P d W d l t i T . M f . ( ® J

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A Simple Model for River Network Evolution

Robert L. Leheny

The James Franck Institute and The Department of PhysicsThe University of Chicago

5640 South Ellis Avenue, Chicago IL 60637

We simulate the evolution of a drainage basin by erosion from precipitation andavalanching on hillslopes. The avalanches create a competition in growthbetween neighboring basins and play the central role in driving the evolution.The simulated landscapes form drainage systems that share many qualitativefeatures with dock's model for natural network evolution and maintainstatistical properties that characterize real river networks. We also present resultsfrom a second model with a modified, mass conserving avalanche scheme.Although the terrains from these two models are qualitatively dissimilar, theirdrainage networks share the same general evolution and statistical features.

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Recently, the idea that drainage basins may be an example of a natural system in a

self-organized critical state has lead to the study of network formation in terms of the attainment of features which reveal critical scaling. Following an approach similar to that

developed by Howard [1990], Rodriguez-Iturbe et al. [1992a] have postulated that rivernetworks arrange themselves to minimize their energy expenditure, and they have

compared the networks modeled on this assumption with natural systems [Rinaldo et al,

1993]. Similarly, Inaoka and Takayasu [Inaoka and Takayasu, 1993] have presented a

model in which erosion drives a landscape with random roughness to a final state where the

water is channeled into a network of rivers. The fully developed networks in each of these

models are static structures with spatial features that show critical scaling and share

statistical properties with natural rivers. However, real drainage basins are not static but

proceed through a characteristic evolution on geological time scales. Therefore, one canask how river networks maintain their statistical properties as they evolve in time.

Clearly, to understand the important mechanisms affecting a river network's

evolution one must consider its interaction with the surrounding landscape. Two new

physical models of landscape erosion address this issue. In these models continuum

equations, solved discretely on a lattice, describe landscape changes due to a variety oferosional processes. Willgoose etal. [1990] have modeled the development of a drainage

network that, initialized at the edge, grows by erosion to extend throughout the lattice. The

rivers, once present at a lattice site, remain fixed there. However, the resulting physically

realistic depiction of network initiation has allowed them to trace the temporal behavior of a

variety of network statistics during this growth period [Willgoose et al. 1991]. Following

a similar strategy, but with a different set of continuum equations, Howard [1994] has

simulated the transformation of a rough surface as it establishes a channeled drainage

pattern through a process of retreating hillslopes and fluvial erosion.While these physical models make a comprehensive effort to describe the relevant

processes in drainage basin evolution, their complexity limits the size of feasible

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h ( x , y ) = I y ( 2 )

where I defines the initial slope. The terrain changes from the effects of a single

precipitation event at each iteration according to the following procedure:1. Precipitation lands at a random site on the lattice.

2. The water flows from the site it occupies to one of four nearest neighbors

until it reaches the lattice edge, y=0. The probability distribution that determines to which

neighbor the water flows is

P(Ahi) = -

exp(EAhi)j = 4£exp(EAhj)

j=l,Ahj>0

0

- l

Ahi>0

Ahi<0(3)

where Ahi is the height of the occupied site minus the height of the neighbor i, and E is a

free parameter in the model. This exponential dependence directs the water down steep

slopes it encounters, but allows it to take a meandering path on more gentle slopes. An

additional rule sets to zero the probability of the water's returning to the site it has immedi

ately vacated if the probability for flow in any other direction is greater than zero.3. Once the water reaches y = 0, each lattice point (x,y) that it has visited loses D

units of height to simulate the water's erosion:

h ( x , y ) - > h ( x , y ) - D ( 4 )

4. Following the erosion, the values of Ah in the landscape are compared against

a critical value, M. If any site differs in height from any nearest neighbor by more than M,

then the higher site (xn,yh) loses Ah/S units to simulate the avalanching of soil:

h ( x h , y h ) - > h ( x h , y h ) - A h / S ( 5 )

In this non-mass conserving version of the model, these units of height are simply elimi

nated as part of the erosion, and avalanches continue until all local slopes are less than M.

After the avalanching has completed, new precipitation enters the simulation at a random

lattice point, and the process begins again at step 1.

The river network draining the landscape can be defined at any time in the simula

tion. In the procedure to define the rivers, every lattice point receives one unit of precipita

tion which traces a path of steepest descent, without eroding the terrain, until it reaches the

lattice edge, y = 0; all points through which at least R units flow define the river network.

(This procedure resembles that developed by O'Callaghan and Mark [1984] to identify

networks on digital elevation maps of natural terrains.)

2.2. The Terrain and River Network Evolution

In selecting values for the model parameters, we typically chose E < 1/1 so that

precipitation on the initial slope had nearly equal probability to move in either x direction ortoward y = 0. Also, we maintained M » D to limit the landscape changes from a single

precipitation event. We examined the river patterns formed for R between 20 and 1280 and

found that, besides a weak effect on the apparent rate of network evolution and the obvious

variation in drainage density, its value did not influence the statistical characteristics of the

river networks. For the avalanche scheme we typically used S = 4.

