List of Figures
List of Tables   Table 1. Frequencies of random walks using 6 cards. Table 2 . Selected probabilities from the binomial distribution. Table 3 . Elevations along the probability profile for Chalmers Creek, Virginia. Table 4 . Data for the Chi-square goodness-of-fit test for a real and simulated longitudinal profile. Table 5 . Elevation differences in probability profiles due to faulting. Table 6 . Information required to run program UDL. Table 7 . Explanation of prompts for control arguments for program UDL. Table 8 . The effect of the number of iterations on final elevations, slopes, and other statistics after 30 m.y. of simulated erosion of the White Mountains.
Introduction
Landforms evolve within the context of tectonic movements and erosion, which can maintain or modify morphometric characteristics such as relief, channel slope, and convexity of valley side-slopes. Davis (1899) described the topography of recently uplifted areas as youthful. The morphologic attributes of such areas distinguish them from areas that have had no uplift for a longer period of time. Davisian concepts of the geographic cycle are still applied to landforms that experience rapid uplift followed by tectonic quiescence. The Davisian scheme becomes cumbersome in areas of slow, continual uplift, where basin morphometry appears to have adjusted to processes operating today (Hack, 1960) , rather than to the decay of an initially youthful landscape. Continuous adjustment of process and form in dynamic equilibrium has led some to conclude that landforms can be independent of time (Chorley, 1962, p. 7) .
Introduction
The present work examines the response of landforms to uplift along nearby faults. Ultimately, landforms may be used in inferring uplift history and uplift rates. We will provide some preliminary theoretical and empirical justification for a probabilistic approach to landform evolution and present a computer program, based on a probabilistic approach, which models landform change through time under various rates of uplift and denudation. Eventually our methodology may show how computer simulation can lead to quantitative procedures for using landforms in discerning the complex interaction between uplift and erosion.
Introduction
Some mathematical approaches to landform evolution Several workers have tried to explain landforms purely on theoretical or deductive grounds (Culling, 1960; Schiedegger, 1961; Leopold and Langbein, 1962; Hack, 1965; Ahnert, 1970a; Kirkby, 1971) . Culling (1960) , for example, uses Fourier theory of heat flow in solids to develop differential equations describing stream profiles and valley side slopes. In a similar study, Devdariani (1967) , examined the interaction of uplift and denudation on a twodimensional anticlinal uplift. Devdariani tried to explain the Davisian cycle of erosion in mathematical terms. In many other similar studies the most common models use equations of continuity or diffusion. Sprunt (1972) developed a computer program to simulate the development of drainage basins, their networks and topography. Sprunt*s work has a distinct advantage over prior attempts at network simulation (Leopold and Langbein, 1962; Scheidegger, 1967) in that the new topography associated with drainage development was graphically displayed. The graphic display permitted more meaningful comparison between real and simulated drainage networks.
Introduction
A random-walk approach to probability modelling Particularly relevent to the present study is the probability model of Leopold and Langbein (1962) , who modelled longitudinal stream profiles by using a random-walk method. In their procedure, a pack containing 5 white cards and 1 black card is prepared. Each white card represents a unit of elevation above base level. A card is brandonturner@example.org. If a white card is selected, the profile is decreased one unit of elevation, but if a black card is selected, the next point in the profile remains at the same elevation. Regardless of which card is selected, it is replaced by a black card. Results of their experiment are shown in Figure 1 and Table 1 . An important property of this experiment is that the outcome of any one card selection depends on the previous outcome. Experiments or processes that have this property are termed simple Markov-chain (Hoel, Port, and Stone, 1972) or first order autoregressive processes (Nelson, 1973) . Leopold and Langbein (1962) assert that Markov-chain generated profiles are comparable to actual stream profiles in overall shape. We can consider this experimental method a model of the process of transporting water and sediment through a system. Because gravity is the driving force, the possible extremes are for material to move down vertically or not to move at all; the most probable material paths lie between these extremes. In actual drainage basins, material is eroded from hillslopes and channels and is eventually transported out of the system. The transport path of a packet of sediment is the topography over which it travels. If the transport paths can be modelled, then topographic form is implicitly modelled as well. Random-walk simulation of stream profiles is useful because in modelling the process the topographic form is predicted. Leopold and Langbein (1962) interpreted the number of possible downward steps as elevation above base level, which leads to the conclusion that the longitudinal profile of a stream is dependent on elevation above base level. Langbein (1964) , perhaps aware of problems in the original random-walk interpretation, tried to show that profiles actually depend on discharge rather than elevation. He also maintained, however, that the number of downward steps is equivalent to elevation above base level, an assumption which precludes a dependence on discharge.
