl; 52) for week i becomes

720
I
I

,I
5:--I

5mL~L.......---_1_125m
I.

.1

2x 180 ~ 360 m
500 rn

'I
5

180 rn

o I~~-----+----FIG. IY.52. Two methods of shovel advancement. (a) Two shovels in line;
(b) two shovels in parallel.

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MINING GEOSTATISTICS

302

d. the definition (II.79) of the auxiliary function atL; l). On average, over
the 50 weeks of production, the term 1'(51,52) can be written as

Yes!> 52) =

50

ex 084 =

0252

## and, thus, for the second mining method,

1'(5,5) = ~[O-087 + 0,252]

0-170.

D 2(s/S)= 0072,

i.e., D(s/S)/m

013 = 13%

## for method 1, and

2(s/S)=

0'040,

i.e., D(s/S)/m

= 0,10= 10%.

for method 2
Configuration no. 2, with the two shovels in parallel, thus provides a
significant decrease in the dispersion variance.

Remark The knowledge of the dispersion over one year may not be
sufficient to control the variablity of grades. For example, it is important to
ensure that weekly mill-head grades will be sufficiently regular over a short
period of time (e. g.. 1 month). For this purpose, the dispersion variance
2
D (5/ M) of the weekly grades within a month's production can be calculated. Krige's additivity relationship (11.37) gives
2

D (sl S)

## = D 2 (s/ M) + D 2(M/ S).

Thus, the distribution of the total annual variability D 2 (s/ S) between the
short-term variability D\s/M) and the long-term variability D\M/ S) can
be evaluated.
If it is desired to further increase the control of variabilities, the deposit
can be simulated, i.e., the spatial regionalization of the grades can be
simulated, ct. Chapter VII. A simulation of the mining method can then be
applied to this simulated deposit to give an idea of the fluctuations of
grades from one period to another, e.g., day per day, month to month, etc.,
d. the case study given in section VII.B.3.

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v
Kriging and the Estimation
of in situ Resources
SUMMARY

The in situ resources of a deposit are distinct from the recoverable reserves which
can be mined within some given technical and economical context. The object of
this chapter is the estimation of the in situ resources prior to the application of any
cut-off; the estimation of recoverable reserves will be treated in Chapter VI.
Section V.A introduces the technique of kriging, which provides a minimumvariance unbiased linear estimator of the mean characteristic of a block of any
given geometry. In the case of non-stationarity, several non-bias conditions are
required and this leads to the technique known as "universal kriging", or unbiased
kriging of order k. When a group of variables are correlated, the cokriging technique can be used to estimate any variable (e.g., silver grade) from the data
Section V.B deals with applying kriging techniques in practice and demonstrates
that it is possible to krige thousands of blocks with reasonable speed and cost. Some
typical subroutines are given along with numerous examples of kriging plans which
can be adapted to the estimation of a large range of deposits of variable
morphology.
Section V.C deals with the global estimation of a deposit or a large zone. The
global estimator is obtained by combining the various local kriged estimators of the
blocks or units which make up the zone. When calculating the global estimation
variance, a distinction is made between the geometrical error involved in evaluating
the limits of the mineralized surface or volume and the quality error due to the
estimation of the mean characteristics (e.g., grade) within a zone of fixed limits.
These two errors are then combined to provide the estimation variance attached to
global estimators, such as quantities of ore and metal and the corresponding mean

303

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MINING GEOSTATISTICS

## V.A. THEORY OF KRIGING

The problem

The problem of local estimation is to find the best estimator of the mean
value of a regionalized variable over a limited domain, the dimensions of
which are small compared to the dimensions of the quasi-stationary
(homogeneous) zones of the deposit, e.g., the mean grade of a block
located well within a zone of homogeneous mineralization. Local estimation differs from global estimation in that global estimation considers
distances larger than the limits of quasi-stationarity and, thus, sometimes
engulfs various heterogeneous mineralizations.
The available information used for local estimation within a quasistationary zone is generally made up of a set of data (e.g., n core grades)
and structural information (e.g., the variogram model characterizing the
spatial variability in the studied zone).
Kriging is a local estimation technique which provides the best linear
unbiased estimator (abbreviated to BLUE) of the unknown characteristic
studied. This limitation to the class of linear estimators is quite natural,
since it means that only the second-order moment of the RF (i.e., the
covariance or variogram) is required, and, in general, it is possible in
practice to infer the moment, cf. section II.B.2.
Of course, when the available structural information includes more than
the simple knowledge of the second-order moment, then non-linear estimators - more precise than simple kriging ~ can be defined, ct. Chapter
VIII, "Introduction to Non-linear Geostatistics". The present chapter will
be limited to linear estimators only.
V.A.1. Equations of krigIng

Let Z(x) be the RF under study. Z(x) is defined on a point support and is
second-order stationary, with
expectation,

E{Z(x)} = m,

## a constant m which is generally unknown;

centred covariance, E {Z (x + h)Z (x)} - m 2 = C(h);
variogram, E{[Z(x + h)- Z(X)]2} = 2y(h).
Either of these two second-order moments is supposed known. When only
the variogram exists, then the RF Z(x) is intrinsic only.
The estimation of the mean value
Zv(xo)=2-

## over a domain V(xo) is required.

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f
V(xo)

Z(x)dx

305

## The experimental data to be used consist of a set of discrete grade values

{Zm a = 1 to n}. These grades are either defined on point or quasi-point
supports, or else they are the mean grades Zu" (x cr ) defined on the supports
Vcr centred on the points x a ; the n supports may differ from each other.
Note that under the hypothesis of stationarity the expectation of each of
these data is m: E{Za} = m, Va.
The linear estimator Z~ considered is a linear combination of the n data
values Z~ = 2::=1 A""Z"". The n weights Ao< are calculated to ensure that the
estimator is unbiased and that the estimation variance is minimal (the
estimator Z~ is then said to be optimal).

## Non -bias condition

To obtain a zero error [Zv - Z~] in expectation (i.e., an average over a

## great number of similar estimations) it is enough to impose the condition

\"'1
L..a~l Aa = 1, an d ,t h us,
E{Z~} =

LA

m = E{Zv},

## Minimum estimation variance

The estimation variance EUZv - Z~ ]2} can be expanded as follows:

E{[Zv - Z~ f}

},

with

E{zt}=~f
V

## dxf E{Z(x)Z(x')}dx'=C(V', V)+m

1
E{ZvZ~d=LAa.

'"

VV a

J dx f
V

V",f-{3

2
E{Z(x)Z(x')}dx'=LAaC(V,va)+m,

Va

E{Z~2}=LLA"AB_l_,
(); (3

u"

'"

dXf E{Z{x)Z(x')}dx'
U(3

'" f3
2

## The terms m are eliminated and, thus, we have

E{[Zv -Z~ ]2} = C(V, V)-2

I
a

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## AaC(V, v",)+ L L AaA/3C(va, VI3)'

c< {3

MINING GEOSTATISTICS

306

(The standard notation C(V, Va) designates the mean value of the covariance function C(h) when the two extremities of the vector h independently
describe the domains V and Va:, respectively.)
The estimation variance can, thus, be expressed as a quadratic form in
Aa , As, and is to be minimized subject to the non-bias condition L" AD: = 1.
The optimal weights are obtained from standard Lagrangian techniques by
setting each of the n partial derivatives a[E{[Zv - Z~ ]2}_ 2M If] ,1/31/ (lA",
to zero. This procedure provides a system of (n + 1) linear equations in
(n + 1) unknowns (the n weights Aa and the Lagrange parameter J.t) which
is called the" kriging system".
tl

i3~1

A13 C (v a ,

VI3)-

J.t

= C(v"" V),

Va = 1 to n,

(V. I.)

written as
tl

## (T~ = E{[Zv - Z~ ]2} = C(V, V)+ J.t -

A",C(va. V)

(V.2)

a = 1

This kriging system can also be expressed'[ in terms of the semi-variogram function y(h), particularly when the RF Z (x) is intrinsic only and the
covariance function C(h) is not defined:

V"

L A13 =

~ 1 to n, }

(V.3)

13

and

(T~ =

L Aa)i(v""

## V)+ J.t - )i(V, V).

t In practice, even when the covariance is not defined, the system (V.l) written in terms of

covariance is preferred for reasons of programming efficiency. For this purpose, it is enough
to define the "pseudo-covariance" C(h) such that YCh)=A-C(h), the constant A being
any positive value greater than the greatest mean value y used in the kriging system (V.3).
The non-bias condition I" A" = 1 allows the elimination of this constant A from the system
(V.3), which then becomes a (V. I)-type system written in terms of the pseudo-covariance
C(h).

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## V THE ESTIMATION OF IN SITU RESOURCES

307

Matrix form
Both kriging systems can be expressed in matrix form as

[K] . [A I = [M2]

(VA)

[A]
and, [A

= [Kr 1 . [M2],

## r being the transposed matrix of rA],

O"k = C( V,

V) - [A It. [M2],

where the matrix of unknowns fA] and the second member [M2] can be
written as two column matrices:
C(VI,V)

Al
A2

[A] =

Aa

An
-/.J-

[M2] =

('(va, V)

C(v", V)
I

## The first member [K], or "kriging matrix" is written as:

[K] =

Main diagonal
Note that the kriging matrix is symmetric, i.e.,

Va, (3.

## Seven important practical remarks

Remark 1 The existence and uniqueness of the solution. The kriging
system (V. I) has a unique solution if and only if the covariance matrix

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MINING GEOSTATISTICS

[C (Va, VI3)] is strictly positive definite and, thus, necessarily has a strictly
positive determinant. For this purpose, it is enough that the point covariancet model C(h) used is positive definite and that no data support
v'" coincides completely with another one. In fact, Va E Va' entails
that ct; v(3)= ci; v(3)' '1(3, and the determinant IC(v"" Vp)1 is, thus,
zero.
This condition for the existence and the uniqueness of the solution of the
kriging system thus entails that the kriging variance (V.2) is non-negative,
d. section II.A.3, relation (II.8).
Remark 2 Kriging, which is an unbiased estimator, is also an exact
interpolator, i.e., if the support V to be estimated coincides with any of the
supports Vc< of the available data, then the kriging system provides:
(i) an estimator Z~ identical to the known grade Z; of the support
V",= V;
(ii) a zero kriging variance, (T~ = O.

## In cartography, it is said that "the kriged surface passes through the

experimental points". Not every estimation procedure has this property,
especially procedures using least square polynomials.
Remark 3 The expressions of the systems and the kriging variances using
the notations C and l' are completely general,
(i) whatever the supports Vet> V{3 of the data and the support V to be
estimated, some data supports may partially overlap, Vet n V{3 ~ 0,
but for {}' ::P {3 it is imperative that V a ?!= Vj3 - some data supports may
be included in the volume V to be estimated, Vet C V;
(ii) whatever the underlying structure characterized by the model C(h) or
y(h), the structure may be isotropic or anisotropic, nested or not.

Remark 4 The kriging system and the kriging variance depend only on
the structural model C(h) or y(h) and on the relative geometries of the
various supports Va, V(3, V, but not on the particular values of the data Za.
Consequently, once the data configuration is known and before drilling is
undertaken, the kriging system can be solved and the corresponding
minimum estimation variance can be forecast. Thus, the kriging variance
can be used to balance the cost of a new drilling campaign with its forecast
utility (new data would decrease the estimation variance, thus providing
narrower confidence intervals) cf. O. Bernuy and A. Journel (1977).
+If the

## serni-variogram is used, then -y(h) must be conditional positive definite.

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309

Remark 5

## The kriging matrix [K] depends only on the relative

geometries of the data supports Va, v(3 and not at all on the support V of
the domain to be estimated.
Thus, two identical data configurations would provide the same kriging
matrix [K] and it is then enough to take the inverse matrix [Kr l only once.
The two solution column matrices [A] and [A '] are then obtained by taking
1
the products of the single inverse matrix [Kr and the respective secondmember matrices:

[AI=[Kr1fM21

and

{A']=[K]1.[M2'].

## In practice, this amounts to using a program for the resolution of linear

systems with two or more second members (such as subroutine GELG in
the IBM package).
If, in addition to identical data configurations, i.e. [K] = [K'], the two
panels to be estimated have the same geometry, then [M2] = [M2 '] and the
resulting kriging weights are identical: [A] = [A ']. It is then enough to solve
the kriging system once, which results in a considerable saving of computing time. For this reason, when evaluating the same stationary or quasistationary phenomenon, it is advisable to ensure, as far as possible, that the
data configurations are systematic and regular. In this way, a single kriging
plan that can be repeated identically over all or part of the deposit, can be
used.
Remark 6

The kriging system and the kriging variance take into account
the four essential and intuitive facts, previously mentioned in section
II.B.2, remark 4, which condition every estimation. These are as follows.
(i) The geometry of the domain V to be estimated, expressed in the
term y( V, V) in the expression of the kriging variance Q"~.
(ii) The distances between V and the supports Va of the information,
expressed by the terms Hv w V) of the matrix tM2].
(iii) The geometry of the data configuration as expressed by the terms
y(vn, V(3) of the kriging matrix [K]. The accuracy of an estimation
depends not only on the number of data but also on their configuration in relation with the main features of the regionalization as
characterized by the structural function y(h) in the various terms
Yev{O v(:J).
(iv) The main structural features of the variability of the phenomenon
under study as characterized by the semi-variogram model y(h).
Consider the estimation in two dimensions of the panel V by the symmetric configuration of the four data A, B, C, D as shown on Fig. Y .1. The
underlying mineralization shows a preferential direction u of continuity

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310

## which appears on the anisotropic semi-variogram y(h", h~) as a slower

variability in the direction u. The kriging system thus gives a greater weight
to the data Band D, although they are at the same distance from V as A
and C.
A

Dx

xB

~c

Direction of continuity
FIG. V.l. Influence of the variability structure on kriging.

## Remark 7 The influence of a nugget effect. In practice, a partial nugget

effect is very often found, i.e., a nested structure, sum of a nugget effect,
and a macro-structure:
C(h) = Co(h) + C 1(h).

The mean values C(v"" VtI) appearing in the kriging system are written,
according to formulae (111.6), as

if Vf3 c v"',
if v'" and vl3 are disjoint,
where A is a constant given by A = SCo(h) dh. (It is recalled - cf. section
III.A.3, formula (IIL7) - that this constant A appears in the nugget
constant value Co~ = AI v of a semi-variograrn regularized over the
support v.)
The kriging system (V.l) can then be written as
A", [AI v",

+ C\(v", va)] + 2:
{3r''''

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Af3 C\

(va ,

va'l

(V.S)

if all supports

Va, V/3'

311

## V are disjoint, and

If all supports v", V/3, V are not disjoint, then each of the terms C of the
kriging system must be calculated.
A case often found in practice is that of a configuration of data disjoint
among themselves, some of which (v c, ) being located within the support V
to be estimated:
Va,C
Va

V,

nV =

but

Vo;,n V/3
Vo;

= 0,

n V{3 =

0,

v,
(V.6)

13~0;

Va

,1/3C 1 (Vo;,

## V.a)-/-L = Cl(V a, V),

Va,

A~ = 1

and

In mining applications, the support V is very often much larger than the
data supports Vo;, V,,(, and the term A/ V is negligible when compared with
the terms Aio; or A/Va" When all the data supports Va, V(3 are disjoint,
system (V.S) is obtained.
If the semi-variogram l'(h) = {Co(O) - Co(h)} + 1'1 (h) is to be used instead
of the covariance C(h), it is enough to substitute (-Yt) for C t in the
preceding systems (V.S) and (V.6).
Note in system (V.S) that the influence of a nugget effect with nugget
constant COli = A/ v affects only the elements of the main diagonal of the
kriging matrix as an additional term A/ Va'

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MINING GEOSTATISTICS

## The influence of a pure nugget effect

If the macro-structure C, (h) disappears, leaving only the pure nugget
effect Co(h), the kriging system (V.S) becomes

A"A/ v" -- u. = 0,

\;fa,

I AJ3 = 1
{3

and, thus,

= Va/~ V(3,

Aa

va,}
(V.?)

I-L

= A /~

V();} Va,

Ve

## V are disjoint, and

CT~ = A [ I

+V

V{3

J.

{3

The weights for the n available data values Zo. are proportional to the
volumes va of their respective supports.
Moreover, if the n supports Vc< are equal (e.g., n core samples of
constant length and volume v), then the system (V.?) becomes
Ao.=l/n,

Va = 1 to n,

if V v. }

The n weights Ac< are all equal and the kriging variance (T~ = COl)/ n is
simply the variance of estimation given by the statistics of spatially uncorrelated variables. The nugget constant COt; is the a priori variance of the
support v data, according to formula (IlL!!):

COt; = A/ v

D (v/ OO )

## Thus, for a pure nugget effect, we have the following.

(i) All data having the same support v (disjoint from V) have the same
weight whatever their distances from the volume V to be estimated.
The close data no longer screen the influence of the more distant
data and the nugget effect is said "to remove the screening effect".
(ii) Within a given neighbourhood of quasi-stationarity, it is no longer
possible to make different local estimations: all estimators Z~ of all

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313

## panels V within the neighbourhood are equal to the unique mean

value LI>: Za./ n of the n data available in this neighbourhood. The
nugget effect is said" to smooth the estimation".
V.A.2. Taking accou nt of a drift

## In all the kriging equations presented heretofore, it is assumed that the RF

'z (x) is stationary or at least quasi-stationary over a given estimation
neigh bo urhood .
In practice, it may be known that a drift (non-stationary expectation
E{Z (x)}) exists within certain zones of the deposit and that there is not
enough data available in these zones to allow quasi-stationary estimations
(by considerably reducing the neighbourhoods of estimation). It then
becomes necessary to take the drift into account when estimating these
zones by kriging. For example, consider the mineralized seam on Fig. V.2,
which thins out towards one end. We want to estimate the mean thickness
(or the ore tonnage) of a panel V located on the fringe of the seam, by
drill-holes all located further along the seam.

Pone I to be
estimated

"-

Available
drill-holes

## FIG. V.2. Drift of a seam thickness.

Kriging under a stationary hypothesis, like any other "stationary" estimation procedure such as inverse-squared distances weighting, will inevitably overestimate the real mean thickness of the panel V. For this reason, it
is essential to take the drift of the thickness intoaccount in the direction of
thinning of the seam.
The following method of "unbiased kriging of order k" (also called
"universal kriging") provides an unbiased linear estimator which takes the
drift into account, provided that both the form of the drift E {Z (x)} = m (x)
and the covariance or variogram of the non-stationary RF Z (x) are known.

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MINING GEOSTATISTICS

## 1. The form of the drift

Let Z(x) be the non-stationary point RF in the studied zone. By definition,
the drift is the non-stationary expectation of this RF:
E{Z(x)}

= m(x).

## The non-stationary RF Z (x) can then be expressed as the sum of a drift

term and a residual term Y(x)=Z(x)-m(x). This residual term Y(x) may
or may not be stationary but has a zero expectation:
Z(x) = m(x)+ Y(x)

with

E{ Y(x)}

= 0,

Vx.

(V.S)

The drift m (x) expresses the regular and continuous variation of the RF
Z (x) at the scale of observation of the experimental data. The RF Y (x)
accounts for the erratic fluctuations around the drift, cf. Fig. V. 3.

m( x)

D(xO)

...

..

xo

## FIG. V.3. Drift and residual fluctuations.

The form of the drift m (x) is supposed known. For example, m (x) may
be a linear combination of any known functions {II (x), I = 1 to k}, the
coefficients a/ of which are unknown, so that the drift m(x) remains
unknown:
k

m(x)=

(V.9)

/= 1

## In mining applications, the estimation of a panel V(xo), centred on point

xo, is made from all the available data within a limited neighbourhood
D(xo) around this panel. It is usually enough to express the drift in this

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315

## neighbourhood D(xo) in the form of a polynomial of the first or second

degree (the first terms of the Taylor expansion of the function m(x), i.e.,
m(x) = al + a2X

(V. lOa)

m(x) =

Ql

+ a2X + Q3 X2,

(V. lOb )

## for a so-called "quadratic" drift.

It should be recalled that x represents a point in the working space. If
this space is one-dimensional (e.g., time with coordinate t), then a quadratic drift would take the form
m(t) = a1 + azt + a3 t 2.

## If the working space is two-dimensional (e.g., horizontal regionalization of

vertical thickness with the two rectangular coordinates u, v), then a
quadratic drift would take the form
m(u, v)= a\ +a2u +a3v +a4u2+asv2+a6uv.

## If the working space is three-dimensional, with the three rectangular

coordinates (u, v, w), then a quadratic drift would take the form
2
2
m(u, v, w) = a1 + a-u + a3V +a4w + asu + a6v
2

## +a7W + agUV + a9UW + aIOVW.

Of course, this drift may only be evident in some directions. For example, the phenomenon may be stationary in the horizontal plane with a drift
in the vertical direction, in which case the previous quadratic drift would
reduce to

## 2. The covariance or oariogram of the non -stationary RF

As in the stationary case, the kriging of a non-stationary RF requires the
knowledge of either one of its structural functions, variograrn or covariance:
C(x, y) = E{Z(x)Z(y)}- m(x). m{y) = E{Y(x) Y(y)},

## =Var{Y(x)- Y(y)}=E{[Y(x)- Y(y)]2}.

The semi-variogram y(x, y) is simply the semi-variograrn of the residuals

## y (r ) = Z (x) - m (x). Because of the unknown drift m (x), it is not possible

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MINING GEOSTATISTICS

to estimate y(x, y) directly from the initial experimental data z (x",). This
would entail the simultaneous estimation of the drift m (x) and the semivariogram y(x, y) from a single realization z(x) of the RF Z(x), which is
not rigorously possible.t
In mining applications, experience has shown that, in most instances, it is
possible to take either of the following approximations for y(x, y):

(i) the quasi-stationary semi-variograrn y(h) calculated on neighbouring zones with a possible correction for a proportional effect;
(ii) more simply, the behaviour at the origin (in general, linear, with or
without nugget effect) of the neighbouring quasi-stationary semivariograrn, l' (x, y) = y(h) = w Ih I, where h = (x - y) is the vector
joining the two points x and y.

## This uncertainty of the structural function 1'(x, y) or C(x, y) means that

the unbiased kriging of order k, as used in practice, will not result in the
minimum estimation variance. This is not particularly important because
the major interest is in avoiding any risk of bias due to the drift, while the
minimization of the estimation variance is a secondary consideration.
Equations of the unbiased kriging of order k

Consider the estimation of the mean grade Zv(xo), defined on panel V(xo),
from n mean grades Z; defined on the supports Va.
The corresponding point support RF Z (x) is non-stationary and, within
the estimation neighbourhood D(xo), it has an unknown drift of the form
k

E{Z(x)} = m(x)=

1=1

## in which the functions f/ (x) are known.

t An approximate iterative solution for the simultaneous inference of the drift m(x) and the
serni-variogram y(x. y) is proposed in G. Matheron (1969). Besides this iterative technique,

Matheron has proposed a model of intrinsic RF, the so-called "generalized covariances",
which can be inferred from a single experimental realization. The advantage of the structural analysis of intrinsic RF of order k is that it can be entirely programmed and is
commercially available in packages such as Krigepack or Bluepack, On the other hand, this
automatic procedure does not have the same flexibility as the "hand-made" structural
analysis presented in Chapters III and IV; in mining applications, all sorts of physical and
qualitative information (geology) intervene in a structural analysis.
The following references are pertinent to these techniques: G. Matheron (1969), Ch.
Huijbregts and G. Matheron (1970), G. Matheron (1971, p, 139) and G. Matheron (1973)
for the theory of universal kriging and intrinsic RF of order k (these publications are of a
rather advanced mathematical level and are not recommended 10 mining engineers or
geologists who have no prior geostatistical background); P. Delfiner (1975), A. Haas and C.
Jousselin (1975) for presentations of the packages Bluepack and Krigepack, respectively.

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317

## D(xo) is a neighbourhood centred on Xu and includes the panel V and

the supports of all the data used in the estir.iation. Within D(xu), the
covariance C(h) or the semi-variograrn y(h) are known.
A linear combination of the n data is then formed to provide the
estimator:

l:"

Z~ =

AaZ".

"=1

Non-bias condition

## The non-bias condition can be written as

E{Zv-Z~} =

E{Zv}- E{Z~} = 0,

where
E{Zv}

= V1

E{Z(x)} dx

V(xol

= L arr:1
I

E{Z~} = L AaE{Z,,} = I

a,

L Au ~
V"

. fi(x) dx

= L alb Iv,

V(xo)

!E(x) dx

= L a,
I

L Aab~",
a

denoting the mean value on the support v of the known function fl(x) by
l .
b v, I.e.,
b Iv = -1

f fT(x)dx.
v

To obtain a zero-error expectation, it is enough to impose the k following conditions, called the "non-bias conditions";

I" A"b~"
~

b ~ , V I = 1 to k.

(V. 1 I)

## The preceding non-bias conditions allow

all the terms involving the unknown drift m(x) to be eliminated from the
expression for the estimation variance, leaving

## E{[Zv -Z~ ]2} = C(V, V)-2

AaC(V, V a )+ I

"

AaAilC(v a, Vi3)'

i3

This estimation variance then appears as a quadratic form in A", A/3, just
as it did in the case of kriging under a stationary hypothesis. However, in
this case the estimation variance must be minimized subject to the k
non-bias constraints, instead of the single constraint for stationary kriging.
Using the standard Lagrangian technique, a system of (n + k) linear equations in (n + k) unknowns is obtained (the n weights A", and the k Lagrange

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MINING GEOSTATISTICS

318

parameters J.1-1)' This system is called the "unbiased kriging system of order
k" or the "universal kriging system".
11

13 ~ 1
11

1= I

At3b~a=blv,

B=

ABC(vm Vi~

/-Llb~"

Va

= ('(v"" V),

1 to n,

(V.12)

VI

= 1 to

k.

## The minimum estimation variance or "kriging variance" can then be

written as
k

ak=E{[Zv-Zkf}=c(v, V)+
minimized

11

J-tlb V

<>: ~

1=1

A(.c~(v(>, V).

(V.U)

## This system can also be established from the serni-variogram y(h)

instead of the covariance C (h). A direct result is obtained by replacing the
C by (~n in system (V.12):

I
t3

## Ai>Y(v,., V(3)+LJ.1-lb~a =Y(v,., V),

\fa

Af3b~13 = b 'v,

~ 1 to n, }
(V.14 )

Vi = 1 to k,

(3

and

O'~

A",')i(Va , V) +

L /-LIb ~ -

ii( V, V).

Remark 1 The kriging system (V. 1) in the stationary case can be seen as
an unbiased kriging of order k = 1, the stationary expectation being
reduced to an unknown constant aj:

Note that the mean value of the constant function I, (x ) = lover any
volume v is equal to 1: b~ = 1, Vv.
Remark 2 Existence and uniqueness of the solution. As was the case under
the stationary hypothesis, cf. section V.A.I, remark. 1, the covariance
matrix [C(v a , VB)] is assumed to be strictly positive definite. It can then be
shown that the system (V.12) of unbiased kriging of order k has a unique
solution if and only if the k functions fl(x) are linearly independent on the
set of the n data, i.e., if the n following linear relations
k

1= 1

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clb~"

0,

Va

= 1 to n,

(V.IS)

## V THE ESTIMATION OF IN SITU RESOURCES

319

imply that ci = 0, V1= 1 to k. For example, consider a data set in which all
the data are aligned perpendicularly to the direction u, as on Fig. V.4. It is
obviously impossible to take a drift in the direction u into account, using
just this set of data, even if this drift is only linear: m (u) = a 1+ a2U.

## In fact, in such a case, b~", = 1 and b~"

relations (V. IS) can be written as

= un,

=1

Va

Va

to n, and the n

1 to n,

## The condition C1 + C2UO = 0 does not imply that c r and

since Cl = -uo and C2 = 1 also satisfy the condition.

C2

must be zero,

Remark 3 Matrix form. The first member matrix [K"J of the system
(V.12) of unbiased kriging of order k can be obtained by adding (k -1)
rows and (k - 1) columns to the first member matrix [K] of the system (V. 1)
for kriging under the stationary hypothesis, i.e.,
C(Vl' VI)

C(v" V,,) 1

(n

C(v n , VI)

c; vn )

b~"

b~,

b~n
(n + 1) columns

~O

b~"

~o~

rows

0
.. 0:
::
(k -1)
"
.
rows
0
0
.. .
0
~
(k -1) columns Main diagonal

## Note that the matrix [Ks ] is still symmetric.

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s:..

+ 1)

MINING GEOSTATISTICS

320

The column matrices of the unknown [.A 1 and of the second members
[M2 k ] are obtained likewise:

C(VIl,V)

[A]

[M2d =

bt.
-p.,k

The system (V. 12) for unbiased kriging of order k can then be expressed
in matrix notation as

[Kd [A 1= [M2d
[A] = [K k

r .[M2

(V.16)
k ],

and, thus,
(T~

[A

## Very often, in practice, we will be required to combine the kriged estimates

of local units to form a "global" estimator of a zone or even of the entire
deposit. Assuming an hypothesis of stationarity for the entire zone, the
theorem for combining kriged estimates shows that the direct kriging of the
required global mean yields the same results as combining the local kriged
estimates. In practice, the second procedure is preferred for its greater
flexibility, and because it does not require the preceding stationarity over
the entire zone.

## Kriging the global mean

Let Z (x) be a point RF supposed to be either stationary of second order
over the entire zone or deposit D, or to have a drift of known form on this
extension D. The covariance or the variograrn of Z (x) is also supposed to
be known.
Instead of estimating the mean value ZV; of each of the panels Vj which
make up the zone D, it is possible to estimate the global mean value

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321

## ZD = (lID)SDZ(x)dx directly over the entire zone D. A linear combina-

tion of the available data can be formed to provide the estimator zt; =
L"A",Z" and the standard kriging systems, (V.12) or (V.I), to be considered can be used by replacing the index V by D (depending on whether
or not a drift is to be considered).
In mining applications, the global mean of an entire deposit D would
very rarely be kriged directly for the two following reasons.
(i) It is not usually possible to assume stationarity or a single drift of
known form over the entire deposit D. but only over limited neigh-

## bourhoods (local quasi-stationarity).

(ii) Even if stationarity could be verified over the entire deposit D, there
are usually too many data in D to construct a kriging matrix and then
to solve the kriging system. Moreover, the construction of such a
kriging matrix would imply that the structural function, C(h) or
y(h), is known for distances lh I of the order of the dimensions of the
deposit D and, as was seen in section III.B.7, rule (111.42), the limit
of reliability of an experimental semi-variograrn is a distance L12,
half the dimensions of the field D.
Thus, to estimate the global mean value] over D, it is usually advisable
to combine the estimates of the component volumes of D. If these local
estimations are done by kriging, then the term "combination of kriged
estimates" is used.

## Theorem of combining kriged estimates

Consider again the point RF Z (z ), which is assumed to be either secondorder stationary over the extent of the zone or deposit D, or to have a drift
with a known form over the extension D.
The kriging systems (V. I) and (V.12) have the remarkable property that
only the second-member matrices [M2] or [M2 k l depend on the support V
1
to be estimated. The solution of the kriging system, [A l = [Kr . [M2], is
thus linear in V.
More precisely, let A,,(x) be the solution of the kriging of the point value
Z(x) at point x from the data configuration {Z",; a = 1 to n}. The mean

t Remark

## The global mean ZD = (l/D)JDZ(x}dx, which has a precise physical meaning,

should not be confused with the stationary mathematical expectation m =E{Z{x)} of the
probabilistic model. ZD is an estimator of the expectation m only when the dimensions of
the deposit D are very large with respect to the actual range of the stationary variogram or
covariance representing the model (ergodic property).

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MINING GEOSTATISTICS

322

## value ZD = (1/ D) JD Z (x) dx on the support D is then kriged from the

same configuration and the sol ution A"" (D) is written as
Va = 1 to n,

(V.I?)

## The kriging of the global mean on D can be expressed as a linear

combination of the point krigings,
This relation also applies to the Lagrange parameters J.L, which are
simply the (n + l)th terms of the sol ution matrix [A]:
MI(D)=

fLl(S) dx,

VI = 1 to k.

## To be entirely rigorous, this theorem is based on the assumption that all

the partial estimations are carried out using the same data configuration,
which means using all the data values available over D. In practice, partial
kriging estimations are usually carried out within sliding neighbourhoods
and the data base for each estimation is limited to the values within this
neighbourhood. This practice can be justified for two reasons:

(i) in most cases, these data neighbourhoods effectively screen out the
influence of data values further away;
(ii) the hypotheses of stationarity or form of the drift are usually limited
to distances Ih I which are of the order of the dimensions of sliding
neighbourhoods.

## Remark Forming the global estimator of a zone or a deposit D of

any form by combining local elementary kriging estimates is done in the
standard way by weighting the estimators by their respective supports.
For example, let {Vi, i = 1 to N} be N units of unequal supports Vi, and
let {zt" i = 1 to N} be their kriged mean values. If the zone D = L~l Vi
consists of the sum of all the N units Vi, then its mean value can be written
as
1
ZD = D~ ViZV i

(V.18)

## * = D1",'-( ViZ *Vi'

ZD
If the units Vi are the mesh-points Xi of a regular grid, the estimator of a
zone D is obtained by taking the mean of the kriged point values z~ (Xi)

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## V THE ESTIMATION OF IN SITU RESOURCES

323

inside the polygonal contour which best approximates the form of the zone
to be estimated, d. Fig. V.5.

## Variance of the global estimator

Kriging variances cannot be combined as simply as the kriging estimators.
Let Z'J; = (1/ D) L
be the combination of the N kriged values z t,
and let (T~V; =E{[Zv; -ZtJ2} be the N elementary kriging variances of
the N units Vi. The global estimation variance of ZD by the combination
Zb can be written as

vizt;

and, thus,
(V.19)

: :6:

~===~~~

FIG.

## If the N elementary supports Vi are all equal, this expression simplifies

to

Each time two elementary krigings zt, and Z't (j,,= i) have considered
] and
common data. the two corresponding errors of estimation [Zv. l
[Zvj-ztJ are correlated. Thus, most of the N(N -1) error covariances
E{[ ZV; - zt, ][Zv; - Z'tJ} are non-zero, and their sum cannot be neglected
with regard to the first term (11N 2 ) Li O"~ v, of the expression of the global
variance (T~D. Theoretically, the calculation of these error co variances is
j

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zt

MINING GEOSTATISTICS

324

## possible if the semi-variograrn y(h) is known for all distances Ih I between

the data in D,
In mining applications, such a calculation is very awkward and, in addition, the serni-variograrn y(h) is usually not defined for the large distances
lhl approaching the dimensions of the deposit D. The solution is then to
calculate the global estimation variance (T~D with the aid of approximation
principles which will be given in section V.C.l,

V.A.4. Cokriging

Sometimes, one variable may not have been sampled sufficiently to provide
estimates of acceptable precision. The precision of this estimation may
then be improved by considering the spatial correlations between this
variable and other better-sampled variables. For example, the silver grade
silver.
From a theoretical point of view, cokriging is no different from kriging;
difficulties arise essentially from notations, since an extra index k is
required to differentiate between the variables.
Consider a regionalization characterized by a set of K spatially intercorrelated random functions {Z, (x), k = 1 to K}. The first- and second-order
moments of these RF, assuming stationarity, are
E{Zdx)}=mk,

Vx;

## E{Zk'(X + h). Zk (x)} - me- . me = Ck'k (h),

E{[Zdx + h) - Zdx )][Zk (x

cross-covarrance ;

## ct. section II.AA.

The estimation of the mean value of the point RF Zk,,(X) over the
support Vko(x,,) is required, i.e.,

## The available data {Z"'k' ak = 1 to n. for each k

the supports {v",J, e.g.,

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= 1 to K}

are defined on

## V THE ESTIMAnON OF IN SITU RESOURCES

325

zt

The estimator
is a linear combination of all the available data values
k
o
of all the K variables in coregionalization:
K

Z'tk O =

tl

L L=! A"kZ"k'
k 1
~

(V.20)

"k

zt

E{Z v, "
- 0
}
k

= E{Z v,

}-

A "k0
E{Z"k
}~
"

OCk"

2: A ukol - L

mka[ 1 -

"k

A"k o

zt, o } =

ko

I L A"'kE{ZaJ
k#k""k

mi; I A"k'
"10

## 0, is then expressed by the K conditions"

= 1 and LA""

"""

0,

\;/k '" k o.