Figures la through lc show a landscape formed from this model at three times in

its evolution, and Figures Id through lh show the corresponding river networks defined at

five times in the simulation. The evolution contains two distinct time scales. During the

first few iterations water typically lands on an uneroded site and takes a meandering path

down the initial slope before intersecting the eroded path from previous precipitation. Once

it has intersected this path, the probability distribution, (3), biases the water to follow it.

This early aggregation process quickly forms a network of shallow incisions resembling

the branched networks of random walk models. With this pattern intact in the landscape,

further evolution follows a largely deterministic process on a much longer time scale as

advantageously positioned branches continue to deepen from the erosion of aggregating

water and to expand their areas of influence through avalanches at the expense of

neighboring branches.

The resulting evolution of the landscape and its corresponding drainage network

contains many elements in common with dock's [1931] theory for the evolution of natural

basins. Glock divided the evolution of a network's morphology into an early period of

"extension" followed by a later period of "integration". During extension the river network

experiences "elongation" by headward growth of main branches into the available land area

and "elaboration" with the addition of tributary networks off of the main branches.

Succeeding the network's reaching maximum extension, integration begins with the

"abstraction", or elimination, of small internal branches of the network as the main

branches increase their own drainage area. Additional minor tributaries suffer "absorption"

during integration, ceasing to exist except immediately after rainfall. A third characteristic

of integration, which Glock suggested may not be easily identified in present day river

systems, is the aggressive adjustment of the main branches to minimize their route to thesea. Glock also noted that, simultaneous with this evolution, stream capture, or "piracy",

causes major changes between networks. In stream capture an aggressively growing basin

intercepts a neighboring stream and diverts it into its own network [Small, 1978].

A variety of observations on natural river systems have supported Glock's model,

[eg., Ruhe, 1952]. However, several of these studies [Schumm, 1956a; Morasawa, 1964]

noted that the integration stage began before extension in the network was complete, as

Glock indicated might occur. In addition, laboratory studies have revealed network

evolution well described by Glock's model, again with pronounced temporal overlap of

extension and integration [Schumm et al., 1987].

In our model the headward growth and broadening of V-shaped valleys plays a

central role in the evolution. The terrain in Figures la to lc demonstrates this headward in

vasion, as the valleys, initiated at the y = 0 edge, grow to fill the topology. As they erode,

the valley sides maintain flat slopes, with slope equal to 3M/4 dictated by the avalanching.

Figure 2, showing the changing cross section of the landscape in Figure 1 at fixed y,illustrates this process. Such parallel retreat of flat slopes mimics the effects of erosion due

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However, because M » I and E > 1/M, these second generation branches align parallel to

one another and show little meandering, simulating structures more reminiscent of hillslope

runoff than stream segments.

2.3. Statistical Properties of the Model Networks

We have also analyzed the simulations' statistical features as they proceed through

their evolution. As described below, the structure of the simulated networks maintains a

close agreement with statistical properties characteristic of natural river networks. This

maintenance of quantitatively correct river network features throughout a realistic

evolutionary process is, we believe, the model's strongest attribute.

2.3.1. Horton's Laws

The hierarchical ordering scheme for the branches of river systems, introduced by

Horton [1945] and later modified by Strahler [1952], has formed the basis for a

quantitative description of drainage network morphology. In Strahler's scheme all river

segments from a head downstream to the first intersection are first order branches, u = 1.When two branches of the same order, u, meet, the downstream segment begins a branch

with order u+1. When two branches of different order meet, the lower order branch

terminates and the downstream segment remains part of the higher order branch.

Applying this ordering scheme to a natural river network, one finds that the num

ber of branches, N(u), and the mean length of branches,<£(u)>, of each order follow geo

metric relations:

N ( u ) ~ e x p ( - i t > u ) ( 6 a )< ^ ( u ) > ~ e x p ( r ^ u ) ( 6 b )

where ib and t? are constants [Horton, 1945]. By defining the network's drainage basin

as the area of land which receives precipitation that ultimately contributes to the rivers, one

finds that the mean area of basins, <A(u)>, also follows a geometric relation:

< A ( u ) > ~ e x p ( r a u ) ( 6 c )

where ra is a constant [Schumm, 1956a].

Using the Strahler ordering scheme, we find that Horton's law of stream numbers,

(6a), holds throughout the simulated river evolution. The inset of Figure 5a gives an

example of the number of branches versus order number at one instant in a simulation, rbevolves with time as shown in Figure 5a: it displays a rapid increase to a peak followed by

a steady decline. While the long time scales for evolution in nature make direct comparison

with this behavior impossible, Abrahams' [1972] study of the correlation between

decreasing ib and declining basin relief in a group of natural mature basins provided an

indirect measurement of their temporal behavior. Noting that basin relief tends to decrease

with age, he identified the correlation as evidence for a reduction in rb with time during the

late stages of basin evolution, similar to the reduction seen in late times in the model.