Introduction
Clearly the number of cards used in the experiment completely determines the probability of a downward step at each point in the sequence of card selection. To illustrate this we can construct an experimental sample space. Figure 2 illustrates the possible choices at each card selection and all possible outcomes.
Introduction
For example, when the first card is selected, choice 1, there are 3 D's (downward steps) and 1 S (same elevation). Thus, the probability of a downward step is the number of D's divided by the total number of possibilities, or 3/4 (0.75). The "elevation" at choice one is therefore 3.0-0.75, or 2.25. When the second card is selected, the probability of a downward step is 9/16 (0.5625), because at choice 2 there are 16 possible outcomes, 9 of which are a downward step (see figure 2) . These probabilities can also be found by using binomial distributions. Table 2 is a modified binomial table where n is the choice number. For example, when n = 1, the first card is selected and when n = 8, the eighth card is selected.
Introduction
The probability parameter, 9 is related to the number of cards in the deck (similar to the transition probability in the Markov sense), because initially there is only one black card. If the initial card deck consists of 10 cards, 6 = 0.10, whereas if only two cards are used, 6 = 0.50. The entries for 9 in table 2 include non-integer deck sizes as well. The fewer the cards the greater the 9, or the smaller the probability of a downward step. For the example shown in figure 2, there are 4 cards, of which one is black, and 9 = 1/4 = 0.25. Under 9 s .25 and n = 1 (see table 2), we find the probability of a downward step is 0.75 and for n = 2 the probability of a downward step is 0.5625, as was calculated.
Introduction
Any geometric meaning of the method relies on assigning a distance quantity to the choice number. If we consider each card selection to represent a fixed length of channel, then larger 9 (fewer cards) results in shorter streams and smaller 9 (more cards) results in longer streams. The shapes of the resulting profiles are directly related to the number of downward steps, which is a measure of elevation above base level. A great deal of empirical data relating stream length and discharge exist (Horton, 1945; Leopold and Haddock, 1953; Leopold, Wolman and Miller, 1964) indicating that discharge increases exponentially with stream length. Thus long streams have larger discharge and lower slopes than shorter streams. Since large discharge streams have lower slopes than small streams, the probability of a downward step must be smaller for larger streams. When each card selection represents a fixed distance, however, large 9 results in shorter streams. This contradiction suggests that each card selection should not represent the same length for all streams.
Introduction
Instead, the stream length represented by a card selection should be proportional to the total length. Consider any length stream to be divided into n reaches. A stream of 100 km length can be divided in 10 reaches so that each card selection represents 10 km, or a 10-km long stream can be divided in 10 reaches so that each card selection represents 1 km. When a card selection is defined in this way, 9 can vary between 0 and 1 and is proportional to discharge. The larger the discharge the greater the 9 and the smaller the probability of a downward step, so that profile shape depends on discharge and elevation above base level. This interpretation of the randomwalk experiment allows a more general application of the method. For example, the original interpretation results in a single longitudinal profile for a given relief. The present interpretation leads to a family of longitudinal profiles, in which each profile can drop a given elevation, but each has a different length. Such flexibility is needed for modelling because, in real situations, discharge and relief both vary.