"10

Note that the condition LakO A"k0 = 1 cannot hold true if nk() = 0. In other
words, cokriging requires at least one experimental value of the main
variable Zk w
The minimization of the variance (T~kn = E {[ Z V k ,) ~ ztkO]2} under the K
non-bias constraints results in a system of (Lk n; + K) linear equations (the
unknown being the Lk ni; weights A"k and the K Lagrange parameters iLk)
called the "cokriging system";
K

"k'

A(3k'

k'=l I'h,'-l

c.;

(V(3".,

v"J- f-tk

Ckok(Vk " ,

v",J,
"ilk

1 to K,

(V.21)

kO

L
(3k
o

A 13k = 1,

= 1

written as
"

O"vk o =

## CkokoCVko, V k ,,) + ,uka ~

"k

L L
k=!

A",,,Ck,,dVk,,, va,,)

(V.22)

"'k~l

t It may happen that one of the secondary variables Z1<" with k' #; k o, has the same meaning
and the same expectation (mk' ~ mko) as ZkO, e.g., ZkO and Zk' are two lead grades, but one is

obtained from core analyses and the other one from cutting analyses. The two variables Zk;o
and Zk' may have the same expectation, but different structures of spatial variability
(generally, their nugget effect would be different due to different measurement errors). The
K previous non-bias conditions are then reduced to
'r:/k #- k o, k'.

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326

MINING GEOSTATISTICS

## (the notation Ck'dvf3k" V"'k) is the standard representation of the mean

value of the cross-covariance Ck'k(h) when one extremity of the vector h
describes the support V/3k' and the other extremity describes independently
the support V"'k)'

Remark 1 Unlike the kriging system, the cokriging system can be written
simply in terms of the cross-serni-variogram 'Yk,dh) only if C'<'kCh)=
Ckk'(h), which entails that Yn (h) = C,,'dO) - Ce(h), i.e., if the crosscovariances are symmetric in (h, -h), cf. section n.AA. The cokriging
system written in terms of cross-serni-variograrns is then simply derived
from system (V.21) by replacing Ck'k by (-)in).
Remark 2 The existence and uniqueness of the solution. It can be shown
that the cokriging system (V.21) provides a unique solution if the covariance matrix [Ck'k (Vl"k" v",JJ is strictly positive definite, for which it is
enough that a positive point coregionalization model [Ck'k(h)) be adopted,
ct. section III.B.3, and that no data value is totally redundant with respect
to another, i.e., that v a k Vf3k" Vak 7'= 13k"
However, as explained above, the non ..bias condition is satisfied only if
the set of data values of the main variable Zk o is not empty, i.e., it is
necessary that n; 7'= O.
Remark 3 Intrinsic coregionalization. It will be recalled from section
IILB.3, formula (III.30), that the variables {Zdx), k = 1 to K} are said to
be in intrinsic coregionalization if their cross-covariances or cross-variograms are ail proportional to the same basic model:
Vk, k',

If the K data configurations of each variable Zi; are identical, then it can
be shown that the cokriging of a particular variable Zk o is identical to the
kriging of this variable from its corresponding data values {Zak o , ako = 1 to
nk o}'

## In the presence of an intrinsic coregionaiization, the complication of

co kriging can be justified only if one of the variables (k o) is under-sampled
when compared with the others (k oj;. k o). This practical remark can be
extended to all other coregionalization models.
In mining applications, a coregionalization study and subsequent cokriging are carried out only if one of the variables to be estimated is
under-sampled with respect to the other variables with which it is spatially
correlated, d. Case Studies 2 and 3 of the following section V.A.5.

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327

## V.A.5. Case studies

The following case studies are simple enough to be carried out by hand,
without the use of a computer. They are intended as a comparison of the
kriging method and the more usual weighting methods, and to demonstrate
the influence of the structural function, y(h) or C(h).

## Case Study 1. Estimation on a square

Consider a two-dimensional regionalization characterized by a point RF
Z (u, v) which, as a first approximation, is supposed to be intrinsic with an
isotropic stationary semi-variogram: y(h) = y(r) with r = Ih [. For example,
Z (u, v) may be the vertical thickness of a sedimentary seam.
The mean value Zv is to be estimated over a square panel ABeD of side
I from the non-symmetric configuration of the four data of support v,
located at 51 (central sample) and 53, 0 4 , 0 5 (peripheral samples) as shown
on Fig. V.6.
v

0t!

Limit of

I~ the deposit

I
I

S3

SI

II

04

I
/
/

## For reasons of symmetry and because the serni-variogram y(r) is isotropic

the two data 0 4 and 0 5 receive the same weight, and they can thus be
grouped together to form the set 52 = {0 4 U Os} of support 2v. On the other
hand, the two data 51 and 53 must be treated separately. The linear
estimator thus considered contains three weights:
3

zt = L
a<

A",Z(S",)

= I

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with

Z(S2)=HZ(04)+Z(05)].

MINING GEOSTATISTICS

328

## The kriging system (V.3) is written as

Alii(Sl, 51)+A 2y(51) 5 2)+A 3y(Sb 5 3 )+ ,u

= y(S1,

V),

(V.23)

and

y:

## y(5 2 , 52) = i'( 0 4,52 ) = Mf(v, v) + )'(2/)]

(the dimensions of the support v are negligible with respect to the length 1),
y(Sl, S3)= y(Sb 5 2 ) = y(I)

and

12

## :2[34 H(31j2; 1/2)- ~ HUI2; 112)J

2

= ~H(3112; 1/2)-~H(l/2;

1/2),

## srv V)=F(l; 1),

cf. in section II.E.2 the definition of the auxiliary functions Hand F.

## Numerical application A linear model with nugget effect is considered:

y(r)={Co(O)-Co(r)}+1XJr. The nugget effect {Co(O)-Co(r)} is characterized on the regularization over the data support v by a nugget constant
A/v = C Ov , cf. formula (III.7). The nugget constant on the regularization of
support V is negligible with respect to that of support v:
Vventails

A/V= C ov ~=O.

The auxiliary functions F and H for the linear model are given by
formulae (H.91). By way of example, we have

## y(V, V)= Co(O)- A/ V + O'5213wl = C o(O)+O'5213ml,

y(v, v) = Co(O)- A/v +O5213we

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= Co(O)- At ,

329

## as the dimension e of the support v is very small compared to the distance

I.
Three cases are considered:
(i) the pure nugget effect with C Ov = 1, m = 0 ~
(ii) the partial nugget effect with C ov = 05, mi =0959;
(iii) the absence of nugget effect with C ov = 0, -osl = 1918.

The previous values of the parameters COl' and wi have been chosen so as
to ensure, in three cases, a constant dispersion variance of data of
support v in the panel V:
2

D (v/ V)

= y(V,

## V)- .:y(v, v) = C ov + aS2l3wl = l.

The solution of system (V.23) is then immediate and the results are given
in Table Y.I and Fig. Y.7.
TABLE

Nugget
effect

## V.1. Estimation of a square panel (stationary case).

Kriging

Poly

10

= 025
A2 = 050
A3 = 025
(T~ = 025

Al = 1
A2 = 0

0484
0344
0172
0324

0727
0182
0091
0553

idem

idem

idem

IT?,

030

0-40

)'1

Pure

)'1

Partial

..1. 2
..1. 3

at.

=0

"-3

O"~ = 1

= 046
= 040
= 0 ' 14
= 029

-,,=()734

ID2

A1=063
Absent

A. 2 = 029

idem

idem

idem

A3 = 008
(T~ = 022

0"1 = 0-468

027

025

A2 / 2

A3

AI

A,z/2
FIG. Y.? Weights used for the estimation of the square panel.

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330

MINING GEOSTATISTICS

## Remark 1 In the case of a pure nugget effect, the conditions of formula

(V.7) are met: (T~ = C ovl4 = 025 and the weights are proportional to the
supports, 11. 1 = 11. 3 = A 2 / 2. There is no advantage linked to the privileged
location of the cen tral datum S 1.
Remark 2 As the nugget effect decreases and, thus, the spatial correlation increases, the influence of the central information becomes preponderant (A 1 increases from 025 to O 63). Note that for 'UJ -:;f:; 0 the weight
of the datum S2 (of support 2v) is always greater than twice the weight of
53 (of support v); this is due to the fact that 52 is nearer to the right-hand
side of panel V, which is less sampled.
Other estimation procedures The standard weighting methods (Poly, ID,
ID2) do not take into account any of the characteristics peculiar to the
regionalization under study and, consequently, provide the same estimator
whatever the structural function l' (h) may be, cf. Table V.I.
The polygon of influence method gives all the weight to the central
datum: A 1= 1 and A 2 = A 3 = O.
Both the inverse and inverse-squared distance weighting methods (ID
and ID2) give the same weight to the three peripheral data: A3 = 11. 2 / 2 . The
central datum Sl is given greater weight by ID2 (A 1= 0,727) than by ID
(A I = O'484). The mean distance from S 1 to the panel has been taken as a
quarter of the diagonal, i.e., lJ2/4.
In the stationary case, these standard methods ensure the unbiased ness
of the estimation, as they satisfy the unique non-bias condition L Aa = 1,
but they do not, by themselves, provide the estimation variances (T~. For
this purpose, it is necessary to characterize in some form the spatial
variability of the phenomenon under study. The geostatistical method is to
use the structural function y(h) with which the estimation variance of any
unbiased linear estimator can be calculated from formula (II.28).
The following remarks can be made on the results of Table V.I.

(i) In all cases, kriging provides the best unbiased linear estimator
(BLUE).
(ii) Depending on the nugget effect (i.e., the degree of spatial correlation), one or the other of the three standard estimators is close
enough to the kriging optimum, but only a structural analysis, i.e., a
geostatistical approach, can tell which one is the closest.
(iii) Generally, the greater the nugget effect (thus, the greater the screen
removing effect), the better are procedures (such as ID) which do

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## V THE ESTIMATION OF IN SITU RESOURCES

331

not give too much weight to data preferentially placed with respect
to the panel to be estimated. On the other hand, the smaller the
nugget effect, the better are procedures (such as ID2) which favour
well-located data (by screening the influence of furthest data).
(iv) In all cases, an unbiased weighted average-type estimator such as ID
or ID2 fares much better than the POLY estimator. Had the
configuration of data been more clustered, e.g. if data 0 4 was located
nearby data O, the advantage of kriging over ID and ID2 would
have been much more decisive.
Non -stationary case Let us consider now a non-stationary regionalization
characterized by a RF Z(u, v) with an anisotropic linear drift which is
evident only in the direction of the coordinate u, i.e.,
E{Z(u, v)}=m(u)=al +a2u.

## For example, Z (u, v) may be the vertical thickness of a subhorizontal

sedimentary seam which thins out towards one end in the direction u,
or it may be the depth of the continental shelf, the direction u being
perpendicular to the coastline, ct. Fig. V.S.

Panel to be
estimated

u.

## As previously, the mean value Z is to be estimated over the square

panel V from the non-symmetric configuration of the four point data
located at Sl, 0 4 , 0 5,53 , cf. Fig. V.6.
For reasons of symmetry, and since the drift does not occur in the
direction v, the two data 0 4 and O will receive the same weight and they

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MINING GEOSTATISTICS

332

can thus be grouped into the set S2 = {04 U Os} of support 2v. And, as
previously, a linear estimator with three weights can be defined:
3

zt = L
Cf=

AaZ(S,,).

## The drift m(u)= Ql +a2u = adl(u)+a2h(u) consists of the two known

functions, f,(u)= 1, 'r/u, and h(u)= u. The mean values of these functions
over the supports S<;>. and V can thus be written as

bL

b~

= 1,

bt

= U<;>.

1
1
h(u)du=V v
1

bt=-

1

udu=-,

EF

C+

1/2

udu=c,

(-1/2

## if the coordinate Ul of the central data S1 is equal to c.

The system (V.14) of unbiased kriging of order k = 2 can then be written
as
3

L AI'Y (S""
.B 1

Va

1 to 3,

(V.24)

and

## With the coordinates Ul = U2 = c and

condition can be expressed as

U3

= c ~ I,

## which gives A3 = 0, i.e., the non-bias condition introduced by the presence

of the linear drift amounts to ignoring the datum S3, whatever the underlying structure y(h) may be.
By putting A3 = and considering the tensorial invariance property of
the kriging systems (d. the addendum at the end of the present case study),
it will be shown that the system (V.24) can be reduced to the system (V.26)
of two linear equations in two unknowns.

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333

## Review of other weighting methods In the absence of a structural analysis

providing evidence of the linear directional drift m (u), the three usual
weighting methods (Poly, ID, ID2) would provide the same weights as in
the stationary case, i.e., the weights given in Table V.l and reproduced in
Table V.2. Consequently, the ID and ID2 estimators are biased, since they
do not fulfil the second non-bias condition (ft. 3 = 0). As for the Polyestimator, it remains unbiased, since it gives a zero weight to the datum 53
(but also - and wrongly - to the information 52).
If, as in the case of the unbiased kriging of order 2, the form of the drift
m (u) = a, + a2U is supposed known, this drift can then be estimated by
least squares (abbreviated to LS). This estimated drift obviously passes
through the value Z(5 3 ) and also through the mean of the three values
2(04),2(5 1) and Z(Os), i.e.,
m*(ut) = HZ(Sl)+ Z(04)+ Z(Os)].
The estimator of the mean drift over V can then be written as

mt =

m*(u) du

m*(ud=~[Z(S,)+Z(04)+Z(05)]'

## If that least-square estimator, mt, is used to estimate the mean value Zv

over the panel V, then the weights are A. 1 = 1/3, A.z = 2/3, ft.3 = 0, and, thus,
the non-bias condition is satisfied. The estimation variance, (J"~ =
E{[Zv - m't]Z}, is calculated from formula (11.28), using the semi-variogram 'Y(h)=~Var{Z(x+h)-Z(x)}-theknowledge of which requires a
geostatistical approach.
It is stressed that the estimation variance (J"~ of the LS estimator mt has
nothing to do with the minimized variance Q of the experimental residues.
In this example, 0 is equal to the experimental variance of the three data
2(5,),2(04 ) , 2 ( 0 5 ) :
Q = 1[[Z(5,)- m*(ut)]z + [2 (0 4 ) - m*(ud]2 + [Z( 0

5) -

m*(uj)]2]

-:;l;

(T~.

## Numerical application As in the previous stationary case, an isotropic

linear model with nugget effect is considered:
y(h) =! Var {Z(x + h)- Z(x)} = {Co(O)- Co(r)} + 'UJf,

Vr =

Ihl.

The same three cases (pure, partial and absent nugget effect) are considered with the same values for the parameters C ov (nugget constant) and eil.
The results of kriging (unbiased of order 2) are compared to those of the
four "non-regionalized" usual methods (i.e., Poly, ID, ID2 and LS) in
Table V.2.

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MINING GEOSTATISTICS

334

## Remark 1 When there is a pure nugget effect (total absence of spatial

correlation of the true residuals Z (x) - m (x)) the results obtained from the
least squares method are the same as the optimal results given by kriging. It
can be shown that this property remains true whatever data configuration is
used (d. G. Matheron, 1971, p. 162).
TABLE

Nugget
effect

Pure

## V.2. Estimation of a square panel in the presence of a drift m(u)

Kriging

LS

Poly

A 1 = 1/3
A z = 2/3

A1= 1/3
A2 = 2/3
A3 = 0
rJ'~ = 1/3

0
0

idem

idem

A3 = 0
a~ = 1/3

Partial

Absent

A1 = 052
A2 = 048
"-3 = 0
rJ'~ = 034
A1 = 066
A2 = 034
"-3 = 0
<T~ = 025

idem

idem

idem

Biased

Biased

idem

(J"t~ = 0,45

0.468

Unbiased

idem
0.734

idem

Unbiased

102

0484
0727
0182
0344
0172
0091
(d. non-defined

o-i, = 0.39

ID

Unbiased

-'*2/ 2
a

A3

AI

Az/2
FIG. V.9. Weights used for estimation of the square panel.

## Remark 2 As the nugget effect becomes smaller, the LS method deviates

more and more from the optimum provided by kriging. In the case of
absence of nugget effect, the deviation amounts to 180% in terms of
estimation variance, 045/025 = 18.

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## V THE' ESTIMATION OF IN SITU RESOURCES

335

Remark 3 In the presence of the directional drift m (u), the ID and ID2
estimators are biased. This bias. E{Zv - zt}, depends on the unknown
drift m (u) and is itself unknown. The existence of this unknown bias thus
prohibits the definition and calculation of the estimation variance (T~ =
Var {Zv - zt}.
Addendum. The tensorial inuariance property of the kriging systems

## (This subsection may be omitted in a first reading.)

Obviously, neither the kriging system (V.24) nor the corresponding kriging variance uk should depend on the arbitrary choice of the origin of the coordinates, i.e.,
of the parameter c. In particular, this system must be equivalent to the following
system (V.25) corresponding to the choice c =:: 0:
Va = 1 to 3,

(V.25)

In fact, with the change of variables J.t; = fJ.l + 11-2C, J..L; = J..L2, the two systems
(Y.24) and (V.25) are equivalent. Thus, A~ =A s, V/3, since the solution of the
kriging system is unique.
This invariance property of the kriging system with respect to changing the origin
of coordinates is related to the polynomial character of the component functions
Mx) of the drift m(x) and is the reason why only functions of this type are used, in
practice, to characterize the form of the drift.
The equation corresponding to a = 3 can be eliminated from system (Y.24) as it
only serves to define the parameter 1-1-2, which has no purpose when c = O. The
system (V.24) then reduces to (V.26) which, in the stationary case, is the same as
kriging panel V by the two data 51 and 52:
Al Y_(5" 51) + (I-A d~(Sj, 5 2 ) +

= ~(51) V), }
A1y (S2, Sd+(l-A 1)r (S2, S2)+j..t1 = y(S2, V)
/-Ll

(V.26)

and

## Case Study 2. Cokriging on a square

Consider the coregionalization of the two stationary RF's Zl(X) and Z2(X)
defined on a horizontal plane and characterized by the matrix of isotropic
cross-serni-variograms [Ykk,(r)], k, k' = 1 to 2.
The mean value Z, v of the main variable Zl(X) is to be estimated ov~r a
square panel V of dimensions I x I. as shown on Fig. V.IO. The available

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MINING GEOSTATISTICS

336

data consist of the central point value Zl(5 0 ) of the main variable and the
five point values Z2(5 0 ) , Z2(A), 2 2(B), Z2(C), Z2(D) of the secondary
variable located at the centre and the four corners of the square. This
corresponds to the case of a variable Zl, undersampled with respect to
another variable (22 ) , and the problem is to evaluate the gain in precision
provided by cokriging (Z1 v is estimated from the two data sets on Z1 and
2 2 ) over kriging (ZI v is estimated from the single data set on Zl)'

8
x

x
o
So

)(------:x

FIG. V.10. Cokriging on the square ABCD. 0, Data set Z); x, data set Z2.

## Kriging amounts to estimating the panel V from its central value 2

The kriging variance is, thus, given by the general formula (II.2?):
(T~ =E{[ZIV-ZI(SO)]2}=2Yll(SO, V)-YI1(V, V)-Yll(SO, So)
=

1(50 ) .

(V.2?)

## 2H l1(lj2, Ij2)-F ll(l, I),

where Hi, and Fi, are the auxiliary functions corresponding to the point
model 'Yl1(r) which characterizes the regionalization of ZI(X),
For cokriging, reasons of symmetry (the coregionalization matrix being
isotropic) allow the four peripheral samples in 2 2 to be grouped together
into one single set 5 R with a grade of

## ziv = )\121(50 ) + A2Z2(50)+ A3Z2(SR)'

The two non-bias conditions of the cokriging system (V.21) can then be
written as
A1 = 1

and

A2 + A3 = O.

## By putting A = A3 , A being the weight of Z2(SR), the co kriging estimator is

to be written as

(V.28)

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## V THE ESTIMATION OF IN SITU RESOURCES

337

The two non-bias conditions have already been taken into account by the
form of the estimator (V.28) itself, and the cokriging system (V.21) reduces
to the following three linear equations in three unknowns (A and the two
Lagrange parameters, f.Ll and f.L2):

~11(50' V)

## 1'12(SO, 5 0) - A1'22(SO, So)+ A1'22(SO, SR)+ J.L2 = 1'12(50, V)

Y12(SR, 5 0 ) - AY2Z(SR, So)+ AY22(SR, SR)+ J.L2 = Y12(SR, V)

and

minimized

## Ykk,(SR, SR)= Ykk{SR, A) = [21'kk(l) + I'kk(lJ2)]/4,

Ykk'(SO, V) = Hkk'(l12; 112),
'Ykk'(S R, V) = 'Ykk,(A, V) = Hkk,(l ~ I),

'5'kdV, V) =

n.. I).

## By simplifying, the preceding system gives the following solution:

A = Y12(SO, SR)+ Y12(SO, V)- YI2(SR, V) = N (numerator)
21'22(5 0 , SR)- Y22(SR, SR)
D (denominator)"
2

(J"CK

UK -

(V.29)

AN,

(J"~ being given by formula (V.27). The gain in precision of cokriging over
kriging is th us equal to

g = (J"~

(J"~K = AN = N

1D.

## This gain is always positive since D is the extension variance of Z2(50 ) to

Z2(SR) and thus always positive or zero:
u~(So, SR) = D ~ 0,

## according to formula (11.27).

Numerical application 1 Consider first the case in which all the semivariograms consist of a linear isotropic model with nugget effect:
Vr>

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## (very small range)

MINING GEOSTATISTICS

338

## The auxiliary functions Hand F of the linear model are given by

formulae (11.91) and, for I s, give

and

## Fkk(l; l) = Cu: + 05 213UJkk'l.

For calculation purposes, I'll (r) will be fixed so that (T~ = 1 and the gain
g = AN will thus be relative: g = g, = AN/(T~. For example, the parameters
e l l = 05 and "UTI I I = 2 OS will be taken, which entails (T~ = 1, according to
formula (V.27).
Similarly, Y22(r) will be fixed so that the variance D is small, which means
2
that the gain, g = N / D, will be large. For example, the model I'22(r) =
~hll(r) will be taken with the parameters Ci = 01 and "U1221 = 041.
According to formula (V.29), D thus becomes

21'22(lJ2/2)-h'22(l)-h'22(lJ2)

## ~C22 + O' 5607"U1221 = 0.355.

With the two direct fixed models ')'11 and ')'22, the cross-model I'12(r) will
be made to vary, taking care that the coregionalization matrix [ - Ykk.(r)}
remains conditionally positive. For this purpose, it is enough that the two
parameter matrices [C kk ,] and [trrkk'] be positive definite, d. section III.B.3,
conditions (111.26). It is thus enough that the two following determinants
be positive or zero:

lee

ll

12

trrll
I 'fJJ12

el21 >0

::>

C12<J(C I l . C n ) = O2236 .

12
UT 1
trr22

::>

## UJ121 < J ('fJJ1l1. UTnl) = O9168.

C 22

>0

Table V.3 gives nine pairs (Cn , "UT 12 1) which satisfy these conditions, and
gives for each one of them the value A of the weight assigned to the
secondary sample Z2(SR), as well as the relative gain gr provided by
cokriging.

Remark 1 When the increments [ZI(X + h)- Zl(X)] and [Z2(X + r)Z2(X)} of the two variables are independent, the cross-semi-variograrn
')'12(r) is zero, and the secondary data on Z2(X) adds nothing to the
estimation of the main variable ZI(X),

## Remark 2 When the cross-variability (i.e., the parameters C ll and m12)

increases, the gain of cokriging also increases. The maximum 77% gain is
obtained with the maximum cross-variability.

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## V THE ESTIMATION OF IN SITU RESOURCES

339

Numerical application 2

## Consider now the silver-lead coregionalization

model adopted for the Broken Hill deposit (Australia) in section IV.C,
Case Study 8. The silver grade is considered here. as the main variable
Z, (x), which is undersampled with respect to the lead grade Z2(X), this
being often the case in silver-lead deposits.
TABLE

12

0
0
0

01
01
01
0223
0223
0223

'UJ12

0
0411
0916
0
0-411
0916
0
0411
0916

A

gr

0
038
084
028
066
112
063
101
1-47

0
005
025
003
015
045
014
036
077

## YI2 is a pure nugget effect

Intrinsic coregionalization
Y12 is a pure nugget effect

## The adopted serni-variograrns Ykk (r) (with a quasi-point support) consist

of an isotropic spherical model "}'1 with nugget effect and range a = 60 ft, cf.
Table (IV.I):

"}'11 (r)

## '}'12(r) = 5 '}'1 (r)(oz Age I )(% Pb).

The auxiliary functions HI and F 1 for the spherical model '}'1 are given by
charts no. 3 and 4 in section ILEA. The following results for I = 100 it. are
thus obtained: for the kriging variance,
(T~ = 2-38(02 Ag e 1)2;
for co kriging, according to formulae (V.29),
N = 4'125(oz Ag

)(% Pb);

= 62.5(%

Pb)";

## hence, A = N/ D = 0066(oz. Ag e )(% Pb)-I, which corresponds to a relativegain gr=ANla~ =O114~11%

Thus, although the silver grade is quite undersampled with respect to the
lead grade, the gain of cokriging over kriging is negligible for the
configuration studied here. This is due to both the fact that the only
available data for silver, Zl(SO) is well located, and that the cross-variability '}'12 is not high.
1

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MINING GEOSTATISTICS

340

## To sum up this case study on cokriging, the following advice can be

given: before undertaking a coregionalization study, first proceed to the
calculation of the possible gain of the planned co kriging, considering a
simple data configuration.

## Case Study 3. Cokriging with systematic errors

In a deposit mined by open-pit, two types of data are available on the grade
of interest.
(i) An unbiased data set derived from drill cores. The analyses of the
cores are considered as the true grades Z 1 (x) of these cores.
(ii) A biased (e.g., because of selective loss of fines being blown away)
and imprecise data set derived from blast-hole cuttings. The analyses
of these cuttings are considered as realizations of the RF Z2(X),
which differs from the true grade Zl(X) by a random error e(x), i.e.,
Z2{X) = Z,(x)+ e (x).

In general, the poor quality of the cuttings data set is balanced by the
large number of blast-holes on any mining bench.
To estimate the true mean grade Z, v of a block with support V, the
following estimator is used:

ziv = I

A<"lZ'(XaJ+

"'1E:SI

A. u.Z 2(Xu. ),

<X2E: S2

51 and 52 being the two sets of core data and cuttings data, respectively.
The supports of the core lengths and cuttings are negligible with respect to
the support V of the block.
As the matrix of coregionalization of the two variables ZI and Z2 is
['Ykk'(h), k, k' = 1,2], the cokriging system (V.2l) is written in terms of
se mi-variograms:

## Ai>I'Y1 1(X", I ' Xi>J+

BlESI

fhES2

A/32Y12(X",p X.aJ+J.Ll

(V.3D)

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341

and
(TZ'K =

## Au 2 1'12{X" 2' V) +,a1 - iill (V, V).

a2 ES2

"lES1

It should be recalled, from section V.AA, remark 2, that the set 5 I of the
unbiased data ZI(X a .) must not be empty. This restriction is quite natural,
since the unknown bias of the data set 52 cannot be corrected unless there
are actually some unbiased data in SI. Under the condition 51 - 0, the
influence of the unknown bias of the data set S2 is made nil by the relation

## and there is no need to evaluate this bias Ei (x)} when cokriging.

However, when this bias is known, i.e., when an unbiased cutting data
set S can be defined, the two non-bias conditions of the preceding system
(V.30) reduce to the unique standard non-bias condition:

"'/31

+ L

I3,ES,

"'/32 =

1.

132ES2

But there is still a need for cokriging, since the two spatial structures
and Y22 of the two types of data may be different.

)'11

Numerical application The three RF's ZI(X), Zz(x), s{x) have the
following characteristics.
First, they are all stationary with unknown expectations:
E{ZI(x)}

= m1;

E{Z2(X)}

= m i:

mi -

ml,

## equal to the unknown bias between the two data sets.

Second, the structure of the true grades Z\(x) is characterized by the
quasi-point model Yl1 (h) derived from drill-core data.
Third, as a first approximation, the error e(x) can be assumed to be
spatially uncorrelated (the loss of fines not being regionalized). Thus, the
semi-variogram of the error E(x) reduces to a pure nugget effect:
yz(h) = ~E{[s(x + h)- e(x)]2} = Var {s (x)} =

E,

'v'lhl> an,

ao being a very small range compared to the distance Ih I of the experimental observation.
Fourth, the error e(x) is supposed to be independent of the true grade
Zl(X) (this is not always true, e.g., in banded hematite quartzite deposits,

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MINING GEOSTATISTICS

342

the quantity of fines is inversely proportional to the iron grade Zl). Thus,
Y12(h) = tE{[Zl(X + h)- Zl(X )][Z2(X + h)- Z2(X)]} = '}'ll(h)

as
Z2(X + h)- Z2(X)

## and the increments of sex) are independent of the increments of Zl.

Similarly,
Y22(h) = !E{[Z2(X

=

Yl1(h)+ C".

## Thus, the following regionalization model is obtained:

Indeed, in practice, the two direct serni-variograms /'11 and '}'22 are
calculated from the corresponding experimental data and they usually
differ only by the nugget effect:
Y22(h)-Yl,(h)= C e,

The cokriging configuration of Case Study 2, above, cf. Fig. V.1 0, will be
taken as an example, i.e., the estimation of the mean grade Z, v over a
square block V of side 1= 20 m by the two data sets 2,(50 ) with weight 1
for the central core analysis, 2 2(So) with weight -A and Z2(SR) with weight
+A for the cutting analyses at the centre So and the corners SR of the block.
The quasi-point serni-variogram YI1 (h) is the model fitted to the
(Mauretania), d. section II.D.S, Case Study 1, i.e., a spherical model
without nugget effect, range a = 40 m and sill C = 133 (% Fe)2.
For the direct estimation of 2 1 V by the only central core grade Z, (So),
the variance O"i, given by (V.27), is
(]"~ = Q185C = 24'6(% Fe)2.

Cokriging, i.e., taking the cutting data into account, gives the following
results from (V.29):
N = 0243C

323

and

1.25 Ce +0451 C.

## By taking the variance of the measurement errors C; equal to the first

experimental value YlI(3), i.e., C, = 15, D becomes 787, and the weight
assigned to the corner cuttings data Z2(SR) is A = N / D = 041. The gain of
cokriging over kriging is thus equal to
g = (]"~ -(]"~K = AN = 133 (% Fe)2,

## i.e., a fairly good relative gain: gr = g/a~K = 054.

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## V THE ESTIMATION OF IN SITU RESOURCES

343

In spite of their bias, taking the cuttings data into account provides a
relative gain of more than 50% in terms of estimation variance.

## V.B. THE APPLICATION OF KRIGING

V.B.1. A comparison of kriging with other weighting methods
If we are limited to the class of linear estimators (the estimator Z*

L: ~ 1 A"Za

## is a linear combination of the n data values) and if the structural

analysis is sufficient to infer correctly the underlying structural function
(covariance or variogram), then, by definition, kriging provides the best
unbiased estimator (in the sense of minimum estimation variance). This
property is irrefutable and is not shared by the more classical linear
estimation methods such as polygons of influence (Poly), inverse distances
(ID), inverse-square distances (ID2) and least-square polynomials (LS), to
cite only the methods more commonly used in mining applications.
The two criticisms most often levelled at kriging are:
(i) the gain in terms of estimation variance provided by kriging is
negligible in practice when compared with more common estimation
proced ures;
(ii) the application of kriging, as well as geostatistics in general, is
difficult and costly in terms of both sampling and computer time.
The following responses can be made to these criticisms.
First, as kriging is optimal, small deviations in the choice of weights
around their optimal values will only result in a second-order increase in
the estimation variance. This means that if, for any given data configuration, the weights provided by any of the usual estimation methods differ
little from the kriging weights, then the gain due to kriging can be neglected, and the more usual method will be preferred for this particular data
configuration because of the simplicity of its application. However, it still
must be known which method most closely approximates kriging, and this
can only be determined by a structural analysis to calculate the variogram
to be used in the kriging solution. When a strong screen effect is present,
Poly will usually be adopted, whereas ID will usually be better when the
screen effect is weak. A modified version of ID2 may prove better when
anisotropies are present, and LS should usually be preferred in cases of
non-stationarity, d. Case Study 1, section V.A.5. Before a structural
analysis is made, it can never be determined which of these methods could,
in practice, replace kriging; it depends not only on the variogram (i.e., on
the structure of variability), but also on the data configuration. Since, in

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MINING GEOSTATISTICS

## mining applications, the configuration of the data used to estimate a panel

may vary considerably throughout the deposit, kriging remains the surest
solution (and not necessarily the costliest).
Second, the understanding of geostatistics and the application of a structural analysis followed by kriging is, without doubt, more difficult than the
arbitrary choice of one of the usual weighting methods.
Is geostatistics really costly in terms of sampling? Geostatistics quantifies
the quality of the available data by means of the kriging variance, and this
allows cases in which the amount of data is dangerously insufficient to be
identified. To reject the determination of variability structure by using a
method which does not have an associated estimation variancet amounts
to ignoring completely the problems involved in estimation.
Geostatistical studies can also result in the conclusion that there are too
many data for a given objective. Used in time, geostatistics can result in
considerable savings in sampling, cf. A. Journel (l973b).
Is the computer cost of kriging really prohibitive? Even though
computer costs are always negligible when compared to the cost of obtaining a few core samples, the answer to this question is yes, if no precaution is
taken when writing and solving the kriging system. Now, if all the symmetries and approximations are taken into account by establishing a "kriging plan", then both the dimension of the kriging system and its solving
cost can be reduced.
The theoretical optimum of kriging is of little practical use if the application of kriging is not competitive in terms of computer cost.
V.8.2. Elaborating a kriging plan.

## Consider the estimation of the mean grade Zv of a panel of support V

from the n data Zcr. defined on supports Vcr.' The kriging system (V. 1) is
written as

~ A/3C(vcr., V(3)-1.l

1L:
/3

and

A/3

C(Vcr., V),

Va = 1 to n,

## (When a drift is present, system (V.12) should be used.)

t Once again, it should be recalled here that the residual variance of the least-squares
polynomial method (LS) is not the estimation variance, d. Case Study 1, section V.A.S and
G. Matheron (1967).

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## V THE ESTIMATION OF IN SITU RESOURCES

345

The various steps in the practical application of the kriging of each panel
V are as follows.
(i) Select the data Z", to be used in the estimation of Z v
(ii) Calculate the mean covariances C(v"" VJ3) between all the selected

data (these are the elements of the first member matrix [K] of the
kriging system).
(iii) Calculate the mean covariances C(vO" V) between each of the
selected data and the panel V to be estimated (these are the elements of the second member matrix [M2]).
(iv] Choose an algorithm for the solution of the linear system.
To carry out a "crude" kriging without any consideration for computer
cost would consist of taking all available data without distinction within a
neighbourhood of quasi-stationarity centred on each panel V, calculating
exactly the various mean values C and then applying a precise algorithm
which would give an exact solution of the linear system (such as S[MQ~SSP
IBM). Such an application would yield very precise results, but would also be
very costly in terms of computer time.
To establish a practical alternative, each one of the preceding steps must
be critically examined to provide an approximate procedure which will
minimize computer time while yielding results that are within acceptable
limits of approximation.
The five essential points to be used in reducing the kriging cost are:

1.
2.
3.
4.
5.

## the reduction of the dimension of the kriging system;

the reduction of the number of kriging systems;
the rapid calculation of the various mean values C (or ii)~
the preparation of a data file adapted to the kriging plan;
the choice of a time-competitive algorithm for the resolution of the
kriging system.

## The dimension (n + 1) of a kriging system is directly related to the number

n of data, Zo<, to which a weight AO' is to be assigned. To reduce this
dimension, either the estimation neigh bourhood can be reduced or data
can be grouped together.
Limiting the estimation neighbourhood The first limitation of the estimation neighbourhood is that of the neighbourhood of quasi-stationarity
defined by the structural analysis: the serni-variogram y(h) is known only

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MINING GEOSTATISTICS

for distances Ih 1< b. Thus, to estimate a panel V(xo) centred on xo, only the
data inside a neighbourhood b(xo) centred on Xu are considered, d. Fig.
V.II. This neighbourhood is not necessarily isotropic.

.,..---.,.
/"

'

"' "-

I
I
\

J
\
\

"

FIG.