The characteristic peak in q? provides a quantitative measure of the variation in

simulations' evolution rate with the model parameters. Not surprisingly, the number of

iterations to the peak, Tp, scales linearly with M/D and linearly with W, the extent of the

lattice in the x direction. More surprisingly, Tp is largely independent of lattice's extent in

the y direction and shows only slow variation with R.

We also find agreement in the simulations with (6c), the geometrical relation for

mean basin area, when we account for an offset to the basin area that the model introduces.

Because the model requires that at least R units flow through any site which is part of a

river, basins formed in the simulations have a minimum area of R. Consequently, we

modify the law of basin areas as

( < A ( u ) > - R ) ~ e x p ( r a u ) ( 7 )

and this form of the law holds well throughout the network evolution. The inset of Figure

5b provides an example of the dependence of (<A(u)> - R) on order number, u, at one time

in a simulation. The variations in ra with time, shown in Figure 5b, correlate strongly with

those of rb, so that the ratio 2rb/ra, plotted in Figure 5c, remains close to 1.8 throughout

10

«

the evolution. Nikora [1994] and Rosso et al. [1991] calculated this ratio for a large

number of river basins. With few exceptions, they found values tightly grouped between

1.6 and 2.0, consistent with the values in the simulations.

The simulations follow the geometrical relation for mean stream length (6b) only

during the early stage of network development and begin to deviate from it at a time before

rb and ra have reached their peak values. These deviations occur as the mean length of first

order branches increases disproportionately to that of second order. This growth in the

mean length of first order branches continues throughout the evolution and is consistent

with experimental observations of networks that have reached maximum extension

[Schumm et al, 1987]. Field measurements also often fail to agree with the law of stream

lengths, particularly when Strahler's ordering scheme is applied, and as with thesimulations the observed deviations result from excessively long first order branches

[Shreve, 1969].

We should note that comparison of the simulations with Horton's Laws is not a

stringent test for the model. Shreve [1966], used general statistical arguments to show thatthe law of stream numbers follows naturally from the ordering procedure in a topologically

random assortment of branched networks. In addition, he derived good agreement with the

laws of stream lengths and basin areas when he assumed realistic forms for the

distributions of lengths and areas that each segment contributes to a network [Shreve,

1969]. However, certain aspects of the simulations, viewed with regard to Horton's

Laws, such as the temporal behaviors of rb and 2rb/ra and the systematic deviations from

(6b), indicate a correspondence with nature that extends beyond what one would expect for

a topologically random set of branched networks.

2.3.2 Hack's Law

Aside from Horton's Laws numerous other scaling relations exist among features

of river systems. Hack [1957] discovered one such characteristic of river network

11

morphology which has become the focus of wide interest [Mesa and Gupta, 1987; Snow.

1989; Robert and Roy, 1990; Rosso et al., 1991; Ijjasz-Vasquez, 1993]. Defining a

principal river in each basin, Hack found that its length scaled with basin area as:

L p - A « ( 8 )with a = 0.6 for a group of small drainage systems.

Again, because of the offset introduced to basin areas in the simulations, we modify

Hack's Law as

L p ~ ( A - R ) « ( 9 )

We find this expression holds throughout the evolution of the drainage networks, where

for Lp we use the longest river in each basin. The inset of Figure 6 shows the scaling of Lp

with A at one time in a simulation. Because of the clear deviations from a power law at low

Lp, we consider only the behavior for Lp > IO2 in evaluating a. Figure 6 shows thevariation of a with time. Initially, while the paths of single units of precipitation on the

featureless terrain dictate the resulting river patterns, a = 0.67, closely matching the value

of 0.669±.001 for the directed walk model on a square lattice [Meakin et al., 1991]. This

result is not surprising since the first units of water transversing the lattice choose among

three accessible sites (x+l,y), (x-1, y), (x,y-l) as they perform random walks down the

slope, mimicking the rules for the random walks in the directed walk model. (Strictly, themodel's early precipitation follows rules matching the directed walk model only in limit D

«I« 1/E; however, the initial value of a appears independent of D/I.) As the landscape

evolves, the value of a decreases to a fixed point near a ~ 0.6, the value originally

measured by Hack. After maintaining this value until a more developed stage, a decreases

to a second fixed point at a * 0.47. Finally, at very late stages the simulations cease to

agree with Hack's law, as the quality of the fits rapidly deteriorates.

Figure 6 illustrates the behavior of most of the simulations. However, in severalsimulations a has settled at its second fixed point closer to 0.5 than 0.47. We also note that

increasing the lattice size in y, relative to its size in x, can enhance the sharpness of the

12

transition in a between 0.6 and 0.47 (compare, for example, Figure 6 with Figure 3 from

Leheny and Nagel [1993]), indicating that the lateral confinement of the basins on thesimulation lattice may play a role in producing this feature.

