Inversion of normal moveout for monoclinic media
Grechka Vladimir Contreras Pedro Tsvankin Ilya





^1 Center for Wave Phenomena, Department of Geophysics,Colorado School of Mines, Golden, CO 80401
^2 PDVSA-INTEVEP, Apartado 76343, Caracas 1070A, Venezuela



Multiple vertical fracture sets, possibly combined with horizontal fine
layering, produce an equivalent medium of monoclinic symmetry with
a horizontal symmetry plane.
Here, we show that multicomponent wide-azimuth reflection data (combined with known vertical velocity or reflector depth) or multi-azimuth walkaway VSP surveys provide enough information to invert for all but one anisotropic parameters of monoclinic media.

To facilitate the inversion procedure, we introduce a Thomsen-style parametrization
that includes the vertical velocities
of the P-wave and one of the split S-waves and a set of dimensionless
anisotropic coefficients. This notation
captures the combinations of the stiffnesses responsible for
the normal-moveout (NMO) ellipses of all three pure modes.
Our parameter-estimation algorithm, based on NMO equations valid for any
strength of the anisotropy, is designed to obtain anisotropic parameters
of monoclinic media by inverting the vertical velocities and NMO ellipses
of the waves P, S_1 and S_2.
A Dix-type representation of the NMO velocity of mode-converted waves makes
it possible to replace the pure shear modes in reflection surveys with
the waves PS_1 and PS_2. Numerical tests show that our method yields
stable estimates of all relevant parameters for both a single layer and
a horizontally stratified monoclinic medium.

Keywords: Seismic anisotropy, anisotropic parameter estimation, monoclinic media, inversion procedure.


91.30.Cd 91.60.Ba 91.30.pc 91.30.Ab
Inversion of normal moveout for monoclinic media
    Vladimir Grechka^1, Pedro Contreras^2, and Tsvankin Ilya^1
    December 30, 2023
==============================================================






§ INTRODUCTION


Natural fractures usually occur in vertical or subvertical sets (networks),
which makes fractured reservoirs azimuthally anisotropic with respect to
elastic wave propagation. In general, two or more sets of vertical
non-corrugated fractures produce an effective monoclinic medium with a horizontal
symmetry plane. Abundant geological evidence of multiple fracture sets corroborates potential importance of monoclinic models in seismic reservoir characterization.

Velocity analysis and parameter estimation for monoclinic media, however, is
a highly challenging task due to the large number of independent stiffness coefficients. To avoid ambiguity in the inversion procedure, we follow
the idea originally proposed for VTI media by <cit.> and attempt to
identify the combinations of the stiffness coefficients responsible for
seismic signatures commonly measured from reflection data. Since moveout
velocity analysis is one of the most reliable tools for mapping the elastic
properties of the subsurface, we use the NMO ellipses of P-,
S_1, and S_2-waves <cit.> from a horizontal reflector to define the anisotropic parameters of monoclinic media. Despite the absence of vertical symmetry planes in our
model, the polarization directions of the vertically propagating shear waves
establish a natural coordinate frame with the vanishing stiffness coefficient
c_45. We specify the anisotropic parameters in this coordinate system
(where the stiffness tensor has 12 independent elements) by analogy with the
Thomsen-style notation of <cit.> for orthorhombic media.
A subset of these parameters
(ϵ^ (1,2), δ^ (1,2), and γ^ (1,2)) is
largely responsible for the values of the semi-axes of the NMO ellipses.
Three newly introduced anisotropic coefficients ζ^ (1,2,3) mainly
describe the rotation of the P- and S-wave NMO ellipses with
respect to the chosen coordinate frame.

We show that the vertical velocities and NMO ellipses of the P- and two split
S-waves can be used to estimate eleven parameters of monoclinic media.
The only parameter not constrained by conventional-spread moveout data from
horizontal reflectors is the anisotropic coefficient δ^(3).
We present numerical tests for a single layer and stratified monoclinic media
which confirm the accuracy of our inversion procedure and its stability with respect to errors in input data.




§ SELECTION OF COORDINATE FRAME

Description of seismic wave propagation in monoclinic media with a
horizontal symmetry plane takes the simplest form in the coordinate
frame where the stiffness coefficient c_45=0. The x_1- and
x_2-axes of this coordinate system coincide with the polarization directions
of the vertically propagating S waves.
Denoting the fast shear wave by S_1 and the slow one by S_2, we find the following expressions for
the vertical component q of the slowness vector
p = {p_1, p_2, q} for waves traveling in the x_3 (vertical) direction:


    q^P = 1/√(c_33)  ,
       q^S_1 = 1/√(c_55)  ,
       q^S_2 = 1/√(c_44)  .