Introduction
Statistical comparison between theoretical and actual profile Theoretical stream profiles must be compared with actual profiles in order to evaluate the adequacy of the random-walk (Markov chain) model. This can be done by testing the null hypothesis that the actual and the theoretical profile elevations have the same distribution. The two profiles can be statistically compared using a Chi-square, goodness-of-fit test (Gibbons, 1976) . The Chi-square test examines how much the theoretical profile deviates from the actual profile. The result of such a test, that is, the equality between the two profiles, can be determined for a desired confidence coefficient, <*.. The test statistic, Q, measures the sum of the squares of the differences between profiles: when Q is very large, the profiles do not match well. If the null hypothesis is accepted, that is if that Q is not too large, then the actual profile is reasonably approximated by the probability model. Acceptance of the null hypothesis does not exclude other probability distributions which may also be adequate. How large Q becomes before the null hypothesis is rejected depends on how sure one wants to be that the two profiles match. To be conservative, the test is performed at a small<* , such as 0.05 or 0.10.
Introduction
To illustrate this procedure, a theoretical profile of Chalmers Creek, Virginia is compared with a plot of the actual profile of the stream ( Figure 3 ; Table 3 ). Elevations (H) for the theoretical profile are calculated from
Introduction
where R is the total drop in channel elevation or relief, p is the probability of a downward step (from Table 2 , using 6 = 0.25) for each unit distance or card selection, and H^n is the lowest elevation of the channel. The Chisquare test was applied to these profiles (Table 4) . The test statistic, Q, is the distributed Chi-square with n-1 degrees of freedom. The null hypothesis, that the two profiles are equal, would be accepted where c< = 0.10. This evaluation shows that theoretical profiles can be reasonable approximations of actual stream profiles. The probability-model profiles of graded streams can be used to statistically test for anomalous or non-graded segments of actual stream profiles. Anomalous stream profiles are those which are steeper or shallower, as a whole or in segments, than one would expect for a given climate, discharge and channel material. Such profiles may result from a variety of independent variables, including tectonism.
An empirical method of profile analysis
A method of stream-profile analysis was developed from a different perspective by Hack (1973) j who demonstrated that in erosional environments, the relative magnitudes of stream power can be approximated by a stream gradient index. When a longitudinal stream profile (or a reach along the profile) plots as a straight line on a graph in which the horizontal axis is the natural log of distance from the headwaters and the vertical axis is elevation, the equation of the line is (Hack, 1973) 
An empirical method of profile analysis
Here H is elevation, L is stream length measured from the headwaters, and C and k are constants. The slope of the actual stream, S, may be expressed as the derivative with respect to L of equation 2 where
An empirical method of profile analysis
The constant, k, is the slope of equation 2 and may be expressed as
An empirical method of profile analysis
where SL is the stream-gradient index, H-|, H2 are map elevations at the end of each reach, and L^, L2 are the respective distances from the headwaters measured along the longest channel.
An empirical method of profile analysis
Real streams may be concave up, straight, or convex up in profile, depending on downstream variations in load, discharge, bed material, hydraulic geometry and tectonism. However, the profiles of most natural streams approximate a series of connected segments of various lengths, each logarithmic in form (Hack, 1973) . For each segment the stream gradient index (SL) is constant, and each segment is also straightforwardly described by probability modelling. Thus, the profile of Chalmers Creek (figure 3) can be modelled with an SL value of 40.
An empirical method of profile analysis
Applications of the probability model and the stream-gradient index model to landform change Assuming a constant rate of increase in discharge downstream and homogeneous lithologies along stream length, a theoretical analog to a natural stream profile can be constructed. The theoretical profile is constructed using the probability parameter 6, which is related to discharge, and to channel relief (the total drop in stream channel elevation), which is given. Tectonic uplift increases channel relief whereas erosion can decrease channel relief. A probability profile that models a stream before uplift is modelling a stream adjusted to hydraulic variables. Similarly, following uplift, a probability profile models the stream that now has greater channel relief as it theoretically will appear when it too is adjusted to hydraulic variables. The time needed for the stream to adjust to hydraulic conditions following uplift depends on how fast the channel can erode its floor.