We have found that the temporal behavior of a in the model corresponds to varia

tions found in nature [Leheny and Nagel, 1993]. Analyzing several thousand data sets,

Mueller [1973] noted that a ~ 0.6 for small basins and that a ~ 0.5 for basins with an

areas between 8000 and 100000 square miles. He also noted a second rapid change to a «

0.466 for basins larger than 100000 square miles. Thus, the early and late fixed points of

a in the simulations match the values for small and large natural basins, respectively. This

correspondence suggests a connection between size and age in drainage basin morphology

that has been the subject of criticism [Mosley and Parker, 1972, 1973]. However, given

the evidence that lateral confinement to the lattice influences the simulations' statistics and

the idea that such confinement may at least crudely model constraints on very large basins

in nature, we find the association between a as a function of size in nature and as a

function of time in the simulations intriguing.

2.3.3. Stream Length Distributions

Recently, the availability of digital elevation maps for large tracts of terrain in theUnited States has motivated methods for identifying river systems from the structure of

natural landscapes. Assigning flow directions between the points on the maps' grids

allows the construction of extensive drainage patterns. Studying networks defined on the

digital elevation maps, Tarboton et al. [1988] have fit the length distributions for Strahlerbranches to a power law at large lengths:

N ( ^ > b ) o c b - Y ( 1 0 )

with y« 1.8 to 1.9. At shorter lengths the distributions deviate from this power law, ap

proaching b = 0 with nearly zero slope. The simulations produce very similar behavior

during their evolution. Figure 7 shows the distribution of Strahler lengths at three times in

13

a simulation. At all times the distribution approaches b = 0 with nearly zero slope. Early, a

power law fit at large lengths gives y~ 1.5. At intermediate times, an excess number of

long branches appears, creating a bimodal deviation from power law decay. However, atlater times the distribution again fits reasonably to the power law decay at large lengths with

y« 1.9, the value measured in natural networks. Given the constraints on the simulation

data both spatially (with finite size effects entering at large lengths and the roll off toward b

= 0 at small) and temporally (with strong deviations at intermediate times), we hesitate to

consider the power law fits more than a guide. Likewise, the data from the digital elevation

maps appears similarly constrained given the limited range of the fits (less than two

decades) and the sparseness of data over much of that region [Tarboton et al, 1988].

However, we find the similarities between the distributions, particularly the approach to

good quantitative agreement at later times in the simulations, encouraging.

2.3.4. Link Length Distributions

Field measurements have revealed that the lengths of external links (first order

Strahler branches) and internal links (segments of higher order branches) in natural river

networks follow either log normal distributions [Strahler, 1954; Schumm, 1956a] or

gamma distributions [Shreve, 1969; Dunkerley, 1977; Smart, 1972]. Figures 8 and 9

display the distributions of external and internal link lengths, respectively, at two times inthe evolution. For the external links, while some simulations have suggested that the

gamma distribution offers a better description, the data does not indicate that either form is

superior during any extended stage in the evolution. The internal link lengths, on the other

hand, clearly follow the gamma distribution at early times. However, as the peak in the

internal link length distribution moves to lower values at later times, the log normal

distribution provides a marginally better fit. The link lengths' maintenance of these

distributions throughout the evolution is a noteworthy indication that the model agrees with

this morphological feature.

14

2.3.5. Area Distribution

Rodriguez-Iturbe et al [1992b], also using digital elevation maps, fit a power lawdistribution to the area drained through each link of a network:

N ( A > m ) ~ n r P ( 1 1 )

where (3 = 0.43. Figure 10 shows the distribution of drainage areas at three times in a

simulation. (To construct the distributions we use every site on the landscape. However, we

find this procedure and that of Rodriguez-Iturbe et al lead to identical distributions for our

data, except at late times when the increased importance of second generation branches gives

disproportionate weighting to links of large area.) Because the finite lattice size leads to a sharpdeviation from any power law at large basin area, we fit the distributions only for m < 10 . As

the simulations evolve, p increases from p ~ 0.38 (in agreement with the result of the directed

walk model [Meakin, et al 1991]) to P = 0.42 (a value closer to the result from the digital

elevation maps). Small but reproducible curvatures exist in log-log representations of these

early time distributions. These curvatures grow at later stages of evolution, as the entire

landscape becomes dominated by the hillslopes of a few large basins. Data from the latest time

in Figure 10 shows such a distribution. During the period in which power law fits seem

reasonable, however, the scaling exponent, p, approaches the one measured from the digital

elevation maps.

2.3.6. Energy Optimization

We have also analyzed the changing morphology of the model networks in terms of

the energy expenditure criteria postulated by Rodriguez-Iturbe et al [1992a]. According to

these criteria branches configure themselves to minimize the network's total power loss,

given by:P = I LjAj0-5

15

where the sum is over all links in the network, and Lj and Aj are the length and drainage

area of the jth link. Figure 11 shows P(t)/P(t=500) averaged over 40 separate 300x200

networks, with R = 100 and R = 200. As the networks evolve from their initial random

walk patterns, their energy expenditure, as expressed by (12), decreases. Thus, the

evolution of this early period drives the system toward an optimal state as viewed by

Rodriguez-Iturbe et al. However, for smaller R the total power increases at later times.

