impacts of surface elevation on the growth and scaling

Ž .Geomorphology 40 2001 37–55www.elsevier.comrlocatergeomorph

Impacts of surface elevation on the growth and scaling propertiesof simulated river networks

Jeffrey D. Niemann a,), Rafael L. Bras b, Daniele Veneziano b, Andrea Rinaldo c

a Department of CiÕil and EnÕironmental Engineering, The PennsylÕania State UniÕersity, UniÕersity Park, PA 16802, USAb Department of CiÕil and EnÕironmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

c Istituto di Idraulica AG. Poleni,B UniÕersita di PadoÕa, PadoÕa, Italy`

Received 1 July 1999; received in revised form 12 January 2001; accepted 13 January 2001

Abstract

We investigate the connection between surface elevation and the growth and scaling of river networks. Three planarŽ .models Scheidegger, Eden, and invasion percolation are first considered. These models develop aggregating networks

according to stochastic rules but do not simulate erosion because the network growth is independent of the surface elevation.We show that none of these planar growth models produces scaling results consistent with observations for natural riverbasins. We then modify the models to include elevation, simulating the effects of fluvial erosion by enforcing the slope–area

Žrelationship. The resulting configurations have scaling properties that still depend on the model Scheidegger, Eden, or.invasion percolation but are closer to natural river networks when compared with those from the planar growth rules. We

conclude that inclusion of the vertical dimension in these three models is critical for explaining the formation andregularities of fluvial networks. q 2001 Elsevier Science B.V. All rights reserved.

Keywords: Drainage networks; Models; Self-similarity; Fractal geometry

1. Introduction

Ž .Primarily since the work of Mandelbrot 1983 ,scale invariant structures have been observed in manyfields of the natural sciences. River networks arewell-known examples of such structures, and proper-

Ž .ties such as Hack’s law Hack, 1957 and Horton’s

) Corresponding author. Tel.: q1-814-865-9428; fax: q1-814-863-7304.

Ž .E-mail address: [email protected] J.D. Niemann .

Ž .bifurcation ratio Horton, 1945 are commonly citedas evidence for some form of scale invarianceŽ .Rodriguez-Iturbe and Rinaldo, 1997 . One approachto understand river basin scaling is through the pro-cess by which river networks grow to fill an initially

Žundrained region we refer to this as the network’s.mode of growth . A classical example is Scheideg-

Ž .ger’s model Scheidegger, 1967 , which developsdirected networks with particular scaling properties.Several other models have been adapted from well-known cluster growth algorithms such as Eden

Ž .growth Eden, 1961 and invasion percolation

0169-555Xr01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.Ž .PII: S0169-555X 01 00036-8

( )J.D. Niemann et al.rGeomorphology 40 2001 37–5538

Ž .Wilkinson and Willemsen, 1983 . For example,Ž .Howard 1971 explored a model for river networks

based on local branching properties that includes theScheidegger and Eden schemes among its variants,

Ž .whereas Stark 1991 proposed an invasion percola-tion model in which a river network grows by cap-turing the adjacent point with the lowest substratestrength.

The above models operate only on the geographi-cal plane and rely solely on the mode of growth todetermine basin structure. In real basins, the rivernetwork configuration may be affected also by thesurface elevation during the process of growth. Theflow directions throughout the network must be con-sistent with the topographic surface and thereforecannot be assigned randomly as is often done inplanar models. In addition, since the expected value

w xof channel slope at a given contributing area E S isw xobserved to vary with contributing area A as E S A

yu Ž .A for some u)0 Flint, 1974 , there is a feedbackmechanism whereby the network configuration af-fects the surface elevation. As a point captures morearea, fluvial erosion decreases the slope thus makingthe point more prone to capture additional area. Thisdynamic can cause changes in the mode of basingrowth and subsequent reorganization of the basinstructure.

In this paper, we examine whether inclusion ofthe vertical dimension impacts the mode of networkgrowth and the resulting basin scaling properties forthree simple models—Scheidegger, Eden, and inva-sion percolation. First, we characterize the scalingproperties of networks developed by the originalplanar models and determine whether they are ade-quate for river basins. We confirm and extend previ-

Žous results for these models Huber, 1991; Takayasu.et al., 1991; Cieplak et al., 1998 and demonstrate

the importance of examining both topological andgeometrical characteristics when comparing withnatural basins. Second, we investigate how the scal-ing properties vary when processes associated withsurface elevation are included in the models. Weaccount for elevation using erosion models similar to

Ž .that of Rinaldo et al. 1993 . Under certain condi-tions, the cluster frontiers of the new models areidentical to their planar counterparts, indicating iden-tical modes of cluster growth. However, even undersuch conditions, the choice of flow direction is made

according to the topography and the elevation field isadjusted using the slope–area relationship, resultingin different river networks. These experiments showthat the scaling exponents depend on both the modeof growth and the process of slope–area enforce-ment.