__

.....A(xo).... /"

## V.U. Neighbourhood of estimation A(xo).

It may happen that the data within b(xo) are either too few, in which case
more distant data may be included in considering a non-stationary kriging
system, or too many, in which case a reduced neighbourhood A(xo) may be
considered. In fact, this procedure amounts to neglecting the weights of
data exterior to A(xo), d. Fig. V.II. This approximation is justified when
the data within A (xo) can be considered as sufficiently screening the
influence of the data exterior to A(xo). This screen effect increases as the
nugget constant of the underlying structural function becomes smaller (d.
section V.A.I. remark 7, "The influence of the nugget effect") and also as
the data within A(xo) surrounds more completely the panel V(xa) in all
directions, without gaps.
A prime objective in defining a data neighbourhood A (xa) should be to
ensure that it avoids all risks of bias in the estimation. Thus, in two
dimensions, for example, when the data configuration is such that one
direction f3 is undersampled, although the structural continuity in this
direction is significant, it is advisable to extend the estimation neighhourhood A(xo) in this direction so as to include more data 2(3, which will take
into account the .a-direction continuity in the estimation, d. Fig. V.12.
Examples of such a procedure are the automatic mapping programs that
partition the space around Xo into equal parts (e.g., four quadrants in R 2 ,
or eight octants in R 3), and only the two or three data closest to Xo are
considered within each of these parts.
It often happens when estimating panels on the border of a deposit that
most of the data lie on one side of the panel, i.e., they are preferentially
located in a rich zone, ct. Fig. V.13. In such a case, it is advisable to do one
of the following:

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## V THE ESTIMATION OF IN SITU RESOURCES

347

(i) limit the extent of the neighbourhood A (xn) to the favoured side of
the panel, so as not to give too much influence to the preferentially
rich data;
(ii) introduce, with due care, a supplementary non-bias condition corresponding, for example, to a linear drift m (x a ) = an + a lX a in the
direction a of impoverishment;
(iii) not carry out the estimation without any additional data - this would
be the proper course if preferential data positioning as well as
heterogeneity of the mineralization were suspected.

xx- - - - - - - -

-~~Z/l

)(

,
I

Xo

)(

J
A(xo)

---------_.--

a
...-..-/'

,/

"- ......

///

x\ x
I x
x t
x
J

"

Border

## Grouping of the data A second method of reducing the dimension of the

kriging system consists in grouping the n data {Zi of support V;, i = 1 to n}
available within the estimation neighbourhood into n' data sets {Zk, k = 1
to n ', n' < n}. Each data set of mean value Z; is defined on a support Sk
which is the union of the supports ViI< of the component data Zil<' Le.,
Sk = U ViI<'

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348

## Thus, on Fig. V.14, the estimation neighbourhood of the square panel V

is taken as the union of nine squares of the same size (the square V itself
plus the eight squares which constitute the first aureole around V). The
available data are classed into nine sets Sk according to their locations
within one or the other of the nine squares. Let {Zk, k = 1 to 9} be the mean
grades of these nine sets. The estimator considered is, thus, a linear
combination of the nine mean grades Zi; i.e.,

k =1

x x

)(

x
8

x
)(

## The datum Z9, for example, which corresponds to the square k = 9,

consists of the union of the two elementary data Zi 9 and Zj9' If these two
elementary data have a quasi-point support and are located at points Xj9
and Xj9' the following relations hold:
Z9 = (Z;9 + Zi.,J/2,

## y(Sg, Sk) = [Y(Xi9 ' Sd+ y(xh' Sk)]/2,

vs;

This grouping of the available data into nine sets allows the initial
dimension (17) of the kriging system to be reduced to 10. The corollary of
this reduction in the dimension of the system is a loss in precision: the
kriging variance resulting from the nine mean grades Z; will be greater than
that resulting from the 16 elementary grades. If the data are distributed
more or less uniformly, as shown on Fig. V.14, then this loss in precision will,
in general, be negligible. To evaluate this loss in precision, the two kriging

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349

## systems can be solved for an average data configuration; it can then be

determined whether the gain in computer time justifies the loss in precision.
However, grouping the data into sets does not always involve a Joss in
precision, particularly when the grouping is based upon a structural symmetry.
Structural symmetry All data that would receive strictly equal weights
from kriging can be grouped together. This grouping reduces the dimension of the kriging system but does not alter the kriged value or the kriging
variance.
Such data must be "structurally" symmetric with respect to the support
V to be estimated. This symmetry will be identical to geometrical symmetry only if the structural function y(h) is isotropic. For example,
consider the estimation of the square panel V on Fig. V.I5 by the
geometrically symmetric configuration of the five quasi-point grades
Z (x d, ... ,Z (x 5)' If the structural function y(h"" h(3) is isotropic, then the
four exterior data will have exactly the same weights and they can, thus, be
grouped into one single set called the "first aureole" with a mean grade
ZA = [Z(X2)+ Z(X3)+ Z(X4)+ Z (xs)]/4. The two-weight linear estimator is
then zT = A j Z ( X l ) + A 2ZA'

x,.

X"

a
FIG. V.IS. Structural and geometrical symmetries.

## However, if the structural function y(h", hB ) is anisotropic with, for

example, a being a preferential direction of continuity, the grouping would
then be along the two perpendicular a and (3 with the two mean grades
Za = [Z(x2)+Z(x)]J2, and ZB = [Z(x4)+Z(xs)]/2. The three-weight
linear estimator would then be zi = A lZ(Xl)+ A2Z" + A]Z{3' Other examples of such symmetrical grouping were given in the case studies, d. section
V.A.5.

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## 2. Reducing the number of kriging systems

In section V.A.l > remark S, it has already been noted that if the data
configuration is repeated from one panel to another, the first member
matrix [K] is the same for each panel and it is necessary to take its inverse
only once. Moreover, if the panels to be estimated are equal and have the
same geometry, the second member matrix [M2] is also the same for each
panel and, thus; there is a unique kriging system which needs to be solved
only once. The system of solution weights, [A] = [Kr 1 . [M2] is unique and
applies to all the panels to be estimated, d. the case study of the kriging of
Donoso copper mine in section V.B.S, Fig. V.27.
Systematic sampling on a regular grid is, in this sense, particularly
favourable for precise and rapid kriging. In addition, such regular sampling
plans facilitate the structural analysis which can be carried out precisely
and without any risk of bias due to preferential positioning of the data.
In practice, it is, thus, very convenient to have a data configuration that is
repeated identically from one panel to another. This may be achieved by:
(i) insisting on a systematic and regular sampling pattern when the
sampling campaign is being prepared;
(ii) defining the panels to be estimated on a constant support, at least
within the central homogeneous zones of the deposit;
(iii) considering a unique "mean" data configuration by using the techniques of random kriging or kriging of gaps in the data grid, d. the
following paragraphs.
Definition of the support to be estimated A constant estimation support is
very frequent in mining applications as it very often corresponds to
constant mining or selection units.
The geometry of the estimation support V should be simple (rectangular
in two dimensions or parallelepipedic in three dimensions) so as to make
the calculation of the terms y(v a , V) as easy as possible. The division of the
deposit into panels to be estimated should be in geometric harmony with
the regular data configuration. Thus, for the two-dimensional case shown
A

)(

## FIG. V.16. Relative locations of panel and information.

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351

## on Fig. V.16, the centred configuration A is preferable to the displaced

configuration B.
For the size of the support to be estimated, the practical rule is to define
a support equal to the grid size or some simple submultiple of it, ct. Fig.
V.I?

## It may sometimes happen that units have to be defined on variable

supports on the borders of a deposit or a lease. The kriging of these units
would then be considered separately, especially as specific problems (nonstationarity, preferential location of data sets) often arise which would
require particular forms of kriging.

Kriging of gaps in the data grid It often happens that certain points on a
data grid have, for various reasons, not been sampled (difficulty of access,
accidental interruption of drilling, not enough time, etc.). Before considering these gaps as missing data, it should be verified (d. section III.C.l,
"Review of the data") that, in fact, they are not zero values nor preferentially located in some way. Thus, consider a regular grid with nonpreferentially located gaps. There are as many different kriging systems as
there are different locations of the gaps, cf. Fig. V.IB. The presence of gaps
in a regular data configuration thus increases the number of kriging systems
to be solved.
In order to return to a unique kriging system corresponding to a
complete data configuration, each gap can be replaced by its kriged estimate. It has been shown, d. A Journel (1977, French version), that the

x
x

XI

X2

Gaps
X

## FIG. V.18. Regular data configuration with gaps.

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MINING GEOSTATISTICS

panel kriged estimator so obtained is identical to the direct kriged estimator, using the configuration with the gaps. Of course, the kriging variance
given by the completed configuration will underestimate that obtained
using the configuration with the gaps. In practice, prior kriging of data gaps
will be of interest only if there is a small number of such gaps. Then, the
kriging of these gaps can often be replaced by simply taking the arithmetic
mean of the neighbouring existing data, d. the Prony case study in section
V.BA.

## Random kriging Random kriging results from a principle of random data

positioning within determined domains.
For example, let Z (x.) be one of the data considered in the estimation of
a panel V. The location of Z(Xi) relative to the panel may vary from one
panel V to another V', d. Fig. V.19, but the datum Z(Xj) remains inside a
domain {3i of fixed location relative to the panel to be estimated. It is
assumed that the location x; in {3; is uniformly random. Similarly, (3i is the
domain of the uniform random location of the datum Z(Xj)'
x

x...1- Z(.Xi)

x
x

jJ,.

v'

v x
x

A
)(

Z(Xj)

## Particular values of Y(Xj, Xj) and y(x;, V) correspond to each particular

location of Xj in 13; and Xj in f3j of the data Z (x.) and Z (Xj), i.e., a particular
kriging system to be solved. The principle of random location of Xi in f3i
and Xj in f3j amounts to replacing the values -r(x;, Xi) for i -:I' j and y(Xi, V)
by their mean values when Xi and Xi describe their respective domains f3i
and (3i> i.e., by yC{3i, (3;) and f({3;, V). This results in considerably reducing
the number of kriging systems to be solved. For each panel V, i.e., for each
particular configuration (Xi, Xi), random kriging provides an estimator
which differs from the optimal estimator provided by strict kriging using
the true y(x;, Xj). What is gained in computing time is lost in precision for
each particular local estimator. This loss in precision can be evaluated by
comparing the estimation variance obtained by random kriging with strict

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353

## kriging using a particular and simple data configuration. In practice,

random kriging is used during the first stages of the estimation of a deposit
when no great precision is required from local estimations. During this
phase, local estimations are used more as an intermediate step before
proceeding to a global estimation by combining all the local kriged estimates; the simplified procedure of random kriging is then entirely justified.
A further justification occurs when there is a nugget effect in the structural
model y(h): the nugget effect removes the screen effect, i.e., it tends to give
the same influence to all data regardless of their distances from the panel
V. In such a case, the precise location of the datum Z (Xi) in {3; is not
important. When there is a pure nugget effect, all data with the same
support will have the same weight; this is the limiting case of random
kriging in which there is only one domain {3; with random locations, and (3
is the entire deposit or the neighbourhood of quasi-stationarity around the
panel V to be estimated.
The diagonal terms in the random kriging system Random kriging
amounts to replacing the rectangular terms y(x;, Xi)' i "" j, by y(P;, (3j) and
the terms )i(Xi' V) of the second member matrix by y({3;, V). However, this
approximation does not affect the diagonal terms Y(Xi, x;) which only
depend on the support Vi of each datum Z (Xi)'
If all the data Z(Xi), Z(Xj), . . . , have the same support v, then
y(x;, x;) = Y(Xj, Xj) = y(v, v)

## and, if the support v can be considered as a quasi-point, then y(v, v) = O.

If the domain (3i contains n, data Z(Xik) with respective supports Vi"
which are not necessarily points, and if each of these supports are located
at random in {3h then the support S, of the data within {3i is the union of the
n, supports V;k and the diagonal term becomes

with
'Vk~k',

and, finally,
(V.3!)

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MINING GEOSTATISTICS

The diagonal terms thus depend on the number and supports of the data
Z (Xjk) in each domain {3;, and the random kriging system is not identical for
each panel V to be estimated.
Some random kriging configurations The domain {3j of random locations
may be a volume, a surface, a line, or a set of discrete points, depending on
the type of sampling. Thus, for example, we have the following.

## (i) In three dimensions, (3; may be a polygonal envelope around the

panel V to be estimated, cf. Fig. V.20(a). The corresponding data
Z (x,J may be the grades of the intersections of drill cores with the
polygonal envelope {3;.
(ii) In two dimensions, (3j may be the surface of the basic regular data
grid, d. Fig. V.19, or the level at the top of a block, this level being
sampled by drifts, cf. Fig. V.20(b).

.---"""0'101

".~~
/1,. <,

[]

~--- Cross-section
( a)

( b)

rDrif'l
x

(c )

)(

( d )

## FIG. V.20. Some random kriging configurations.

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## V THE ESTIMATION OF IN SITU RESOURCES

355

(iii) In one dimension, (3; may be one of the sides of the panel V to be
estimated; this side may correspond to a drift sampled by channel
samples, d. Fig. V.20(c).
(iv) In two dimensions, (3i may be the discrete set of the eight nodes of
the first aureole of a regular grid. Gaps may occur at random in any
of these eight locations, ct. Fig. V.20(d).

## 3. Rapid calculation of the mean values for

Before being able to solve a kriging system, the system itself must first
be written down, i.e., the terms y(v"" V(3) and y(v"" V) which make up
the matrices [K] and [M2] must be calculated. Algorithms for the rapid
calculation of these mean values f were presented in section II.E,
and it is recalled that there were two possibilities: either (i) to calculate the
required mean value y(v,,", V(3) by approximating each of the two supports
Va and Vf:' by a set of discrete points, or (ii) to split the calculation of
the multiple integral y(v,,", v.s) into successive steps, some of which being
calculated in advance and presented in the form of auxiliary functions or
graphs.
Some authors, ct. J. Serra (1967a), Royle (1975), have constructed
kriging graphs which give the weights and kriging variances for various
particularly simple panel data configurations (two-dimensional and
rectangular) and for various isotropic structural models. Experience has
shown that such graphs are only useful as demonstration exercises; in
mining applications there is no reason for any particular case to conform to
the necessarily very limited frame of these graphs. It is certainly more
precise, and often faster, to construct a kriging plan adapted to the particular problem rather than trying to compel the reality into the prerequisites
of a preconceived graph.
The smaller the number of different mean values y(v a , V(3) to be
calculated, the faster the calculation of the y terms will be. Again,
the advantage of a systematic data file on a regular grid is obvious. It
may very often be possible to calculate in advance all the values y(v a , vs)
that will be required by kriging; these values could then be stored in computer memory and read out at the appropriate stage of solving each kriging
system.
Note that the random kriging procedure allows a considerable reduction
in the number of different relative geometries (v,,", V(3) to be considered. In
fact, the random kriging system retains only the value y({3i, (3j), whatever
the particular locations XiI< and XiI<' of the available data are in the two
distinct domains {3. and (3j.

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MINING GEOSTATISTICS

## 4. Preparation of a file for a kriging plan

A considerable amount of the cost of kriging is spent in defining the
estimation neighbourhood and searching for data within these neighbourhoods. This cost can very often be cut down by establishing a data file
adapted to the particular kriging plan. The set of data would be classified
and organized into a file which would permit rapid access to the data
required for solving each kriging system with a minimum of intermediate
calculations.
Consider the example on Fig. V.14. In the two-dimensional space, the
plan for kriging the square panel V consists of grouping all the data on the
nine grid squares {Vk , k = 1 to 9}, and considering the linear combination
of the nine mean grades on the nine domains V k , i.e.,

In order to build a data file adapted to this kriging plan, the space is divided
into a certain number of grid squares, each square being indexed by a pair
of integer coordinates (iUk> jVk)' For each domain V k the following
information is recorded in the data file.
(i) The number n of data (with support supposed constant and quasipoint) within Vk .
(ii) The coordinates (Ui,., Vi,.) of the location Xi,. of each of these data
(ik = 1 to nk). This information allows the calculation of the mean
values y of the first member kriging matrix [KJ:

## S being the support of the data within the domain V k

(iii) The mean grade of the nk data, i.e.,

## This information is used to construct the kriging estimator Z*.

The following section, V.B.3, details a typical program, CLAS, for the
classification of data from any given set of drill-holes (in all directions)
within a regular network of parallelepipedic blocks. This program is by no
means universal, it simply prepares a data file adapted to the kriging plan
used by program KRI3D. This second program for three-dimensional
kriging is also given as an example.

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357

## 5. Choosing an algorithm for solving kriging systems

An important element of kriging costs is the time involved in solving
the system of kriging equations. For a three-dimensional deposit sampled
by drill-holes in all directions on an irregular grid, it is common to have to
estimate a number of blocks of the order of 10000, each of which requires
a particular kriging system which may consist of five to 12 equations. Thus,
it is essential to make use of a solution algorithm that is both adapted to the
kriging plan and is particularly competitive in machine time.
The kriging (stationary or otherwise) system of linear equations has two
important properties that should be taken into account.
(i) The first member matrix of the system is symmetric.
Va are compact, which implies no grouping of the
data, and if the structural model does not present any hole effect, the n
first equations of the system written in terms of the covariance t are
such that:

## (ii) if all data supports

Va,f3.

The symmetry property of the first member matrix [K] suggests the use
of the square roots method (the matrix [K] is expressed in the form of a
product of two triangular matrices, one the transpose of the other: [K] =
[Tf [T]), or the Cholesky method ([K] = [R] . [D] . [R]\ where [D] is a
diagonal matrix). Both these methods can be found in B. Demidovitch and
I. Maron (1973, pp. 287-294). These two methods were tried and rejected
at the Centre de Geostatistique of Fontainebleau, because of the machine

If the structural analysis has been carried out in terms of the serni-variogram y(h), the
kriging system can be written in terms of the covariance C(h) = C(O) - y(h) = 1'(0) - y(h),
provided the covariance exists. If the covariance does not exist (e.g., a serni-variogram
model without a sill, i.e., I' (co) = co), a pseudocovariance C'(h);; A - y(h) can be considered, where A is a constant with a value greater than the greatest value of Y(v", v/l) that
can be encountered in the various kriging systems. Because of the non-bias condition
L.a A(3 = J, A is eliminated from the kriging equations, which are then written as

L A.aC'(v", VIJ)- I
fJ

1J.1b ~a =

C'(v", V),

'ria

1 to n,

and provided there is no data grouping, nor hole effect in the model:
'rIa,{3.

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MINING GEOSTATISTICS

time involved. Two other methods are used at Fontainebleau: the standard
pivotal method and the method of successive approximations when the
number of unknowns is greater than 20 (which is not often the case in
mining applications).
Program RELMS (due to P. Delfiner) The speeding up of the pivotal
method is due to the fact that there is no search for the maximum element
on each line of the first member matrix [K], the pivots being taken in order
along the main diagonal of the matrix [K] written in terms of the covariance.
For a stationary kriging system (with only one non-bias equation) the
pivots chosen in this manner are effectively the maximum elements. For an
unbiased kriging system of order k (with n + k equations of which k are
non-bias equations) all pivots after the nth are not necessarily the maximum elements. However, it can be shown that these pivots are never null
because of the existence and uniqueness of the solution of the kriging
system, ct. relations (V.1S), The listing of RELMS is given in section
V.B.3.
Program REITER (due to A. Marechals The method of successive
approximations, cf., in particular, B. Dernidovitch and I. Maron (1973, pp.
294, 316, 447) consists of putting the matrix equation [K] . [A] = [M2] in
the form [A] = [B] + [A] . [A]. Let [A (0)] be the initial approximation; for
example, the second member column matrix could be taken as the first
approximation, [A (0)] = [M2]. Successive approximations are then derived
as follows:

[A (I)]
[A (2)]

=
=

## [B] + [A] . [A (0)],

[B] + [A] . [A (I)],

[A (k+1)]

## If this sequence of approximations has a limit, [A] = limk-H'O [A (k)], then

this limit is the solution of the system [A] = [B] + [A] . [A], and thus also of
the kriging system [K] . [A] = [M2].
This iterative procedure will converge to a unique solution if any of the
canonical norms Iiall of the matrix [AI is less than unity. Considering the
usual three canonical norms, this sufficient condition for convergence is
written as

Ilall=max
j

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I"

i> 1

ja iil<

359

or
,..

L la;jl<l

Ilall=max
i

i= I

or

"

,..

liaW = ;=1
L L

laill

< l.

i~1

## An equivalent sufficient condition is that the greatest eigenvalue v of

matrix [A] is such that jvl< 1.
The Lusternik method for improving the convergence consists in correcting the last solution fA (k)] using this maximal eigenvalue v:
(V.32)
The maximal eigenvalue v can be estimated after each iteration (A (k l J by
the relation
(V.33)
or

## L [A ~k -I) - Aik -2)] .

(A

=- - - - - - - )-

A~k -I 11

i

## if the matrix [A] is symmetric.

The {A~kl, i = 1 to n} are the n values of the column matrix [A (klJ. The
imprecision in the estimation variance due to the approximation error
[A]- [A (k)] can be evaluated using formula (V.32). Let a~ be the kriging
variance corresponding to the exact solution [A] and
that corresponding to the approximation [A (k)]. The imprecision in the estimation variance,
A 2
2
2
b
UaE = (1"E - (1"K, IS given y

at

6,(1" E k =

v" [ (k)]2 ;
1_ v '7 k ii LiA i

(V.34)

## the {k ii , i = 1 to n} are the diagonal elements of the first member kriging

matrix [K].

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MINING GEOSTATISTICS

Using this last relation, the iterations can be stopped as soon as the
imprecision ~(Tlk is less than a given limit.
The listing of program REITER is given in section V.B.3.
As it is REITER works only for simple kriging systems, d. section
VIlLA.!, i.e. for kriging systems without any unbiased ness condition.
REITER should therefore be modified before being used in KRI3D.
For the sake of comparison, the solution times for an n-dimensional
stationary kriging system, written in terms of covariance, are given in Table
VA for each of the following programs.
(i) GELG (standard IBM subroutine) pivot method with search for
maximum elements.
(ii) RECHOL, Cholesky method.
(iii) RELMS, pivot method without search for maximum elements.
(iv) REITER, method of successive approximations with Lusternik's
6
correction. The precision on the variance calculation is dO"t < 1 0for a kriging variance close to 1. The number of iterations is 2 for
n ~ 10 and 10 for n = 40.
TABLE VA. Solution times

## for solving a system of linear

equations

Dimension,
n
5

10

15
20
30
40

GELG

RECHOL

RELMS

REITER

002
009
024
050

0-01
003

0-01
003

366

272

008
0-17
0-48
105

006
010

157

0-02
0-07
018
0-36
1-20

020
036

The times given in this table are only relative (in IRIS-SO seconds), the
same FORTRAN programs translated by another computer would result
in different absolute times.
Note that, for the most common dimensions (n :s; 15), RELMS is three
times faster than the standard GELG, and, for large dimensions, (n ~40)
REITER is more than 10 times faster than GELG. Moreover, the performance of RELMS and REITER can be improved if they are rewritten
taking account of the particularities of the machine language of the
computer used.

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## V.B.3 Computer subroutines

The following subroutines of data classification (CLAS), of three-dimensional kriging (KRI3D), of linear system solving (RELMS and REITER)
should be considered as initial elements of a kriging packaget that must be
constructed according to each practitioner's requirements. It is clear that a
unique kriging program, even a three-dimensional one, cannot suit (within
reasonable cost limits) all the possible data configurations and kriging plans
that could be considered in mining practice.
Program CLAS

## This program is designed for the classification of drill-hole data within a

regular network of parallelipedic blocks. The drill-holes are of any length
and direction in the three-dimensional space. For each block of the
network and, according to the index, when INC = 1, the number of distinct
samples having their centre of gravity within the block is calculated. The
characteristics of each of these samples (coordinates, length of the sample,
number of the drill-hole from which it originates, mean grade, etc.) are
then put into memory. Thus, for the block in Fig. V.21 which is intersected
by two drill-holes, two distinct samples are considered on drill-hole no. I,
and four distinct samples on drill-hole no. 2.
When INC = 0, within each block the samples belonging to the same
drill-hole are grouped, thus constituting pieces of core originating from

Four samples

Drill-hole 2

Drill-hole I

## FIG. V.21. Sorting of the data in a network of blocks.

tThe Centre de Geostatistique of Fontainebleau gives out (under certain conditions) its own
Mining Geostatistics package, d. Ch. Huijbregts (197 5b).

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MINING GEOSTATISTICS

362

distinct drill-holes (e.g., two pieces of core for the block of Fig. V.21). The
characteristics of these pieces of core are put into memory.
If required, the program CLAS provides a listing of this classification
once completed.

## Locating the blocks All coordinates are referred to orthogonal axes

parallel to the three main directions of the parallelepipedic network of
blocks, ct. Fig. V.22. The network is constituted of NX * NY * NZ = NB
blocks of identical sizes. These blocks are numbered in standard matrix
form (integer coordinates IXC, IYC, IZC). The origin of the network, of
coordinates XYZ(3), is located at the centre of block no. NY(IYC = NY
and IXC = IZC = 1).
Each sample is defined on its drill-hole by the coordinates (XF, YF,
ZF) - end of the sample.
Block no I

IZC = I

BlOCk no NY

IYC e NY

f--+-+----------(
IZC" NZ
Origin of the block
network XYZ (3)

o
Reference

IXC~

IXC~

NB

NX

axes

## FIG. v'n. Locating the network of blocks.

Organization of the data file Only the informed blocks (which contain at
least one sample or piece of core) have a memory reservation on the file.
The size of this memory reservation is given by the parameter NIMB maximum number of data per block; each datum being characterized by
nine values (the 2 x 3 coordinates of head and end of sample, the length of
sample, the number of the drill-hole and the mean grade); there are
9 x NIMB words reserved per block.
The location within the file of this memory reservation is read from the
integer table NUM (NX, NY, NZ). The number of data contained in each
informed block is given by the integer table NPB (NX, NY, NZ), with
1 < NPB >s; NIMB. If NPB;2< NIMB, only the NIMB first data are stocked in
the file.

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## V THE ESTIMATION OF IN SITU RESOURCES

363

The CLAS version given here corresponds to a direct word coding on file
(ell French computer). This type of coding can easily be adapted to the
particular coding required.

SUBROUTINE CL~SINS,XYZ,N)~ltD8,INC,Il,IFl,IF2
l,NIMB.NUM,NPB,IS'
C

## SORTING OLl CF ORIlL-~OlE DATA

WITHIN A REGULAR SYSTEM OF BLGCKS (CEFI~EC BY
ORIGIN ANr GRIO):SCRTING C~N EE SA~FlE B~ SA~PlE
DR BY DRILL-HOLE INTERSECTIO~ RESULTS A~E WRITTEN
C ON A DIRECT ~CCESS FILE
C

C
C
C

C
C PARAIitETEPS
C
C
C
C
C
C
C
C
C

C
C
C
C
C

C
C
C
C

C
C
C
C
C
C
C
C
C
C

N\$

## NUMBER OF CRIlL-~CLES Te EE SORTEe eLl

XVZ(3) CCOROINATES Of CRIGl~ OF 8lCC~ SYSTE~
fLOCKS ARE RANKED It-. INCREASI~G )C
It\ CECRE.6SHG 'y
F~C~ lOP TC M~ll IN l
~XYI(3J
GRID OF BLOCKS AL(~G OX.OY,C2
ORIGtN CF BLOCKS 15 CENTRAL lC BLOCK:
LINE NY,COLUMN 1,lE~EL 1
LINE 1 CORRESPONDS 1( ~AXt~u~ ~
OB(3) OI~E~SION OF BLOCK AlCt\G AXIS eX,CV.Cl
Il UNIT NU"8EA OF CRIll-HOLE DATA FILE
IFI(20)
FORMAT Of DRIll-HClE COll~R I~FCR~AT[C~
:DRILl-HOLE NUMeER,~L~BER CF SA~PlES,
COCRDINATES OF PCINl ~HERE S~~FlING
ST~RTS

IF2C20J
FOR~AT CF SAMPLES:EN( COORDl~AlE,~ALUE
NIMB MAXIMUM NU~8ER OF INFC~~AT[CNS (O~SIDEREO
IN A BLOCK:OEFINES RE(C~O LENGT~
NUM(NX*N~.Nl)
ACCRESS ON FILE IE CF eLCC~.IF
NUM=Q ai oc s (O",HIN\$ ~o Ir-.FCR"'ATIC~
~PBCNX.~Y*Nl)
INCICATES ~u~eER OF DATA OF BLOCKS

CPTIOt\S
INC.EQ.O All DISTINCT CRlll-relE INTE~5ECTIO~S
ARE CO~PUIED AND ~RITTEN
INC.EO.J SCRTlhG AND FllIN( SAMPLE 8~ SA~PlE
IS.NE.l
SORTING RESULTS ARE PRIN1EC B~ SVBRCU1INE

C ..... COMMOf\S
LINE PRINTER UNIT ~lMeER
C lOUT
INFERICR BCUNCARY CF EXISTI~G DATA
C TEST
IF
T.lE.TEST
MISSING OR El(~I~ATEC DA1A
C
C
C

C
C

IWRIT/1E,Ifl,IE2

## UNIT NU~BER~ OF CA FILES:

SORTED CA14,ARRAYS ~u~ AND

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NPB

364

MINING GEOSTATISTICS
OIME~SION

NUMIlI,NPB(l J,IFJ(20),IFi(iO),

lXYZ(3),N~YZ(3),OB(3)

(OMMeN ICUT,TEST

COMMO~/~RIT/IE,IE1,IE2

c
c

NX=NXYZC 1)
NY=NXYlt2t
hl=NXYl(31

NB=NX'\YNl
~OB=XYl(

YOB=XYZ(2)

## loe= XYZ (3)

00 10 l=l,NB
NPB(I)=Q
10 NUMCI}=O
ICl=O
NECT=C
~CUT=C

NTUT=C
IRE~NIMB9
.~ITE(IOUT,2000)XYI,NXYZ,rE

.RITE(IO~T,2CC1)NS,NIMB

C
C
C

SORTING

ORIll-~ClE/D~lll-~(lE

CO 4 ISO"'~l,NS
REAO(Il,IFIINUS,NECH,XC,'C,ZD
N081=C

NECT=f\ECT+NECtJND=l

00 31 IECH=l,NECh
REAOIIL,IF2JXF,YF,IF,T

IFIT.lE.1EST}GO TO 3
C

XC=(XF-XC)/2.+XO
VC=IYF-YCa/2.+YO

lC=(ZF-ICI/2.+Z0
HC:SOPT(XF-XC).I~F-XO)+C'F-YOI*(YF-'D}+

l(IF-IC)*(IF-IC)}

IXC=INTXC-XOBJ/OBIIJ+l.!)
IYC:I~T(YC-YCB)/OB(2)+1.!)

IIC=INTt(Z08-1CI/C8(3J+1.!J

c
c

IYC=NY-Iye.!

If(IXC*CNX+I-IXC).LE.OIGC 10 3
IFIIYC*INY+I-IYC}.LE.OJGC fC 3
IFtllC*(NZ+l-IIC).lE.OJG( 10 3
NOBL=IYC+NY*IIXC-ll+NY*NX*IIlC-!J

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## V THE ESTIMATION OF IN SITU RESOURCES

IFIIA[.GT.01GC TO 11
ICL=ICL-tl
NUM(NCBl)=ICL
11 IFlINC.EC.01GC TC 2

SORTI~G

AND

W~ITING

SA~PLE/SAMPlE

NPBCNCBL1=NPB(NOBl)+1
IF(NP8lNCBL).GT.NIMB)GC l( 13
12 NCUT=NCUT.l
C

## ON DIRECT ACCESS FilE IE AT

IDA IN WORDS

C
C

W~ITE

~C(RESS

C
IDA=IRE*(NUM(NOBl)-1)+9.(~Fe(N08l)-lJ

WRITE DISC

IE,IDA,NUS,XO,~C.IO.XF,'F,ZF,HC,T

GC Te .:
13

~RJTECIO~Tt2002)hCBl~I~C,J)C,IIC,NFB(NCBl)

GO TO 3
C

## COMPUTATION CF DRllL-HCLE INTERSE(TICh

C
C

2 IFCNOEl.EQ.OIGC TO 23
IF(NOEl.NE.Noel1GO Te 22
If(NPECNC8L).GT.NIMB)GC TC 3
I\CT=NCT"l
TV=TV+f-iC*T
t1l=Hl+t;C
XTF=XF
YTF=YF
ITF=Zf
IF(IEC~.lT.NECH)GO TO ~

20

END OF DPIlL-hCLE

C
21 JND= 1

IFCHl.lE.O.JGC TO 4
TV""T\I' IHL

NCUT~'CU1+NCT
IDA=IRE*(NU~("OBl)-1)+9.(~Fe(NOBL)-1)

c
c

C
C

WRITE DISC
lZTF,Hl,T'V

JE.IDA,NUS.JTtt'TO.ZID.)lf,~TF,

GO TC 4

## .RITE ORILl-HClE INTERSEClION

22 IFlNPECNOBl'.GT.NIMB)GC lC 23

XTF=)(C

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365

366

MINING GEOSTATISTICS
YTF=YC
ITF= Z r
TV=lV n"L

NCUT .. to.CUT+NCT

C
C

## WRITE ON DIRECT ACCESS FILE IE AT

IDA I~ WCROS

~OCRE~S

C
IOA=IRE*CNUM(NOBl)-ll+9*C~feCNOBl)-lJ

## WRITE DISC IE,IOA,NUS,X1C,'TO,Z10,)lF,YTF,ZTF,HL,TV

23 NOB1=t..OBl
t..P8(NCBl)=NPB(NOBL)+1
IFCNPBCNOBl).GT.NIMBJGC Ie 13

## INITIALize NEW INTERSECTICt..

C
C

NCT= 1
NTUT=NTUT+l
JND= 1
IF(IEC~.EQ.NEC~)GC

TO 12

JNO=O
XTD=XC
YTD=YC
ZTD=Z r

XTF=Xf
YTF=Yf

ZTF=: If
TV=HC*T

HL=HC
3 CONTlt..UE
XO=XF
YD=YF
ID=Zf

31

caNTI~ue
IF(INC.EC.O.A~C.JNO.EQ.OJG(

TO 21

4 CONT I to.UE
DO 41 I J l =1 , NB
41 NPB(IJL)=MINOCNPe(IJL),Nl~E)

IAO=O
WRITE DISC IEl,IAO,(~U~(I),I=l,NB)
IFCINC.EQ.OlGO TO 42
WRITE(IOUT,2003)ICl,NECT.~CUT

GO

42

c
c
c

TC

43

W~ITECIOUT,2004)ICl,NECT,NCUT,NTUT

43 IFCIS.EO.I1GO TO 5
PRl~l

wRITEIICUr,2010)

co 41 L=l,NI

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RESULTS IF IS.to.E.l

## V THE ESTIMATION OF IN SITU RESOURCES

367

11=IOE-CE(3)*(L-l)
~RITE(IOUT,2C05)L,ZZ,TESl

DO 47
00 46

I=l,N'I'

J=l,NX
IJL=I+NY*(J-l)+NX*NY*{l-])
If(NP8IIJL).EC.OlGO TO 4t
VY=YOe+08(2J*(NY-I)
WRITE(IOUT,2006II,YY
r AD= NlM ( (Jl J
XX=XOE+OS(I)*(J-l)
WRITECIOUT,20C7IJ,XX,NFBCJJl),IAC
NO=N P E ( I Jl )

DO 45 1(=I,NG

WRITE(IOUT,2008'NUS,XO,VC,ZO,XF,YF,IF,Hl,T
IDA= ICA+C;
45 CONT II\UE
46 CONTItl:UE
47 CONTI~UE
C

2000

FORMAT(l~l,'SCRTING eN
l'INFO~MATIONS

2'XO=

## FJlE OF CRIll-hOLE "

'IIH ,5X,'BlCCK GRID: CRIGIN "

',FIO.3,' VO=

',FI0.~,'

ZO= '.FIO.3/lH ,

## 317X, , NUfiIlBER IOX='.I4,' IC'-=',14,' /(1=',14

4/lH ,11X,'
SIZE OBX=',FIC.3,' OB~=',FI0.3,
5' OBZ=',FIO.))

2001

FORMAT(l~ ,'NUMBER
l'MAXI~UM NUMBER Of

OF DRIll-HOLES: ',I4,lOX,
INFOR~~110NS TA~E~ IN A '.