We identify this increase with the formation of second generation branches on large

hillslopes, as shown in Figure 3b. As discussed by Rigon et al[1993], such straight,

parallel flow, characteristic of hillslope runoff, should not optimize (12).

3. Mass Conserving Model

The avalanche scheme plays the central role in producing the temporal changes in

the simulated landscapes. Therefore, we have varied the way the avalanches occur in the

model to test the sensitivity of the evolution to their precise form. The simulations with this

modified model follow the same iterative procedure outlined in section 2.1, except that (5)

in step 4 is replaced with:

h ( x n , y h ) ^ h ( x h , y h ) - A h / S ( 1 3 a )

h ( x i , y i ) - > h ( x i , y i ) + A h / S ( 1 3 b )

thereby conserving the avalanched soil by redistributing it to the low side, (xi,yi), of the

steep slope.This alteration in the model introduces the complication that localjninima form in

^ ~~^ the interior of the lattice from which the probability distribution, (3), makes no allowancepY<?4> kv„, for escape. Therefore, when (3), with the proviso that the water not return to the site it

immediately vacated, leads to zero probability of movement in all directions, the water

automatically flows from the occupied site in the same direction as the last unit of

precipitation to visit that site. In the escape of a minimum, this rule leads to flow to a siteof higher elevation. When recording the erosive effects of the water in step 3, the model

16

does not apply (4) to those sites from which the flow is uphill, so that the erosion of

subsequent precipitation quickly removes any local minima formed upstream of avalanchedsoil. Occasionally (less than 1% of all iterations), the water enters a pathologically

configured minimum for which this escape procedure fails. In such cases the waterfollows the path of its immediate predecessor not only out of the minimum but to the edge

of the lattice, y = 0. Again, because sites from which the flow is uphill do not erode, these

pathologies are short lived. The definition of the rivers is the same in this model as for thenon-mass conserving model with the stipulation that, if a test unit enters a minimum, it

leaves following the direction of the last eroding precipitation to visit the site.

In choosing parameter values for simulations with this version of the model we

followed the same procedures described in section 2.2. However, we typically used a

smaller value of M, while maintaining M > D, to limit the size of barriers forming minima

in the landscape. Also, we used S = 8 in the avalanche scheme so that the relative change

in slope after an avalanche matched that in the simulations with the non-mass conserving

model.

Figures 12a through 12c show an evolving terrain simulated with the mass

conserving model. Comparison with Figure 1 reveals qualitative differences between these

landscapes and those formed using the non-mass conserving model. Both simulations

begin with V-shaped valleys initiating at y = 0. However, this formative stage comprises amuch smaller fraction of the total time of evolution when avalanched mass is redeposited.

In simulations with the non-mass conserving model, the valleys quickly broaden near their

mouths, so that the region near y = 0 becomes dominated by a few large basins before the

terrains' upper reaches suffer any significant erosion. In contrast, when avalanched soil

replenishes the valley bottoms, the headward growing valleys fill the landscape before the

region near y = 0 reaches more advanced stages of erosion. These two morphologies

represent two extremes with respect to the ability of the river system to transport avalanchedebris. In the original model the rivers can fully support the instantaneous removal of

17

avalanched mass. In the mass conserving model, however, the sedimentation erodes no

more easily than the undetached terrain beneath.

Despite these differing landscape features, the river networks created in the mass

conserving model, shown in Figures 12d through 12h, have the same general features as

those in Figure 1. In particular, the dominant characteristics of their evolution remain:

abstraction and stream capture as well as the straightening of main branches and the growth

of second generation streams at late times. However, because the mass conserving model

produces a more spatially uniform valley development, its simulations do not share the

pronounced temporal overlap of the headward invasion of valleys and the abstraction of

tributaries that the non-mass conserving model displays.

The statistical properties of river networks produced with each model reveal

slightly different temporal behavior, reflecting the differences in the landscape evolution.

Figures 13a and 13b show the variation with time of rb from Horton's Law and a from

Hack's Law, respectively, in a simulation produced with the mass conserving model.

Compared with the original model the rapid increase in rb and decrease from 0.67 to 0.6 in

a comprise a much smaller portion of the total time of evolution. However, the exponents

vary in the same general manner as they do in simulations with the non-mass conservingmodel: rb shows a steady decrease during most of the evolution and a has two fixed points

at 0.6 and 0.47. Indeed, we have repeated our analysis for the mass conserving model of

all of the statistical properties of river networks described in section 2.3 and have found

that, despite the differences in landscape features, the behavior for each shows agreement

with the original model.