The organization of the paper is as follows. AŽ .background section Section 2 describes the proper-

ties used to characterize the scaling of real andsimulated basins. The planar Scheidegger, Eden, andinvasion percolation models are briefly presented inSection 3. Their simulated basins are compared withtypical natural basins in Section 4. Section 5 presentsthe models with elevation included, and Section 6analyzes the results from these models. The final

Ž .section Section 7 summarizes the main results andstates the conclusions.

2. Methods of analysis

To characterize the scaling properties of rivernetworks, two sets of parameters are used. The first

Ž .set Tokunaga, 1978; Peckham, 1995 quantifies thetopological characteristics of the network. Strahler’sstream ordering is used to assign an order to eachbranch of the network. Branches that begin at astream source are labeled as order one. When twostreams of equal order merge, the resulting stream isgiven an order value that is one higher than itstributaries. When two streams of differing ordermeet, the resulting stream is assigned the larger of

Ž .the two upstream orders Strahler, 1957 . The topol-ogy of the tree can then be described through a lowertriangular matrix T in which the elements give thev,k

average number of side tributaries of order k flowinginto a stream of order v. Topological self-similarityrequires that the elements of any given diagonal beidentical. Specifically, T 'T for all v. It hasv,vyi i

been further suggested that the T values can bei

related as T sbciy1 where b and c are constantsiŽ .Tokunaga, 1978; Peckham, 1995 .

The second set of parameters includes three scal-ing exponents which describe the topological andgeometrical scaling of river networks: b , h, and HŽMaritan et al., 1996; Rigon et al., 1996; Veneziano

.and Niemann, 2000 . The exponent b characterizes

( )J.D. Niemann et al.rGeomorphology 40 2001 37–55 39

the distribution of contributing area A for pointsregularly spaced over the basin. One can write:

w x ybP AGa sa f ara 1Ž . Ž .max

Ž .where f ara describes the deviation from log-max

linearity as a approaches the area of the basin amaxŽ .Maritan et al., 1996 . b is largely determined by

Ž .network topology de Vries et al., 1994 . The secondŽparameter, h, is the exponent in Hack’s law Hack,

. w x h w x1957 , which states that E L AA where E L isthe expected main stream length for a sub-basin ofarea A. Using a finite-size scaling argument, Rigon

Ž .et al. 1996 have extended the mean value relationto moments of any order q:

w q x w qy1 x hE L rE L AA 2Ž .where q is an integer exponent. The third parameteris the Hurst exponent H, which relates the Euclideanwidth of a basin L to its Euclidean length L .H IHere L is defined as the straight line distanceIbetween the mainstream source and the basin outlet,but other definitions are possible. One can writew x 1rŽHq1.E L AA or, in terms of the moments,I

q qy1 1rŽHq1.E L rE L AA 3Ž .I I

Hs1 is required for self-similarity of basin shapes.The three parameters, b, h, and H, can be estimated

Ž . Ž .by plotting the relations described by Eqs. 1 – 3 inlog–log space, and performing linear regressions.

3. Planar models

Ž .Scheidegger 1967 proposed one of the firstquantitative models for the planar development ofriver networks. The model operates on a hexagonallattice and develops channels from an edge of thesimulation domain that forms a line of basin outlets.At each iteration, an entire row of adjacent neighborsis simultaneously captured, and each of these pointsis randomly assigned a flow direction toward one ofthe two neighbors already within the basin. Thismodel was originally presented as a way to developriver networks, but it may also be viewed as a simplecluster growth rule. The open boundary serves as aseries of seed points, and the clusters, which corre-spond to the river basins, grow by a pixel layer at

each iteration. Fig. 1a shows an intermediate snap-shot and the final configuration for a simulation withthe Scheidegger model. The model produces uniformheadward growth and the aggregate cluster frontierremains smooth during development. Each channeltends to follow a relatively straight path between itssource and outlet.