2'BlOCK =',12)
FORMAT(l~ ,' BlOCK
',IS,' IYC;',13,' IZC=',
113,' IIC='tI3,4X,'NU~8ER (F DATA 'tI3,'A80~E "
2'AllOwEO MAXIMUM')
2003 FORHAT(I~O,' NUMBER OF BLOCKS COhT~I~IhG DATA'
1,' .',15/lh ,'TOTAL NUMBER OF SA~PLE5 =',16,
2' NUMBER OF SAMPLES TAKEN ~',I6)
2004 fORHAT(l~O,' NUMBER OF BLeCKS COhT~I~ING DATA'
1,' ;',I5JIH ,'TOTAL NUMBER OF SA~PLE5 :',16,
2' NUMBER OF S,tMPLES TAKEr. =',16,' I" THE "
316,' INTERSECT IONS WRllTe .. f
2005 FORMATflllHO,' LEVEL ',15,' ElE~ATJOh,
l'lz ',FlO.3,' LIST OF \ALUES PER BLOCK*.'
2/1H ,Ell.5,'sMISSING ~Al~E5'.
2006 FORMAT(lHO,'.lINE ',15,' '=',FIC.3)
2007 FORMATCIH ,'COLUMN ',15,' X=',F1C.3,' hU~BERI
1,' OF DATA .',12,5X,' FilE AOORESS~ ',16)
2008 FORMATCIH ,'DRILL-HOLE ',18,' ST,tRT AND END'
I,'COQRDINATES '.6(2X,F8~3),SX,'LENGT~ =',FIO.3
21tH ,'VALUE ',FIO.3.
2010 FORHAT(llllHO.

2002

5 RETURfIi
END

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368

MINING GEOSTATISTICS

Program KRI3D

## This program is designed for the kriging estimation of parallelepipedic

units from three-dimensional drill-hole data, previously sorted by the
program CLAS.
The unit Uk to be estimated is a submultiple of the block B of CLAS
network. There are NSB units to be estimated in each block B. The
location of these units Uk are deduced from the location of the block B by a
system of integer coordinates similar to that used in CLAS (the unit no. 1
within B is located at the superior left corner of the upper level of B, cf.
Fig. V.22).
The same estimation neighbourhood is considered for all the units Uk
comprising block B. The form of this neighbourhood is parallelepipedic,
centred on block B, and consisting of an integer number of blocks B;. The
data within the estimation neighbourhood are grouped into NL sets. Each
of these sets consists of the data which are within a certain number NP
(L = 1 to NL) of blocks B, comprising the neighbourhood. A particular
kriging weight will correspond to each of these NL sets of data. All the
units Uk of the same block B are estimated from the same data configuration; their kriging systems differ only by their second member matrices.
The mean value C(LI, L 2 ) calculations, between two sets L 1 and L 2 , are
carried out taking into account the precise locations of each datum (sample
or piece of core) constituting L 1 and L z . For the calculation of the mean
values C(L;, Uk), the unit Uk is approximated by a regular set of discrete
points.
The algorithm used for solving the system of linear equations is RELMS,
cf. following section.
The total volume to be estimated is contained in the network of blocks B
defined by the program CLAS, and is constituted by an integer number of
blocks B; (those indicated by the index IEXP = 1). All the units Uk, comprising a block B to be estimated, are estimated.
Subroutines used KRIGMS+RELMS, for the writing and solving of the
kriging system, When kriging systems with large dimensions (more than 20
equations) are to be considered, RELMS (pivot method) should be
replaced by a modified version of REITER.
CBARS, for the calculation of the various mean values C. The structure
of this subroutine is similar to that of FUNCTION GBAR given in Section
II.E.1.
VOIS, for the data search within the estimation neighbourhood of each
block B,
FUNCTION COVA, for the calculation of the covariance function C(h)
defined as a nested sum of basic models (exponential or spherical). Except

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## V THE ESTIMAnON OF IN SITU RESOURCES

369

for the definition of the nugget effect, the structure of this subroutine is
similar to that of FUNCTION GAM given in section III.B.5.
t

C
KRI-30 KRIGING

C
C KRIGING

O~

T~REE-DIMEN~lCN_l

PROG~~~

SYSTEM Cf

B~OC~S

## C DEFINED BY ORIGIN ANt GRID F~C~ ORILl-~(lES WITH

C ANY CRIENTATION .. CATA HAve BEEP! INllIAlLl' SORTED

C IN BLOCK SYSTEM

B~

elOC~

## C MIGHT BE SUBDIVIDED IN A SVSTE~ OF SUB-SLeCKS wITH

C SAME KRIGING NEIGhBOURHOOD
C

C PARA"ETE~S
C
C
C
C

C
C
C
C
C
C

C
C

C
C
C
C
C

C
C

C
C
C

C
C
C

C
C
C

C
C
C
C
C
C
C
C
C

## :SORTING OF CATA lIN BLOCK SYSTEM)

XOB,Y08,ZOB ORIGIN OF BLOCK !'STEM : CE~TRE OF
BLOCK LOCATED DOWN lEFT ON TOP lE~EL
tax,DeY,CBI SIZE OF BLOCKS ALONG AXI~ (X,OY,Ol
NX,NV,NZ BLOCK SYSTEM 15 AA~'NGED AS I~ <ClAS>
HIMB SEe SUBROUTINE ClAS
TEST,JS SEE COM~ON AND PR!P!l (PTICN
,.S8 ,NSY ,HSZ SUB-BLOCt< SYS 1EII :lJ,J TH IN E'CH BteCK
:KRIGING PLAtNL
NUIlIBER OF DATA GROUPINGS (KRIGING I.E IGIHSJ
IJV(3) NUMBER Of lINES,COL~~~S,lE~El~ (f
PARAllELEPIPEOIC K~IGING NEIGHBOURHOOO
NUMBER OF BLOCKS COP~ESPONDI~G TO EACH
"PCf~l)
CAT' GROUPING
NoeIMAX.NP) LOCATION OF THE!f BLOCKS .ITHIN
NEIGHBOURHOOO.BteCKS ARE NlMBEREC
ACCORDING TO LINE ,COlU~~,lE~El
POSITION IN Nel'~BOURHOOC
:STRUCTURAL P~~AMETERS
NST.cO NUMBER OF STRUCTUAES,~UGGET EFFECT
C(NST)
SILLS
A.eNST) RANGES:<O. EXPCNEN~I'L,>O. SPHERICAL
C'XtNST,3J,CAYtNST.3J,CAIIN!1,3) SEE <C(~A>
NBX,NBV.NBl SUBCIVISION Of SUB-8LeCKS fOR
COVARIANCE CALeltATICN
:tIRECT-ACCE~! fILES U~IT ~U~BERS
IE: SORTED DATACPROVIDED BY <elAS
lEX: AR~AY IEXP
LOCATION Gf KRIGEC BLeCKS
I EXPCNX*NV*NIJ =1 KAJ~EO BLOCK =C NO KRIGING
IN SA"E OAeER AS BLCCK SYSTE~
lEI. AR~AY NUM FROVIOEO ev LAS>
IE2: ARRAY NPB PROVIDED fy LAS>
IE2: ARRAY NPB PROVIDEO BV eLAS
10:
OUTPUT FilE OF RESULTS.ONlY BlCCJS SUCH
THAT IEXP=l ARE WRIITE~ IN SEQUENCE.wITHIN
EACH BLOCK SUB-BLOCKS APE NU~8E~EC ACCORDING
TO lI~e,CCLUMN.lEVEl

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MINING GEOSTATISTICS

370

C MEMOR~

peCUIRE~EN1S

CB,CBl,CB2

~S8=NSX.NSY.~5Z

C
C
C
C
C
C
C
C
C

XOB.YCB,ZOB
IP

3.N8T=3.(~P'1)+NPC2)+

NIH

~SB.NBX*N8Y.~EZ

+NPC~L))

~l.l
~8T.NIMB

XA,YA.ZA
TA
TK,SK
NUM,NPB
R,RR,X
A

~l

NX*NY*NS8
NX*NY*IJV(3)
(NL+l'*NSe
(HL+l)*CNl+c)J2

C (0I4MC,..5
LINE PRINTER UNIT
C lOUT
C

C
C
C

C
C

~lMBER

TEST
lNFERICR BOUNDA~Y CF EXI5TI~G DATA
IN CASE Cf fLOCK WITH NO INFCPMATICN
VALUES A~E THEN =1E51.
PARA~&lERS FeR ~(IS
l'JOtSII
PARAMETERS FOR C(~A
ISTRUtl

CPTIOfl,S
IS.EQ.l STATISTICS Of ~RJGE( BLOCKS _RE PRINTED
C IS.NE.l ALSO RE5UlTS BLOCK/BLOCK ARE P~I~TEO
C
C

C
C SUBROUTI~ES CALLED
C MRIGMS+RElMS KRIGING
C
C
C

VOIS
tBARS
COVA

SYS1E~+~OlUTION

## NEIGHBOUPHOCO OF ESTIMATEC flOCK

AVERAGE COVARJA"CE
COVARIANCE FU~ClICN

C COMME"TS:
BLOCKS ARE NUMBERED AS I~ SteFoUTINE CLAS:BLOCK
C
C ~UMBER '+NY.(J-l)+NV*NX*(L-I) IS eN LINE I,
C COLUMN J,LEVEl l lINE 1 CC~RESPONCS lL~"S Te
C ~AXIMUM ~ COORO[NATE,COlU~~ I to CPIGIN,lEvEL 1
KRIGING PLA~:~E[GHBCU~H(C[ IS
C TC TOP lEVEL

## DEFINED AS ~ PARAlLELEPIPECIC ENVELOPfE MADE UP

CF BlOCKS INFOR~ATIONS BeL(~(ING TC ELeCKS OF
TtE ESTI~ATION NEJGHeOU~HCCC IPE GPCUFE[ IN NL
C SUB-SETS EACH OF WHICH Will t'VE A PA~TICULAR
C KRIGING WEIGHT (HOICE OF T~E5E SLe-SET~ SHCULD
C BE ~AOE ACCCRDING re lOCATICh,SVM~ET~\,EIC TO
C DEFINE A PAPTICUlAR SUB-SET L THE NUMEE~S OF THE
C BLOCKS DELINEATING IT ARE GJ~EN AS A lI5T OF ~P(L)
C ~AlUES (ARRAY NCe) THESE BLeCKS ARE ~U~8eRED
C ACCORDING TO THEIR RELATIVE LINE (1 TC IJVCIJ)
C POSITION 11 (LINE 1 CORRESP(~[S ALWAY5 IG Y
C ~AXIMUM),COlUMN (1 TO [JV~2) AND LE~EL (1 TO
Co
IJV(3)) POSIT10NS 12 AND 13 B' 11+JJVCll.(12-1'+
c JJV(2).IJV(1)*CI3-1) SUC~ ~U~BERS ARE CC~VERTEO
C INTO ~ElATIVE POSITIONS USEC EY SU8ROlTl~E VOIS
C CSEE AR fl:.A YIP J
C
C
C

.",,,.,.......

..

.. ~,"""ebook.blogf
..com

## V THE ESTIMATION OF IN SITU RESOURCES

371

c
c ........
C

c
c

OIME~SlCN NP[lO),NOB(75)
DIMENSION C(5),A~t5),CA~C!,3),CA~[~,~),CAlI5,3J
DIMENSION CB(S),CBICSJ,CB2IBJ
DIMENSION XOf(64),YOB(641.ICBC64J
DIMENSION IP(225),NINIIIJ.1A(lC)
OIMENSIO~ XA(150),YAI150),ZAt150)
DIMENSION TKIEOOJ,SKISOOJ,JEXPIIOQ)
OlMENS10N NUM(300),NPBI3CC)

DIMENSION R(SBJ,RR(SB),XIEJ,A(33C)
DIMENSION
CO~MCN

IJ~e3J,N~Yl(3J,)'1(3),OBe3J,JlI3)

ICUT,lEST

COMMO~/VC(SI/~IMBtXB,Ye,lE,IJVL
COMMON/STRUC/NST.CO,C,~A,('~,CAY,C_l

## READ INPUT PARAMETERS AKC FRINT TIlLE

C
INP~

IOUT-=3

INl,TEST,NIM8,IS

cSX=CeX/NSX
CSY""C8YJfliSY
CSl:CEI/NSZ
"SB=N5X.~SY.NSl

WRITE(tOUT,2000)OBX,OB~,Cel,XOe,YCE,lCe,~y,~X,

C
C

INI,OSX.CSY,CS1,NSe

CEFINE

NEIG~B(URHOOO

A~O

CJ1A

G~GLFI~G

~EAO(JNP,l003.(NP(l),l:1,~l)
~RITEIIOUT,20Cl)Nl

Nl=O

NLJ=[JV(l)
NlC=IJV(1)*IJ~(2)

IJVL=JJV(3)IZ-tl

NU~8ERS

DEFINI~G

GROUPING L

C
CO 100 l=l,NL

N=NP ( l )
REAOe INP,l003J(NOB( IJhIJ:I,N)
W~ITE(lOUT,2002)L,(NOe(IJ.,IJ=1,N)

DO 100 1=1, N

Nl=Nl+1

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372

MINING GEOSTATISTICS
N2=3(~1-1)

lB=(NCB(I)-lJ/NlC+1
JB=CNCB(I)-Nl(.llB-l)-l)/~ll+l

IB=NOe(I)-NLI*(JB-l)-NlC*(lB-lJ
IPtN2+1'=IB-IJV(lJ/2-1
IP(N2+2J=JB-(JV(2)/2-1
IPIN2.3J=lB-IJV(3J/2-1
100 CONTIr-UE

ST~UCTUP~L

FARA~EIEFS

C
~RITE(IOUT,20C3)NST,CO

DO 103

~=l,NST

lCAZtN, 1),1=1,3)
103

WRITE(IOUT,2004JC(NI,A~(~J,(CA)l(N,IJ,I=I,3Jt

## INPUT OUTPUT CIRECT

ACCE5~

FILES

C

NXYZ t l':::NX
NXVI12J=NY
NXYZ(?I=NI
XVZ(l}=XCa
XZ(2J=YCB
XY l ( 3 ).:;.ICB
CB(lJ=CBX
[;8(2)=08'1

CeC3J=CBI

SUBCIVIDE SUB-RLeCKS IN

FCI~TS

C

NCB=NeX*~BY.NBZ

11=0
00 14

lS=l,NSZ
Iso=ceI12.-0SI/2.-0SZ*tlS-lJ
DO 13 IS=l,NSY
YSO=C8V/2.-D5Y/2.-0SY*(IS-I)

Co 12 J\$=l,NSX
XSO=-D8X/2.+0SX/2.+0SX*(J~-1)

Jl=-l
J2=-1
XO=XSC-DSX/2.+0SX/(4.*Ne~J

YO=YSO-O\$Y/2 0SY/(4.*hB'I)
ZO=ZSO+CSZ/2.-0SZ/(Z.*hBZ)
L=O

DO 11 ll=l,NBI
J 1=- J 1

XO*XC+Jl*OSX/CZ.*NBXl
00 11 l2:::1.2
L=l+ 1

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## V THE ESTIMATION OF IN SITU RESOURCES

IFCL.~T.~BZ)GC Te 12
J2=-J2
YO=YO+J2*OSY/C2.*NBV)

00 10 J=l~NB)(
00 10 l=l,NB'1'
r 1= 11 +l

X08(Il)=XO.(J-1)*OSX/NBX
YOBCll)=YO.(!-l)*OSY/NBY
10 loBt j 1)= 20
lO=lO-OSl/NBl
11 CONT If\UE

12

CONTI~UE

13

CONTI~UE

14 CONTIhUe
C

## COMPUTE VARI.hCE Of A FOINT

~ITHIN

SlB-BlOCK

C
NINt 1)=1

NIN(2)=NC8+1
CALL

c
C
C

CBARS(XCE.YCe.IOB.l~,NIN,XCe,YCBtIDB,

IhOB,NSB,CBJ
KRIGI~G

## BLeCK PER BLeCK

"Bl= a
UK=O.
IIK=O.
~K=O

"O=NX*~Y

IFtIS.NE.IJWRITECIOUT,20CtJ
00 1 lB=l,NI
le=IOE-C~I*(Le-1J

IF(IS.~E.l)W~ITE(IOUT,20Ci)L8~Ze

NBK=O
lO=MAXO( 1, lE- IJV (~J/2 J
ll=MI~O(~I,LB.IJV(3J/2J

C
C
C

PEAC IEXF,NuM,~pe
ON RESPECTIVE UNJ1~ IEX,IEl,IE2

~1=NX*NY*LI-IAC2.1

## READ CIS C IE 2 , I A0 2, NU,.. ( J J J, I J'" 1 , ~ 11

COMPUTE LINE BY Lll\E
CO 2 I8=l,NY
CO 3 JE!"'l,NX

## IJB= I EHa* (JB- U

IfCIEXPCIJB).EQ.O)GO Te 2

f\ e1(= N E! K + 1
~BL=NEl+l

XB=XOf+ceX*CJB-IJ

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373

374

## MINING GEOSTA TISTICS

YB=YOf+OBY(N~-Ie)

Jl ( l ) = I 8
JLC21=J8
JLC3J=L8

c
C

SORT CUT

INFOR~ATICN

WJT~Jh

NEIGHe(U~hCOO

CALL

VOIS(Nl,NP,IP,Jl,NX~Z,NPB,hU~,IE.)A,~A,

lZA,NI~.TA.NAJ

If(NA.EQ.OJGO TO 37
If(NA.GT.IJGO TO 31
(NlY C~E JNFCRM~TJ(N fCLNC
CAll CBARS(XA,YA.ZAtl,1,~I~.XDB.YCf,2CB.NCB,

INse, CEll

CALL CeAPS(XA.YA,ZA,l,C.NI~.xoe,~CE,lCB,NDe,
INSB,Ce2)
DO 30 /C.=l,NSB
J=I(.N S8. CNBK-l)
TK(J )=TAC 1)
\$KlJI=C8CIJ-2*C82(K).CBICIJ
30

CONTI~UE

GO TO 34
C
31 CALL

## SOLVE KRIGIk SYSTE~

KRIGMS(NA,XA.YA.ZA,NI~.)Oe,YCe,lOB.Noa,

INSB,R,RR,A.X,IER,CBll
IFlIER.NE.OIGC TO 36
C

SCLUTION

DO 33 K=l,NSB
J=K.NSB*tNBK-l)
1TK=0.
lK={N,A+l)*K
SSK=cetl,-x( IK)

DO 32 l=l,NA
lK=I.(~A+l).CK-l)

TTK=TTK+X(IK)+TA(I)
32SSK=SSK-)(IKj*RRCIKI
TKIJJ=TTK
33 SK(JJ "5SI<

34 00 35 K=l,NSB

J=K"'Nse* (NBK-l)
UK=UK+TK(JJ
VK=VK+1KCJI*TKlJJ
35 NK=NK+l
[F(IS.EQ.l)GO TO 3
PRINT OUT If IS.NE.!
WRITEtIOUT.200~)18,JB,)B,~f,NA

CO 40 K-l,NSB

## JzK+f\I sa* lNBK- L

40 WRITEtIOUT,2010)K,TKlJ),SKCJJ

GO

rc

3
ERRO~ED

S~SlE~

36 WRITElIOUT.200SJIB,JB

GO TC 38

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~o

INFORMAT]C~

## V THE ESTIMATION OF IN SITU RESOURCES

375

37 WRITE(IOUT,2012tJ8,JB

38 00 39 K-" NSB
,J-K+NSS* 'N8K-l)

TK(JI-1EST
39 SKCJ.-TESt
:3 CONTINUE
2 CONT INUE
N2-H8K*NSB

RESULTS ON FILE

W~ITE

IAC=I.6C+2*NZ
1 CONT I t>.UE
C

P~INT

## STATISTICS CF KRJGEC BLOCKS

C
WRITe(IOUT,2013JCSX,DS~,CS2,NBL

IFtNK.EQ.O)GC TO 4
VK~(V~-UK.UK/"'K)/NK

UK=UK/f\K
WRITE(IO~T,2014JUK,VK,NK

## 1000 FCRMAT(6F8.3,313,F8.3, 11)

1001 FORMAT(3IZ)

FORHATl4I2}
1003 FORMAT(4012}

100Z

1004

FQ~MAT(IZ,F10.4J

laOS FORMAT(fl0.4,F8.3,9F4.2J
1006 FOAMATI5IZ)

## 2000 FORMAT( UH, 4t I ),4EX, '****Kf<IGI hG Of fLOCKS* '

111.*O~IlL-HOlE DATA SO~TING GRID (I'" BLOCK S',
2'VSTE~).'II.
sIze : 0)& ',Fe.3,' tv ,Fa.3
3,' oz. I,FB.3/tH " ORIGIN OF BLOCKS: XO-',
4FII.3,' YO- ',FII.3,' lea ,fll.!/lH ,
5' GRID OF ',13,' LINES, ',13,' CQlL~NS AND "
613,' lEVELS 'I' .~RIGee 8leCKS. SIZE : OSX- "

NUMBER',

## 8 'OF SUB-BLOCKS: ',(3)

2001 FOA"AT(l~
,'KRIGING PLAN: ',12,' .eIGNTS'/IH ,
2'WEIGHT:CORRESPONCING BLaCKS WITHJfI~fJGt180UAt,
3'HOOD (NUMBERS)')
2002 FOAHAT(l~ ,2X,12,2X,':.~~,40tl),I2J)
2003 FORMAT(lHO,' STRUCTURAL PAPAMETERS OF THE ' t
l'ESTJMATEO VARIABlE'/Ui ,'''UfltBER OF !TRUCTURES'
Z,t - ',II,' NUGGET EfFECl ,F12.~/'
SIll I ,
3'

RANGE

It' I

tAX

CA'I

.,

CAl' )

## 2001t FORHAT(lH .Fl0.4,lX,FB.3,1),9('I',F4.Z))

2006 FORHAT(lHO,4(!',48X,'.*KRIGING RES~lTS BLOCK "
1'PEA ELOCK**'II)
20~1 FORMAT(l~O,/tlH,'
.*lEVEl ',12,'
Z-',
IF8.311)

VtIWW.minebook.blogfa.com

MINING GEOSTATISTICS

376

2008

FO~MAT(l~O,.BlOCK
l'ERRO~ IN SYSTEM.')

IB=

',I:,'

J8= ',13,

## 2009 FORM.T(lNO,'.*.BlCCK IE= ',13,' JEs ',I),

l' XB= ',FII.3,' YB= ',FIt.2,' ~LMBER CF '
2,'WEI<: ... T5 NA=

',[),'***fJ

## ,'SUB-BlOCK ',1,' VALUE: ',FIC.3,

I' VARIANCE OF ESTIMATICN = ',FI2.5)
2012 FORMATClt- ,'*BLOCK IS= ',I?,' JB= ',{3,
l'
NO INFORMATION*')
2013 FORMAT(1"l,4(1),25)(,'***.I(I'IGI~GOF ElGCKS "
I' DX= ',F8.3,' DY= ',F8.~,'
D1: ',FS.3,
2' ****'/lHO,2X,I5,' KRIGEC BLOCKS SlATISTIC\$'
31lH "
AVERAGE
ClSPERSTCt- "
3' NUMBER OF elOCKS')
2014 FORMAT(l~ ,9X,FIO.3,2X,FIC.3,S),I4J
2010

FC~~AT{l~

STOP
END

SUBRCUTINE

KRIGMSINA,X~,~J,lA,NIN,)CE,YCe,lCB

1,NDB,~SB,R,RR,A,X,IER,CB)

C
CO~PUTE KRIGING MATRIX At-C SClVE S~STEM

C
(

C PA~A"'ETE~S

c
c
c
c
c
c

c
c
c
c

G~OUPI~GS

## XA,VA,IA COCROINATES OF O.TA ~ITHI~ NEI(~BOUR~OOO

hlN LOCATION OF tAlA GROUPING IN XA,'A,ZA
NSB NU~BER OF SUB-BLOCKS
R(CNA+ll*NSBl RIGHT-~ANO SITE MATRIX
'CCNA.ll*(N'+21/2) COEFFICIE~l MATRIX
XNA+ll*NSBl SOLUTION
KTlll
ERROR INDICATOR IF KIILT.NE.O

SUBROUTINES C'llED
(BARS
COVARI~NCE COMPUTAl](N
PElMS
SYSTEM SOLUTION

C
C
C
OIME~SION

xAtl),YA(l),IAll),XOBllJ,~CB(lJt

IIDIHlI,NIN(l)
o1ME NSI 0 ~ C8 0

RClJ,RRIU,AU ),)0'
NEQ=N ... 1
hN=(NEC+!.*NEC/2

OIME""SIO~

COMPUTE COEFFICIENT

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~ATP]~

## V THE ESTIMATION OF IN SITU RESOURCES

377

IHuO
CO 1 J=l.NA
00 2 l:l.J
IN:J N+l
CAll CaAPS(XA.YA.ZAtl,J,~Ih,X08,YCe,lD8,NOB.
INSB,CeJ
"C INJ=CaiU
2 CONT I'~UE
C.t.ll C8ARSCXA,YA,lAtJtC.NI~tXDB,YCetlCB,NCB.
INSB.CE)
00 10 l""l,NSB
f(=-J+NEQ+(l-lJ
R (K)-CBCL)

10 RR(K)-R(K)
1 CONTINUE
00 3

J=l.,NA

INaIN+l
3 AlIN)a}.
IN:I:IlJN+t
UJNI=O.

CO 4 L=l.NSe
K-NEQfoNE,. fl-H
RU:)-l

.. RR(K):ltl.
NSHaNSe
M...NEO

CALL

SYSTE~

SCLUTICN

RCU11~E

CALL REl~SIX,"'.R.~.NSM.IEPI
RETURN
END
SUBROUTINE

REl~SIX,A,8'~t~,KTllTJ

## OBTAIN SOlUllCN OF S~!TE~

OF SIMULTANEOUS LINEAR EQUATI(~S ~ITH ~~~METRIC
C COeFFICIE~T ~ATR[X UPPER TRIA~GUlA~ PA~T BEING
C ASSUMED tc BE STORED COLUMNWI5f
C

C
C PARAMETERS
C

C
C
C

C
C
C
C
C

## X(M*NJ FINAL SOLUTION

CM+ 1 J /2 J UPPER TRIAN~UL_F PART OF
COEFFICIENT MA1PIX DESTRO~EC BY
TRIANGULATION
BCM*N) RIGHT-HAND SIDE HATRI) DESTRO~EC
~
NUMBEP OF EQUATIONS
N NUMBEP OF RIGhT-~AND SIOE~ECTCRS
KTILT RESULTING ERROR PARA~E1ER
xr n, T:O NO ERROR
A

"'*

VtIWW.minebook.blogfa.com

378

MINING GEOSTATISTICS

c
c
c
c

## KTIL1=-1 NO RESULT M<I

~T[LT~K ZERO PIVOT WAS FOUND AT
ELI~INATION STEP K

c
C
C
C
C
C
C
C
C
C

.... COMMe~TS:
SOLUTION IS CONE BY MEANS ~~ GAUSS-Ell~INATJON
WITH PIVCTING IN ~AIN OIAGO~'l HehCE Nt SEARCH
OF ~AXI~UM ELEMENT EQUATIONS ARE KEPT IN ORDER
AND RIGHT-HAND SIDE ~ATRI~ IS FIRST ~EP1 SYSTE~
IS THEN SOLVED BY SI~PLY WORKING e'CKh'~OS
~TILTK I~OICATES HIGHEST RANK MINOR CE1(GCKJ)
IS ZERO IN GENERAL AFTEA T~lANGULATICN THE MAIN
DIAGONAL ELEMENT OF L[NE I N.! THE V'lue
'CI.C[+1./2J-OETCGCI/CET(GCI-l))
e REl~S SHCUlC BE USED ONLY FC~ SYSTEMS WITH NONe ZERO HIG~EST RANK MINORS

DIMENSION XCI),AC!),eCl)
IFCM.lE.C) GO TO 14
IFCM.GT.IJ GO TO 1
IFCAIIJ.EQ.O.) GO TO 20
00 19 I-1,N
19 XCI )-I!C I )lAC 11
KIll TaO
RETU~N

20 KTIll-1
RETU~N

INIT UlllE

C
C

1 TOl .. 0.0

KTIlY-O
NM,.N*fIl
/I( 1-M-l

KK-O

START TRIANGULATION

C
C

00 7 )(:II:1,M1
KK-KKiK
,61(#A (I( t< )

IFtAI(-TOll3,2,3
2 KTll T:I(
3

RETU RN
PlV-1./AK
II=KK

LP-O
K"'1""K-1
CO 6 I-K,M1
Ll .. II
11-[ 1+[

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## V THE ESTIMATION OF IN SITU RESOURCES

R-At ll)*PIV
lP=l P.. l
IJ" I [-KM 1
CO ~ J"J,Ml

IJ-I J"J

ll""ll+J
it

~CIJ)-~(IJ)-R*~CLl)

Lle-K,NM,flI
IN"'LLB+lP
5 BtIN)-81IN)-R*BellBJ
6 CONT I~UE
DO 5

CONTl~UE
~:cA(IJI

IFCR-TOL19,8,t;
8 KTIlT-M
lleTU~N

C
C
C
C
C

END

TRIA~GUl.TION

## START BACK SOLUTION

9 PI V-l./R

DO 10

llB.M,N~,M

10 XCllBJ-BtlLB)+PIV
taM

KK-IJ

DO 13 11*1 , M1
IlK-K.,-(

PIV-I.I... U(IO
1-1-1

DO 12 LLB-I,NM,M
IN-LLB
R-Be ltd
IJ-KIC

DO 11 J-J.Ml

IJ-IJ+J
IN-INti
11 R-R-A IIJ

'.JI(

12 Xlll8)-R*PJV
1'3 CONTINUE
C
C
C
C
C

IN)

END SOLUTION
RETURN
ERROR RETURN
14 KTJLT--!
Q:ETURN

END

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379

380

MINING GEOSTATISTICS
3~BRCLTINE

Ce~RSI~A,YA.l~.l,J,~IN,)Cf,~Ce,lCB

l,NOB,~SB,C8)

C
C
CCMPUTE AVERAGE CC\lRIA~(f eEl~EE~
C SETS OF CAT~ I A~C J OR Bla(~ IF J=~
(

C
C

P/IRAMETEPS
CBeNSB)

AVERAGE
CT~ER

COVARI~NCE

PftRA~ETERS

SEE

KRIG~S

C
C
C
C

SUBRCUTI~ES CftLlEC
COVA COVARIANCE FU~CTIrN

.................................................
o I ME'" 5 ION X /J. ( 1 ) , YIi 1 ) , 1A 1 It N IN ( 1 ) ,C E 1 J
DI~E~SION

XDBC11,~08(1),Z[e(1)

NC=O
CC 100 I=l,Nse
100 (811)=0.
IFe I .H:.CIRETliRN
Nl=NHdIJ
N2=NHdt+U-l

IFCJ.EQ.O)GO TO 3

AVER~~E

C
fIIl=NHdJJ
lQ=NUdJ+U-l

00 1

I2;:;Nl,~2

DO 2

J2=p.ll,~2

CB ( l

## =CB ( l ) + COVA, ( xAI 12) , '1 A ( 12 J ,I A 12 ) XA J 2 J ,

lYA(J2J.ZAI J2J)
NC=NC+l

2 CONTIJ\UE
1 CONT INUE
CBIl)=CB{lJ/~AXO(l,NC)

flETURt\

AVERAGE

COVARIA~CE

BET~EEN

~ET

I _NO

StB-BLOC~S

C
3 DO 4 Kl*l,NSB
Nt*a

00 5 J2*hl,~2
00 5 Jl*l,Noe
Jl*J l+ttOS* IK2-1)

C8(K2)-tBIKl'+COVAIXA(12),'1AfJ2),Z.(12',
lX08(J2),YOBCJ2),lOBIJ2))
Nt-Ne.l

## 5' CONT INUE

CB(K2)ceIK2J/~AXO(1,Nt)

tONT It<iUE
END

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## V THE ESTIMATION OF IN SITU RESOURCES

381

SUBROYTI~E VCISINL.NP,IP.4l.NXYI.~PB,~LM,IE,XA
1,YA,1~.N(N,TA,NA'

C
SORTS OUT I~F(RMATIC~~
C
C BELONGING TO NEIGHBOURHOOD OF ESTf~~TEC ELCCK
C AND GROUPS T~ESE BY SETS WITh SAME ~PI'I~G WEIGHT
C ACCQRDING TO KRIGING PLAN CA1' ARE READ ON FILE
C PROVIDED ev <CLAS>
C
C
C
C

C
C
C
C
C
C

PARA MEIE~ S
IPI3*N8TI LOCATION OF 8l0CK~ ~ITHIN NEI~HBCURHCOO
XAINIMAI,YAfNIMAJ,lAINIMA) C(CRDI~ATES CF DATA
NIMA=Nel*~I"B
I~

NINe NL+l1

## LOCATION OF EAC~ GROUPING

XA,VA,lA
TAINt) AVERAGE VALUE OF EAC~ GROUPI~G
COMMONS
SEE ~AIN PRCGPA~
IVOISll

ARRAYS

C
C

c
c

CIMENSION NPIIJ,JLlIJ,JP(lJ
DIMENSION NXYZ(l),NPB(l'.~l~IIJ
DIMENSION XAIIJ,YA(lJ.IA(J),NIN(lJ,T~ll)
COMMON/VOISI/NIMB,XB,VB,Ze,IJVl
"X=NX'tZC 1)

N't=NX'tlC2J
"'Z=NXYlI3J
NA"'O
IA=O
J::zQ

NINCl)=l
N=O

NEW

G~OUPING

CONSIOEREC

00 1 Jlal.Nl
NB=NPHLI
IA=NA.l
t"A=O.

TACI.&'=O.

NEW BLOCK CF

l~(S

GROLPI~G

DC 2 JII':l,NB
N~N+ 1
Nl=3.1I\-U

IO=JLCll+IPINl+!)
JO=JL(2).IPINlt2J
lO=JL(3J+IPtNl+3J
l8=IPC~1+3)+IJVl

IF(IO*(NY+I-IOJ.lE.OlGC lC 2
IFCJO*(NX+I-JCI.lE.OtGC lC 2

'#NNtI.minebook.blogfa.com

,_lJJi ...-'..J
t...... ' J
.-

,'~, ~ ~

. . . - ..,............

382

MINING GEOSTATISTICS

IF(LO*CNZ+I-lO).LE.O)GC lC 2
IJL*IO+NV*CJO-l)+NX*NY*(LB-lJ
I A0" NFe ( I J U

IF(IAC.EC.OJGC TO 2
IDA=9*CNUMCIJLJ-ll*NIMB

c
C

CO 3 I"l,IAC

t~TA

ON FILE IE AT

~CCRESS

IC~

lh WCROS

IEt]CAINUS.XC,'C.ZO,~f.YFtZf.HLtl

IF(T.lE.TEST)GO TO 3
J:J+l
XA(J)=~D+(XF-XC)/2.-XB

YAIJ)*YO+IYf-YCl/2.-VB
ZAtJ)=IC+(lf-lC)/2.-Z8
HA""'HA ..... L
TA(IA)=TA{lA)+T*~l

IOA-I[A+C;

3 CONT I"UE
2 CONTINue
IFCHA.EO.OlGO TO 1
C

G~OUPING

NINIIA+l)-aJ+l
TAU ,I TAIIAI/HA
NA-NA+l
1 CONTltoiUE

kITH tATA

FlETURN
END

c
c
c
c
c
c
c
c
c

COVARIANCE F~~CTION
DEFINED AS ThE su~ OF EXPO~EhTIAL CR SPHERICAL
STRUCTURES W~IC~ ARE ISOTRCPIC AFTER LINEAR
TRANSFORMATION Of THE INITIAL COOROI~A1E~ CF
T~E VECTOR (XI-X2.YI-Y2,Zl-Z~I

COMMONS
ISTRuel
NST NU~8ER OF ELEMENTARY SlFllCTURES
c co NUGGET EFFECT
c C(NST) SILLS
c AACNST RANGES <~. eXPONENTI_l
c
>0. SPHERICAL
c C~X(NST,3)tCA~(NS1.3 CAZ{N~1,3) llNeA~
c
TRANSFOR~ATICN OF ]~ITIAl CCCJCJ~ATE\$

c
c

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383

## OIMENSICN A(1),CCl),CA)Cl CAY(I.,CAIC1)

COMMOh/STRUCINST,CO,C,AA,Cj).CA~,CA2

COVA=O.
"X=X2-Xl

HY;Y2-Yl
HZ=Z2-Z1
H=SQRT(HX*HX+~Y.HY+HZ*HZ)

IFIH.GT.I.E-03)Gn TO 2
COVA=CO

2 00 1 IS-t,NST
IJS= IS
IJSlaJJS+NST
IJS2 aJJSl+NST
DXaH~*CAXtIJS)+HVCAX(lJS1'+HZ.CA)(IJS2)

DY-HX*CAYtJJS)+HV*CAYIIJSl)+HZ*CAYCIJS2)
DlaHX*CAZ(IJS)+HY*CAZIIJS1J+HZ*CAZtIJS2)
HaSQ~TIOX*OX+CY.OY+OZ*OIJ

IftACIS)lO,12.11
10

COVACOVA+C(IS)*eXPC~/AA(JS)

GO TO 1

11 JFtH.GE.'.IISGO TO 12
COVACOVA+C(IS).ll.-l.5*~I'A(lS).

## 10. 5*.,*tI*t11 CAA ( IS) -AAl IS) tAA .~J S U )

GO T(! 1

12

CONTJ~UE

1 CONTINUE
JlETUAN
END
SU8ROYTl~E

RelTERIX,A.~,~,EPS,[ER)t

c
c ~
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C

## OBTjIN jPPRCXIMATE SOLU1ICN

OF SIMULTANEOUS LINEAP EQUATIC~S AX~R LFPER
lRJANGUlA~ PART CF COEFFICIE~T MATRIX IS ASSUMED
Tt BE SlOPED COlUMNWISE

PARAMETEPS
XCM) VECTOR OF FINAL SOLUTJC~S
AIM*(M+11/2) UPFER lAIANGUL'~ PART QF
COEFFICIENT MA1~IX ILNM(CIFIEOJ
RIM) RIGHT-HAN[ SICE VECTOR
EPS TOLERANCE eN SOlUTIO~ (GI~EN '5 FRECISION
of I KRIG IN GI VARIA Ne E)
IER DIVERGENCE INDICATOR
IER:O Goeo SOLUTICN
IER=-1 DIVERGING S~SlE'
COMM ENTS:
THE MET~OO IS BY

successive

'PPRCXI~'TICNS

t As it is REITER works only for simple kriging systems (without the condition

NNI.minebook.blogfa.com

La A.. == 1).