4. Conclusion

Throughout this discussion we have stressed the good correspondence between the

spatial and temporal features of the simulations and those observed in nature. Despite thesesuccesses we note that the model may serve less as a concise enumeration of mechanisms

18

driving network evolution than as a warning about the ease of producing branchedstructures that possess certain statistical characteristics. As Shreve demonstrated for

Horton's Laws, many bifurcating branched networks could, as a general property of their

form, possess statistical properties that are considered distinguishing characteristics of river

networks. However, a noteworthy attribute of the models' performance is that the

simulated networks improve their compliance with several statistical features, such as

Hack's Law and the stream length and area distributions, as they evolve from their initial

configuration of random walk aggregates. Thus, with only simple procedures for erosion

and avalanching, the model produces a realistic network evolution that maintains or

improves the networks' agreement with natural river systems. This temporal improvement

strongly suggests that the model indeed offers insight into the mechanisms of networkformation.

Finally, along with its successes the model's limitations offer insight, particularlywhen the missing elements that would extend its correspondence with natural systems can

be identified. For example, the use of the threshold parameter, R, in defining the rivers

prevents the model from simulating network extension, which a more physical criterion forchannel initiation, such as that of Willgoose et al [1990], might allow. In addition, only

with a more realistic description of the transport and sedimentation of soil eroded via (4)

could we hope to compare some of the statistical properties of the simulated terrains against

natural features, such as the variation of stream gradient with drainage area [Flint, 1974] or

the roughness of the erosional landscapes [Dietler and Zhang, 1992; Czirok, Somfai and

Vicsek, 1993]. As a first approximation to such a model, we have introduced [to be

published] to the rules a minimum height to simulate the presence of a bedrock or a base-

level, below which erosion via (4) ceases. Preliminary results indicate that this minimum

generates interesting changes in landscape features that require further study. Given the

simplicity of the model in its present form, the breadth of comparisons we have been able

19

to make in both its temporal and statistical behavior with natural river networks provides

encouragement for future work.

Acknowledgments: I am grateful to Sid Nagel, with whom I worked in close consultation

throughout this project, for his support and guidance. Also, I thank Sue Coppersmith,

Alan Howard, Frank Richter, and Tom Witten for helpful discussions. Funding was

provided by NSF DMR 91-11733 and DOE DE-FG02-92ER25119

20

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Czirok, A., E. Somfai, and T. Vicsek, Experimental evidence for self-affine roughening in amicromodel of geomorphological evolution, Phys. Rev. Lett., 71, 2154-2157, 1993.

Dietler, G., and Y. Zhang, Fractal aspects of the Swiss landscape, Physica A, 191, 213-219,1992.

Dunkerley, D. L., Frequency distributions of stream link lengths and the development of channelnetworks, J. Geol, 85, 459-470, 1977.

Flint, J. J., Stream gradient as a function of order, magnitude, and discharge, Water Resour. Res.,10, 969-973, 1974.

Xjlock, W. S., The development of drainage systems: a synoptic view, Geograph. Rev., 21, 475-482, 1931.

Hack, J. T., Studies of longitudinal stream profiles in Virginia and Maryland, U.S. Geol Surv.Prof Paper, 294B, 45-97, 1957.

orton, R. E., Erosional development of streams and their drainage basins; hydrophysicalapproach to quantitative morphology, Geol Soc. Am. Bull, 56, 275-370, 1945.

Howard, A. D., Simulation of stream networks by headward growth and branching, Geogr.Anal, 3, 29-50, 1971a.

Howard, A. D., Simulation model of stream capture, Geol. Soc. Am. Bull, 82, 1355-1375,1971b.

Howard, A. D., Theoretical model of optimal drainage networks, Water Resour. Res., 26, 2107-2117, 1990.

, /rtoward, A. D., A detachment-limited model of drainage basin evolution, Water Resour. Res., 3CL y^ 2 2 6 1 - 2 2 8 5 , 1 9 9 4 . " v

Ijjasz-Vasquez, E. J., R. L. Bras, and I. Rodriguez-Iturbe, Hack's relation and optimal channelnetworks: the elongation of river basins as a consequence of energy minimization, Geophys. Res.Lett., 20, 1583-1586, 1993.

r~Inaoka, H. and H. Takayasu, Water erosion as a fractal growth process, Phys. Rev. E, 47, 899-910, 1993.

, Zeheny, R. L., and S. R. Nagel, Model for the evolution of river networks, Phys. Rev. Lett., 71M470-1473, 1993.

Leopold, L. B., and W. B. Langbein, The concept of entropy in landscape evolution, U.S. GeolSurv. Prof. Paper, 500A, 1-20, 1962.

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Morasawa, M., Development of drainage systems on an upraised lake floor, Am. J. Sci. 262, 340-354, 1964.

Meakin, P., J. Feder, and T. J0ssang, Simple statistical models for river networks, Physica A776,409-429, 1991.

Mesa O. J., and V. K. Gupta, On the channel length-area relationship for channel networks,Water Resour. Res., 23, 2119-2122, 1987.

Mosley, M. P. and R. S. Parker, Allometric growth: a useful concept in geomorphology?, GeolSoc. Am. Bull, 83, 3669-3673, 1972.