The second planar model used here is a well-Ž .known variant of Eden cluster growth Eden, 1961

with the addition of a simple scheme to assign flowdirections as points are added to the cluster. Themodel operates on a square lattice and begins with acluster seed point. This point is located at the borderand represents the outlet of the growing basin. Ateach iteration, a single point is randomly selectedfrom the neighbors of the existing basin. The point iscaptured and given a flow direction toward a ran-domly selected neighbor already within the basin.Fig. 1b shows the growth of an Eden network withan outlet specified at the corner of a square domain.At all stages of growth, the basin is quite compactwith growth occurring at a well defined but irregularbasin frontier. Advancement of the cluster frontier ismore erratic than in Scheidegger’s model, and onlythe very large channels are directed toward the out-let.

Both the Scheidegger and Eden models conformto Howard’s concept that river networks expandheadward through an intense Awave of dissectionBŽ .Howard, 1971 . In fact, if one alters Scheidegger’smodel to operate on a square lattice and to allow aseed at any given location, then this Scheideggervariant and the Eden model above differ only by thenumber of random neighbors added at each iteration.The two models are therefore end members ofHoward’s numerical growth procedure in which thenumber of points added to the cluster at a giveniteration is controlled.

While the above models suggest that channelsgrow essentially by chance, invasion percolationchannels grow by following paths of least resistance.This process is simulated by assigning an indepen-dent random number to each point on a square latticeat the beginning of the simulation. After the initial

Ž .outlet or outlets has been specified, the basin growsby capturing the adjacent neighbor with the highestnumber. In order to define a drainage network, theadded point is assigned a flow direction toward a


Ž . Ž . Ž .Fig. 1. Channel network growth using the models of a Scheidegger, b Eden, and c invasion percolation networks. Left panels show thecluster and network configurations after 80 iterations for Scheidegger, about 25,000 iterations for Eden, and 8000 iterations for invasionpercolation. Right panels show the final spanning networks. Points outside the clusters are shaded gray, and points within the clusters areshaded according to their contributing areas. The domain size in all panels is 200=200 pixels.

randomly selected neighbor within the basin. Fig. 1cshows an example basin growing by invasion perco-lation. At intermediate stages, the basin is not com-

pact and has a very erratic frontier. The pattern ofgrowth differs considerably from Howard’s concep-tual model because the main channels develop first,


cutting well into the simulation domain before thesmaller tributaries are defined. Instead, this style ofgrowth is consistent with Glock’s concept that flu-vial networks first AelongateB to capture new terri-tory and then AelaborateB the minor tributariesŽ .Glock, 1931 .

4. Results for the planar models

Using each of the previous models, a suitableensemble of runs was generated for analysis. In thecase of Scheidegger’s model, five runs were made ona 500=2000 pixel lattice with an open boundaryalong one of the shorter sides. For the other twomodels, five runs were made on a 500=500 pixeldomain with an outlet specified at one corner. Thesedomain sizes were selected so that power laws couldbe observed over at least two orders of magnitude.For clarity, the example networks shown in Fig. 1

Ž .have smaller domains 200=200 .Fig. 2 shows results of the topological analysis for

the three simulation ensembles. Each line gives the

values of T for fixed i and variable v. Topo-v,vyi

logical self-similarity requires that such lines behorizontal, which is observed with good approxima-tion over a range of stream orders v. For v)6, asignificant decrease in the number of side tributaries

Žis observed the highest stream orders are not shown.in the figure . The decrease is due to the domain

shape, which narrows as one approaches the outlet.This causes many high order streams to join near theoutlet, and consequently the higher order streamshave short lengths and few lower order tributariesŽ .see Fig. 1 . There is also an excess of first ordertributaries for streams of any order. Streams of orderone are likely to depend heavily on the numericaldiscretization of the domain, which largely controlsthe number of stream sources. Over the range oforders that display self-similarity, the T values arei

highest for Scheidegger’s model and lowest for inva-sion percolation. Assuming the model T sbciy1,i

Ž . Žone obtains bs1.3, cs3.3 for Scheidegger, bs. Ž .1.3, cs2.9 for Eden, and bs1.0, cs2.7 for

invasion percolation as best-fit values. In all threecases, only a small number of tributary orders areavailable for comparison, but the model fits the data

Fig. 2. Topological scaling for the ensembles of Scheidegger, Eden, and invasion percolation networks. Each line gives the coefficientsT for different Strahler orders v and a fixed value of i. The vertical axis has a logarithmic scale.v,vy i


Ž .well. Peckham 1995 analyzed two natural basinsand found bs1.2, 1.2 and cs2.4, 2.7. Theseresults indicate that the Scheidegger and Eden mod-els develop basins with more side tributaries thanreal river networks, whereas invasion percolationnetworks are more similar to real basins in thisrespect.