MINING GEOSTATISTICS

384

## c IN WHICH THE FINAL SDlUTICN 15 IMPROvED (REF.

c tE~IOOVIT(H/MARON ELEMENTS CE (AleUl ~U~ERIQ~E
c ED. DE ~OSCOU 1913, THI5 ~El~ao ~AS BEE~
c ACAPTEO TO THE SOlUTICN OF A ~RIGI~G 5Y5TEM THE
c SCLUTION BEING CNlY ~PPRaXI~~TE T~E IIE~ATI\tE
c PROCESS IS STOPPED WHEN TkE KPIGING VARIANCE
c IS OBTAINED WITH A GOOD PRECI5ION (PAPA~ETER
c EPS E.G.:A GOOD LEVEL OF PFECISION 15 CBTAINED
c WITH EP\$=I.E-04 OR I.E-06 FCR A KRIGI~G \tAR[A~CE
c ABOUT 1.) ~AXIMl~ ~UMBER OF 11ERAIIC~S IS 2C.
c
c c.a PAC IT Y:
C

C
C

c
c

(SEE ARRAYS T

50 ECUATIONS

~~o

DT)

.................................................
OiMENSIO~

TiSO),[l(SC)

DIMENSION A{IJ,RIII,X))
INI1I~l

\$OlUTICN

c
c

CO I 1=I,M
lIU""O.
1 XIl)=R(l)

NEW ITER.aTION
1ER=O
fHTER=O
IT=4
2 CO 5 f(=l,IT
l=O
PRECEOI~G

SCllTION

DO 3 1=1,1'4

l=l+ I
~=XII)/A(LJ

eTC I )=V-l( I)
Til J =v

3 XCI)=FlIO
C

## CCMFUTE NE" SClUT rcs .

C
C

l=O
00 4 J=2,M

LENDaJ-l
l"'L+ 1

CO It I:zl,lENO
L=L+ 1

X(I)s)(IJ-A(l.*Y(JJ
4 X(JJ:xtJ,-AllJ*TII)
5 CONT I NUE

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## V THE ESTIMATION OF IN SITU RESOURCES

TEST

OF ~CCURACY

385

AND G(N\EFGENCE

C
L=O

fIoI1ER=N ITER. IT
CNMU =0.
ODMU =0.

CSIG=C.
DO 6 I:l,M
L=L+ [

## Y::XC I)fA IL J-I( I J

CNfllU=CNMUt-V*Y
COMU=CCMU.~.OT(IJ

CSIG=(SIG"~*V.A(lJ

ANU= O(~U

IOM~ll

CIVERGI~G

\$Y~lEM

IF(ABSCAhUJ.lT.I.OIJGO IC ~
DS:DSIG/CANU*(ANU-IJEPSJ
GIVEN ACClR~(' ACHIEvEC
IFeOS.LT.l.JGD TO 7
IfINITER.GE.20JGO TO 9

n-2

GO TO 2

FINAL SOLUTICN

1 ANU=l./IJNU-l.}
l=O

CO 8 l-l,M
L=<L+I
V""'XIIJ/AIl)
8

X'I)~V.(~-l(I))*ANU
~ETUflN

9 JER=-l
~ETUP~

END
V.BA. Typical kriging plans

Numerous complete geostatistical studies of actual deposits, from structural analysis to kriging, can be found in the bibliography. In this section,
some of these studies have been selected because they are typical examples
of kriging plans and their presentation will be limited to this kriging aspect.
Three types of deposit can roughly be distinguished: stratiform deposits
proven by vertical drill-holes; massive deposits (e.g. porphyry-copper)
proven by drilling on an irregular grid and in all directions; and vein-type
deposits proven by drilling or by drifts with channel samples.
This rough classification, based on geostatistical practice, does not correspond to any precise metallogenic limits. From a geostatistical point of

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386

MINING GEOSTATISTICS

## view, a thick vein (subhorizontal or not) can be treated as a stratiform

deposit in the plane of a vein. A porphyritic mass may also be regarded as a
stratiform deposit on each horizontal bench. Conversely, for a deep sedimentary deposit sampled by drives and percussion-drill fans it may be
advisable to use kriging plans analogous to those used for three-dimensional porphyritic masses, etc.
1. Stratiform deposits
In geostatistical applications, the techniques of stratiform deposit are relevant when the structure of the mineralization and the layout of the available data, in the three-dimensional space, favour one dimension with
respect to the other two. The most characteristic example is a subhorizon tal sedimentary deposit sampled by vertical drilling: the vertical
direction is the best known, and it is often the direction of preferential
variability (due to surface leaching by water for example).
The study of these deposits can very often be reduced to a two-dimensional problem, by considering, for instance, the horizontal regionalizations of variables accumulated over the total vertical thickness.
Prony nickeliferous laterites (New Caledonia), after Ch. Huijbregts and A.
Joumel (1972), A. Journel (1973b).

The deposit was sampled by vertical drills on a regular grid. The drill cores
were cut into constant I-m lengths and analysed for Ni grade and MgO
impurity. From the experimental profile of the Ni grades of each drill-hole
5;, a new, more regular and representative grade file of a small panel
centred on the drill 51 was defined, using an unbiased kriging of order 3
(with a quadratic vertical drift), d. Fig. V.23 and Fig. ILl in section II.A.I.
100 m

51

5,

1%

!Ie (% Ni)

Service
varia bles

(Q)

( b)

## FIG. V,23. Prony deposit (vertical cross-section).

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387

## The horizontal dimensions of this small unit panel are 10 m x 10 m.

Various nickel cut-off grades were applied to the small panel profile, d.
Fig. V.23(b); gc=O, 08, 1,1'2% Ni, A thickness t(x) and two accumulations in Ni and in MgO impurity correspond to each cut-off grade gc. An
accumulation a(x)= t(x).z(x) is defined as the product of the thickness
t(x) at point x and the corresponding mean grade z(x). These variables,
called service variables, are regionalized in the horizontal two-dimensional
space. These horizontal regionalizations are sampled on a systematic
regular grid of 100 m x 100 m and by two 20~m in-fill drilling crosses to
detect small-scale variability structures. The fitted variogram models are
isotropic and consist of the nested sum of a nugget effect and two spherical
models.
The data on the regular 100 m x 100 m grid were used to estimate by
kriging the service variables at each node of a 10 m x 10 m grid, cf. Fig.
V.24. This kriging is not intended to provide estimates of such small
10 m x 10 m unit panels, since the relative estimation standard deviation
would vary between 25% and 95% depending on the variable and the
location of the unit. The object of this kriging is to condition a simulation
of the deposit, cf. section VIII.B.l, or to provide estimates of larger panels
of various shapes by combining the kriged estimates of the 10m x 10m
unit panels.
The kriging plan The zone that was studied covered 176 squares of the
100 m x 100 m grid, I.e., 17600 small 10 m x 10 m units. It is, of course,
out of the question to solve 17600 kriging systems for each of the 12
studied service variables (three variables for each of the four cut-off
grades). First, note that provided the regular data grid is complete, the data
configuration is the same for all 1OO-m grid squares. This limits the number
of kriging systems to be solved to 100 for each service variable; there are
100 unit panels within each lOO-m grid square, to which correspond 100
kriging systems with a unique first member matrix [K] and 100 second
member matrices [M2] (v), v = 1 to 100. Furthermore, because of the two
symmetries with respect to the medians and diagonals of the 100-m grid
square, the 100 second member matrices [M2](v) can be reduced to 21, cf.
Fig. V.24. Thus, for each service variable, the problem of solving 17600
kriging systems is reduced to that of solving a single system with 21
different second member matrices. In such a case, kriging proves no more
costly than any of the more traditional estimation methods.
There are several non-preferential gaps in the 100-m data grid due to
nine accidental breaks in the drilling. These gaps were filled in, before
kriging was undertaken, by simply taking the arithmetic mean of the eight
data values surrounding each of them on the 100-m grid, cf. kriging of gaps

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MINING GEOSTATISTICS

388

in section V.B.2. All 100 unit panels within the 100-m grid square were
estimated considering the 16 data {Za, a = 1 to 16} of the first and second
aureola, d. Fig, V.24. The kriging estimator is thus a linear combination of
these 16 data,
16

Z*(V)=

Aa(v).ZO'

0'=1

for each of the 100 unit panels (v = 1 to 100). The dimension of the
stationary kriging system is, thus, 17 (16 weights A", plus the Lagrange
parameter J.L).
e

<XI

12

16

II

15

..

III

..

...

9-

"

..

..

"

..

II

,.

..

..

::/

'"
r&i~- ~.-.7"'~6"
5
.. "43 111 ...

4!I

~i

/34
./.:2.~

II

6 "

..
..

.
..

ill

## 21 31 41 5' 61 71 8' 9'

~-+-Iom

Unit to be es Ii mated

2i/I~
I~

...

/'4

10

100 m

.. I

<Xl

<XI
9

/~

lSI
13

## Togo sedimentary phosphates (West Africa), after A. Journel and H. Sans

(1974)
The deposit was sampled by vertical pits on a regular 50 m x 50 m grid and
analysed in O'5~m sections for percentage recovery p(x) of a granulometric

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## V THE ESTIMATION OF IN SITU RESOURCES

389

fraction and for the tricalcic phosphate grade z (x) of this fraction. The two
studied regionalized variables are the percentage p(x) and the accumulation a (x) = p (x). z (x). These variables are defined on the constant su pport, O 5 m x horizontal section of the pit, and are regionalized in the
three-dimensional space. In addition to the 50-m regular square grid,
data were also available on two diagonal lO-m in-fill grids to detect
micro-structures.
The three-dimensional variograms that were fitted consist of three nested structures:
y(h". h.. h w ) = Co + 'YI(.J(h~ + h~), h w )+ 'Y2(.Jh~ + h~))

## where Co is the nugget effect.

is a structure characterized by a marked geometric anisotropy: the
range in the vertical direction h.; of sedimentation is 1 5 m, while the
isotropic range is 25 m in the horizontal directions (h", h,,).
"Y2 is an isotropic horizontal macro-structure with a 30G-rn range corresponding to the horizontal silting of beds. The problem consisted of
kriging a 25 m x 25 m x 05 m panel slice P, d. Fig. V.25. The object of this
kriging was to produce a vertical mean grade curve for each mining unit of
25 ill x 25 m horizontal dimensions. Cut-off grades would then be applied
to this curve so as to define optimal roof-wall profiles, taking into account
various constraints such as ore quality control and accessibility to mining
equipment.
The kriging plan was essentially determined by the structure of the
Only the structure 1'1 differentiates the vertical distance h ; and, therefore, the DS-m sections in any given pit. The horizontal range of this
structure 'Y1 is 25 m, which is not greater than half the distance between the
pits. Thus, when kriging the slice P which is wedged against the pit 55, it
serves no purpose to differentiate between any of the vertical pit section
other than those of 55' Furthermore, since the vertical range of "II is 15 m,
it is only necessary to distinguish those data in 55 which are less than 1 5 m
from P; in the example shown on Fig. V.25, only the first seven sections of
pit 55 should be distinguished.
As the horizontal structure
is isotropic, the data from 52 and 56, 51
and 59, 54 and 58 can be grouped together because of their symmetrical
locations in relation to P. It was shown that, in regard to the structure "2,
the first aureole (pits 5 10 52, 53, 54, 56, 57, 58, 59) forms an almost total
screen again the influence of more distant data.
Eight different weights were distinguished in the kriging plan, as shown
on Fig. V.26: the five weights {3t, {32, {33, {34, /35 for the eight pits in the first
aureole (grouped into five sets, 53, 5 2+56 , 51 +59,54+58,57 ) ; and three

"1

"2

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390

MINING GEOSTATISTICS

weights, al, a2, a3, for the vertical data of pit 55 which runs down on one
side of the panel slice P.
Of course, if any of the data corresponding to the weights f3i or Cj are
missing, the dimension of the kriging system will be reduced accordingly.
The eight weights resulting from kriging the accumulation variable for
the particular configuration considered are also given on Fig. V.26. Note
that the value a 1 = 011 is significantly different from the value 1 which
I

\$,

52

53

r-

, ~
55 I

\$4 I

~--

--

57

56

8
---+-

_..J

56

..

\$9

5Cm

8
A

,
2

3
4

5
6
7
8
9

50 em

## FIG. V.25. The kriging data configuration in Togo.

would have been obtained if the value of each vertical section had been
assigned to the corresponding slice P (polygons of influence method). Note
also the significant influence of the eight pits of the first aureole, L f3 =
067; kriging of slices P results in little vertical differentiation. Although
kriging causes a smoothing effect (d. Chapter VI, relation (VI. 1, it is

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391

## normal for the mean characteristics of slices P of 25 m x 25 In x Q. 5 rn ::::

3125 rn3 to be much less dispersed than the characteristics of data defined
on pit sections of Q5 x 0 -pit =04 m 3 , and, indeed, production data did
verify this remark.

s
al'II%,
a2' 12a 0/0

A 54

113'9-3%

_.--...c--------5r:-6 B

## FIG. V.26. Togo's kriging plan.

2. Massive deposits
In geostatistical applications, a massive deposit is one in which the
geometry of the mineralized mass has no preferential direction of

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392

MINING GEOSTATISTICS

## elongation or flattening. In general, the mineralization is diffused and the

scale of the heterogeneities is small when compared to the scale of mining.
Estimation usually consists of evaluating a mean grade or a mean recovery
rather than determining the exact geometry of each are lode. The most
characteristic examples of this type of deposit are porphyry-type brecciated
masses (copper or molybdenite). Such masses may be proven on any sort of
grids with drills in all directions.

## Los Bronces copper deposit (Chile), after 1. Damay and M. Durocher

(1970)

The first campaign consisted of vertical drilling and horizontal drilling from
drives. All drill cores were cut into constant 2-m lengths and analysed for
copper.
The structural analysis, given in section IV.A, Case Study 2, resulted in
an isotropic structural model consisting of an exponential model with a
nugget effect:

'Vr E [2,60 m]
with r = Ihl = h~ + h~ +h:'.
A second campaign was then carried out with vertical drilling on a
regular 25 m x 25 m grid determined by the previous practical range 3al =
25 m. The object of this second campaign was to estimate the upper part of
the deposit (Donoso zone) which was to be mined by open-pit. The mining
units were cubic 12 m x 12 m x 12 In blocks, the horizontal dimension
being approximately equal to one quarter of the 25 m X 25 m square grid,
d. Fig. V.2? A single data configuration was used for the kriging of these
blocks. This configuration consisted of the following.
2

## (i) The estimated global mean grade of the Donoso zone.

(ii) the 4 x 24 grades of core lengths of 2-m support, located on the four
drill-holes Si> 52, 53, 54 in the estimation neighbourhood of each

## block. Taking symmetry and structural isotropy into account, the

dimension of the single kriging system can be reduced to 38 and
consists of:
one weight for the global mean grade;
12 weights for each of the three cores 51> 52, 53 (because of
symmetry, core S4 receives the same weights as SJ;
one Lagrange parameter required for the non-bias condition,
37

L x, = 1.

a =1

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## V THE ESTIMATION OF IN SITU RESOURCES

393

Figure V.27 shows the weights of each core as a function of the distance
to the median horizontal plane of the block to be estimated. Note the
following:

## (i) the rapid decrease of the curve 51;

(ii) the screen effect of cores 52, 54 on 53, which means that enlarging
the estimation neighbourhood would serve no purpose.
(iii) the relative importance of the weight assigned to the global mean
grade, Am =0143, which is due to the presence of the relative
nugget effect Csl C, = 028.
12 r-------~-------__.

10
8

6
4

I---i--+-'--~----~--~---"'

12

54

10

-2

-4

4
2

r-,

I'"

-2
-4

f\
K

-,

- __ .1_
I
I
I

"

-6
Weight (%)

FIG.

## Chuquicamata copper deposit (Chile), after I. Ugarte (1972)

This deposit is mined by open-pit and estimated by two types of sampling.

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394

MINING GEOSTATISTICS

(i) Drilling in all directions. Drill cores are cut and analysed on l'5-m
lengths. These samples are used essentially for long term estimation.
(ii) Vertical blast-holes analysed over the thickness of l3-m benches.
These samples are used for short-term estimation of mining units.
The regularized variograrns calculated from the lS-m core lengths were
compared to the horizontal variograms graded over the constant 13-m
bench thickness calculated from the blast-hole data. The three-dimensional regionalization of the copper grades, with the l'5-m support assimilated to a point, is characterized by a model of anisotropic, nested
variograms, specifying two horizontal directions and the vertical direction.
The presence of a proportional effect means that neighbourhood of quasistationarity must be considered.

## Short-term estimation of a mining unit (20 m x 20 m x l S m block) by

blast-hole data on previously mined neighbouring blocks is illustrated in
Fig. V.2B. Note that the blast-hole data available do not completely surround the block V to be estimated. On the Western front, with mining
advancing from East to West, all the blast-hole data is located on the
Eastern side of block V, and the reverse is true for blocks on the Eastern
front.

Block V to be estimated

## Ave ilable information

over nine blocks, a = I to 9

## FIG. V.28. Short-term kriging plan at Chuquicamata.

For kriging purposes, the blast-hole data are grouped together inside
each of the nine neighbouring blocks already mined, d. Fig. V.28. Since
the number and exact arrangement of the blast-holes within each of these
blocks are variable, it is assumed when calculating the mean values 'Y of the
kriging system that each block contains, on average, six blast-holes on a
regular 6 m x 10 m grid, the grid being positioned in a uniform random
manner within the block, cf. Fig. V.29(a).

~~-

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## V THE ESTIMATION OF IN SITU RESOURCES

395

Thus the terms Yea, f3) between two blocks a and (3 for which data is
available, each with six blast-holes, are approximated by the mean value
1

## -yea, (3)= 36 i~l i~l

Block

Y(Xi -Xj).

={

I to 30
or
I to 6

Block JJ
j : I to 6

; : I to 6

(a )

( b)

FIG. V.29. Discrete approximation of the blocks (a) blocks for which data is
known; (b) block V to be estimated.

If the object is to estimate the mean grade of the entire block V, then the
terms Ha, V) are calculated using a discrete approximation of V consisting of 30 blast-holes on a regular centred grid within V, d. Fig. V.29(b),
On the other hand, if the object is to estimate the arithmetic mean of the
six blast-holes within V, then the terms yea, V) are calculated using a
discrete approximation of V consisting of six blast-holes on a regular
6 m x 10 m grid positioned at random within V:

Yea,

V)

## ="6 j~l ii(x;, V)

with

y(x;, V) =

Using the blast-hole data file from the benches already mined, the
experimental arithmetic mean
of the blast-holes within V can be
calculated and compared with the kriging estimator Z ~ corresponding to
the discrete approximation (k = 1 to 6). Note that the same experimental
value zt. can be compared to two kriging estimators Z:<'w and Z~E
depending on whether the block V is considered as coming from the
Western front (the data a are then located East of the block) or from the
Eastern front (the data a are located West of the block), The comparisons

zt

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MINING GEOSTATISTICS

396

## (Z~w, zt) and (Z~E, zt) are given

Z~w ), and (zt - Z~E) on Fig. V.30.

by

## the histograms of errors

(zt-

If the results corresponding to the East and West data configurations are
grouped together, the following remarkable verifications can be made:
(1) the non-bias of kriging, the mean error being 000% Cu;
(ii) the representativity of the kriging variance, (T~ = 0117, which
very close to the experimental variance of 0118.

W front

IS

E front

01

>.

<J

'"
c;
::I

005

001
-05

05

% ell

FIG_ V.30. Histograms of errors for the stationary case in which the kriging system
is of dimension 10.
For the Western front, mean error = ~011 % Cu; experimental estimation
variance = 0-125 COlo Cu)"; number of values=410. For the Eastern front, mean
error = +011 o Cu; experimental estimation variance = 0111 (% Cu?; number of
values = 400.
Grouping the two fronts, mean error = O 00 % Cu; number of values = 810;
experimental estimation variance = 0-118 (% Cul; kriging variance O'~ = 0-117
(% Cu?; mean grade of the considered zone = 2% Cu.

On the other hand, if a distinction is made between the East and West
configurations, there is a significant bias of modulus 011 % Cu, the sign of

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397

## which depends on whether the block is considered as coming from the

Western front or from the Eastern front. This bias can be explained by a
local East-West drift that cannot be corrected because of the anisotropic
data configuration, More precisely, as shown on Fig. V.31, the Chuquicamata deposit is bounded by a large fault oriented N-S, and the variability
of the grades is related to a network of small faults with the same main
direction. On average, over the whole deposit, this E-W variability appears
on the mean stationary variogram as a marked geometric anisotropy in the
horizontal plane. Locally, at the scale at which the comparison (Zi, zt) is
made, this E-W variability may appear as a local drift of the copper grades
(non-stationarity). In fact, as all the data are located on only one side of
block V, any separation of these data by a mineralized fault will result in a
locally biased estimate; if all the data are located on the other side of the
block, this bias will be reversed.
w

'Moil1
fault

## FIG. V,31. Local East-West drift.

The solution consists in taking this local E-W drift into account in the
short-term kriging system, which means introducing more non-bias conditions than the single condition,
9

Aa = 1,

a=1

presen t for the stationary case. The variogram model 'Y remains
unchanged, and a linear drift is considered which, in the three-dimensional
space, amounts to introducing three additional non-bias conditions:
9

Aau a = Uv,

a=l

where (u"", Va, wa ) and (Uy, vv, Wy) are the three-dimensional coordinates

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398

MINING GEOSTATISTICS

of the centre of gravity of the block a for which data is known and the
block V to be estimated, d. relation (V.lO) in section V.A.2.
Using the same data file coming from the same already-mined benches
considered above, the comparison of the kriging estimates Z~w, Z~E and
the experimental values
yields (cf. Fig. V.32):

zt

(i) an almost null bias, whether the East or West data configuration is
used;
(ii) an excellent agreement between the kriging variance ai = 0261
and the experimental estimation variance. Note that the value 0261
is greater than the value 0117 previously obtained for the stationary

W front

E front

0-1

>.
u
c:

'"
cr
J

Lt'" 0-05

0-01

-05

0-5

% Cu

FIG. V,32. Histograms of errors for the non-stationary case in which the kriging
system is of dimension 13,
For the Western front, mean error = -0,02% Cu, experimental estimation
variance = 0247 (% CU)2, number of values = 397. For the Eastern front, mean
error = +001 % Cu; experimental estimation variance = 0273 (% CU)2; number of
values = 397.
Grouping the two fronts, mean error = 0,00% Cu; number of values = 794;
experimental estimation variance = 0261 (% Cuf, kriging variance. ui = 0261
(% CU)2; mean grade of the considered zone = 2% Cu.

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399

## case: what is gained in correcting the bias is lost in the kriging

variance.
An important aspect of this short-term estimation is the simple manner
in which the local.driftwas taken into account. It was enough to introduce
several additional non-bias conditions in the stationary kriging system
without modifying the structural model. It is advisable to use such a
procedure when the data is anisotropically positioned and if a local drift is
suspected, e.g., at the edges of a deposit.

## Long-term estimation of blocks B (60 m x 60 m x 13 m) from the core

samples only, is illustrated in Fig. V.33. The estimation neighbourhood
consists of a set of 3 X 9 = 27 blocks of the same size, centred on the block
B to be estimated, i.e., nine blocks on each of the three horizontal benches.
Each intersection of the drill cores with each of the three blocks of the
vertical column containing B is considered separately; on Fig. V.33, these
intersections correspond to the five weights 1, 2, 3, 5, 6. Each intersection

60 m

## FIG. V,33. Long-term kriging plan at Chuquicamata.

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MINING GEOSTATISTICS

## of the drill core with the aureole of 27 - 3 = 24 remaining blocks is also

considered separately; on Fig. V.33 these interesections correspond to the
three weights 4, 7, 8. A length la corresponds to each of these intersections, identified by its centre of gravity X a and a mean grade Z"". The
stationary kriging system can be written in the standard form:

..

Va = 1 to n,

(3=1

## where there are n intersections fa (n = 8 in the example of Fig. Y.33).

For the calculation of the terms y(la, t{3), a ~ (3, the lengths ia are
assimilated to their centres of gravity: y{la, f(3)= y(x a -x(3), Va {3.
Similarly, .y(t, B)::::" y(x a , B) where the block B is represented by a set of
discrete points. On the other hand, the length fa must be taken into
account when calculating the diagonal terms

## where F is the standard auxiliary function.

A certain number of blocks already mined, and thus containing blasthole data, were estimated by this long-term kriging. The resulting kriging
estimates Z[ were then compared to the corresponding arithmetic mean
Z~ of the blast-hole data within B. The resulting histogram of errors
(Z~ ~ Z~) has a mean of 0004% Cu (non-bias) and a variance 0093,
which falls within the interval of kriging variances [0045 to 0209
(% CU)2].

## Michiquillay copper deposit (Peru), after D. Guibal, (1976)

This massive porphyry-copper deposit was estimated by 135 vertical drills
on a pseudo-regular 100 m x 100 m grid. The variable studied was the
grade of total copper defined on the support of a core sample of 15 m
length, which is equal to the height of a bench. The zone studied here is the
upper enriched zone (secondary sulphur mineralization).
The structural analysis resulted in the following structural model:

## y(h u , h.; h w)= C o+ CtYt(r)+ C2Y2(.J(h~ + h ~,

where Yt is a three-dimensional isotropic model with a range at = 105 m
and Y2 represents an horizontal macro-structure with a range a z = 300 m.
As the data grid was not regular, but only of constant density (on average
one drill per 100 m x 100 m square and assumed to be located at random

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## V THE ESTIMATION OF IN SITU RESOURCES

401

within this square); a random kriging plan was adopted for this estimation
of 100 m x 100 m x 15 m blocks. The estimation neighbourhood consisted
of 27 blocks: the block B to be estimated and the 26 blocks of the first
aureole surrounding it; cf. Fig. V.34. These 27 blocks were divided into six
domains {Dcl'; a = 1 to 6} within which it was assumed that the data were
randomly located within each 100 m x 100 m square.
D} is the domain consisting of the block B to be estimated; D 2 and D 3
are the blocks located above and below B, respectively; D 4 is made up of
the eight blocks of the first aureole surrounding B on the same level; Ds
and D 6 consist of the eight blocks located, respectively, above and below
the eight blocks of D 4
The data inside each domain D a consist of no; drill intersections, all
vertical, 15 rn long and of mean grade Zo;o In the general case, the six
domains Do; will all contain some data (no; > 0, Va), and the kriging
estimator will be the linear combination of the six mean grades Zo;:
6

Z~ =

I
0;

= 1

x.z:

6

## A{{y(Du , D(3) + f.L = y(D at , B),

Va

1 to 6,

,B=!

A,a

= 1.

{3

100m

Lf------...---"~4__~
4

~I-p~5

15m

## FIG. V.34. Six domains with random positioning of the data.

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MINING GEOSTATISTICS

402

As the three intersections S1, S2~ 53 come from the same vertical drill
intersecting the block B, cf. Fig. V.35, then y(D lo D z) = y(Slo Sz), and
similarly for the other two rectangular terms y(Dl~ D 3 ) and y(D z, D 3 ) .
This relation holds, strictly, only when there is a single vertical drill intersecting each block B. For this reason, only one intersection per block was
retained in the classifying program of the data (nl; nz. n3 ~ 1); in any case,
only rarely did more than one drill intersect a block.

\$2

\$1

\$'3

## FIG. V.35. Data within domains D 1 , D 2 D 3

For the calculation of the other rectangular terms y(Da , D(3), a :-13~ the
hypothesis of random locations was used. Thus, for example y(Dl~ D 4 ) is
the arithmetic mean of the eight mean values 1(B, B k ) , k = 1 to 8, between
the block B = D 1 and the eight blocks Bk of the first aureole that make up
D4
The diagonal terms are

Similarly,

## The domain D 4 consists of eight blocks {R k , k = 1 to 8} and

intersections (0 < n4 ~ 8) and, using a formula similar to (V.31)~
1

y(D 4 D 4 ) = -F(15)+
n4

n. ".

L L

n4

drill

ii(Bk~ B Ic ,) .

k=1/('''''1

k'..=k

## The terms y(Bk; B k ,) correspond to the blocks in D 4 that are intersected

by drills. The same applies for y(D s D s ) and Y(D 6 D 6 ) . The hypothesis of
random locations is again used for the calculation of the terms y(Da , B) of

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## V THE ESTIMATION OF IN SITU RESOURCES

403

the second member matrix. Thus, for example, -y(D j , B)= y(8, B) and

1 8
y(D 4 , B) ="8 "~1 y(Bk' B).
In the case of a complete configuration (na > 0, 'tIa = 1 to 6, and
n4 = ns = n6 = 8), the following results are obtained:

A 4 = 0 ' 16 7,

A.s=A 6=0'215

fTi = 00212 (%

## Cui, for a mean estimated grade of 07 (% Cu).

Note that while the three intersections 51, 52, 53 of Fig. V.35
represent only 11 % (3/27) of the volume of the total data used in kriging,
they receive 40% of the total weight (A 1 + A2 + A3 = 0,403).

and

## El Salvador copper deposit (Chile), after M. Didyk and A. Marechal quoted

by J. Walton (1972)
The deposit (porphyry-copper mass) is mined by block caving and, for the
structural analysis, two types of data were available.
(i) Drilling in all directions. The drill cores were cut and analysed in
constant 3-m lengths.
(ii) Continuous channel samples along drifts, which were also analysed
in constant 3-m lengths.
The structural analysis resulted in a single structural model for both sets of
data. The model adopted is isotropic with a nugget effect and a hole effect.

'dr=lhl>A
Two types of mineralization can be distinguished from the geology of the
deposit. The first one is a massive breccia which corresponds to the
exponential model (1- e-a,), cf. section IV.A, Case Study 2, the breccia
structure of the Los Bronces deposit. The second mineralization is due to
intensive enrichment within fractures, having a pseudo-periodic character
corresponding to the hole effect (1 - (sin br)/ br).
In the kriging of a cubic 100 m x 100 m x 100 m mining block B, the
following data sets were used, cf. Fig. V.36:

## (i) the 10 clusters of drill intersections within the 10 horizontal slices of

the block, each having a thickness of 10 m;
(ii) the 10 clusters within the 10 slices of the aureole of a 300 m x
300 m x 100 m volume, surrounding the block B;

VtIWW.minebook.blogfa.com

300m

~.

Ie
I-

/_---~reOle
-----

/ /

---

/
/

0"'"

100m

0-"- /

,"- /

'i;>-~/

")0

Block B

//

a<\$'

Zx= 172
o-}=O-006%2

,0

/
/

z. 2-IBo/o, ,I' 21

A,o:; 33,

I
\

A,05-01, Z

ol-B3 %

A~"605,

02-I?o/0,,,=19

A7 6-93, Z, 2 20 %

, ,, '

\\'

11:

17

no 15

/
/

## All' 10-31,,2'01'46%, n> 15

AI~'7-77, Z.j-23 %

'"'

, ,, =19

E
0

, n o I5

'1-54%,n
~jBo~;:'/7

___

8
9

10

FIG.

~~~e

":>.,'//
1i,o/

v'/

'i;>-/
/

08

~~

1/
A
..... : I
I
J

## V.36. El Salvador's kriging plan. n = number of core samples within each

slice, Z = mean grade of each slice, A = weight in %.

## V THE ESTIMATION OF IN SITU RESOURCES

405

(iii) the set of channel samples taken along the drifts which run along the

base of block B.
This provides a total of 21 weights for the 21 data sets {Om a = 1 to 21},
and a stationary kriging system of dimension 22. As the results of this
kriging were to be used for production planning, it was essential that they
be as accurate as possible. For this reason, the number and exact location
of each in tersection S, constituting the cluster G, were taken into accoun t
when calculating the terms ii( Go:., G/3) rather than assuming random locations of the clusters Go: in their respective domains:

Similarly,

## where the cluster C; consists of no: intersections S, of length l; This

precision in calculating the various terms ii of the kriging matrix will result
in an increase in computing time over the method of random positioning;
all these terms must be recalculated for each block B to be estimated.
However, for block-caving mining, the number of such blocks B is limited.
The kriging results for a particular block are given on Fig. V.36: Zk =
l'72(217/(=O'15)% Cu. Note the small weight, "-21=0'055=55%,
given to the data on the level at the base of the block, even though this
level is extremely well sampled by drifts and channel samples. The number
of channel samples was shown to be excessive by studying their influence
on the kriging variance 17~; nothing is gained by increasing the number of
channel samples and drifts if more drilling is not carried out within the
mass of block B. This is a common characteristic of massive deposits
proven by drilling and channel samples along drifts, d. the similar study of
the EI Teniente data configuration in A. Journel and R. Segovia (1974).
3. Vein -type deposits

## As far as geostatistical estimation is concerned, the difference between vein

deposits and stratiform or massive deposits is the significant influence of
the geometric error on the error of estimation of the mean ore characteristics. Thus, in a deposit which consists of a single vein affected by
tectonics, the estimation will consist more in locating the vein and determining its walls than in estimating the mean quality of intra-vein are. These
geometric problems often cannot be formalized probabilistically, which

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MINING GEOSTATISTICS

406

## explains why geostatistical applications to vein deposits are not as

developed as those for stratiform or massive deposits.
The first problem is to determine when, from a geostatistical point of
view, a deposit should be considered as vein-type. A brecciated deposit
with multiple interlaced veins can be regarded as a massive deposit. The
estimation of the mean width of a single vein estimated by drifts can
sometimes be reduced, in the plane of the vein, to a two-dimensional
problem of the stratiform type, cf. the following case study of Eagle vein. If
the vein is thick enough, intra-vein estimations can be made: the geometric
problem vanishes and kriging plans used for massive-type deposits can be
applied.
Eagle copper vein (Canada), after A. J. Sinclair and J. Deraisme (1974)
This single quasi-vertical vein forms an undulating surface of small amplitude. The curvilinear distances h were measured along the vein, which
amounts to flattening it out, d. Fig. V.37. We can thus speak of the plane
of the vein, this plane being vertical.

-.J:.!91l!-

-=:--

of the vein
-I

I
_ _ _ ____

I
I

~l:-

I
I

I"

"I
x+h

## FIG. V.37. Plane of the vein.

Data for this study consist of vein width and copper assay values taken at
approximately 10ft intervals along three main horizontal development
drifts located in the plane of the vein. The two regionalized variables used
in the study were the vein width and the corresponding copper accumulation.
Nested structures, y(h)=Yl(h)+Y2(h), were fitted to the variograms of
these two regionalizations. The two structures are spherical with respective
ranges for the width of a 1 = 20 ft and a2 = 100 ft. These two ranges can be
interpreted as pseudoperiods of the successive bulges in the vein, as shown
by the thickness profile obtained on one of the development levels d. Fig.
V.38.

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## V THE ESTIMA nON OF IN SITU RESOURCES

407

In the plane of the vein, the three levels are about 600 ft apart, which is a
much greater distance than the largest range az = 100 ft. Thus, local estimations of ore resources can be made only within 100 m of workings for
which samples are available, i.e., the zone of influence of the levels. This

Smoll st ructure
,

Lorge

--------._- -----

s truer ure

Small structure

... ---

## zone of influence is a narrow 200-ft band centred on each level, d. Fig.

V.39. Beyond these three bands of influence, no differentiated local estimations are possible.

1200

ft
Le~el

6930

Le~e~
~~-l-4-+-1-4-44-l--I-4-1-~~-+-+-+-+-++-+-+-+-t-1~=
6360

## FIG. V.39. Local estimation of Eagle.

The three vertical bands of influence are split into panels of vertical
dimension 100 ft x 200 ft. For the estimation by kriging of the mean
horizontal thickness and the corresponding mean copper accumulation of

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408

MINING GEOSTATISTICS

each panel, the information on the central level is grouped into three data
sets, S1' 52, 53, d. Fig. VAO. 51 is the data set within the panel to be
estimated; 52 and S3 are the next data sets taken along the central level.

Existing
drift

100 ft
FIG.

VAO. Data configuration for the estimation of a 100 ftx 200 ft panel.

As the mining company planned to carry out additional sampling for the
local estimation of all the mining panels, several kriging plans were
established to predict the improvement to be gained by different additional
sampling plans, d. Fig. VAL It was shown that, with an additional level
between each of the existing ones, the relative standard deviation 0"/ m, of
the global estimation of the ore-tonnage, would be reduced from 48% to
23%, the relative standard deviation (TKI m, of the local kriging of the
ore-tonnage of 100 ft x 200 ft unit panels, would be reduced from 17% to
15%. This latter decrease is not especially significant; however, all the
units in the deposit could be differentiated by kriging, while, with the three
existing levels, only 40% of the panels were differentiated (had different

G1

--If-----------+--

200
II

(a)

FIG.

( b)

VAL Kriging plans with additional data. (a) Two additional drifts; (b) one

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## V THE ESTIMATION OF IN SITU RESOURCES

409

kriging estimates). Similar results were obtained for the estimation of the
quantity of metal and the mean copper grade.

## Uchucchacua argentiferous vein (Peru) after E. Tulcanaza (1972)

The data for this study consist of samples taken along drifts and raises. The
samples were taken at l-rn intervals perpendicularly to the direction of the
workings on an approximately constant support. The mineralization
essentially consists of argentiferous pyrite with a little galena, and the
problem was to determine the intra-vein variability of the silver grade. The
regionalization of this grade is characterized by a marked zonal anisotropy:
the horizontal variability as determined on the drifts is greater than the
vertical variability as determined on the raises. This is explained by the
vertical direction of the main faults. The chosen structural model is anisotropic and consists of the nested sum of two horizontal structures (with 4-m
and 70-m ranges) and a vertical structure (with a vertical 20-m range).
The problem was to estimate the mean silver grades of blocks of dimensions 50 m x 50 m or 40 rn x 20 m in the plane of the vein (the constant
thickness of the vein is fairly well known). The kriging configuration simply
consists of two data sets, cf. Fig. V.42: (i) for the 50 m x 50 m block
sampled by drifts and raises, the two mean grades Zd and Z, of the drifts
and the raises were used; (ii) for the 40 ill x 20 m block sampled by a single
central drift, the two mean grades Zdl and Zd2 of the samples in the section
of the drift inside the block and the two symmetric sections outside the
block were used.
For the calculation of the y terms of the kriging matrix, the supports of
the channel samples were assimilated to line segments; thus, for the third
configuration, d. Fig. V.42, the support of the datum Zd is the line d 1 of
40 m length. Thus, f(dt, d l ) = F(40) and y(d h B) = X(20; 40), where F
and X are the standard auxiliary functions.
This assimilation of the support of the discrete set of channel samples to
a continuous line amounts to assuming that the mean grade of the line
j

Drift

dZ

Block

Raise

[50m

I
I

Block

40m

50 m

## FIG. VA2. Kriging configuration at Uchacchacua.

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I
I

I.~---~I
d

dZ

dl

~~~
I
I
I
1

20 m

410

MINING GEOSTATISTICS

## segment is perfectly estimated by the mean grade of the channel samples

within it. It is then necessary to add a correction term to the preceding
kriging variance. This correction term corresponds to the estimation of the
line segments by their component channel samples; thus, for the third
kriging configuration, zk = AZd j + (1- A)Zd2 , the three segments of 120 m
total length are estimated by 120 channel samples spaced at 1-m intervals,
and the variance correction is cr;/120, where
is the extension variance
of a central point to its segment of influence of 1 m length, d. section
V.C.2, the principle of independence of elementary errors. For this third
configuration, a~ = 118 and cr~/120=O'3, which gives an estimation
variance E {[ ZB ~ Z~ f} of the mean grade ZB of block B equal to 11 8 +
03 = 121.

a;

## V.C. VARIANCE OF GLOBAL ESTIMATION

V.C.1. Defi nltion of the 9 10 bal esti matio n va ria nee
A global estimation differs from a local estimation in two important
aspects.