Mosley M P and R. S. Parker, Re-evaluation of the relationship of master streams and drainagebasins:' discussion, Geol Soc. Am. Bull, 84, 3123-3125, 1973.

Mueller J E., Re-evaluation of the relationship of master streams and drainage basins: reply,Geol Soc. Am. Bull, 84, 3127-3130, 1973.

Nikora, V. I. , On self-similarity and self-affinity of drainage basins, Water Resour. Res., 30,133-137, 1994.

O'Callaghan, J. F., and D. M. Marks, The extraction of drainage networks from digital elevationdata, Comput. Vision Graphics Image Process. 28, 323-344, 1984.

Rigon, R., A. Rinaldo, I. Rodriguez-Iturbe, R. L. Bras, E. Ijjasz-Vasquez, Optimal channelnetworks: a framework for the study of river basin morphology, Water Resour. Res., 29, 1635-1646, 1993.

/ftinaldo, A., I. Rodriguez-Iturbe, R. Rigon, E. J. Ijjasz-Vasquez, and R. L. Bras, Self-organized^fractal river networks, Phys. Rev. Lett., 70, 822-825, 1993.

Robert, A., and A. G. Roy, On the fractal interpretation of the mainstream length-drainage arearelationship, Water Resour. Res., 26, 839-842, 1990.

Rodriguez-Iturbe, I., A. Rinaldo, R. Rigon, R. L. Bras, A. Marani, E. Ijjasz-Vasquez, Energydissipation, runoff production, and the three-dimensional structure of river basins, Water Resour.Res., 28, 1095-1103, 1992a.

Rodriguez-Iturbe, I., E. J. Ijjasz-Vasquez, R. L. Bras, D. G. Tarboton, Power law distributionsof discharge mass and energy in river basins, Water Resour. Res., 28, 1089-1093, 1992b.

Rosso, R., B. Bacci, and P. La Barbera, Fractal relation of mainstream length to catchment area inriver networks, Water Resour. Res., 27, 381-387, 1991.

Ruhe, R. V., Topographic Discontinuities of the Des Moines lobe, Am. J. Sci., 250, 46-56, 1952.

Scheidegger, A. E., A stochastic model for drainage patterns into a intramontane trench, Int.Assoc. Sci. Hydro! Bull, 12, 15-20, 1967.

Schumm, S. A., Evolution of drainage systems and slopes in badlands at Perth Amboy, NewJersey, Geol Soc. Am. Bull, 67, 597-646, 1956a.

22

Schumm, S. A., The role of creep and rainwash on the retreat of badland slopes, Am. Jour Sci254, 693-706, 1956b.

Schumm, S. A., M. P. Mosley, and W. E. Weaver, Experimental Fluvial Geomorphology ,413pp., John Wiley and Sons, New York, 1987.

Seginer, I., Random walk and random roughness models of drainage networks, Water ResourRes., 5, 591-607, 1969.

Shreve, R. L., Statistical Law of Stream Numbers, J. Geol, 74, 17-37, 1966.

Shreve R. L., Stream Lengths and Basins Areas in Topologically Random Channel Networks, /.Geol, 77, 397-417, 1969.

Small, R. J., The Study ofLandforms, 502 pp., Cambridge University, New York, 1978.

Smart, J. J., Channel Networks, Adv. Hydroscl, 8, 305-346, 1972.

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Strahler, A. N., Hypsometric (area-altitude) analysis of erosional topography, Geol Soc. Am.Bull, 63, 1117-1142, 1952.

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^/Tarboton, D. G., R. L. Bras, and I. Rodriguez-Iturbe, The fractal nature of river networks, WaterResour. Res., 24, 1317-1322, 1988.

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1685-1696, 1991.

23

FIGURE CAPTIONS

FIGURE 1: The evolution of the landscape: (a) at t = 1 x IO4 iterations, (b) at t = 5x IO4

iterations, and (c) at t = 9xl04 iterations; and the corresponding river networks (d) at t =

lxlO4 iterations, (e) at t = 3xl04 iterations, (f) at t = 5xl04 iterations, (g) at t = 7xl04

iterations (h) at t = 9xl04 iterations, in a 300x200 site simulation. The model parameters

were: D = 10, E = 0.05,1 = 1.0, M = 2000, and R = 100.

FIGURE 2: Cross sections of the landscape elevation at y = 50 in the simulation in Figure

1: (—) at t = lxlO4 iterations, ( ) at t = 3xl04 iterations, ( ) at t = 5xl04

iterations, ( ) at t = 7xl04 iterations, and (••••) at t = 9xl04 iterations. The parallel

retreat of the valley sides mimics effects of rainwash erosion observed in nature [Schumm,

1956b].

FIGURE 3: Details from Figures 1(e) and 1(0 illustrating changes in the simulated

networks with time. In (a) "A" labels streams that will suffer abstraction and "S" marks

locations for stream capture. In (b) "2" labels newly formed, second generation branches.