Here it is assumed that the x_1-axis points in the direction of the
polarization of the fast shear wave S_1, which implies that
c_55 > c_44.



§ NMO ELLIPSES FOR HORIZONTAL REFLECTORS


Grechka and Tsvankin (1998) showed that  azimuthally varying NMO
velocity V_ nmo(α) of pure (non-converted) modes is represented by the following quadratic form that usually specifies an ellipse in the horizontal plane:

    V_ nmo^-2 (α) =
                  W_11cos^2 α +
             2   W_12sinαcosα +
                  W_22sin^2 α  .

For a single horizontal layer of arbitrary symmetry, the matrix
W is given by <cit.>


    W =  -  qq^_,11 q^_,22 - q_,12^2 (
              [   q^_,22 - q^_,12; - q^_,12   q^_,11;          ])

where q ≡ q(p^_1, p^_2) denotes the vertical slowness
component,
. q^_,i≡∂ q/∂ p^_i|_p_i=0,
and
. q^_,ij≡∂^2 q/∂ p^_i∂ p^_j|_p_i= p_j=0.

Equivalently, the NMO ellipse (<ref>) can be written through the eigenvalues λ_1,2 of the matrix W as
<cit.>

    V_ nmo^-2(α) = λ_1 cos^2 (α - β) +
                                  λ_2 sin^2 (α - β)   .

Here
β is the rotation angle of the ellipse with respect to the horizontal coordinate axes:

    tan 2β = 2   W_12W_11 - W_22  .


The matrix W for horizontal events in monoclinic media is obtained from
equation (<ref>) by substituting the corresponding
values of q equations (<ref>) and the derivatives q^_,i and
q^_,ij  which can be determined from the Christoffel equation. The exact
expressions for the matrices W^P, W^S_1, and
W^S_2 in a monoclinic layer in terms
of the stiffness coefficients c_ij are rather lengthy. They do show,
however, that the “purely monoclinic” coefficients c_16, c_26, and c_36, which vanish in orthorhombic media, contribute to the diagonal elements
W_11^Q and W_22^Q (Q = P, S_1, or S_2) only through the
products c_i6 c_j6 (i,j=1,2,3).

In contrast, the off-diagonal matrix elements W_12^Q are approximately linear in c_i6. Since the rotation angle
β of the NMO ellipse is almost proportional to W_12^Q [equation (<ref>)], the coefficients
c_i6 are primarily responsible for the rotation (but not for the semi-axes) of the NMO ellipses.



§ ANISOTROPIC PARAMETERS OF MONOCLINIC MEDIA AND LINEARIZED NMO ELLIPSES


Below is a list of our Thomsen-type parameters for monoclinic media with a horizontal symmetry plane defined through the
density-normalized stiffness coefficients c_ij. Note that the vertical
velocities and anisotropic coefficients ϵ^(i), δ^(i) and
γ^(i) are introduced exactly in the same way as the corresponding
<cit.> parameters for orthorhombic media with the vertical symmetry
planes [x_1,x_3] and [x_2,x_3].

V_P0 – the P-wave vertical velocity:
      
    V_P0≡√(c_33)  ;

V_S0 – the vertical velocity of the fast shear wave S_1 polarized in the x_1-direction:
      
    V_S0≡√(c_55)  ;

      
    ϵ^(1)≡c_22 - c_332   c_33  ;

      
    δ^(1)≡(c_23+c_44)^2 - (c_33 - c_44)^2/2   c_33 
                           (c_33-c_44)  ;

      
    γ^(1)≡c_66 - c_552   c_55  ;

      
    ϵ^(2)≡c_11 - c_332   c_33  ;

      
    δ^(2)≡(c_13+c_55)^2 - (c_33 - c_55)^2/2   c_33 
                  (c_33-c_55)  ;

      
    γ^(2)≡c_66 - c_442   c_44  ;

            
    δ^(3)≡(c_12+c_66)^2 - (c_11 - c_66)^2/2   c_11  (c_11-c_66)  ;

      ζ^(1) – the parameter responsible for the rotation of
      the S_1-wave NMO ellipse:
      
    ζ^(1)≡c_16 - c_36/2   c_33  ;

      ζ^(2) – the parameter responsible for the rotation of
      the S_2-wave NMO ellipse:
      
    ζ^(2)≡c_26 - c_36/2   c_33  ;

      ζ^(3) – the parameter responsible for the rotation of
      the P-wave NMO ellipse:
      
    ζ^(3)≡c_36/c_33  .