An empirical method of profile analysis
When the rate of increase in discharge does not constantly increase downstream, however, or when resistant and non-resistant rocks crop out alternately along the stream channel, several probabilities are needed to construct a theoretical profile. If discharge actually decreases downstream, the probability of a downward step becomes higher downstream, resulting in less concavity in the profile or even convexity. Coarse load introduced by resistant lithologies along a channel may affect discharge by increasing the channel roughness, which would also result in a steepening of the channel. Under these complicated situations, shorter reaches must be modelled separately along the length of the stream. Meyer and Kramer (1969) have shown that hillslopes also tend towards concavity, a feature analogous to streams. Material can either move down slope or not move. We assume therefore, that hillslopes may be modelled in a similar manner (Meyer and Kramer, 1969) , keeping in mind that variations in features such as vegetation or aspect may necessitate a complex probability assignment. For present purposes, a constant rate of increase in discharge is assumed.
Base-level fall
Random-walk stream profiles derived from probability considerations can be used to predict changes in stream profiles caused by uplift and erosion. Figure 4 shows a hypothetical, initial stream channel whose relief is increased 100 meters by block faulting located at base level. Since the stream has not changed in map length, 6 remains the same. Change of the faulted profile over time may be modelled by comparing the initial profile with a profile that has adjusted to the new base-level. The major modifications in the initial stream profile (figure 4; table 5) that followed faulting and subsequent stream downcutting occurred where stream power increased the most, that is, in the area of the greatest modification in channel slope. If the initial channel were preserved as a strath terrace, the familiar divergence of strath terrace from the present channel towards the fault would be apparent. Such terrace divergence is a clue that a tectonic perturbation has taken place.
Base-level fall
The stream gradient index (SL) can also be used to determine where change is likely to occur. Ideally the SL value is constant if the stream slope is adjusted to discharge. Tectonism (as well as outcrop of resistant rocks and coarse material) can cause a steepening of the slope that is reflected as an increase in the SL value for that reach of the profile. For example, a fault offset across a channel may produce a steeper slope and thus a higher SL value for that reach of the stream. Because work expended on a channel is proportional to the SL value, it will be concentrated at this tectonic nickpoint. With time, the nickpoint will be flattened until SL values for each reach become uniform and work is distributed more evenly along the channel. The even distribution of work has been described as the tendency for least work (Langbein and Leopold, 1966) . Thus the trend towards a uniform stream gradient index, a crude indicator of stream power, indicates that the tendency for least work is analogous to a minimization of stream power.
Base-level fall
Base-level stability Probability profiles can also be used to model change under the restriction that base level is fixed but relief is reduced. Figure 5 illustrates a situation where the initial relief of 1100 meters was reduced to 1000 m. The profiles, constructed using 8 = 0.5, show that the maximum change in channel elevation occurs at the headwaters and progressively decreases towards base level. This result is consistent with the observations of Bull (1979) , who deduced that the tendency for downcutting under conditions of base-level stability is a maximum in the headwater reaches and progressively decreases downstream.
Three-dimensional simulation
Three dimensional simulation of landform change is preferable to two dimensional simulation, because many landform parameters can be measured only when three dimensional data are available. Also, three dimensional displays allow more critical evaluation of the simulation results.
Three-dimensional simulation
A computer program called UDL, written to simulate the evolution of topography, produces perspective diagrams of three-dimensional surfaces for plotting on pen-plotters or interactive display on CRT-terminals. In addition, the program calculates parameters such as the hypsometric integral, slope and relief, and plots histograms displaying the frequency distributions of slope and elevation.