FIGURE 4: Histograms of s, the length in steps of the straight line sections that comprise

the river networks' main branches (Strahler order >1), at three times in the evolution: (A) t

= IO4 iterations, (X) t = 5xl04 iterations, and (O) t = 1 lxlO4 iterations. The data is a

collection from fourteen 300x200 site simulations generated with the same model

parameters that produced Figure 1 and is compiled in bins spaced equally in log steps.Each point is the number of straight sections with lengths that fall within its bin divided by

bin size. As time progresses, the number of long straight paths increases.

24

FIGURE 5: The time development of rb and ra from Horton's Laws and of their ratio

2rt/ra. The steady decrease in ib at late times, shown in (a), matches a trend that

measurements suggest exists in natural mature basins [Abrahams, 1972]. The temporal

development of ra, shown in (b), reveals a strong correlation over time with rb- The

roughly constant ratio, 2rt/ra, in (c) maintains a value consistent with data from natural

networks [Nikora, 1994; Rosso et al. ,1991]. As an example, the inset of (a) shows the

scaling of the number of branches, N, with order number, u, at t = 90xl04 iterations. Theinset of (b) shows the scaling of the mean basin area minus the offset that the model

introduces, (<A> - R), with u at the same time. The data results from a 1500x1000 site

simulation with model parameters: D = 10, E = 0.05,1 = 1.0, M = 2000, and R = 70.

FIGURE 6: The time development of the scaling exponent in Hack's Law for the same

simulation that produced the results in Figure 5. The fixed points at a = 0.6 and a = 0.47

match those values measured by Mueller [1973] for small and large drainage basins,

respectively. The inset shows the scaling of the principle river length with basin area at t =

25x104 iterations.

FIGURE 7: The distribution of Strahler branch lengths at three times in same simulation

that produced Figure 5: (A) t = 5X104 iterations, (•) t = 40x104 iterations, and (X) t =

70x104 iterations. The distributions at t = 5xl04 iterations and t = 40x104 iterations have

been offset for clarity. At early and late times the distribution at large lengths approximates

a power law decay with scaling exponent y ~ 1.5 at early times and y « 1.9 at late times. At

intermediate times, such as t = 40x104 iterations, the distribution deviates from this

behavior. The late time distribution with y ~ 1.9 shows quantitative agreement with data

obtained from digital elevation maps [Tarboton et al, 1988].

25

FIGURE 8: The distribution of external link lengths, Le, at two times in a 1500x1000 site

simulation: (a)t= 5xl04 iterations, and (b) t= 60xl04 iterations. The solid (dashed)

lines are fits to a gamma (log normal) distribution. The model parameters used in the

simulation were: D = 10, E = 0.05,1 = 1.0, M = 2000, R=70.

FIGURE 9: The distribution of internal link length, Li, at two times in the same simulation

that produced Fig. 8: (a) t = 5xl04 iterations, and (b) t = 60x104 iterations. The solid

(dashed) lines are fits to a gamma (log normal) distribution. At early times the gamma

distribution provides a superior fit.

FIGURE 10: The distribution of drainage areas, A, for all sites at three times in same

simulation that produced Fig. 8: (A) at t = lxlO4 iterations, (X) at t = 15xl04 iterations,

and (•) at t = 30X104 iterations. The distributions at t = lxlO4 iterations and t = 15X104

iterations have been offset for clarity. At early times power law scaling provides a good

description of the distribution with the scaling exponent approaching with time P = 0.43,

the value measured from digital elevation maps [Rodriguez-Iturbe et al, 1992b].

Deviations from the power law scaling become pronounced at later times.

FIGURE 11: The variation with time of the total power loss, P, as defined by (13),

normalized to the power loss after 500 iterations. The data, compiled from forty 300x200

site simulations with model parameters D = 10, E = 0.05,1 = 1.0, M = 2000, are for two

values of the threshold parameter: (A) R=100and(«)R=200. The steady decrease in

power loss with time indicates that the evolving networks are approaching an optimal

configuration, as described by Rodriguez-Iturbe et al, [1992a]. The increase at late timesfor R=100 reflects the contributions to the networks of second generation tributaries on the

hillslopes.

26

FIGURE 12: The evolution of a landscape: (a) t = 400 iterations, (b) t = 4x104 iterations,

and (c) t = 25x104 iterations; and corresponding river networks: (d) t = 400 iterations, (e)

t = lxlO4 iterations, (f) t = 4xl04 iterations, (g) t = lOxlO4 iterations, and (h) t = 25x104

iterations, produced by the mass conserving model. The model parameters used in the

300x200 site simulation were: D = 10, E = 0.05,1 = 1.0, M = 80, and R = 100.

FIGURE 13: The time developments of rb from Horton's Law, shown in (a) and a from

Hack's Law, shown in (b), for a 1200x1000 site simulation produced with the mass

conserving model. The model parameters used were: D = 10, E = 0.05,1 = 1.0, M = 80,

and R = 70. The variations in time of these statistical features repeat the behavior seen in

the non-mass conserving model, shown in Figures 5a and 6, although over a greatly

extended time scale.

27

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