(i) The distances involved in global estimation are usually greater than
the limits of quasi-stationarity defined by the structural analysis, and
sometimes include several heterogeneous mineralizations.
(ii) The global estimation does not have the repetitive character of the
local estimation. As the deposit is unique, there is only one global
estimation error and it is, thus, difficult to probabilize it.
The first aspect means that a global estimator should be built by combining various local or partial estimators, as shown in section V.A.3. In
general. kriging techniques can be used to provide local kriged estimates
within each zone or neighbourhood of quasi-stationarity; the global estimator is then defined by the weighted sum of these local estimators,
according to formula (V.18).
The problem of the definition and calculation of a norm of the quality of
a global estimation remains. In local estimation, the estimation configuration for a particular block Vi can be considered as repeating itself identically for a large number of blocks V within a zone of homogeneous
mineralization. In such a case, the probabilistic interpretation of the local
error (z Vi - Z t as a particular realization of the random error (2 v; - z t; )
has a physical meaning, and the local estimation variance E {[Zv; - zti
of a block V was then defined. In global estimation, the deposit D is
i

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f}

## V THE ESTIMATION OF IN SITU RESOURCES

411

unique as well as its mean grade ZD. The probabilizationt of the global
error (zv - zt) is obtained, no longer by considering various particular
deposits DJ, but by considering various locations of the data configuration
as a whole over the unique and fixed deposit D: a particular global
estimate z~ corresponds to each particular position i of the origin of this
data configuration, cf. Fig. V.54, the example of two surface estimates
S*(xo) and S*(xo) linked to two particular locations Xo and xb of the data
grid origin. Thus, the global estimation error (z D - Z 0) has been probabilized, and the global estimation variance is defined as E{[ZD - zt, ]2}.
However, the distribution of the error (ZD - z}b) will always remain
inaccessible since, in practical applications, only one particular location io
of the data configuration over D is available. Hence, all confidence intervals (e.g., the 20" Gaussian standard) attached to the global estimation
variance will have but a mere conventional meaning.
Choice of the structural model y(h) or C(h), used in the calculation of the
global estimation variance

## Consider the "local variogram" on D, i.e., the variogram 2 Y{D)(h) that

would be available if the domain or deposit D were perfectly known, ct.
section III.B. 7, formula (111.40):

In practice, this local variogram is estimated by the experimental variogram, cf. formula (111.39),

where N' is the number of pairs of data (distant of vector h) available over
the whole domain D.
As this experimental variograrn 2y'b(h), and, thus, its fitted model
2YD(h) are defined over the whole deposit D, they may differ from the
quasi-stationary models 'Yi(h) used in the local estimation of each particular zone or block Vi of the deposit D. Often, in mining practice, these
t Note The global estimation error probabilization as well as the conceptual basis of the
global estimation variance calculation is related to an approach different from the approach
of regionalized variables considered in this book. This approach is called "transitive
theory", cf. G. Matheron (1965, p. 17) and (1971, p. 9), and results in calculation algorithms
and solutions remarkably similar to those provided by the regionalized variables theory. In
the present text, this transitive theory approach is briefly outlined in section V.CA, which is
devoted to the surface or volume estimation problem.

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412

MINING GEOSTATISTICS

## quasi-stationary local models 'Yi(h) are linked by a proportional effect, ct.

section IILB.6: the local model 'Yrn*(h) corresponding to the estimator
m * = Z b of the mean grade of D should then be taken as the global model
to be used in the calculation of the global estimation variance.
If the deposit D comprises various zones of truly heterogeneous
mineralizations, it is advisable to proceed to a global estimation in each
homogeneous zone (homogeneous at the mining scale, not at the scale of
the geologist's pick), and then to combine the various estimation variances
assumed to be independent.
In general, two types of error are present in global estimation:
(i) a geometric error, due to the uncertainty of the exact limits of the
mineralization or of the zone considered as the deposit;
(ii) a quality error, due to the estimation of grades or other characteristics of the ore within a geometrically and volumetrically defined
domain.
In the following two sections, V.C.2 and V.C.3, the study will be limited
to that of the quality error, assuming that the geometry and volume of the
domain or deposit to be estimated are known. In section V.C.4, the
geometric error and its influence on the global estimation of ore and metal
quantities and mean grades will be quantified.
V.C.2. Global estimation on a domain of known geometry

unit blocks Vi,
N

D=

Vi.

;=1

## The global estimator of the mean grade Zo is obtained by a weighted sum

of the N local estimators zt;> i.e., Z't = (1/ D) Ii v;zti The error of
global estimation is then the weighted sum of the N local errors, i.e.,

(T~D = EUZv =

Z'b ]2}

j -

j ]} ].

## If Z, was estimated by kriging, the local estimation variance, (T~ Vi =

E{[Zv; - Zt,]2}, could then be obtained directly from the kriging system.

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413

## The difficulty lies in the calculation of the N(N - 1) error covariances

E{[Zv; -ztJ[Zv -zt
Such a calculation quickly becomes tedious
when N is great. The practical solution consists in expressing the global
estimation error [Zo - Z b] as the sum of n (with, in general, n ~ N)
elementary independent errors:
j

]} .

ZD -zt =

A.k(Zk -zt).

(V.35)

k=l

## The global estimation variance then takes the simplified form

(T~D = E{[Zo -

zt, ]2} = L

A ~E{(Zk -

zt ]2}.

(V.36)

k= 1

The problem thus consists in distinguishing n successive stages of estimation, the errors of which are approximately independent. These n successive stages will depend on the available data configurations; in mining
applications, there are essentially two cases, depending on whether the
elementary errors come from the same estimation configuration (d.
subsection 1, "Direct combining of elementary errors", immediately
following) or from different configurations (cf., hereafter, subsection 2,
"Combining the terms of a global estimation error").
1. Direct combining of elementary errors

## First, consider the two-dimensional example of the regionalization on the

horizontal plane of the vertical thickness of a sedimentary bed. The deposit
is sampled by vertical drilling, and three cases will be considered, according
to whether the drills are on regular, random stratified (i.e., constant
density) or irregular grids.
Regular grid Consider the regular square grid of side I shown on Fig.
VA3. The surface S of the deposit is divided into N elementary grid
surfaces s; each of which has a positive central drill-hole (i.e., with a
non-zero mineralized thickness Z (Xi The mean thickness ZS; of each of
these elementary units is estimated by kriging from the thicknesses
measured 011 the central drill-hole and, for example, the eight holes of the
first aureole surrounding s; If the deposit is assimilated to the surface S of
known geometry, then the estimator of the mean thickness Zs of the
deposit is the average of the N local kriging estimators, i.e.,
=

z;

(lIN)IiZ~,

Any two units s, and s, which are separated by a distance less than 3! will
have common drill-holes in their estimation neighbourhood and, as a
consequence, the two estimation errors [ZSI - Z:'l and [ZSj - Z~] cannot

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414

MINING GEOSTATISTICS

## be considered as being independent. On the other hand, if the global

estimation error [Zs - Z~] is considered as the sum of the n = N elementary errors of extending the thicknesses Z (Xi) of the central drill-holes to
their respective surfaces of influence si; then the simplified formula (V.35)
can be used:
Zs-Zs* <z, - N1 " Z(Xi)= N1 ~ [Z,;; -Z(Xi)].

## As a first approximation, the N elementary errors [Z,;; - Z (Xi)] with a

variance (T~s, can be considered as independentt because none of them
have data in common. The global estimation variance can then be expressed simply as

E {[ Zs-Zs]

2}

1
=N 2

,2:N

,~1

1 2
E{fZsi -Z(x,)] 2 }=-<TEs

(V.3?)

ut

..

with 01s = 2H(lj2; 1/2)- F(l; l), according to the formula for
corresponding to Fig. II.50 in Section II.E.3.
Note that this sum of elementary independent errors, Zs, - Z (Xi)' is
related to the approximation

1
* =N,-Z(Xi)'
1"
Zs* =-IZs;
N

## This approximation improves as the number N of elementary units s,

increases. For a local kriging neighbourhood limited to the first aureole;
N> 50 gives an excellent approximation.
o 0 0
x x x x
x x x x
x x x x
x x x

0
0
0
0

x~x
X X X

r---------x
x

'{.

L
0(

W W
x

X
L-_~~~

Xj

Xi

si

_______

FIG.

## V,43. Direct combining of elementary errors from a regular grid.

t The N elementary errors [Z.. - Z (Xi)J are independent but only as a first approximation,
although they do not involve any common data. This is because the spatial regionalization
entails a certain degree of correlation between the values ZS!' ZSj and their estimators Z(Xi),
Z(Xi)' However, formula (V.37) provides a good approximation for the estimator variance. as
will be shown later. in the addendum, on Figure V.47.

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415

## A random stratified grid, d. Fig. V.44, is one in

which the sample grid, while not regular, has a constant density with an
average of one sample in each elementary grid surface s ~ the location of
this sample within s is assumed to be random and uniform (i.e., the sample
is likely to be on any point of s).
Consider the preceding sedimentary deposit with the N positive drillholes on a random stratified grid. As before, the global estimator z1 is the
weighted sum of a certain number of local estimators obtained, for example, by local kriging, and the global estimation error (Zs - Z~ 1 is considered as the sum of the N independent elementary errors of extending the
thickness of each positive drill-hole, Z(x;), to its zone of influence s; (in this
case, s, is the grid unit square):
and

E{[Zs - Z s]

N 2 ~ O"Es;'

In this case, each of the elementary errors [ZSi - Z (x;)] has its own
estimation variance O"~s, which depends on the particular position Xi of the
drill-hole in the grid square 5j. Over a large number N of such grid squares,
x will take all possible locations inside the grid square s and the elementary
estimation variances
will have the foIlowing mean value:

at

N i~l

2
O"Es j

If

## = --; E{[Zs - Z (x)]

} dx

= D (01 s).

Indeed, from formula (11.33), the mean value over s of the variance of
extension E{[Zs -Z(x)f} of a point datum Z(x) to the mean value Z; is,
2
by definition, the dispersion variance D (Ol s ) of this point datum in s. The
global estimation variance is then written as
1

*2

E{[Zs-Zs]}= N2~(jES' =

x
X

1
N

D (Ols).

sI
X

xI

X
X
X

X
X

## FIG. V.44. Random stratified grid.

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(V.38)

416

MINING GEOSTATISTICS

## The dispersion variance D 2 (Ol s ) is expressed very simply by means of the

auxiliary function F:
2
D (Ol s ) = y(s, s)= F(1; I),
if s is a square of side I.
The principle of uniform random location of the data within the grid unit
can be applied in a similar manner to data configurations in one, two or
three dimensions. In three dimensions for example, for a parallelepipedic
grid unit v containing, on average, one datum of support o' v and located
in a uniform random manner within v, the global estimation variance of a
large volume D consisting of N grid units v can be obtained by direct
combination:
(V.39)

Irregular grid Consider the same sedimentary deposit, this time with the
N positive data on an irregular grid, d. Fig. V.4S (the N surfaces of
influence s; are unequal). Note that under no circumstances can a preferential grid, with all its associated risks of bias, be assimilated to an
irregular grid, cf. section III.C.l.
The global estimator Z~ is the weighted sum of a certain number of
local estimators obtained, for instance, by local kriging; note that these
local estimators may not be defined on the surface of influence s, of each
positive hole. By expressing the global estimation error [Zs - Z~ J as the
sum of the N independent elementary errors [ZSi - Z (Xi)], of extending the
thicknesses Z (x;) to their respective surface of influence Si, the following
formula is obtained:
(VAG)

## where S = (11 N) Li s.; and a~s; is the variance of extension of a central

point datum to its surface of influence. This variance is expressed by means
of the two auxiliary functions Hand F, cf. section II.E.3, if the surface s, is
a rectangle. If s, is not rectangular, it can, for instance, be assimilated to a
square of the same surface area s, = ti, and, thus,
(T~sl

## = H (1;/2; 1;/2)- F(1;; Ii).

Note that the previous expression (V.40) of the global estimation variance is related to the approximation
1 ~ SiZs;*
Zs* = S

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S1 ~ SiZ(Xi),

417

zt

## i.e., to the assimilation of the global estimator

weighted sum of local
kriging estimators, to the polygons of influence estimator. When N is large,
there is very little difference between these two global estimators. Nevertheless, at the local level, the kriging estimators should always be preferred
to the polygons of influence estimators (because the local kriging variances
are, by definition, minimum).

## Other configurations for directly combining elementary errors

In one dimension, let L be a line sampled by N points regularly spaced at
intervals I, cf. Fig. V.46(a). This line may represent, for example, a drift of
length L with a negligible transverse section which has been sampled by
vertical channel samples spaced at regular intervals I. The line L thus
consists of N segments I each having a central, centred point datum. The
global estimation error [ZL 1of the mean characteristic on L is considered as the sum of the N elementary errors of extension of the central data
Z (x) to their segments of influence l. As a first approximation, these errors
can be considered independent, resulting in the following expression
similar to (V.37):

z:

(VAl)
with

## ukl = E{[ZI - Z (x )}2} = 2X(112) - F(l).

The same line L = NI may be sampled by (N + 1) points in a closed
arrangement, each segment l, having two extreme data Z(Xi) and Z(Xi+l)'
as shown on Fig. V.46(b). Two contiguous segments I, and l, have one
datum in common and, as a consequence, the two corresponding local

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418

MINING GEOSTATISTICS

## estimation errors can no longer be considered as independent. It can be

shown that, for N:::::: 6, a closed arrangement of (N + 1) points can be
assimilated, with an excellent approximation, to a centred arrangement of
N points, and formula (VAl) can then be applied. This practical rule can
also be applied for closed arrangements in two or three dimensions (with
N~36).
L

( a )

Z(x)

Ii

ii
~

II

II

...-----------.......

Z(Xi)

FIG.

II'

( b)

Z(Xi+ l)

V.46. Direct combining of elementary errors on a line. (a) Centred arrangement; (b) closed arrangement.

## If the N points are in a centred arrangement, but on an irregular grid

(Ii -;j;.lj), each segment l, being informed by a central point datum Z(Xi),
then the weighted average of the elementary errors of extension of the
Z(Xi) to their respective segments of influence l, results in a formula similar
to (VAO):

(V.42)

where L

L. l, and
ukl,

= E{[Zjj -

Z (Xi)]2}

= 2X(//2) -

F(li)'

On this simple example of centred arrangement in one dimension, it is advisable to
test the robustness of the approximation underlying formula (V.41), i.e., the
influence of the hypothesis of independence of the N elementary errors of extending the central data to their segments f of influence. For this, a spherical scheme
with range a and unit sill and without nugget effect, was considered, d. formula
(IIL1S). The estimation variance of a line L =NI surveyed by N point data
regularly spaced on a central arrangement is then calculated from the exact formula
(II.27) and compared with the approximation given by formula (VAl). These
compared results are shown on Fig. V.47 (by courtesy of Michel David) for various
values of N and the quotient lJ a. The approximation is excellent for all small values
of the spacing (lla < 05, which is a common case in practice); for greater values of

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## V THE ESTIMATION OF IN SITU RESOURCES

419

the spacing (if a> 0-5, the number N must be great (N > 30) to provide a good
enough approximation.
007
/
/

006

I
~"

"-

005
/
/
,'\,
~'

004
NlI)

003

002

0-01

-0-2

01

0-3

0-4

0-6

05

0-7 08 0-9

ro

110

FIG. VA7. Estimation variance of a line L = Nl by N data on a centred arrangement. Spherical model, sill 1, range a.
, Exact values: - - - -, approximate
values.

## In two dimensions, consider the estimation of the mean grade over a

surface S divided into N rectangles Si = L; x l;, each containing a median
drift assimilated to a line of length L; as on Fig. V.48. The variance
of

crt;

I-----~--------\

I
I

~----------------~

Li

I SI

I
~

I
I

~ - - - - - - - - - - - - - - - __ I
I

!
J

## FrG. Y.48. Direct combination of elementary errors from a configuration of

median drifts.

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420

MINING GEOSTATISTICS

## the elementary error of extension of the mean grade of a drift L; to its

rectangle of influence 5; = L; x I; is calculated using the auxiliary functions X
and F, or read directly from charts no. 8 and 18, d. section II.E.3. The
global estimation variance over the surface
N

S=

s,

i=1

## can then be obtained by directly combining the elementary estimation

variances (Tt;. The result is a formula similar to (VAG):

EUZs -

Z~ ]2} = ;2 ~ S;l7L"

(VA3)

where

## In three dimensions, consider the estimation of the mean grade over a

volume V divided into N rectangular parallelepipeds V; = s, X hi, each of
which being informed by its median section of surface 5;, cf. Fig. V.49. The
variance l7~v, of the elementary error of extension of the mean grade of the
median level s, to its parallelepiped of influence Vi = s, X hi is calculated
using the auxiliary functions X and F, or read directly from charts no. 9 and
19, d. section II.E.3. The global estimation variance over the volume
N

V=

Vj

;=1

-,+--5,.

## FIG. V.49. Direct combining of elementary errors from a configuration of median

sections.

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421

## can then be obtained by directly combining the elementary estimation

.
2
vanances (J" s-, :
(V.44)

## where, if the median section s, is assimilated to a square of the same area,

l ,~ = s."
(T~", =2x(h;/2; l~)-F(h;; l~)-F(l;; ';).

Note In the last two examples, corresponding to Fig. VAS and V.49, the
elementary extensions of the mean grade of a median drift or a median
level to its rectangle or parallelepiped of influence were considered. But
both the median drift and the median level have been estimated with a
certain estimation error. Thus, a term corresponding to this estimation
error must be added to the estimation variances given by formulae (V.43)
and (V.44); this results in combining the various terms of the global
estimation error.
2. Combining the terms of a global estimation error

## The data available for the estimation of a three-dimensional phenomenon

is very often concentrated along alignments (e.g., pieces of core along
drill-holes, sampling along drifts) and planes (e.g., planes of vertical
sections along profiles of vertical drill-holes or horizontal levels of the
drifts). Thus, the data available for the estimation of the mass shown on
Fig. V.50 are vertical channel samples taken along horizontal drifts on each
of the three horizontal levels.
The procedure for estimating the mean grade of this mass can be considered as consisting of three steps corresponding to three terms of the global
estimation variance.

L;
- Level 2

"

Chennel
scmples

-lev~1

FIG.

Drift

## V.50. Combining the terms of the estimation of a mass.

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422

## MINING GEOSTA TISTICS

(i) The estimation of the drifts by the channel samples. The estimation
variance corresponding to this step is called the "line term", T/.
(ii) The estimation of the levels by the drifts which are supposed to be
perfectly known. The estimation variance corresponding to this step
is called the "section term", T s .
(iii) The estimation of the slices of the mass by their median levels,
which are supposed to be perfectly known. The estimation variance
corresponding to this last step is called the "slice term" , Tv.
As a first approximation, the three estimation errors corresponding to
these three steps can be considered as independent, and the global estimation variance of the mass then consists of the sum of the three preceding
terms:
(VAS)
The following example of the calculation of these three terms is based on
Fig. V.50. Another example is given in the study of Ity Mount Flotouo gold
deposit in section V.C.S.

## Line term If each drift is assimilated to a line L, along which channel

samples are taken at regular intervals, then the estimation variance of this
line can be expressed using the direct combining formula (VAl):
2
(TEL,

12
= N-

(TEl>

## where L, = N, x I and (T~l = 2X(lj2)- F(l).

The estimation of all the drifts (of total length L = I. L, = NI) by the set
of N = Ii N, channel samples, regularly spaced at intervals I along these
drifts, is thus characterized by the line term
2
1
T 1 = (TEL = N

2
(TEl

Section term The three levels are assimilated to the sum of n rectangles
s, = hi X L; each estimated by a median drift L, supposed to be perfectly
known. The estimation variance of these three sections (of a total surface
S = I~=l 5,), known as the section term, is then obtained from the direct
combining formula (V.43):
T, =

(rES

= 52 ~ S i (TE,-"

## where (Ttl is the variance of extension of the median L, to its rectangle of

influence 51 = hi XLi.

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423

Slice term

## The final term is the error of estimation of the three slices of

the mass by their median levels supposed to be perfectly known. Each slice
may, for example, be assimilated to a rectangular parallelepiped estimated
by its median level Si, and the slice term is then obtained from the direct
combining formula (V.44):

where

## and (T~vk is the extension variance of the median rectangle of surface S to

its parallelepiped of influence Vk.

## Remarks The terms combining to give formula (VA5), cr~ =

T 1 + Ts + Tv, express the intuitive notion that the estimation would not be
improved by increasing the density of channel samples in the drifts (T1
decreases) without also increasing the number of drifts on each level (Ts
decreases) and the number of levels within the mass (Tv decreases). The
line term is often negligible with respect to the section (Ts) and slice (Tv)
terms. If the costs of channel sampling and drift development are known,
the two-term formula, T 1 + Ts. can be used to make optimum use of the
sampling budget on any particular level.

Section

\v
Vertical
drills

## The term-combining formula used in the preceding example can be

applied to configurations other than channel samples in drifts on levels. For
instance, it could be applied to the global estimation of a deposit from
vertical drilling along vertical sections, d. Fig. V.51.

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424

## V.C.3. Estimation of a quotient or a product

The quantity to be estimated is often in the form of a quotient or a product
of two quantities estimated separately. Consider, for instance, a subhorizontal sedimentary deposit of surface S and variable vertical thickness
t(x) as shown on Fig. V.52. For each vertical drill at point x, two additive
variables are defined: mineralized thickness t (x) and accumulation a (x) =
t(x). z(x), where z(x) is the grade measured along t(x). This grade z(x) is
not an additive variable because it is defined on a variable support t(x), and
so the regionalization of t(x) cannot be studied directly, d. section IILC.2.
The mean grade Zs of the deposit is estimated by the quotient of the two
global estimators of mean accumulation and mean thickness:
Ssa(x)dx.
z, = l/S
I
f (
estimated by
1 S stx)dx

* A~
Zs =~.
Ts

## Similarly, if the mineralized surface S is estimated by S*, the estimator

of the global volume of ore is in the form of the product of the two global
estimators S* and T~:
V = S x Ts =

Is t(x) dx

estimated by

v- = S* x

T~.

## The global estimation variances of mean accumulation and thickness can

be calculated by means of the regionalization of the factor variables a (x)
and t(x). The problem is to deduce the global estimation variance of the
mean grade estimated by the quotient Z} =AVT~. More generally, the
estimation variances of the product P = A . B and of the quotient Q =
A/B, estimated by P* = A * . B* and Q* = A */ B*, respectively, are to be
evaluated knowing the estimation variances, O"~ = E{[A - A *]2} and (T~ =
E{[B ~ B*]2}, of the two factors A and B.

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425

## The product and quotient of two random functions underline

difficult problems of existence and convergence. Thus, the quotient of two
normally-distributed variables is a variable with a Cauchy distribution
which has no moments (no expected value, no variance); the expectation of
a limited expansion is not always the limit of the sum of the expectations of
the terms, etc. The following demonstrations are more symbolic than
rigorous, which should be sufficient for the practitioner of mining estimation. All the following results can be established rigorously for the product
and quotient of two lognormally distributed variables.
Warning

## Estimation variance of the product P = AB

The true global values P, A, B are considered to be known, and their
estimators P*, A*, B* are interpreted as random variables:
P" -<Prr e;

## where CA, CB, cp are the random estimation errors.

The estimations of the factors A and B are assumed to be unbiased, i.e.,
E{cA} = E{eB} = 0, and their estimation variances, cr~ = E{e ~} and
=
E{e 1}, are known.
Thus,

a1

P* =A* B*

and

## Therefore there is a bias, E {p* - P} = E {cAE B}' in the estimation of the

product P by P* = A *B*. This bias is null if the estimation of the two
factors, A and B, can be considered as independent, i.e., if E{eAcB} =
E{EA}. E{cB} = O. In practice, the term EACS corresponds to a secondorder error, and the bias term E{SAEB} is negligible with respect to the
value AB.
By neglecting the third- and fourth-order errors, the quadratic estimation error of the product P is written as

## (P * -P)2 = A2 en2 + B 2 fA2 + 2 ABeACB+'"

and the relative estimation variance of P by P* is

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426

MINING GEOSTATISTICS

Putting PAB = (E{[A - A *][B - B*]}/O"A . 0"8) for the correlation coefficient
of the two estimation errors of the two factors, the formula can be written
as
(V.46)
In practice, the true values A and B in formula (V.46) are replaced by
their unbiased estimators A * and B*. The calculation of PAB will be
detailed later.
Estimation variance of the quotient Q = AI B

## Using the preceding notations, the estimation of the quotient can be

expressed as
Q*=A* =A+CA=A+A
1
B* B+B
B
"1+f818'
If the relative error fBI B is small with respect to 1 - and this is generally
the case for global estimation - the following limited 'expansion is obtained:
Q *=

( Q+A)
B

B e 1 E 1
. [ 1--+-..,--+
...
3

B- B

(VA?)

and
(0*) =

o[

1+

cr~ B

E{EAEB} + ...

AB

J.

There is, thus, a relative bias, ((T~I B2)-(E{fAC8}! AB)+ . . " .which persists
even when the two estimations are independent, i.e., even when E{cAB} =
E{A} . E {fa} = O. In mining applications, this relative bias is negligible
when the relative estimation variance 0"1/ B 2 is less than 0.01, which is
often the case in global estimation. Using (V.4?), the relative estimation
variance of the quotient Q can be expanded up to the second order as

crb _ EUG Q2 -

- A 2
B2
AB
" .. ,

Q2

i.e.,
(V.48)

PAB =

## E{[A - A *][B'- B*]} /(O'A'

(TB)

Note first that if the two estimations of the two factors A and Bare
independent, the covariance EHA - A *][B - B*]} of the two errors is null,

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## V THE ESTIMATION OF IN SITU RESOURCES

427

and the approximate formulae (V.46) and (VAS) reduce to the standard
formula of addition of relative variances:
2

Q2

crp

(TQ

or

p2

(T A

2
(TB

A 2 + B2 + ....

(V.49)

If the two variables a(x) and b(x) are intrinsically coregionalized, i.e., if
the two structural functions Ya(h) and Yb (h) are proportional, d. section
lILB.3, formula (IIL30), and if the two configurations for the estimation of
their mean values A and B are identical (similar in practice), then it can be
shown that the coefficient PAB is equal to the standard correlation
coefficient pal> between the two variables a (x) and b (x).
If the two variables a (x) and b (x) are not intrinsically coregionalized,
then the covariance of the two errors E{[A - A *][B - B*]} must be calculated, i.e., the coefficient PAB is calculated from the cross-structural
function

## Yab(h) = !E{[a(x + h)- a(x )][b(x + h)- b(x)]}.

When the two configurations for the estimation of the two factors A and
B are identical (or very similar), a "reduced difference" variable can be
defined, d (x) = a (x)1 A - b (x)1 B. Its structural function can then be
determined:
2Yd(h) = E{[d(x + h)- d(X)]2} =
2Yb(h)
= 2Ya(h)
+--..-".-.2
2
A

## E{ [a ex; h) _ a~)J-[ hex; h)

b~)J}

4Yab(h)
AB'

If the two configurations for the estimations of the two factors A and B
are supposed to be identical, then the two estimation variances (T~ and a1
are obtained by applying the same linear operator g to the two structural
models Ya and Yb, respectively, cf. formula (111.2):
(T~ =E{[A-A*]2}= g'(Ya)

and

a1 =E{[B-B*]2}= %'(Yb);

similarly,
E{[A - A*][B - B*]} = gCYab).

as

ub

Q2=

## AI B can then be written

1
1
2
A 2 g'(Ya) + 2 ?E'(Yb)- ABg(Yab)+ ...
B

= g'(Yd) + ....

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(V.50)

428

MINING GEOSTATISTICS

## The relative variance ubi Q2 is obtained directly by applying the operator

g to the structure Yd of the "reduced difference" variable.
In the case of the estimation of the product AB, the "reduced sum"
variable, s(x) = a (x)/ A + b (x)/ B would be defined, its semi-variogram
yAh) would be calculated, and the relative estimation variance of the
product could then, similarly, be written as
a~
1
1
2
p2 = A 2 g (Ya) + B 2 ~n Yb) + AB g (Yab) + ...
=

(V.51)

g(y,,)+ ....

## V.CA. Global estimation on a domain of unknown geometry

Up to this point, the geometric error has not been considered, i.e., the
error due to the ignorance of the exact limits of the mineralization. The
volume of the area to be estimated was assimilated to the sum of N positive
blocks or zones of influence Vi, i.e., the estimator of the total mineralized
volume V was taken as the sum
N

V* =

Vi

i=l

of the volumes of these blocks. In doing so, an error (V - V*) was incurred
and must be evaluated, for example, through a variance of the geometric
error. This geometric error (V - V*) not only affects the volume and the
are tonnage, but also the estimations of metal tonnage and mean grade of
the deposit D. Thus, it is equally important to combine the influence of the
geometric error with that of the different quality errors, e.g., the quality
error due to the regionalization of the grade variables in the total volume
V.

## The determination of the limits of the mineralization is a constant problem

in global estimation of a deposit. It sometimes happens that some of these
limits are known a priori, e.g., the limit of a lease, a fixed depth for drilling
and estimation, clearly defined vein walls, etc. Other limits require estimation by one of the following methods:
(i) all-or-nothing-type drilling, e.g., the limits of an underground
reservoir or a mineralized lens;

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429

## (ii) economic cut-off grade, e.g., in deposits such as porphyry-copper,

all zones with grades greater than 04% Cu will be considered as
mineralized zones;
(iii) with the aid of geological interpretations made on planes or crosssections.
When this estimation of the geometry of the deposit is directly related to
geological hypotheses (metallogenesis, tectonics, etc.), the geometric error
is of a deterministic type not favourable to its quantification (this does not
mean that the error will be large or that such geological methods of
estimation should be disregarded). In such a case, it is advisable to examine
these geological hypotheses and possibly consider different ones. There
will be a different global estimator of the ore tonnage for each interpretation of the geology which returns a certain random character to the
estimation, Thus, even if the variance of the geometric error cannot be
evaluated, at least its limits can be.
However, the geometric error may intrinsically have a purely or partially
random character which can be modelled.
(i) From the experimental knowledge of similar deposits (e.g., random
location in space of mineralized lenses of random volume and
geometry). The morphology of the deposit can be simulated several
times. The available data configuration can be applied to these
various simulated deposits, and an estimation variance can, thus, be
determined. It should be stressed that such a variance depends
heavily on the model adopted for the morphology of the deposit.
(ii) There are, however, some cases, frequently encountered in mining
applications, which allow an approach entirely free from any a priori
hypothesis or model. This is the '<transitive theory" approach, d. G.
Matheron (1971, p. 9), the main aspects of which are outlined in the
following subsection.

The transitive approach Let k(x) be the indicator variable of the mineralized volume V to be estimated, This indicator is an all-or-nothing variable
defined as
k(x) =

{~

if the point x is in V,
if not,
xe V.

## The mineralized volume V is then the triple integral,

dimensional space, of this indicator variable:
V =

-e-co

k(x) dx

-co

+00

dr,

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-co

(V.52)
III

the three-

+00

dx,

LXl

k(x", xv, x w ) dx w .

(V.53)

430

MINING GEOSTATISTICS

## In the transitive approach, the structural function "geometric cova riogram"

of the volume V is defined as
K (h) =

k (x)k (x + h) dx,

(V.54)

## where h is the standard notation for a vector in the three-dimensional

space for the estimation of a volume V, or in the two-dimensional space for
the estimation of a surface S.
The real unknown volume (or surface) to be estimated is simply the
value of the geometric covariogram at the origin: V = K(G).
In the two-dimensional case, when the surface S is estimated by a
regular grid (aI, a2), the origin of which is located at the point Xo of
coordinates (xou , xov ) , the estimator S* of the surface area can be written,
ct. Fig, V. 53, as
+00

+00

L
Pu =

~-co

k(xo u

## + Pu. ai, XOv + Pv az),

Pt'! =-00

Pu. and P being two integers defining the various nodes of the regular grid.
If n(xo) is the number of positive drills, the estimator S*(xo) is then written
as S*(xo) = a laz . n (xo). Note that this estimator depends on the choice of
the origin within the elementary grid surface alaZ, ct. Fig. V.53.