The anisotropic parameters ζ^(1,2,3) depend on the stiffness
elements that vanish in orthorhombic media. The coefficient ζ^(3) is
analogous to the anisotropic parameter χ_z introduced by <cit.>.
The parameters ζ^(1) and ζ^(2) are different from their
counterparts in <cit.> notation because the latter is
based on the approximate phase-velocity function.



§ APPROXIMATIONS FOR THE NMO ELLIPSES

To demonstrate that the anisotropic coefficients defined above are useful
in describing the NMO ellipses in monoclinic media, we linearize the exact
equation (<ref>) in terms of ζ^(1), ζ^(2), and
ζ^(3). Also, additional linearizations in the parameters
ϵ^(1,2)  δ^(1,2) and γ^(1,2) were performed for
the elements W_12^Q of all matrices W^Q.
This yields the following approximate expressions for the matrices W in a monoclinic layer:

    W_11^P = 1/V_P0^2   (1 + 2  δ^(2)) 
        W_12^P =  - 2  ζ^(3)/V_P0^2  , 
    
        W_22^P =  1/V_P0^2   (1 + 2  δ^(1))  ;



    W_11^S_1 = 1/V_S_1^2   (1 + 2  σ^(2))  ,
         W_12^S_1 = - 2  ζ^(1)/V_S_1^2 ( V_P0/V_S_1)^2   , 
    
         W_22^S_1 = 1/V_S_1^2   (1 + 2  γ^(1))  ;



    W_11^S_2 = 1/V_S_2^2   (1 + 2  γ^(2))  ,
        W_12^S_2 = - 2  ζ^(2)/V_S_2^2 ( V_P0/V_S_2)^2   , 
    
        W_22^S_2 =  1/V_S_2^2   (1 + 2  σ^(1))  .

Here
σ^(2)≡ (V_P0/V_S_1)^2 (ϵ^(2) - δ^(2)),
σ^(1)≡  (V_P0/V_S_2)^2 (ϵ^(1) - δ^(1)),
V_S_1 = V_S0, and
V_S_2 = V_S0√(1 + 2  γ^(1)1 + 2  γ^(2));
V_S_1 and V_S_2 are the vertical velocities of the fast and slow shear waves, respectively.

The approximate diagonal elements W_11^Q and
W_22^Q in equations (<ref>)–(<ref>) are identical to the
corresponding exact expressions for orthorhombic media given in <cit.>.
The “monoclinic” coefficients ζ^(1,2,3) contribute to W_11^Q and
W_22^Q only through their products that were dropped during the linearization procedure.
In contrast, the off-diagonal matrix elements W_12^Q in
equations (<ref>)–(<ref>) are linear in the coefficients
ζ^(1,2,3) and quadratic in the other anisotropic parameters.
In monoclinic media, all three W_12^Q 0 and, in general, the NMO ellipses of the P-, S_1- and S_2-waves have different orientations.



Numerical tests show that the accuracy of the
approximations (<ref>)–(<ref>) depends mostly on the parameters
ζ^(1,2,3) and is less sensitive to the other anisotropic coefficients.
The example in Figure <ref> demonstrates that for typical values of
ζ^(1,2,3) up to 0.05–0.06 equations (<ref>)–(<ref>) (dotted lines) yield a qualitatively
adequate approximation for the exact NMO ellipses (solid).

As predicted by equations (<ref>)–(<ref>), the axes of the NMO
ellipses of  S_1-, and S_2-waves in Figure <ref> are
parallel neither to each other nor to the coordinate directions (i.e., to the S-wave polarizations).
This is a distinctive feature of monoclinic
media that can be used in the inversion for the anisotropic parameters.



§ PARAMETER ESTIMATION



 §.§ Analysis of the weak-anisotropy approximation


Although equations (<ref>)–(<ref>) lose accuracy with increasing
|ζ^(1,2,3)|, they provide a useful insight into the influence of the
medium parameters on normal moveout and help to design the inversion
procedure. Note that only one of the anisotropic coefficients, δ^(3),
is not contained in any of these equations and, therefore, cannot be estimated
from NMO velocities of horizontal events.
In the weak-anisotropy approximation,
the inversion of W_11^Q and W_22^Q is completely analogous to the
parameter-estimation problem in orthorhombic media. As discussed by <cit.>,
the vertical velocities, W_11^Q and W_22^Q can be inverted for the anisotropic coefficients ϵ^(1,2),
δ^(1,2), and δ^(1,2).
The remaining anisotropic parameters ζ^(1,2,3) of monoclinic media
can be estimated from the off-diagonal matrix elements W_12^Q, which
yield three
more equations for the three unknown parameters. We confirm these conclusions by performing numerical inversion based on the exact NMO equations.