Three-dimensional simulation
Input for the program is a two-dimensional array of elevations, or an elevation matrix. Elevation arrays for parts of the U.S. at a scale of 1:250,000 are available on computer tape from the National Cartographic Information Center of the U.S. Geological Survey, or the user can construct his own array by cell digitizing or by line digitizing followed by interpolation to cell format. H^j refers to any specific elevation in the array where i is the row coordinate and j is the column coordinate.
The algorithm used in program UDL
The basis for the three-dimensional simulation is an array of elevations that is modified through time. Alternatively, three-dimensional modelling could be accomplished by modelling a network of profiles and then contouring the predicted elevations. Hillslope profiles would have to be included as well. This requires modelling a large number of separately digitized but spatially coordinated profiles, using a modified stream-gradient index to predict where change would take place. Digitized stream profile information is not available, however, and would have to be generated specifically for use in the model. Digitized topographic data are available, so the program is structured to use it. The algorithm makes several crude approximations which are needed in order to use elevation arrays. Maximum elevation in the array is considered analogous to the elevation of the headwaters (that is, the drainage divide). The distance between each point in the array and the drainage divide (the denomination is 4) is crudely approximated by dividing the difference in elevation between the highest point and each point in the array by the mean slope .
The algorithm used in program UDL
A logarithmic model can then be applied to the data by In where K^j is a function analogous to the stream gradient index. Hmax is the maximum elevation in the data set used to approximate the divide, and S is the mean slope of the area above base level. The denudation rate at each point is assumed to be proportional to K^J. K.y is used as an indication of the relative contribution of a specific cell to the overall denudation rate.
The algorithm used in program UDL
Given an average denudation rate, the total volume of material removed per increment of time is Vd = AT D(t)A t (5) where V^ is the total volume removed, A^. is the total area denuded, D(t) is the average denudation rate, during time t, and At is the increment of time over which the region is denuded subject to erosion. V^ is assumed to be proportional to the sum total of K j_ .. , so KIJ (6) i = 1 j = 1 where C is a constant of proportionality. For any location in the elevation array, the denudation rate may be expressed as where D^ *(t) is a cell denudation rate, and AJ • is the area of the cell.
The algorithm used in program UDL
The program is iterative; the total time of deformation and erosion is divided into segments of equal length (At). During an iteration the mean slope (S) of an eroding landscape decreases and the total area above base level (A,j,) also decreases. These decreases affect the denudation rate. After each iteration, the denudation rate is recalculated to reflect the magnitude of these changes. Because the denudation rate is modified after each iteration, it can be considered a function of time.
Uplift
Uplift is incorporated into the algorithm by adding an increment of elevation to each cell in the elevation array, every iteration. In the case of no erosion this is simply AHij = UAt (8) where AH^ is the increment of elevation, U is the average rate of uplift and At is the length of the iteration. Either the entire area or a part of the area can be uplifted as a simple fault-bounded block with no tilting or warping. The faults across which uplift occurs are located by specifying an elevation; uplift occurs at all cells with elevations greater than that specified. If the entire terrain is raised, these faults lie outside and below the area.
Uplift and Erosion
When both uplift and erosion are applied to a landscape, the elevation increment added to each cell is calculated by j = UAt -D1J (t)At .
Uplift and Erosion
(9) As noted above, when the net effect is lowering a landscape, the mean denudation rate is decreased after each iteration. Similarly, if the net effect is uplift, the mean denudation rate increases. The increase in denudation rate is considered to be proportional to mean slope; as uplift increases the mean slope, the denudation rate will increase in direct proportion. When very high relief develops, this proportionality to slope becomes less responsive, so that as some threshold in relief is reached, the mean denudation rate is considered to be proportional to relief (Ahnert, 1970b) . The proportion used is D»(t) = 7.35 x 10~5 (relief).
When D*(t) becomes greater than D(t), D*(t) is the denudation rate used.
For landscapes with a hypsometric integral below 0.5, the distribution of denudation is weighted towards the higher elevations, so that the denudation function for each cell, K.JJ, is proportional to relief. The hypsometric integral is calculated by the method of Pike and Wilson (1971) .