)(

,-0

I
I
L_

I
I

_____________ I

a2(~

~Origin

of the grid

~
1

## FIG. V.53. Estimation of a surface from a regular grid. x, Positive hole;

0, negative hole; - - , real unknown limit; ----, polygonal estimated limit.

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431

xo

## For each new position

of the grid, there is a corresponding estimator
S*(x&) and, thus, a new realization [S - S*(x&)] of the estimation error of
the surface S, d. Fig. V.54. One possible way of defining the quality of the
estimator S* is to take the mean value of the quadratic error [5 ~ S*(XO)]2
when the origin Xo of the grid takes all possible locations inside the
elementary grid surface alQ2. This mean quadratic error is expressed as

Note the similarity of this expression for the mean quadratic error to that
of the estimation variance E {fS - S* ]2}. Using this similarity, even though
no probabilistic interpretation is involved in this "transitive" definition, the
notation O'~ for the estimation variance can be used, i.e., in one-dimensional notation:
2
1
O'E(a)=-

' ] 2 dxo.
[S-S>i'(xo)

(V.55)

This mean quadratic error, called the "variance of the geometric error", is
a function of the available data grid; here, a = (a 1, a2).
If S*(Xo) is replaced in the integral of (V.55) by its one-dimensional
expression
S*(xo)=a

p=-oo

k(xo+Pa),

## then, after some calculations, the following formula is obtained:

+00
J+co
(T~(a)=a p='2:. K(pa)~ -00 K(h)dh.
oo

(V.56)

The variance of the geometric error linked to the grid spacing a can be
expressed uniquely with the geometric covariogram K (h) of the surface 5
(or volume V) to be estimated. However, this covariogram is just as
unknown as the geometry and surface area of S (or V). It is at this point
that an important property of geometric covariograms is introduced, which
allows both the formula (V.56) and the transitive approach to geometric
errors to be of practical use.
Whatever the geometry and the extent of the domain to be estimated (S or
V) are, their geometric couariogram K (h) exhibits a linear behaviour at the
origin:
K(h)=K(O)-w"lhl+ ... , when Ihl~O.
The direction subscript a indicates that this linear behaviour may not be
isotropic.

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MINING G EOSTATISTI CS

432

Using this property, the first term in the limited expansion of the
expression (T~ (a) can be evaluated when a tends towards zero. This first
term is sufficient to give the order of magnitude of the geometric error
variance.
For the two-dimensional problem of the estimation of a surface from a
regular, rectangular grid (all az), it can be shown (d. G. Matheron (1965,
p. 108)), that the relative estimation variance can be expanded up to the
first order as

u~
S

[1

(N 1)2
+ ... ,

2=-2 6 N Z+ 0 0 6 -

. J

(V.57)

## where n ~ lOis the number of positive drills (the estimator is then S* =

n . a laZ), and Nz:<:S; N 1 ; 2N 1 and 2N z being the number of elements parallel
to the two sides of the grid (a l , az) which make up the perimeter of the
reunion S* of the n positive zones of influence. The perimeter of any
possible gaps in the mineralization must be taken into account in the
calculation of the values 2N l and 2Nz .
Taking the example of Fig. V.53, with n = 14 positive drills, the values
for the perimeter of the estimated surface S* are:

## 2N 1 = ten elements of length a 1 parallel to the side a 1 of

the grid;
2N z = eight elements of length az parallel to the side a : of
the grid.
Thus, the order of magnitude of the relative estimation variance of the
surface S is

(T.~

1 [4
2S ]
52=1426+0-064 =0005,

## i.e., a relative standard deviation of 7%, us/ S = 0,07, and a standard

confidence interval equal to SE[1421alaz. It will be recalled from
section V.C.1 that this interval should be considered more as a comparative norm of the quality of the estimation (S, S*) than as the true 95%
confidence interval. Figure V.54 gives two further examples of the use of
formula (V.57); the same surface S is estimated from the same grid
(a 1, az), but with two different locations, Xo and x(), of the origin of the grid.
Note that there are gaps in the mineralized surface.

## Remarks The relative estimation variance of the surface, given by

formula (V.57), is proportional to the number of elements N 1 and N:
which make up the perimeter. This means that, for a constant surface area

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## V THE ESTIMATION OF IN SITU RESOURCES

433

5, the more jagged (i.e., the longer) the perimeter, the more difficult is the
surface estimation.
When a portion of the perimeter is perfectly known (e.g., the limit of a
lease), this portion must not be considered in the calculation of the values
2
2N j and 2N2 , and this will reduce the relative variance (T;,/5 by approximately the ratio of known perimeter to total perimeter.

---'-.j--+----+-

, ; - 11
-- ---t-+;""1'---1'--+-

......--+----'t-""'t---i

'r-

(a )

( b)

FIG. V.54. Estimation of the same surface from two distinct origins of the data
grid.
(a) n = 37, 2N I = 26, 2N 2 = 22, s* = 37ala2' crl s = 45%, s = (3733)a 1a2'
(b) n = 39, 2N j = 32. 2N 2 = 28, s* = 39aj Q 2 , (T/ S =4 7",{" s = (39 37)alQ2'

## In practice, S is estimated either by the sum of positive zones of

influence, or by drawing a plausible contour, taking into account any
knowledge of the morphology of the domain under study.

Three-dimensional case For the estimation of volumes by a parallelepipedic grid of sides (a I, a-, a3) and when the distances (a h a2, a3) tend
towards zero, there is a first-order expansion formula similar to (V.S7), cf.
G. Matheron (1965, p. 105). In mining applications, such a parallelepipedic grid is rarely available and volumes are usually estimated by
combining the surface estimations of different sections, or sometimes by
the sum of prisms of influence of positive drill-holes.
First, let V = Ii Ii S,. be the volume estimated by V* = Ii 1St, where I; is
the thickness of the section S;(vertical or horizontal), d. Figs V.51 and 55.

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MINING GEOSTATISTICS

434

## The surface Sj is estimated by st with an estimation variance of U~i =

EUS; -St]2}.
The elementary estimations of the surfaces S; are usually independent
and the estimation variance of the volume V can then be expressed as ut = E{(V - v*f} = L; L1cr1;- The relative estimation variance is
2
(Tv

~ 1

V2

V2

2eTS;

~ v; 57 '

(V.58)

where u, = 1;. S;
--~---~--~-~~-

BJ

___________

~_~

I'

t,

5,.

a
FIG. Y.55. Estimations of a volume by combining sections.

## Second, let V be a volume containing n positive vertical drill-holes of

mineralized length l.. These n drill-holes are on a regular horizontal grid of
sides (ai, a2), cf. Fig. V.56. The horizontal projection S of the mineralized
volume V is estimated by S*, e.g., the union S* = n . a I az of the n positive
zones of influence. The mean vertical mineralized thickness L is estimated
by

L=-

11

I t;
;=1

## The estimator of the volume V is then V* = L * x S*.

The first term of the limited expansion at the origin (when al and a2 tend
toward zero) of the relative estimation variance of V by V* is given by the
formula

at
S*
V = L
2

*2

1 [
n 2 0, 06A

1T

1]

+ 90 A z +, .. ,

(V.59)

## where A = a-] al ~ 1. For such a three-dimensional estimation of a volume,

this formula only provides a rough order of magnitude of the estimation

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435

## variance. In mining applications, if there are sufficient data to characterize

the regionalization of the vertical thickness l(x) at each point x, it is better
to calculate the estimation variance of the mean thickness E{[L - L *f},
then to combine this variance with the variance of estimation of the surface
E{[S - s*f}, using formula (V .63).

.-

Positive
hole

S1f~

C12

170102

ii

## 2. Influence of the geometric error on global estimation

Up to this point, a distinction has been made between the geometric error
due to the determination of the limits of the mineralization and the quality
error due to the estimation of a mean characteristic (e.g., grade) within a
field of known limits. In practice, these two errors simultaneously affect the
global estimations.
Consider, for instance, a subhorizontal sedimentary deposit sampled by
vertical drill-holes, ct. Fig. V.57. A contour of the estimated surface S* is
defined, using the positive drill-holes. By means of the transitive approach,

+
} /(x)

oLd

+
+
+
+

+
+
+
+

+
+
+
+
+

+
+

+
+

+
+

5*

## FIG. Y.57, Estimation of a subhorizontal deposit from a regular grid of vertical

drill-holes.

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436

MINING GEOSTATISTICS

## the order of magnitude of the relative estimation variance of the surface

can be evaluated from formula (V.57), i.e., (j~/S2, with (]'~ = E{[S-S*J 2 } .
On the estimated field S* of known limits, the following unbiased estimations can be made:
(i) the mean mineralized thickness

Ts~=~l
t(x)dx,
S s~
estimated by T~* ;
(ii) the mean metal accumulation

estimated by A~*;
(iii) the mean grade Zs* = As*jTs*, estimated by zt* = A~*ITt*. (The
relative bias due to the quotient of the factors A~* and T~* is
neglected, d. section V.C.3, formula (V.47).
By means of the study of the two regionalizations, the thickness l(X) and
the accumulation a (x) = l(X) . z (x), defined as the product of the thickness
t(x) by the corresponding grade z (x), the corresponding estimation variances can be calculated, i.e.,
for the estimator T~*,

(j~A

= E{[A s* -

A~* J2}.

## These variances correspond to the estimation of the mean characteristics

(mean thickness and accumulation) over the estimated field 5*. In fact, the
real field is 5 = S* + [5 - S*], and it is the mean accumulation (or thickness)
on this true field 5 that is required, i.e.,
As

=~

a (x) dx.

## The difference between As and A s *, i.e., dsA = As - As", corresponds to

the impact of the surface error.
Thus, estimating the mean accumulation As on the true field S by A~",
the error involved is [As - A~*] = dsA + [A s* - A~*], the variance of which
is

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437

## The hypothesis of internal independence Often (but not alwayst) in mining

applications, the surface or volume estimation is made independently of
the estimation of the mean quality; the estimation error [As .. - A~.. l on the
field S* of known limits can then be assumed to be independent of the
impact dsA of the geometric error. The covariance of these two errors is
null and, therefore,

## The global estimation variance of the mean accumulation As is then

reduced to the sum of two terms:
(i) a term (T~A for the estimation on the known field S*. provided by
the standard study of the regionalization of the accumulation a (x );
(ii) a "border" term E{(ds A )2} due to the geometric error.
It can be shown that, under the hypothesis of internal independence,
2

(Ts
2
E{(dsA)}= S2' Da(OjS),
2

(V.60)

## where (T~/ S2 is the relative estimation variance of the surface, and

D~(O/S) is the dispersion variance in S of the accumulation a(x) of
quasi-point support. In practice, provided that the data a (x;) are distributed in an approximately uniform manner over the field S*, this dispersion variance D~(OjS) can be estimated by the experimental dispersion
variance of the n data {a (Xj), i = 1 to n} available in S*, cf. section II.C.l.
Finally, the global estimation variance of the mean accumulation As can
be expressed as

(V.61)
Similarly, the global estimation variance of the mean thickness T s over
the true surface S by the estimator T~.. , defined on the estimated surface
S*, can be expressed as

(T~ = E{[Ts ~ T~ ..
+When

## (T~T+ ~i, D;(O/S),

the limits of the mineralization are economic limits linked to the quality of the ore
(e.g., cut-off grade), the geometric error and the quality error cannot be considered as being
independent, particularly at the borders of the deposit. But if the extent of these border
zones is negligible with respect to the extent of the deposit (of surface S), an hypothesis of
internal independence can be made as a first approximation.

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438

MINING GEOSTATISTICS

## The estimation variance of the mean grade Zs = Asl T s by the quotient

Z~" = A~" I T~* of the previous estimators defined on the known surface
S* is obtained from formula (V.48):
222
<Fz
<FA
<FT

(FA crT
Z2= A2+T2-2PATA'y:+"
The quantity of metal in the deposit is the product Q = As. S of the
mean accumulation As and the true mineralized surface S. The global
estimation variance of Q by Q* = A~* . S* is obtained from formula
(V.49):

## Under the hypothesis of internal independence, the correlation PAS

between the estimation errors of the mean accumulation and the surface is
null. If <F~ is replaced by its expression (V.61), the following formula is
obtained:
(Fb _ <F~A <F~
22 + 2
Q
As S

(V.62)

## Similarly, the relative variance of global estimation of the are tonnage

V d = Ts x S x d by V* d = T~* . S* . d is
(V.63)

where the density d of the are is supposed to be constant and known. If this
density, d(x), is regionalized and sampled along the drill-holes, it should
then be studied by means of a new accumulation variable b (x) =
a(x). d(x).
Note All the preceding formulae correspond to a geometric error of
surface (S - S*). If a volume is involved, the border term E{(dvAf} =
E{[A y - A y * ]2} is written similarly to (V.60):
2

2
(Fv
2
E{(dvA) } = V 2 Da(Oj V),

where

(V.64)

## D~(OIV) is the dispersion variance of the point accumulation a(x) in V,

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439

## Global estimation of fty Mt Flotouo deposit, after Ph. Formery (1963)

This case study is well known to geostatisticians and has been used in
courses at the Centre de Geostatistique of Fontainebleau since 1968. It
concerns the Ity Mt Flotouo deposit in the Ivory Coast, which consists of a
gold-bearing crust of laterite and clay. The thickness of this crust varies
from 3 m to 5 m with a slight overburden. The deposit was sampled by
vertical pits on a regular 20 m x 30 m grid, d. Fig. V.58. Channel samples
were taken down each pit along the entire mineralized thickness.

b z " 30 m
~

) b l ~ 20 m

)0

FIG.

CJ
x

)(

)(

## Estimation of the mineralized surface The locations of the positive pits

(non-null mineralized thickness) are shown on Fig. V.58. The estimated
mineralized surface consists of the sum of the grids of influence of the
n = 58 positive pits. Note the gap in the mineralization at the NW corner
and the rather jagged outline of the polygonal contour of the Northern
end, The influence zones of the positive pits towards this Northern
boundary are debatable, but on such a small deposit, no better estimation
of the Northern boundary can be made from geology. The transitive

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440

MINING GEOSTATISTICS

## approach by means of formula (V.57) is used to characterize the surface

error:
(i) the estimated mineralized surface is S* = nb ib-; = 58 x 20 x 30 =
34800 m";
(ii) the polygonal contour of 5*, including the NW gap, consists of
2N, = 44 and 2N2 = 20 grid elements ib-, b 2 ) ;
(iii) the order of magnitude of the relative estimation variance of the
surface is then

u1 12 [10
22 J
-==--+O06~ = 00014
2

52

58

10

'

## Estimation of the mean accumulation The regionalized variable (in the

horizontal two-dimensional space) is the accumulation a(x)= t(x). z(x),
defined at each point x by the product of the mineralized thickness t(x) and
the mean grade z(x) measured over this thickness. If A* is the mean
accumulation measured over the 58 positive pits,
1

A*=n

L a (Xi ),

;=,

## then the relative accumulation a (x)/ A * can be defined. By defining this

relative regionalized variable, the value of A * can be left unknown (for
proprietary reasons). By means of this relative variable a (x)/ A *, the
various relative variances can be calculated directly; indeed,

## D 2{a (x )/ A *} = D 2 {a (x )}/ A *2.

The N-S and E-W serni-variograms of the variable a (x)/ A *, calculated
from the 58 positive pits, showed structural isotropy, and the two experimental curves were combined to give a single mean serni-variogram
characteristic of the isotropic horizontal regionalization of the accumulation. A spherical model with a range a = 66 m and a nugget was fitted:

Vr=lhl>o,
where Co = 030 and C 1 = 075, giving a sill value of Co + C, = 105, which
is very close to the experimental dispersion variance of the 58 positive
accumulation values, S2 = 10.
The estimation variance of the mean accumulation A s * over the estimated known field 5*, U~A =E{[A s . - A * ]2}, can be calculated either by
direct combining of elementary errors, assumed to be independent, or by
combining terms of the error [As - A *], cf. section V.C.2.

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441

## Direct combining of elementary errors. As the estimator A * is simply the

arithmetic mean of the n = 58 positive accumulations, the global estimation errors [A s * - A *] is made up of the 58 elementary extension errors of
a central accumulation datum to its grid of influence (b1, b2 ) . Thus, from
formula (Y.3?), the global estimation variance of the mean accumulation
on the known field S* is

with

For a spherical model of range a and sill 1, the auxiliary functions Hand F
are given on Charts no. 3 and 4 in section ILEA:
(T~blb2 = Co + C x 011

= 038

and

## (T~A = 038/58 = 00066.

Combining terms. In this case, the global estimation variance is calculated as the sum of a line term and section term. The line term (corresponding to the N-S lines which have the greatest data density, b, < b 2 )
measures the extension error of the 58 central accumulations to their
segments tb , = 20 m) of influence. This line term is expressed by formula
(VAl) as

The value of the variance of the elementary extension a~bl is read from
Chart no. ? and gives (j"~bl = C o+ C x 0077 = 0.36. (Note the preponderance of the nugget effect Co = 0,30.) The line term is then T1 =
036/58 = 00062.
The section term measures the extension of the mean accumulations of
the lines L;, j = 1 to 6, to their sections of influence (rectangle L, X b 2 ) . The
elementary extension variance of a median line L, to its rectangle of
influence L, x b: is calculated using Chart no. 8, and, for the first line
(L I = 160 m), this gives
a~Llb:l = C

001 = 00075.

In this case the nugget effect has no influence, d. formula (111.9); this is due
to the fact that the nugget constant COLI' regularized over the length L 1 ,
can be considered as zero because L 1 is much greater than the diameter of
the sampling pit, which is the support on which the constant Co = O 3 was
defined.

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442

MINING GEOSTATISTICS

Table V.5 gives the variances (T~Lh as well as the variances weighted by
the square L of their length of influence, for each of the six lines L; The
section term is expressed by formula (VA3):

7,

. (jEL;b2'

## where 5* = b2 LjLj; i.e.,

7"L~

1534

I (TEL;b2

Ts =

(~ L

TABLE

= (1160)2 = 00011.

j)

Lj

1
2
3
4
5
6

160
240
200
240
160
160

(T~L;

b:

Lf(T~Li X b 2

00075
00060
00067
00060
00075
00075

192
345
268
345
192
192

L: = 1160 m

L: = 1534

## The global estimation variance of the mean accumulation on the known

field S* is the sum of the line and section terms:
(j~A = T+ T s = 00062+00011

00073.

Thus, the combining-terms method provides the same order of magnitude for the global estimation variance as the direct combining method
(0'0066). To be on the safe side, the higher value (T~A = 00073 is retained.
Calculation of the border term The preceding variance, (T~A =
E{[A s * - A *]2}, corresponds to an estimation on a known field S* and
takes no account of the surface error (5 - S*). The impact of this surface
error on the estimation of the mean accumulation As over the true surface
S is characterized by the border term, cf. formula (V.60),
2

E{(dsAf} =

~~ x D~(O/S),

## where (T~/ 52 = 00014 is the relative estimation variance of the surface,

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443

## and D;(O/S) is the dispersion variance of the point accumulation in S, and

is estimated by the a priori experimental variance of the 58 positive
accumulation data, i.e., D~(O/S)= 10. The border term is then
E{dsA)2} = 00014.
Taking the surface error into account, the relativet estimation variance
of the mean accumulation is finally given by formula (V.61):
2

(TA

A *2 =

(TEA

## i.e., a relative standard deviation of 9-3%, O"A/A*

0093.

Estimation variance of the quantity of metal The quantity of metal available in the deposit of surface S is equal to the product Q = As. S. This
quantity of metal is estimated by the product Q* =A*. S*. Under the
hypothesis of internal independence, the estimation variance of Q by Q* is
given by formula (V.49):
2

UQ

-2 =-2

as

## i.e., a relative standard deviation of 10%, aQ/O = 01.

2
Note that the accumulation error a~/ A is quite significant with respect
to the surface error U"~j S2.

t Recall that, since the regionalized variable is the relative accumulation, the calculated
variances are also relative.

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VI

## Selection and Estimation of

Recoverable Reserves
SUMMARY
This chapter is intended as a study of the influence of the regionalization of the
selection variable on the recovery of the resources of a deposit.
Section VI.A is concerned with the respective influences of the support of the
selection unit (mining selection is made on blocks and not on core samples) and of
the level of available information (in general, selection is made on estimates while
real grades are recovered and treated in the mill).
Section VI.B is a study of hypotheses of permanence of distribution laws, by
means of which one histogram - e.g., the histogram of block grades used for mining
selection - can be deduced from another with a different variance - e.g., the

## Because of the variability of mining and treatment costs added to the

spatial variability of are quality, the entire resources of a deposit are rarely
mineable. Out of these "in situ resources", the mining company must define
a certain tonnage of mineable are, or "recoverable reserves", within a given
economic and technical context.
Whilst these in situ resources are determined by the geological
environment of the deposit, the recoverable reserves will depend, not only
on the in situ resources with their particular characteristics (spatial variabilities of grades, mineralogy, tectonics, etc.) but also, and essentially; on
the economic and technical context of the mining project, i.e., as follows.
(i) On the criterion of selection, e.g., maximization of some benefit
function or supply of a market.

444

-~

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.

445

## cut-off grade or a limiting mineable thickness.

(iii) On the technological constraints of the mining project, e.g., in an
open-pit operation, a block V can only be mined if all the blocks
above it, in the cone with its vertex on V, have been previously
mined.
(iv) On the support (size and geometry) of the selection unit (block of
several hundred or thousand tormes). In general, this support will be
considerably different from the support of the exploration unit
(core samples for example).
(v) On the information available at the time of selection. In general,
once the selection is made, real values are recovered (the mill
The influence on selection of these last two factors, support and level of
information, is expressed by the geostatistical variances of dispersion and
estimation, and this represents one of the main contributions of geostatistics to the techniques of mining estimations. The influence of economic and
technical factors on the recovery of the resources of a deposit, while
essential, will not be treated in this work. These factors are extremely
variable from one deposit to another, and they involve a wide range of
fields which have little or nothing to do with regionalized variables, such as
sociology, politics, fiscal studies, mining techniques, haulage, etc. Since the
object of this book is the study of regionalized variables, we shall limit
ourselves to the study of the effect of regionalization of the selection
variables on recovery. It is the ultimate job of the mining economist and
mining engineer to conclude the feasibility study of the deposit by using
information such as grade/tonnage curves supplied by the geostatistician.

## Consider a deposit G sampled, for example, by drills, the cores of which

are cut into constant lengths c and analysed. Let v be the mining unit on
which future selection will be made.
The in situ resources are often characterized by histograms of dispersion,
and the histogram of the grades z; (x) of the core lengths of support c is
shown on Fig. VI. 1; the abscissa values are for the grades and the ordinates
are the frequency of occurrence of these grades. It is supposed that this
histogram is representative of the entire deposit or zone G, which, in

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446

MINING GEOSTATISTICS

practice, amounts to saying that the sampling of G by the set of drill cores
is unbiased ~ no particular zone of G is preferentially sampled.

## It is very important to ensure that the experimental histogram of the grades

of the available core samples is truly representative of the histogram (called
"local over G" from the terminology of section I1LB.?) that would result
from the set of core samples obtained by dividing the entire deposit G into
drill cores. Thus, the experimentally available core samples should be
uniformly distributed in the volume G; this will be approximately so if the
drilling grid is regular over G. But if the drilling grid is irregular, with, for
example, significant gaps or clusters of drills, it is advisable to assign an
will prevent a non-representative mode appearing in the rich-grade area of
the histogram if the drills tend to be concentrated in rich zones. The
weighting of the grade values will, of course, influence the mean and
variance of the histogram. In particular, the mean of the weighted histogram will no longer be the arithmetic mean of the data set available over G,
but it will be closer to the true mean grade over G (correction of the bias
due to the preferential concentration of drills). It should be ensured that
the new variance agrees with the theoretical variance of dispersion
2
D (c/ G) of the grades of support c in G; if the mineralization is globally
2(c/G)
homogeneous over G, then D
can be calculated from the structural
function (covariance or variograrn), ef. section II.C.2. Among other possible
declustering weights, one may consider to take as weighting factor for each
of the data values z; either
(i) its volume of influence in G, in which case the mean of the histogram
is considered as the polygonal influence estimator (unbiased but not
optimal) of the mean grade over G; or
(ii] the weight attributed to it in the global kriging of G, in which case
the mean of the histogram is considered as the kriged mean grade
of G.

## Selection of units of support v

Consider the histogram of Fig. VI.l, which is supposed to be representative
of the dispersion in G of the grades z; of core samples of support c. This
histogram has
(i) an experimental mean m *, which for the moment is supposed to be
equal to the true mean m of the deposit; this amounts to neglecting
the global estimation error of the deposit;

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447

## (ii) an experimental dispersion variance S2(C/G);

(iii) a certain shape, asymmetric for example, which could be fitted to a
lognormal distribution.
Consider the grade z 0 on the abscissa: the hatched area represents the
proportion of core samples with grades z; ~ Zoo
Suppose now that the true grades z,,(x) of all the mining units of support
v are known; the histogram of dispersion of the grades z,,(x) can then be
constructed as on Fig. VI. 1. This histogram has

## (i) a mean equal to the mean of the core sample grades;

(ii) an experimental dispersion variance S2(V/G). (Note that s\v/G)<
s2(cIG) since, in general, the support v is much larger than the
support c.);
(iii) a certain shape, symmetric for example.
Consider the cut-off grade Zo; the dotted area represents the proportion
of panels v with true grades z, ~ zoo

## Just as the serni-variograrns Yc and y" regularized on core-sample

lengths c and panels v, respectively, are different, so are the two histograms
and the two hatched and dotted areas of these histograms. There may be a
non-negligible proportion of core samples c with grade z; ~ Zo = 5 % Cu,
but there are no panels v with a mean grade z, ;;3 5% Cu. For high grades,
zo> m, the hatched area of the experimental histogram of core grades may
seriously overestimate the possible mined recovery in both tonnage and
mean grade and, correspondingly, underestimate the tonnage of metal left
in the waste, i.e., underestimate the area corresponding to the panels with

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448

MINING GEOSTATISTICS

## In mmmg, the actual selection is made on panels v and not on core

samples c, and for the estimation of reserves, it is essential to take the support
v of the selection unit into account. Otherwise there is a risk of bias with
serious economic consequences.
This remark is not so trivial as it may sound. For example, consider a
sedimentary deposit sampled by vertical drills analysed over the total
mineralized thickness. If the mean mineralized thickness is t, the
support c of the data is a core of section 4> and mean length i as shown
on Fig. VI.2.

I
I
I
I

I
~

L __

I..
FIG. VI.2. Estimation using polygons of influence.