One can also examine the bifurcation ratios RbŽ .Horton, 1945 . The R values for Scheidegger,b

Eden, and invasion percolation are R s5.2, 5.0,bŽ . Ž .4.6, respectively Fig. 2 inset . Tarboton et al. 1989

reported bifurcation ratios for nine natural basins,with a maximum of 4.7 and an average of 4.1.

Ž .Peckham 1995 found R s4.5, 4.8. These resultsb

confirm that Scheidegger and Eden networks havemore side tributaries than natural channel networks.

Figs. 3, 4 and 5 show the distribution of contribut-ing area for the three models. At large contributingareas, finite-size effects occur for all three models,but these effects are observed over the widest rangefor the Eden model. Over a range of intermediateand small areas, power laws are observed with bs0.34, bs0.41, and bs0.39 for Scheidegger, Eden,

and invasion percolation, respectively. These valuesare in good agreement with those previously reportedin the literature. bs1r3 has been analytically de-

Žrived for Scheidegger Huber, 1991; Takayasu et al.,.1991 and bs0.40, 0.38 has been observed else-

where for Eden and invasion percolation, respec-Ž .tively Cieplak et al., 1998 . The proximity of the b

values for the Eden and invasion percolation modelsmeans that b does not capture the morphologicaldifferences between the networks that are so appar-

Ž .ent to the eye. For natural basins, Rigon et al. 1996have found values of b in the range 0.40–0.46 withan average of 0.43. Hence, the values of b for Edenand invasion percolation are at the low end of therange observed in nature but are not unreasonable. Incontrast, the exponent for Scheidegger does not agreewell with real basins.

Figs. 6, 7 and 8 show Hack’s law for the simula-tions associated with the three models. Large devia-tions from log-linearity are again observed for theEden model at large scales. The reason is that thelargest tributaries are forced to converge near theoutlet thereby rapidly increasing the contributing area

Fig. 3. The distribution of contributing areas within basins developed by the Scheidegger and headward models. Three cases are shown forthe headward model as described in the text. Exponents are estimated over the ranges spanned by the offset regression lines.


Fig. 4. The same as Fig. 3 with data from the Eden and random models.

Fig. 5. The same as Fig. 3 with data from the invasion percolation and ranked models.


Fig. 6. The scaling of the moment ratios of main stream length with contributing area for the Scheidegger and headward models. Three casesare shown for the headward model as described in the text. The h values represent averages among the four ratios of moments. They areestimated over the ranges of data spanned by the offset regression lines. The moment ratios are also offset for clarity.




Ž .and flattening the plots see Fig. 1 . At smaller areas,power laws are approximated with hs0.67 forScheidegger and hs0.61 for both Eden and inva-sion percolation. For each model, the exponentscalculated from various moment ratios vary onlyslightly. The similarity of the h values for Eden andinvasion percolation is not surprising given the simi-

Ž .larity of the b values. Maritan et al. 1996 observedthat bqhf1 for natural networks and derived thisresult under reasonable assumptions on the geometryof basins. The same value hs0.61 has been previ-

Ž .ously measured by Cieplak et al. 1998 for invasionpercolation.

Ž .Rigon et al. 1996 reported values of h fornatural basins between 0.52 and 0.60 with an aver-age of 0.55. Here again, the Scheidegger model is inpoor agreement with natural river networks. Theother two values of h fall at the upper edge of therange but are not unreasonable. High h values meanthat the length of the main stream grows morerapidly with contributing area than it does in mostreal basins. Such an increase in length can be dueeither to fractal sinuosity of the streams or elonga-

Ž .tion of the basin shapes geometric self-affinityŽ .Rigon et al., 1996 .

Finally, the scaling of the basin shapes for thethree models is shown in Figs. 9, 10 and 11 byplotting moment ratios of L against A. The plotsIobey power laws over two orders of magnitude withmarkedly different exponents for the three models.The slopes of the plots give Hs0.50, 0.69, 1.04 forScheidegger, Eden, and invasion percolation, respec-tively. The differences in the H values are substan-tial and indicate that geometric scaling of the sub-basin shapes is a key discriminant for these models.While the power laws considered earlier were verysimilar for the Eden and invasion percolation mod-els, the H values of these models differ substan-tially. Like Scheidegger basins, Eden basins areself-affine because they become more elongated astheir areas increase, whereas invasion percolation

Ž .basins come close to self-similarity H close to one .The self-affinity of Scheidegger and Eden basins

can be understood by considering their modes ofgrowth. Scheidegger’s basins grow uniformly head-ward, giving large channels little opportunity to cap-