When D*(t) becomes greater than D(t), D*(t) is the denudation rate used.
The material eroded from the uplifted block is not treated by the algorithm. Thus transport and accumulation of sediments is ignored, and basin elevations remain unchanged. Because parameters such as relief and slope are calculated at each iteration, it would be possible to use published statistical relationships between sediment characteristics and source-terrain characteristics to model aggradation in the sedimentary basin as well.
When D*(t) becomes greater than D(t), D*(t) is the denudation rate used.
The algorithm is an approximation of the probability model discussed above. At present it is uncertain how well the algorithm follows the probability model. The simulation of landscape evolution is an initial test. Further work may be aimed at directly incorporating the probability model into the algorithm by transforming the elevation data array into a form more convenient for analysis.
Results of Simulation Uplifted Mountain Block
We applied program UDL to three specific landscapes -a Basin-and-Range mountain block, a plateau escarpment, and a basalt-capped mesa. The White Mountains, California, were selected as an example of a fault-bounded mountain range. The topographic data were digitized from a standard 1:250,000-scale topographic map with a contour interval of 200 feet (Mariposa, California). The present topography is shown in figure 6 . The 5900-foot contour was arbitrarily chosen to represent both the fault and base level. Figure 7 shows how UDL applies uplift along a fault.
Results of Simulation Uplifted Mountain Block
To examine the possible outcome of continual denudation of the White Mountains without any uplift, program UDL was used to simulate 30 million years of erosion with base level at 5900 feet. The initial denudation rate was 0.1 ft per thousand years, a rate suggested by Marchand (1971) for that area. After 30 million years of erosion and tectonic quiescence, landforms associated with tectonic stability were formed (figure 8). The simulation shows well-developed pediments on both fronts of the mountain block, but better developed on the less steep, eastern front than on the steeper western front. Figure 9 shows the differences in elevation between the initial condition (figure 6) and after 30 million years of erosion over the entire area. The most denudation occurred at the higher elevations. This is what the probability model predicts for a stream profile under conditions of baselevel stability (figure 5).
Results of Simulation Uplifted Mountain Block
The results of simulation are consistent with Bull and McFadden's (1977) field observations and quantitative characterizations of inactive mountain fronts. For example, the simulation shows pediments, inselberg, pediment pass and a slightly increased mountain-front sinuosity.
Plateau Edge
A small area along the Mogollon Rim in central Arizona was selected to test the UDL algorithm in a plateau-edge setting. The topographic data were digitized from a 1:62,500-scale topographic map with a contour interval of 50 feet (Pine Mountain, Arizona). Figure 10 shows the present Mogollon Rim. Program UDL was used to simulate 10 million years of erosion of the escarpment. Base level was fixed at 5800 feet, approximately the elevation of an erosional surface adjacent to the Rim determined by geologic and altimetric analysis (Mayer, 1979) . Estimates of the denudation rate along this section of the Mogollon Rim by Mayer (1979) are a minimum of 0.17 feet (5 cm) per thousand years and a maximum of 0.38 ft (11.5 cm) per thousand years.
Plateau Edge
To achieve conservative results a denudation rate of 0.13 feet (3 cm) per thousand years was used. Figure 11 illustrates the results of the simulated erosion. The average rate of pedimentation and scarp retreat, determined from the simulation, is consistent with the minimum estimate of scarp retreat proposed by Mayer (1979) , 350 m per million years.
Mesa and Canyon
Buckhead Mesa in central Arizona was selected to test UDL on a complex mesa and canyon topography. The topographic data were digitized from the 1:24,000-scale Buckhead Mesa quadrangle, with a contour interval of 40 feet. This sample has the finest grid resolution of all the simulation trials (figure 12). In order to see where material is eroded, the elevations along the perimeter were fixed. Erosion at an initial rate of 0.10 feet (3 cm) per thousand years was used to simulate 10 million years of denudation with no constraint on base level. Figure 13 shows the results. Note that the highest erosion rate occurs at the highest elevation and adjacent to the canyon. The mesa appears to be dissected by downcutting of streams but still has some planar surface.