If the polygonal influence method is used, then the mean grade of the
central drill core S, is attributed to each polygon Vi. Now, if the cut-off
grade Zo is applied to the estimates obtained by this method, the estimated
recovered tonnage will correspond exactly to the hatched area of Fig. VI. 1,
with all the associated risks of bias. The results of the case study given in
section VLA.4 are particularly good illustrations of this point.
Another typical example concerns the method of estimation by inverse
squared distances and, more generally, all estimation methods which do
not take account of the particular geometry of the panel to be estimated.
For a given data configuration, such a method would give the same estimated value regardless of the estimation support v. Accordingly, if selection is made on these estimated values, the result will not depend on the
size v of the selection unit. Now, it is obvious that mining a deposit with a
small hammer (very selective mining with a very small support c) will yield
results very different from block-caving (the mining unit v being very
large), see P. Switzer and H. Parker (1975).
Within any homogeneously mineralized zone G (at the scale of mining),
geostatistics can be used to calculate the variance of dispersion D 2 (v/ G) of

~~~~~'

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""""'"

## VI SELECTION AND ESTIMATION OF RESERVES

449

the grades Zv of the mining units of support v (d. section ILC.2), either
formally by the general formula (I1.36), D 2(v/G)= y(O, G)-f(v, v), or
experimentally by means of Krige's additivity relation (11.37), D 2 (v/ G) =
D 2(cIG)-D 2(c/v). The two dispersion variances of the core grades in G
and v can be obtained experimentally.
The mean over G of the grades of the units v can be estimated, for
example, by the mean m * of the grades of available core samples. But the
first two moments, mean m and dispersion variance D\v/ G), are not
sufficient to determine the distribution of the grades Zv. It is also necessary
to know the type of function.
(i) In some cases, this distribution may be known a priori; for example,

## as a result of mining, or from blast-holes on benches, or from zones

that have already been mined, cr. section II.e.3, the case study of the
(ii) In other cases, an hypothesis of permanence of law can be postulated, and the shape of the histogram of experimental data Z; can
be adopted for the distribution of Zv. Thus, if a lognormal distribution can be fitted to the histogram of the Zc, the Z; are
supposed to be lognormally distributed with the variance D 2 (v/ G).
The problems involved in determining one distribution (Zv) from
another (Zc) defined on a different support by means of
permanence-of-law hypotheses will be treated later, in section VLB.

## In the preceding section, VI.A.l, we were concerned with the estimation of

the histogram of the true mean grades Zv, defined on the support v on
which selection will actually be made. But is it really useful to know this
In fact, except in rare cases, these true grades Zv are unknown at the time
of selection, and selection is made on the estimates z~. This means that the
actual recovery will consist not of the units with true grades z, ~ zo, but of
the units with estimated grades -: ~ Zo corresponding to the dotted area in
Fig. VI.3.
This dotted area, corresponding to the units actually recovered (i.e.,
those for which z ~ ~ zo), differs from the hatched area corresponding to
perfect selection made on the true grades (z, ~ zo). This difference
becomes more important as the deviation between the estimators z~ and
the true values z, becomes greater, i.e., as the estimation variance E{[Zvf} becomes larger. Now kriging minimizes this estimation variance.

z;

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450

## In the preceding selection, it was supposed that it was possible to recover

all units v with estimated grade z~ greater than the cut-off grade Zo,
regardless of the position of v in the field G. In practice, this is not always
possible; the actual recovery of a unit also depends on its accessibility and
the proposed mining method. In an open-pit operation, for example, a
block can be mined only if all the blocks inside a cone with its vertex on this
block have already been mined; in underground mining, a new shaft will be
sunk only if sufficient global reserves are accessible from it. In fact, technological considerations define a first rough selection, from which a second
and more precise selection (by cut-off grade, for instance) can be made.

FIG.

## VI.3. Distributions of true and estimated grades of units v.

Thus, on each level, the pit design defines the accessible surfaces; one then
proceeds to a second stage of selection which consists in distinguishing
waste from ore on the basis of the mining unit.
In this chapter, we limit ourselves to a one-stage selection only, made on
the basis of a mean characteristic (grade, for example) of the unit v,
independent of its location in the deposit or in the zone G. The difficult
problems of estimation of the recoverable reserves after a two-stage selection are still in the domain of geostatistical research. One approach to these
problems may consist of simulating the deposit and applying a simulation
of the mining process to this numerical model: the influence of technological conditions on the recovery of reserves can then be observed, d. J.
Deraisme (1977) and section VILB.3.
Smoothing effect
If the estimator Z~ used in the actual selection is a kriging estimator, then
the variances D 2(vjG) and Dk*(vjG) of the true and estimated values can

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----------------------------------

451

## be related by one or the other of the following two relations, called

"smoothing relations".
If the dimensions of the field G are large and if the available information
is sufficient to provide a global estimation variance (T~G which is negligible
with respect to the mean local estimation variance ~V, of a unit v, then
D\vIG)=D~*(vIG)+(J'~v - 2ii

(VI.l)

## (the notations ~o and ii are defined in the following paragraphs).

If the kriging proced ures that provide the estimators Z~lJ can be assimilated to a complete kriging procedure, i.e., to a unique kriging procedure
using the same data configuration comprising all the data available in G,
then
(VI.2)
2

where D (vI G ) is the dispersion variance of the true grades Zv(x) in the
deposit or zone G, and D~* (vi G) is the dispersion variance of the corresponding kriged values Z~li(X). From section II.C.l, it is known that
these two variances, D 2(vIG) and D~*(vIG), are estimators of the two
experimental dispersion variances s\vl G) and s~* (vi G) of the two histograms of the true and kriged grades of Fig. VL3.
2

IN

(TKv= -

N;=l

2
(TKVj

= -

L E{[ZVi -

*2

Z Kli,] }

is the mean value of the kriging variance of a unit v. There are N such units
v in the zone G. If the grid of available data over G is not regular, then the
various local kriging variances O"~Oi will be different. Ii = (11 N) L f-I-; is the
mean value of the Lagrange parameter used in the kriging system of each
unit v, d. system (V.l).
(T~G = E{[ZQ ~ Z'b J2} is the kriging variance of the mean grade ZG over
the zone G. In the case of a complete kriging system, it can be shown (d.
section V.A.3, "Theorem of combining kriged estimates") that the global
kriged estimator Z~G is the average of the N local kriged estimators Z~Vi'
i.e.,

ZKG

N1

~ Z *KV;,
7

The proofs of these two smoothing relations VLI) and (VI.2 are
given in the addendum at the end of this section, VI.A.2.
The smoothing relation (VI.2) implies a smoothing effect: D~* (vi G)~
2(v/G).
D
Indeed, the global kr~ing variance is always less than the mean
local kriging variance: O"~G ~ (TKv' On the other hand, in the local kriging

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452

MINING GEOSTATISTICS

## system ((VI.1 or (V.2)), there is no restriction that (T~1Ji ?3 2JLi, so that it is

not necessary that ~,,~ 2,iL. However, the smoothing effect D~* (vI G)~
D\vl G) has always been observed in all data configurations encountered
in mining applications and for the standard variogram or covariance
models.
The kriged dispersion is smoothed with respect to the true dispersion
and the amount of smoothing increases as the mean ~" of the local kriging
variance increases, i.e., as the estimators zt,(x) of Z,,(x) become worse.
This smoothing effect means that the histogram of the kriged values
zt,(x) will display more values around the mean (close to m*) but less
extreme values than the histogram (unknown in practice) of the true grades
Z,,(x), d. Fig. VI.3. This may be awkward when the cut-off grade is high,
zo>m*.
Note that the smoothing effect expressed in relations (VI.1) and (VL2) is
characteristic of the kriging estimator. The linear estimator obtained from
the polygonal influence method, for instance, has a tendency to overevaluate the real dispersion D 2 (v/ G), d. Fig. VI.9 in section VI.A.4.

## Practical estimation of the recovery

At the time of selection, the true values z v are generally unknown, and the
actual selection is made on the estimators z ~ (preferably on the kriging
estimator z L). It is, thus, imperative to evaluate the recoverable tonnage
from the histogram of the estimators zt, i.e., by the dotted area (z~ ;:: zo)
on Fig. VI.3.
1. If, at the time of the evaluation of the reserves, the ultimate estimaupon which selection will be based, are known, then the histogram
tors
of the z~ is experimentally available and the recoverable tonnage (z~ :;:;: zo)
can be estimated without any problem. In the next section, VLA.3, it will
be shown that the kriging of the selected units (those for which z~v;;::: zo)
provides a fair estimator of the mean grade of this recoverable tonnage.
From the histogram of the kriging estimators z~", it is possible to obtain
an estimator of the histogram of the true values z, by the following means.

zt,

## (i) Use of a permanence-of-law hypothesis to extend the type of law

observed on the distribution of the z~" to the distribution of the z".
(ii) Use of a correction of variance: the variance of the distribution of
the z, is n 2 (vl G). This variance can be deduced from Dic* (vi G) by
one of the smoothing relations (VI.1) or (VI.2), or calculated from
the general formula (II.36).
t Formulae for deriving recovered tonnage and average grade from a known distribution are

## given in the addendum VI.B.3.

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453

## It is then possible to evaluate the increase in the recovery of the

resources of a deposit by improving local estimation: a decrease in
will
result in the histogram of the z ~v (used by the actual selection) becoming
closer to the histogram of the true values z.,
2. Very often, at the time of the study of the evaluation of reserves, the
ultimate data (e.g., blast-holes in an open pit) from which the estimators
z~* used in selection will be constructed, are not yet available. This means
that the only histogram available at this stage is that of the estimators z ~v,
constructed from the data available at the time of the study, rather than the
histogram of the z ~* on which the actual selection will be made. This first
set of data is often very inferior to the ultimate set of data available for
selection, e.g., the study of evaluation must be performed from drill-hole
core data, while actual selection is based on a dense grid of blast-hole data,
plus important visual information from the mining front. As a
consequence, the two estimation variances cr~* = E{[Z.., - zL ]2} and
E{[Z.., - zt: flo as well as the two histograms, may differ considerably.
Note that from the fact that u~* > E{[Z.., - Z~* ]2}, the dispersion variance Dh*(v/ G) of the ultimate estimators zt* will generally be greater
than the dispersion variance D~* (vi G) of the kriging estimator Z~.., corresponding to the primary information.
The tonnage recoverable from future selection (z ~* ;:3 zo) corresponds to
the dotted area on Fig. VIA.

;rv

## FIG. VIA. Distributions of estimated values corresponding to two levels of

information.

In practice, at the time of the study, both the histogram of the core
grades z; and that of the kriged grades zt, are available, these two
histograms having respective variances D 2(c/ G) and D~* (vI G). The
problem is to obtain an estimator of the histogram of the ultimate estimators z ~* (to be used for selection) from these two experimentally available
histograms. To do this, it is necessary to act as follows.

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MINING GEOSTATISTICS

(i) To evaluate the level of ultimate information which will be used for
future selection and to calculate the corresponding mean kriging
variance ~~*. Remember that a kriging variance can be calculated
in advance, once the corresponding data configuration is known. d.
section V.A.!, remark 4. However, the prior calculation of (T~~* is
often approximate, since it must take into account future information which may be difficult to quantify, e.g., the experience of the
shovel operator or information about the stope face.
(ii) Using one of the smoothing relations (VI. 1) or (VI.2), to deduce the
dispersion variance D 2**(vIG) of the ultimate estimators Z~*.
Thus, for example, if the following approximate smoothing relation
is used:

## D **(vl G) = D~* (vi G) + [~~ - ~~*].

2

(iii) To deduce an estimator of the histogram of the Z~* from one of the
two experimentally available histograms by a variance correction
and assuming permanence of law.
If the core grade histogram is considered, the variance correction is a
2
smoothing, D **(vI G )< D \ cI G ). It is then advisable to verify that the
same permanence-of-law hypothesis will also reproduce the histogram of
the z; by correcting the variance D2(cjG)~D~*(v/G), with
2
D~*,vjG)< D **(vl G)< D 2 (cj G ).
If the histogram of the kriged grades Z~u is considered as the starting
point, correcting the variance then amounts to increasing the variability:
I 2*
D 2 **(V IG) ';3 D K2 * (10)"
V
, Since, III genera,
O"Ku::2: 2**
0- Ku .
The case study of section VLBA gives an example of the development of
such permanence-of-law procedures.
Note that, sometimes the sought after histogram of the ultimate selection
estimator Z~* can be obtained directly from data coming from zones or
Addendum. Demonstration of the smoothing relations
(This demonstration does not provide any new insight into the problems of selection and can be omitted on a first reading.)
Let V be a stationary field made up of the union of N units, all with the same
support v, centred on the points {Xi, i = 1 to N}. The global mean grade over V can

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455

## be written as a Sum of the N true grades of the units v:

The dispersion variance of the true grades of the units v in V can be written
according to formulae (11.31) and (Il.35):

Consider now the N unbiased estimators Z~ (Xi) of the grades of the units v. The
global mean grade estimated over V is then
Z '"v (x ) = -I '\'
L. Z*
._" ( Xi ) .

N,

In a similar way. we can define the dispersion variance of the estimated grades of
the units v in V:

(v/ V) =

E{

~~ [Z~ (x,)-ztex )1

## = ~ ~ E{Z;2 (Xi)} - E{Z't} (x)}

The case of kriged estimators If the estimators used above are kriged estimators,
then Z~ (Xi) = Z~v (Xi) is a linear combination of the n, data {Z,,(Xi)' a '" 1 to nil in
the neighbourhood of estimation, i.e., zt" (Xi) = L" A",Z",
and the kriging
system (V. I) is written as

ex, ),

## : A.C(a, /3) ~ C(v" a)+ ""

{ IA s

\:fa

= 1 to n;

= 1

13

and

Considering that C(h) is a centred covariance, and that the stationary expectation is m = E{Zku (x)} = E{Z~v(x)}, the following relation can be expressed:
E{[Z~UeXj}-m]2}=

## l: L. A"AsC(a, (3) = L A",C(Vi' a) +!-t,

"

{3

"

(VI.3)

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MINING GEOSTATISTICS

Now, by putting

and noting that C(Vj, Vi)= C(v, v), 'Vi (all units Vi have the same support v), the
following relation is obtained for the dispersion variance of the kriged grades:

## Di* (vi V) = C(v, v)+ 2,iL - a~u - E{[Z~v(x)- m f}.

By substituting D\vl V)+ C(V, V) for C(v, v), this relation becomes
D 2 (vl V)

= D~*(vIV) +aL, -

## 2.u - [C(V, V) - E{[Z~v(x)- mf}].

(VIA)

The term between the brackets is of the order of magnitude of the global
estimation variance over the field V, and can generally be disregarded when
compared to mean local estimation variance o-L. The first approximate smoothing
relation (VI. I) is then obtained:
D (v/ V)= D~* (vi V)+uL- 2ji.
2

The case of complete kriging The term "complete kriging" is used when the N
krigings Z~,,(Xi) have been carried out using the same data configuration comprising alit the available data in V. From section V.A.3, "Theorem of combining
kriged estimates", it can then be shown that the sum zt = (1/ N) L Z~" (x.) cor-

responds exactly to the direct kriging of Zv(x) from the previous data configuration. For E{[ Zt. v (x) - mY} we then obtain an expression similar to (VI.3):
E{[Z~v(x)-mf} = C(V, V)+2tLv -(T~v,

where

## d. relation (V. I?).

Substituting (T~ v - 2/.Lv) for the term between brackets of formula (VIA), relation (VIA) gives the second smoothing relation (VI.2):
D\v/ V)= D~* (vIV)+I7~u-(T~v

The mean local kriging variance t:T~u is always much greater than the global
kriging variance akv, and, thus, the previous relation reduces to
D (v/ V)= D~* (vi V) + o-i",
2

## Di* (vi V).,; D 2 (v / V).

t In practice, the information used for the kriging of a unit v is limited to a neighbourhood

surrounding v. However. and provided there is no strong nugget effect (since the nugget
effect removes the screen, cf. section V. A.l, re mark 7), this nearby informatio n screens the
influence of more distant data and, thus, the kriging of v can be assumed to be complete and
taking all the available information over V into account.

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457

## VI.A.3. Bivariate distribution of the estimated and true values

In the previous section, it was shown that the actual selection is carried out
on the estimators z~ and not on the unknown true values z., The recoverable tonnage (z~ ~ zo) must thus be estimated from the distribution of the
estimator Z ~. But once this selection is made, the mill receives the true
values, not the estimates; more precisely, it receives the true values z,
conditioned by the fact that their estimators z~ are greater than the cut-off
Zoo The set of recovered values is denoted by {zv/ z ~ ~ zo}.
By interpreting the true and estimated grades z; and
as realizations
of the random functions Zv(x) and Z~ (x), the bivariate distribution of the
two variables Z; and Z~ can be conveniently represented by the scatter
diagram (or correlation cloud) on Fig. VI.5. The true grades appear along
the ordinate axis and the estimated grades along the abscissa axis. The
distribution of the true grades, with variance D 2(v/G), can thus be seen as
the projection of this diagram on to the ordinate axis; similarly, the
distribution of the estimated grades, with variance D 2*(v/G), can be seen
as the projection of the diagram onto the abscissa axis.
A regression curve f(z), giving the mean of the true grades of the units
for which the estimated grades are equal to a fixed value Z~ = z, is also
shown on the same figure. In probabilistic terms, fez) is the "conditional
expectation" and is denoted by f(z) = E{Z,j z~ = z}, Note that the
recoverable tonnage for a cut-off Zo, {Zu! Z~ ~ zo}, corresponds to the
dotted area on Fig. VI.5.

zt,

--1f-----t- m

==-_z~

FIG. VI.S. Bivariate distribution of the true and estimated grades, Z" and Z~.

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MINING GEOSTATISTICS

458

## Conditional non -bias

The true grades {Z,j Z~ = zo} of the selected units with estimated grades
Z~ = Zo are distributed along the vertical chord with abscissa zo, and their
mean conditional expectation is [(zo) = E{Z,j Z~ = zo}. In practice, this
true mean grade is unknown and is estimated by the cut-off value zoo It is
very important that this estimation Zo be as close as possible to the true
value f(zo). The ideal case would be f(zo) = Zo, VZ o, i.e., an estimator
such that its conditional expectation curve E{Z,) zt = z ] is identical to the
first bisector. The conditional non-bias relation can then be defined as

Z:

(VI.5)

Vz,

## which can also be written as

E{[Zv-Z~]/Z: =z}=O,

Vz.

Note that this conditional non-bias condition is more severe than the global
non-bias condition E{Zv - Z~} = O. Note also that the conditional nonbias (VI.5) entails the unbiased ness of the estimation of the true mean
grade of the recoverable tonnage by the mean of the estimated grades of
the selected unit, i.e.,

z:

## If selection is made from the estimators

verifying the conditional
non-bias relation (VI.5), the problem of the estimation of the recoverable
reserves is thus resolved, The recoverable tonnage, i.e:, the number of
;?:: 20, is evaluated from the histogram of the
estimated values z t; the true mean grade of this recoverable tonnage is
estimated without bias by the mean of the estimated grades of the selected
units.
In practice, there is no reason why the estimator used, Z~, should verify
the conditional non-bias relation (VI.5); the conditional expectation
E{Zv/ Z~ = z} = f(z) will be different from the first bisector: f(z) r= z, cf.
Fig. VI.5.
An estimator
should then be found, which would minimize the mean
quadratic deviation [f(z) - Z]2, i.e. the variance E{[f(zt) ]2}. This
criterion characterizes the accuracy of the effective selection; this selection is
said to be accurate if the mean recovered grade f(z) is close to the value z of
the selection parameter.
Accuracy, however, is not sufficient: the selection made on the estimator
(Z: ;::. zo) must also be as close as possible to the ideal selection carried out
on the true values (Z, ;::. z 0)' The tonnage recovered by this ideal selection
corresponds to the hatched area {Zv! Z; ~ z o} on Fig. VI. 5. This hatched

z:

Z:

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z:

459

z:

## area differs in two ways from the dotted area {Zul

~ zo} corresponding
to the tonnage which can be recovered in practice from the information
Zu
*
(i) A certain number of units estimated as being too poor when they
are, in fact, above the cut-off Zo; these units are wrongly rejected and
correspond to the hatched area only.
(ii) A certain number of units estimated above the cut-off grade when
they are, in fact, too poor; these units are wrongly accepted and
correspond to the dotted area only.
The problem is to minimize these two areas corresponding to wrongly
estimated units by reducing the length of the vertical chords of the scatter
diagram (z., z ~), i.e., to minimize the dispersion of the selected true grades
around their mean f(z) = E{Zj
= z}, which amounts to minimizing the
conditional variances E{[Zu - f(z )]21 Z~ = z} for all values z,
The desired estimator Z~ is, thus, the one which minimizes both
the accuracy criterion E{[f(Z~) - Z~
and the dispersion criterion
E{[Zv )]2}.
Consider the variance of estimation of Z, by Z~ :

z:

f}

tiz:

+ 2 {[Zv -

z; )][f(Z~ ) -

Z~

f}

]}.

## From the definition of the conditional expectation f(z) = E {Zvl Z~ = z},

the cross-term of the preceding expansion is zero and, thus,
E{[Zv - Z:

E{[Zv -

Z~ fl.

(VI.6)

za,

## Now, the kriging estimator

is precisely the linear estimator which
minimizes the estimation variance, i.e., the sum of the accuracy criterion
and the dispersion criterion. By minimizing the sum of these two criteria,
kriging represents a compromise between satisfying one or the other, and,
because of this, it is preferred to any other linear estimator for selection
problems.
VI.A.4. Case study on a simulated deposit
A simulated deposit, as opposed to a real one, is perfectly known; in
particular, all the distributions necessary for the solution of selection
problems can be readily obtained (distribution of sampling grades, bivariate distribution of true and estimated grades of selection units, etc.). On a
simulated deposit, it is thus possible to determine exactly the effect of any
estimation procedure on the recoverable reserves.

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MINING GEOSTATISTICS

The deposit
This simulation can be considered either as a zone of sedimentary deposit
of variable vertical thickness (such as the lateritic or silicated nickel
deposits of New Caledonia, d. section IV.E, Case Study 13), or as a
horizontal mining bench of constant vertical thickness in a massive deposit
mined by open pit (such as a porphyry - copper deposit).
The simulated zone is a rectangle of dimensions SOu x lOu, made up of
500 square blocks Vi (of dimensions u xu), cf. Fig. VI.6. A total of 60 500
samples were simulated on a regular u] 11 x u] 11 grid over this rectangle,
giving a total of 11 xlI = 121 samples per block. These 60 500 samples are
considered as the real deposit G to be studied.
IOu

I'

"1
u

I'
:w;

:w;

x x

vI

x x x x

r----t----t:,..--.;;- - - -

'I
:w:

:-------:
S,.

II

x II drill- holes

## Centro I drill- note 5,.

Block

v,.

1----+--+ - - - - Deposit G

FIG.

## If G is a subhorizontal sedimentary deposit, then these 60 500 simulated

values represent the accumulation a (x) = t(x). z (x) at each point x, i.e., the
product of the thickness t(x) by the corresponding mean grade z (x). If G is
a bench in an open pit, then the simulated values represent the grade z (x)
over the constant vertical height of the bench. There are, thus, 500 true
mean grades z,,(x;) of the blocks Vi, available over G (the true mean grade
of each block being taken as the arithmetic mean of the 121 grades of the
samples that fall within it). The samples S; at the centre of each block are
assumed to be the result of a sampling campaign providing 500 grades z (Xi)
on a regular u x u grid, d. Fig. VL6.
The grades z(x) were simulated according to the techniques of simulation presented in Chapter VII, i.e., in such a way that they reproduce the
following.

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## VI SELECTION AND ESTIMATION OF RESERVES

461

(i) A given distribution - with mean 957% and variance (T~ = 22,4%2
- which is, in fact, the experimental histogram of the nickel accumulations (thickness x grade) of the Mea deposit (New Caledonia).
Obviously, if z(x) is interpreted as a grade, the value m = 957% is
divided by 10.
(ii) A given model of spatial variability; in this case, a spherical model
with a nugget effect and isotropic in the horizontal plane:
!E{[Z(x +h)-Z(X)]2} = y(h)= Co+C,y,(r),

'rtr =

Ihl> u/ll,

## where Cs> 6%2, C'Yl(r) is a spherical model with sill C 1 = 18%2

and range a, = lOu.
Kriging of selection units
A model y(h) was fitted to the experimental semi-variograrn y*(h) of the
500 samples from the sampling campaign of the u x u grid. These samples
were then used to estimate each block V; by kriging, using the estimation
configuration on Fig. VI-7, i.e.: one weight Ai for the grade of the central
sample; one weight A2 for the mean grade of the four median samples of
the first aureole; one weight A3 for the mean grade of the four corner
samples of the first aureole.

x~x

## FIG. VI.7. Kriging configuration of the selection units.

This kriging configuration is complete only for the 384 blocks Vj inside
the zone 0' of dimensions Su x 48u, defined as the erosion of 0 by a unit
distance u. The following study will be confined to these 384 "blocks for
which the kriging system is identical.
The results AI, AZ, A3 ,
of the above kriging system are given in Table
VI.1. Since the true grades of these 384 blocks are available, the kriging
variance u~ = 1 32 % 2 can be compared to the experimental estimation

ai

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MINING GEOSTATISTICS

## variance (T~* = 135%2. The kriging system thus provides an excellent

prediction of the error variance.
If each of these 384 blocks had been estimated by the polygons of
influence method (Poly), by assigning the grade of the central sample to
each unit v, i.e., A1 = 1 and A2 =) J = 0, the experimental estimation variance would have been (T~* = 692% 2. Using geostatistics, this variance
would have been predicted, according to formula (II.28), as cr~ =7,31%2.
VI.l. Compared results of the kriging and
polygons of influence methods of estimation

TABLE

Krig
Al
..1. 2
..1. 3

O"~, theoretical

0'1, experimental

0204
0-458
0338
1323
1348

Poly
1

o
7307
6919

## Estimation of recoverable reserves

Suppose, now, that at the time of mining, the only information available on
which to carry out selection is the u x u grid of samples. Recovery consists
of all units Vi which have an estimated grade z~; greater than the cut-off
The tonnage of recovered ore T(zo), i.e., the number of units N(zo)
recovered, is determined from the histograms VL9(a) and (b) of the 384
estimated values, Z~Vi by kriging or zt by polygons of influence. Note
that histogram VI.9(b) of the estimators zt is that of the 384 grades of the
central samples; thus, using the polygons of influence estimator
amounts to ignoring the influence of the selection support v, d. section
VLA.1. The histogram of the 384 true grades of the blocks Vi is given on
Fig. VL9(c).
Once the selection is made, ZftJ; ;3 Zo or zi ;3 zo, the conditional nonbias condition must be verified, d. section VLA.3, formula (VI.5), i.e., the
true mean grade E{ZtJ/ Z~ ;3 zo} and the estimated mean grade E{Z~/ Z~ ;3
zo} of the recovered tonnage must agree. For each of the two estimators
considered, Fig. VI.8(a) and (b) give the two curves E{Zvl zt ;:;. zo} and
E{Z~/Z~ ~zo} as functions of the cut-off value zoo Note that kriging
ensures conditional non-bias, whereas selection carried out on polygonal
influence estimators is subject to a serious bias (systematic overestimation of
the actually recovered grades); this bias increases with the selectivity (i.e., as
Zo increases).

z:

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463

## The global quantity of metal actually recovered, O(zo) =

T(zo). E{Z,j Z~ ~ zo}, is proportional to N(zo).E{Zv/ Z~ ~ zo}. The
difference between the two quantities of metal recovered, corresponding to
the two types of selection estimator considered, is a direct measure of the
economic benefit to be obtained from a geostatistical approach to selection
problems, i.e., an approach which takes into account the two fundamental
influences of the selection support and the level of information.
The results of a hypothetical selection based on the true grades ZVj of the
384 selection units (i.e., perfect knowledge of the deposit) can also be
given. This ideal selection will obviously lead to the best results.
For the three selection hypotheses (based, respectively, on true grade,
gives, for the three cut-off grades zo=4, 9 and 14%:
1. the number N(zo) of recovered blocks;
2. the true mean grade of the recovered tonnage, E{Zvl z; ~ zo};
3. the estimated mean grade of the recovered tonnage, E{Z~I Z~ ~ zo};
4. the conditional bias, E{Zv -Z:;zt ~zo};
5. the experimental estimation variance of the grades of the recovered
blocks, E{[Zv ~Z~ ]2/Z~ ~ z.};
6. the global quantity of metal actually recovered, N(zo).E{Zv/Z: ~

zo};
7. the estimation of this quantity of metal, N(zo).E{Z7IZ7 ~ zo}.
TABLE

Zo

True

selection,
Zv
Kriging,
Z*

"

Poly,

Z*

(1)

(2)

(3)

(5)

0
0
0

(6)

(7)

U
U

370
203
67
365
205
65

1013
12-82
16-04

1005
12-66
1591

1-36
1-36
156

3698
2628
1043

3668
2593
1034

{1~

337
199
67

10-51
12-52
14-93

## 10-68 -0-17 6-94

13-52 -1-00 708
16-95 -2'02 781

3542
2491
1000

3599
2690
1136

- 4

10'08
1297
1619

(4 )

0
008
016
0-13

3730
2633
1085

Note that the selection made on kriged estimates results in a better recovery from the deposit, in terms of the global quantity of metal actually
recovered, Also note that the standard polygons of influence estimator
always overestimates the effectively recoverable quantity of metal, which is
particularly dangerous during a feasibility study. Selection based on the

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MINING GEOSTATISTICS

464

## standard estimators of inverse distances (ID) or inverse squared distances

(ID2) would have yielded better results than selection based on polygons of
influence estimators (Poly), but these results are always strictly inferior to
those obtained from selection based on a kriged estimator.

20

(0 )

15

10
>,

<.>

c:

<u

:I
C'

lL.

(b)

:>?
o

/
/
/

20

/
/

/
/
/

'"
'"
...
.... ....
,,-

15

,/

"..-

IOc...__--

..... ".......

.... ....

""

9
15

20

FIG. VL8. Conditional bias of the estimators. (a) Kriging. ~--, E{Zv/ zt, ~

## zo}; ----, E{Z:'v/Zkv ~zo}. (b) Polygons of influence.

----, E{Z~/Z~ ~zo}.

, E{Zv/Z~ ~zo};

If, instead of a regular grid, the 384 blocks were estimated from data on
a pseudo-regular grid with clusters of data and gaps (which is often the case
in mining applications) the results obtained from kriging would be still even
better than those obtained from ID, ID2, and Poly.

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465

## The smoothing relation due to kriging

The smoothing relation (VI.2) due to kriging can be verified from the
experimental dispersion variances of Fig. VL9(a) and (c):
D

2(v/G')

= Dk* (V/Gf)+(T~v

(the global estimation variance (T~c:J' is disregarded). Indeed, the experimental values show

## 1614 = 1471 + 135 = 1606.

20
(a)
,--

--

>-

<.>

"

'"0u:'"
~

10 f-

( b)

'--;.-' - ,--r--

--

,--

I--

I
18

14-4

rh

I--

20

144

r..

20
20

(c)

----

rh

1-8

I--

144

20
FIG. VI.9. Histograms of true grades and of their estimators. (a) Kriging: m* =
9,72, a 2*=1471. (b) Polygons of influence: m*=9'70, a 2*=2313. (c) True
grades: m* = 9,82, (T2 = 1614.

## The bivariate distribution of true and estimated grades

It was shown on Fig. VL5 of section VLA.3 that the conditional non-bias
E{Zv/ Z~ = z} = z, Vz, is verified when the conditional expectation curve
f(z) = E {Zv/ Z~ = z} is identical to the first bisector. The best linear estimator of this regression curve f(z) is the regression line D 1 of the true
values Z, given the estimated values Z~. It is thus advisable to verify the
proximity of D 1 to the first bisector on the two scatter diagrams (z., z~)
corresponding to the two types of estimator used.

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466

MINING GEOSTATISTICS
Zv,----------------------

-----..,

(a)

r-

l
I

:3.

I 2

I
I

## I... ,.. ....

1..... 1 I

E 1"-3
I"'-:!

....'6. 2

'5M
1

i! i:.
2:
4

I!
I

2'

.... 2" I

<

"

~ .... 1

r-z

I
I

I I

I
I

"

I I

I ~
.2 <
~

I ~

2" I
I 2

III
E Z

I .....

2: 2'2

"

,/1

1';'.':')'

"'1
,.,

f l r st

....

* I dose to

I Z l<t'

bisec tor

'"

M
(b)

,I
t

I I
,
, I

I I
I

,,

Z"
v

FIG. VI. 10. Scatter diagrams. true grade/estimator. (a) Kriging; (b) polygons of
influence.

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467

## For the kriging estimator, the equation of D, is Z; = 0074 + 1003 zt.,

with a residual variance approximately equal to the kriging variance 1 348.
Thus, there is no significant difference between this line D 1 and the first
bisector, d. Fig. VI.10(a).
For the polygons of influence estimator, the equation of D 1 is Z, =
3040+0699Z~, with a much greater residual variance (4-829). This
regression line D 1 is significantly different from the first bisector, as can be
seen on Fig. VI. 1O(b).
On the other hand, if we consider the second regression line D z of the
estimated values
given the true values Z; (i.e., the best linear estimator of the second regression curve g(z) = E{Z~/ Z; = z}), then kriging gives
the equation of D z as zt, = 0744 + O' 911Zv , with a residual variance of

Z:,

1229.
The polygons of influence estimator gives the equation of D z as Z~ =
-0-144 + 1002ZtI , with a residual variance approximately equal to the
estimation variance 6919, cf. Table VLl. In this case, line D z corresponding to the Poly estimator is closer to the first bisector. In fact, the mean of
the grades Z* of the samples within a block v of true grade Z; = z is equal
to z. However, since the actual selection is made on the estimator Z~ and
not on the unknown true grade Zv, regression line D, is the line of interest
and, thus, the kriging estimator Z~v is retained and conditional non-bias is
ensured (D t is close to the first bisector).
It was D. G, Krige (1951) who first distinguished the practical influence
on mining selection of these two regression lines, and for this reason the
preceding scatter diagrams (Zv, Z~) are sometimes referred to as "Krige's
ellipse" the term ellipse is used because the experimental scatter diagram
takes an elliptical shape when the bivariate distribution is Gaussian.

## VI.B. PERMANENCE OF DISTRIBUTION

It was seen in the preceding section that the estimation of the recoverable
tonnage requires the inference of the histogram of the estimators zt*
which are ultimately used in the future selection. For example, this
inference is made from the histogram of the grades Z; of the available core
samples, or from the histogram of the estimators Z~ available at the time
of the study.
More generally, we have the experimental histogram of a stationary
variable Z, from which it is required to deduce the distribution of a variable
Y that has the same expectation, but a different, known variance: E{ Y} =
E{Z} = m and Var{ Y};:= Var{Z}. This problem as posed has no solution,
since the type of the distribution of Y is undetermined. However, the

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468

MINING GEOSTATISTICS

## problem can be solved if we assume that the distributions of Y and Z have

the same "shape". The following section will show how this permanence of
shape can be interpreted in practice.
VI.B.1. Simple hypotheses of permanence

1. Permanence of normality
Let Z be a Gaussian stationary variable (in practice, a variable the histogram of which can be approximated by a normal distribution), with expectation m and variance <T~. Let U be a standard normal variable with zero
expectation and unit variance. These two variables are related by the
standard formula Z = m + Uz. U.
The hypothesis] of permanence of normality means that Y is considered
to follow a normal distribution with expectation m = E{ Y} = E{Z} and a
variance <T~ #- (T~. Y can be expressed in terms of the standard normal law
as

Y= m+Uy. U.

(VI.7)

## The distribution of Y can then be readily determined from tables of

standard normal distribution.

2. Permanence of lognormality
Let Z be a lognormal variable with expectation m and variance a~. The
Gaussian transform function 'Pz which transforms a standard normal variable U into this lognormal variable Z is
Z

cpz( U) = em'+a~u

(VI.8)

or, conversely,
U = cP 2/ (Z) = (log Z - m')j <T~ ,

where m' and cr'l are the expectation and variance of the normal variable
log Z. These logarithmic parameters are deduced from the known arithmetic parameters m and (T~ by the standard relations
}
m = E{Z} = em'+a~/2,
{ u~ = Var {Z} = m 2(e(T] -1).

(VI.9)

t If Y can be expressed as a linear combination Y = Let AaZa of normal variables, and if the
distribution of the RF Z(x) is multivariate Gaussian, then its distribution is normal, and the
permanence of normality is a property of Y and not an hypothesis. However, normal-type
symmetrical distributions are not very frequent in mining applications; the histograms of
grades in particular are usually asymmetric of the lognormal type.

~-~~

, VNNI.minebook.blogfa.com
.

Z

469

({Jz

(VI. 10)

## The hypothesis'] of permanence of lognormality amounts to considering

that the variable Y follows a lognormal distribution with the same expectation m = E{Y} = E{Z} and a variance
lT~. The Gaussian transform
function cpy which expresses Y as a function of the standard normal
variable U is thus written as

O"t

Y = 'PY (V) = m . e

variance
by the relation

O"t

O"t

IT'y

---o,2j2
Y

e ,r Y'-

(VI. 11)

Var {Y}

= m 2(e'T':j ~

1).

## It is thus possible t to construct the required distribution of Y from the

standard normal distribution of U. The distribution function is then
F y (y ) = Prob { Y

where:

< y} =

y=cpy(u)

or

Prob { U < u} = G( u ),

(VI.12)

U=tp1}(Y),

## 3. Indirect correction by permanence of a two-parameter distribution

The standard two-parameter distributions, such as the normal, lognormal
or gamma, have the advantage of being tabulated, and an immediate
correction of the variance O"~ ~ a~ can be made. However, as these distributions depend on two parameters only, they are often unable to take
the major characteristics of an experimental histogram into account; a
bimodal histogram, for instance, requires at least a three-parameter
function in order to obtain a good fit. The lognormal distribution fitted to
the experimental histogram on Fig. VI. 11 (a) cannot take account of the
two significant modes. However, the major interest lies in an estimator of
t Contrary to the normal case, a linear combination Y = L:,,, A",Z", of lognormal variables Z",
lognormal distribution, then, to be theoretically rigorous, the grades Z; of blocks cannot
follow a lognormal distribution. In mining applications, however, the histogram of the block
grades z" very often has the same lognormal type of asymmetry as the histogram of core
sample grades Zc' For this reason, a- working hypothesis of permanence of lognormality is
adopted. A case study involving permanence of lognormality is presented in section II.C.3,
"Grade dispersions at Chuquicarnata mine", ct. Figs. 11.22 to 11.24.

t The lognormal recovery functions formulae are given in the addendum to section VLB.3.

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470

MINING GEOSTATISTICS

the distribution of the variable Y with the same expectation but different
variances (T~;t a~, and not in the quality of the fit of the histogram of Z.
If the two modes of the Z histogram are to be exactly reproduced on the
estimated histogram of Y, then, under certain conditions, t the following
procedure can be used.
0/ , . . . - - - - - - - - - - - - - - - - - ,
0

(b)

(a)
Experimental
histogram

(m,cr})
-',

)IC
\.

Log normal fi I

"

:; I

"' -, \.

(m,cr})

-,

"-

o
FIG.

"

## variance through a lognormal

(a) Lognormal fit; {b) correction of variance.

model.

(i) Using the simple two-parameter distribution which fits best the
experimental histogram of Z, make the variance correction a~ ---;. (Tt,
cf. the two lognormal distributions on Fig. VI. 11(b).
(ii) From these two two-parameter distributions, read the recoveries
corresponding to the cut-off grade zoo The quotient of these two
recoveries gives an estimate p of the correction coefficient, to be
applied to the recoveries deduced from the experimental histogram
of Z to obtain the recoveries corresponding to the histogram of Y.
Thus, if the recovery read on Fig. VI. 11(b) from the lognormal distribution (m, a~) is 20% (hatched area) and if the recovery read from the
lognormal distribution (m, a~) is 25% (dotted area), the correction
coefficient is 25/20= 125. The hatched area on Fig. VI. 11(a) gives the
recovery read from the experimental histogram of Z, say, for example,
18%. The estimator of the recovery of Y for this cut-off grade Zo is, then,
Prob* {Y;2< zo} = 18 x p = 225%.
t This indirect correction procedure retains all the secondary modes observed on the histogram of Z As a consequence, and in mining applications, this procedure will be used if these
secondary modes are significant and therefore should be reproduced on the histogram of Y.
For this, the variance correction (T~ ~ 17~ must not be significant; in practice, the relative
differences l(Ti-l7~I/u~ must be less than 30%. In fact, if this variance correction
represents a significant smoothing (l7~ (T~), this strong smoothing should mask the
secondary modes, which then should not appear on the Y-histogram.

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## VI SELECTION AND ESTIMATION OF RESERVES

471

This procedure has the advantage of being simple and of reproducing the
major characteristics of the experimental histogram of Z on the estimated
histogram of Y. However, the histogram of Y estimated in this way will not
have the exact required values tm, O"~) of the mean and variance. In
practice, the deviations from the required values (m, at) can usually be
corrected by a slight modification of the few extreme values of the histogram of Y. In any case, no permanence-of-law procedure is robust enough
to be used for these extreme values (y greater than m + 3ay for instance).
4. Affine correction of variance

## Instead of the preceding indirect correction, it is also possible to make a

direct variance correction (T~ --7 cr~ by means of the very simple affine
relation

Y-m
. ..
Z-m
---has same distribution as - - - Oy

(VI.13)

Oz

## This simple variance correction on the standard variable (Z - m)/CTz

results in a variable Y with the required mean m and variance {T~.
Knowing the distribution of Z (or its experimental histogram), it is a
simple matter to deduce the distribution of Y:
Fy(y)= Prob {Y

(VI. 14)

where

az
z=-(y-m)+m.
cry

## This affine variance correction results in a distribution for Y, the shape of

which is very similar to that of Z, i.e., all secondary modes, significant or
not, of the Z-histogram are reproduced on the Y-histogram, d. section
VI.BA, Fig. VI.16. However, if CTy/(Tz is greater than 1, there is a nonzero probability of obtaining negative values of Y, which can be awkward if
Y represents a grade. If they are few, these negative values can be corrected by hand (set equal to zero, for example) on the Y-histogram,
In practice, this affine variance correction produces results very close to
those provided by the indirect correction by permanence of a twoparameter distribution. Either one of these two procedures can be used
when the variance correction is small, in practice when la~ - a~1J (T~ <
30%, and when a great deal of confidence can be placed in the experimental Z-histogram. Note that the indirect correction, as opposed to the affine
correction, guarantees that no value of Y will be negative.

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472

MINING GEOSTATISTICS

## As the simple two-parameter distributions are often insufficient to take the

various characteristics of an experimental histogram into account, a more
general solution would be to use n-parameter distributions or, more precisely, order n expansions with Hermite polynomials.
Let Z be a stationary variable with any distribution, represented by an
experimental histogram with mean m and variance (T~. We can then define
the Gaussian transform function 'Pz which expresses the variable Z in
terms of the standard normal variable U, i.e., Z = 'Pz( U). This transform
function is fitted to the experimental Z-histogram, and the variance {T~
appears as one of the parameters defining the function 'Pz.
One possible hypothesis of generalized permanence of law consists in
considering th at the Gaussian transform function 'PY of the variable Y is
identical to 'PZ, with the variance parameter u~ replaced by (T~. The
application of generalized permanence then consists in:
(i) determining the transform function 'Pz of a variable Z from a
representative experimental histogram;
(ii) determining how this function !Pz depends on the variance (T~ so as
to make the variance correction (T~ -+

ut..

## In general, the distribution of Z is given in the form of an experimental

histogram which is often too complex to be correctly represented by a
simple two- or three-parameter model. A way around this problem is to fit
an expansion up to an order n to this histogram; more precisely, to its
Gaussian transform function (f)z. The order n increases with the complexity
of the histogram. It can be shown that the Gaussian transform function 'Pz
of a variable Z which follows any distribution with a finite variance can be
expressed in the form
+CO

'Pz(u) =

l/J-

,L ~H;(u),
,=0 1.

where the H;(u) are the Hermite polynomials. The definition and main
properties of these polynomials will be presented in the following section,
VLB.3.
In practice, the Hermite expansion is carried out up to an order n, i.e.,
(f)z(U) =

.L

l/J;

--:-;H";(u).

'=0 I.

(VI. 15)

## VI SELECTION AND ESTIMATION OF RESERVES

473

The order n is such that the distribution of the variable fPz( U) correctly fits
the experimental histogram of the data z. The method of fitting the (n + 1)
coefficients l/Ji of the expansion of (fJz from the experimental Z -histogram
is given in the following section, VI.B.3. It can be shown that

~ IjI~

1/10 = m,

L, -'f

j~l

I.

(VI. 16)

(Tz,

where m and O'~ are the mean and variance of the experimental Zhistogram.
Applying the hypothesis of permanence to this Hermite expansion
consists of considering that the Hermite expansion of the transform
function (fJy of the variable Y can be written as
n

!/J.e.

i=O

I.

I, ~ ~(u)

(,Oy(u)=

(VI. 17)

where
n

0/0 . Co =

i.e., Co = 1,

rn,

"
c...

.=1

.1,2 2
o/iCi _

~l--(Ty

l.

Thus, (fJz and cpy have the same Hermite expansion, except for the variance correction coefficients c; The variance correction is written c, = a' (a
to the power i) which entails
2

I/ITa 2 i

,= 1

1.

O'y=.2: -.,-=!(a).

(VI.18)

The justification for adopting the form c, = a' for the variance correction
is given later, in remark 1.
Since the n parameters {I/Ii, i = 1 to n} are known, the function f( a) can
be plotted as on Fig. VI.12. The known value (T~ then determines the
parameter a of the variance correction. Formula (VI.17) gives the expression of the Hermite expansion of ([iy, i.e.,
n IjIja i
fPy(u)=.2: -.-, H;(u).

.=0

(VI. 19)

1.

Remark 1

## If Z is a normally distributed variable with expectation m and

variance (T~, the Hermite expansion of its Gaussian transform function is
reduced to two terms:

## z = (fJz(u) = 1/10+ I/IIH1( u ) =

Knowing that H1(u)=

~u,

0/0 = m

+ O'z

u.

## relations (VL16) can be verified:

and

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## ifij = -(Tz, i.e.,

o/f =

O'~.

474

MINING GEOSTATISTICS

2
Uy
=

2
1/112 a,
i.e., a

= cry/ Uz,

y

+ o-y.

u.

f( a)

0-

2
Y

o<
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