 §.§ Numerical inversion


Input data for the parameter-estimation procedure include the vertical
velocities and NMO ellipses of the three pure modes (P, S_1, S_2)
determined from either wide-azimuth reflection data or walkaway VSP's.
The coordinate frame needed for moveout inversion can be established by performing Alford (1986) rotation of small-offset shear-wave data to identify the S-wave polarization directions at vertical incidence.
The anisotropic coefficients ϵ^(1,2), δ^(1,2), γ^(1,2) and ζ^(1,2,3) are obtained by inverting the exact equation (<ref>) for the matrix W.
Figure <ref> displays the inversion results for a monoclinic layer with
the parameters specified in Figure <ref>.
The NMO velocities of
the three modes were computed for azimuths 0^∘, 45^∘, 90^∘,
and 135^∘ from equations (<ref>) and (<ref>).
To simulate errors in measured data, we added Gaussian errors with a variance of 2% to the vertical and NMO velocities. Then we reconstructed the NMO ellipses for each realization of the data (distorted by noise) and carried out the inversion.




Overall, the stability of the inversion algorithm is quite satisfactory, but there is a substantial variation in the results from one anisotropic parameter to another. In particular, the error bars for ζ^(1) and ζ^(2) are much smaller
than those for ζ^(3) and the other anisotropic parameters.
This can be explained by the structure of equations (<ref>)–(<ref>) for W_12^S_1, W_12^S_2 and W_12^P. The expressions for W_12^S_k (k = 1,2) have scaling factors (V_P0/V_S_k)^2 that reach 4.0 and 6.5 for the model used in our numerical test. Therefore, the elements W_12^S_k are much more sensitive to ζ^(1) and ζ^(2) than W_12^P is to ζ^(3), which helps to recover ζ^(1,2) with a higher accuracy.

The inversion technique introduced above was also extended to horizontally
layered monoclinic media using the generalized Dix equation of <cit.> that allows one to obtain the exact interval NMO
ellipses. Although both the Dix differentiation and polarization layer stripping of shear waves have known limitations with respect to the vertical resolution, they give stable results for coarse intervals with sufficient thickness.



§ CONCLUSIONS

Effective monoclinic media with a horizontal symmetry plane represent a general anisotropic model of hydrocarbon reservoirs with two or more vertical fracture systems. An analytic study of normal moveout in monoclinic media, presented here, leads to a Thomsen-style notation that captures the combinations of the stiffness coefficients responsible for the NMO ellipses of P- and S-waves. Natural horizontal coordinate directions for monoclinic models are associated with the orthogonal polarization vectors of the vertically traveling shear waves.

The anisotropic coefficients of monoclinic media can be separated into two distinctly different groups. The first group contains seven parameters (ϵ^ (1,2), δ^ (1,2,3), and γ^ (1,2)) defined identically to the corresponding <cit.> coefficients for orthorhombic media. While δ^ (3) has no influence on the NMO ellipses of P- and S-waves, the remaining six coefficients control the normal-moveout velocities in the coordinate directions x_1 and x_2.

Three additional anisotropic coefficients ζ^ (1,2,3) (the second parameter group) depend on the elements of the monoclinic stiffness tensor which vanish in orthorhombic media. The ζ coefficients determine the rotation of the NMO ellipses with respect to the S-wave polarization directions.

Our algorithm, based on the exact NMO equations, is designed to invert the vertical velocities and NMO ellipses of the three pure modes (the shear waves can be replaced by mode conversions) for the anisotropic parameters ϵ^ (1,2), δ^ (1,2), γ^ (1,2) and ζ^ (1,2,3).
Numerical tests for a single layer and stratified monoclinic media indicate that the inversion procedure is sufficiently stable, and all eleven parameters are well constrained by the vertical and NMO velocities.




§ ACKNOWLEDGMENTS


We are grateful to members of the A(nisotropy)-Team of the Center for
Wave Phenomena (CWP) at CSM for helpful discussions. Pedro Contreras thanks
PDVSA-INTEVEP for giving him the opportunity to work at CWP.
V. Grechka and I. Tsvankin acknowledge the support provided by the members of the Consortium Project on Seismic Inverse Methods for Complex Structures at CWP and by the United States Department of Energy (award #DE-FG03-98ER14908). I. Tsvankin was also supported by the Shell Faculty Career Initiation Grant.

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