Discussion
Simulation of prolonged erosion of a Basin-and-Range block uplift shows that pedimentation indicates tectonic stability and only occurs during periods of stability. The occurrence of pediments and uplifted pediments would require episodic rather than continuous uplift. The period of time represented by the pediments can be estimated by simulation of erosion. For the Wfcite Mountain example, pediments formed at a maximum rate of 0.25 km/10 1 rate.
Discussion
km/10 yrs, although this rate is very sensitive to the average denudation The program could be modified to generate information on morphometric characteristics of drainage basins under specified conditions. For example, drainage basins with a known denudation rate may have a morphometry including stream channel slope, average valley side slope, and a shape that depends on uplift rate. The program may be useful in determining the rate of uplift that best matches the observed morphometry. The estimated rates of uplift could then be plotted on a map for many drainage basins. If a spatial consistency in these rates emerged, the sources of the uplift might be determined.
Discussion
Simulated erosion of the White Mountains implies that asymmetry in a mountain block caused by tilting persists for millions of years. Asymmetry of ranges that have long been tectonically inactive thus can be used to infer tilting where asymmetry presently exists. This is especially useful in granitic ranges where there are no sedimentary units to document tilting. Simulated erosion of the Mogollon Rim suggests that the program can be used to bracket the age of fault-generated escarpments, a useful tool where stratigraphic control is lacking.
Discussion
Desirable improvements in the program are the direct incorporation of the probability model and treatment of the aggradational system. Modelling sediment, for example, will yield idealized trends of particle size decrease in the downstream direction under various climatic (denudation rate) and tectonic (uplift rate) regimes. In the southwestern U.S. this would allow more comprehensive interpretation of alluvial fan sedimentology. In the humid and subhumid areas of the U.S., the sedimentology of terrace deposits may provide information on past uplift rates.
Summary
Through Markov-chain random-walk modelling it has been shown that theoretical profiles can be constructed to approximate actual stream profiles. Such theoretical profiles can be described probabilistically. To determine how well the probability profiles approximate the actual profiles, the two can be compared statistically. Such tests have shown that the models work well for streams constantly increasing in discharge downstream. The degree of accuracy can be improved by dividing the stream into segments, each modelled separately.
Summary
The stream-gradient index can be used in much the same way. The index allows two stream reaches to be compared, on the same stream or on different streams, without the construction of entire profiles. Both methods of modelling can be used to identify anomalous reaches of stream channels. They can also be used to predict how erosion will be distributed if the channel downcuts in response to a drop in base-level.
Summary
The techniques for modelling in two dimensions, modified for simulating erosion and uplift in three dimensions, were incorporated into a computer program called UDL. This program proved useful in evaluating the significance of various landform features. It also shows promise as a tool that could lead to the development of quantitative procedures for using landforms in discerning the complex interaction between uplift and erosion.
Summary
Computer Program UDL Computer program UDL was written in MULTICS Fortran, on the Honeywell Multics system at the U.S. Geological Survey in Menlo Park, California. The program is not portable, but can currently be used by other MULTICS users who can link to or copy the program (table 6). The program is interactive and prompts the user for information it needs (Table 7) .
Summary
The program can produce tabular and-or graphic output. Frequency distributions of slopes and elevations can be listed or plotted on histograms (Figure 14) . The elevation array can be plotted on the screen as a view of a three-dimensional mesh-covered surface, or the data can be stored in fileOS and then plotted as a contour map or as a view of a three-dimensional, contoured surface. Two different graphics packages, Surface Display Library (SDL) (Dynamic Graphics Group, 1975) and Display Integrated Software System and Plotting Language (DISSPLA) (ISSCO, 1975) were used to produce the graphs, maps and views of the landform.
Summary
DISSPLA can produce graphs and mesh views on a CRT, wheras SDL produces the contour maps and views of the landforms on the Versatec plotter. The user need not restrict himself to uplift along a fault and erosion only above that fault elevation; the entire area can be uplifted and eroded. Alternatively, one can construct a data set that contains a fault and then erode the entire area. This can be done by specifing a rate of uplift, a fault elevation, and no erosion for one iteration and saving the data. These data can then be reentered into the program and uplifted and eroded with a new set of parameters.
Summary
The number of iterations has an effect on the final elevation array even though all other input parameters are identical. The White Mountains elevation array was used to illustrate this. In each of two cases the initial denudation rate was 0.1 feet/1000 years, with no uplift, for a total time of 30 million years. In the first case, there were 30 iterations, each 1 million years long; in the second case there were 3 iterations, each 10 million years long. The statistics generated by the program (table 8, figure 14) show that although the final denudation rates are about equal, the elevation arrays are different. In the second case (3 iterations), the maximum elevation is almost 900 feet higher than in the first case (30 iterations), and the relief is also greater, resulting in steeper slopes. More iterations may produce a landscape closer to reality, however increasing the number of iterations also increases the cost.
Data and Data Format
The maximum size of the elevation array is now 30 rows by 64 columns. This is easily expanded by changing the dimension statements at the beginning of the program and by changing the call to the subroutine "surmat" in the final plotting section of the program. The data must be in file07. The first record contains the number of columns and the number of rows, with the format (3x, i2, 1x, i2). The second record is the grid cell size, the map length of a side of the cell used in digitizing the elevation data, or the map distance between points in the array expressed in feet. All subsequent records contain the values of the elevation array. The format is 8 (f8.1, 1x) ; thus, the number of columns of data must be an integral multiple of eight. The program was written by Larry Mayer in the summer of 1979 and is listed in Table 9 . 1  1  600  550  50  2  2  350  275  75  3  3  225  138  87  44  163  69  94  55  131  34  97  6  6  116  17  99  77 108 9 99 The rate of denudation in feet per 1000 years for the initial landscape. This number is recalculated after each iteration.
Data and Data Format
The rate of differential uplift in feet per 1000 years for that part of the area above the faults. This rate remains fixed for the entire calculation.
Data and Data Format
For erosion with no uplift, this number is equal to the base level. If a rate of uplift has been specified a fault will occur, the trace of which follows a contour at the given elevation. Erosion will then take place only above the fault.
Data and Data Format
This number determines how often the denudation rate and the elevation array are recalculated. For example, for a factor of 10, each iteration will be 10,000 years long.
Data and Data Format
This number, the output frequency, determines how often the elevation array is plotted on the screen. To calculate the number of years between plots multiply the iteration factor by 1000 by the output frequency. This specifies the number of bars in the histogram. Any integer between 1 and 50 is allowed but 10 to 20 gives the best looking results.
Data and Data Format
If you do not have a Tektronics terminal you must type "1", if you do want a plot type another integer. If a plot is not generated on the screen, the elevation array may still be saved and plotted at some future time on a contour map or view of a 3-dimensional, contoured surface can be made with the contour.pit program on the Versatec plotter.
Data and Data Format
The view distance from the center of the plot and the height above the plot are fixed, but the user can rotate the plot by specifying the view angle in degrees.
Data and Data Format
This is a scaling factor 0.1 to 1.0 which determines the vertical exaggeration of the mesh-view plot on the screen.
Data and Data Format
This reduces the effective denudation by 1/5 (Schumm, 1963) . To ignore this type another integer.
Data and Data Format
The last elevation array calculated can be saved in a file called fileOS. The file should be renamed immediately after the run and has the same format as the input data. 

{'paper_id': '5328604', 'title': 'Probability models and computer simulation of landscape evolution', 's2_url': 'https://api.semanticscholar.org/CorpusID:5328604', 'source': 'S2ORC'}