Fig. 9. The scaling of the moment ratios of Euclidean distance from the main stream source to the outlet with contributing area. Data arefrom the Scheidegger and headward models. Three cases are shown for the headward model as described in the text. The exponents shownare averages among the four ratios of moments and were estimated over the ranges of data spanned by the offset regression lines. Themoment ratios are also offset for clarity.




ture width from smaller neighboring basins. Eden’smodel produces some deviation from uniform head-ward growth, causing the cluster frontier to develop

Ž .self-affine irregularity Vicsek, 1992 . This irregular-ity gives the channels that have extended further anadvantage in capturing additional side area, thusmaking the sub-basin shapes more self-similar. In thecase of invasion percolation, the main channels growmuch earlier than the smaller tributaries, and self-af-finity reduces to self-similarity.

The values of H noted above invalidate Schei-degger and Eden growth as models for river net-works. Of the 13 basins analyzed by Rigon et al.Ž . Ž1996 , only one has a value of H below 0.88 with

.a value of 0.75 , and the average value is Hs0.93.While the value of H for invasion percolation ismore realistic, the simulated river courses have fartoo much sinuosity. The deviation between h and

Ž .1r Hq1 can only be due to the sinuosity of theŽ .channels. Thus, one can use h Hq1 to describe

Ž .the stream sinuosity Rigon et al., 1996 . For the setŽ .of basins analyzed by Rigon et al., h Hq1 is on

average 1.06 and never exceeds 1.08. For invasionpercolation, it is around 1.25. This result is also

confirmed visually, since invasion percolation chan-nels are far more sinuous than natural river coursesŽ .Fig. 1c .

5. Models with elevation

The previous section has shown that the mode ofcluster growth affects the scaling properties of thefinal network configuration. It has also shown thatthe three planar models considered develop networkswhose scaling characteristics differ from those ofreal river networks. In this section, we examine theeffects of including elevation in the three models.We consider a scenario in which the network growsby cutting into an initially high and undrained plateau.Erosion is assumed to reduce elevations until thelocal shear stresses reach a threshold value. Thesereductions lead to elevations that agree with the

Ž .slope–area law see below . Unlike the planar mod-els that operate only at the cluster frontier, theextended models have erosion events that can mod-ify elevations anywhere in the simulation domainŽ .either inside or outside the cluster , and a particular


point can be eroded multiple times. The erosionalgorithm is based on the work of Rinaldo et al.Ž .1993 , which provides both a criterion for determin-ing which points are vulnerable to erosion and a rulefor lowering the surface elevation at eroded points.The present model differs from the one by Rinaldo etal. because both the sequencing of the erosion eventsand the initial conditions are controlled to reflect theplanar growth models. In addition, we do not perturbthe resulting basins.

Our algorithm is as follows.Ž .i Specify the initial conditions. Initial elevations

Žare assigned to all points on the lattice hexagonal or.square . The elevations of outlet points are set to

zero, and the remaining elevations are high enoughthat the growing basins will eventually capture theentire domain. If the initial elevations were set toolow, the small slopes encountered at the clusterfrontier would result in shear stresses below thestability threshold. Stability at all frontier pointswould halt the network growth. In the simplest case,the initial surface is flat except for the outlets. In

Ž .some cases see below , white noise is superimposedto define flow directions, but the amplitude of thenoise is insufficient to induce erosion. The combinedcluster frontier is defined by an elevation contourjust below that of the initial flat surface. To extendthe invasion percolation model, an independent ran-dom value is also assigned to each point in thedomain. This value is unrelated to the surface eleva-tion and is used to determine the sequence of erosionevents.

Ž .ii Calculate flow directions and contributingareas. At all points, the flow direction is assumed tobe the direction of steepest descent. On the squarelattice, eight neighbors are used. If no neighbor islower than the point under consideration, the point islabeled a Apit.B Accordingly, all points on a perfectlyflat surface are considered pits. Once drainage direc-tions are assigned, contributing areas are calculatedby summing the number of grid cells upstream ofeach point and multiplying by the cell area. Con-tributing area summations are terminated at pits,where discharge is assumed to infiltrate or evaporate.

Ž .iii Identify unstable points. Erosion can occuronly at points where the shear stress t is above a

Ž .critical threshold t Rinaldo et al., 1993 . The shearcr

stress is evaluated as tskAuS, where S is the

gradient slope and the contributing area A is used asa surrogate for the geomorphically significant flowŽWolman and Miller, 1960; Wolman and Gerson,

.1978 . In all our simulations, we set us0.5, whichŽ .is a reasonable value for river basins Flint, 1974 .

The constants t and k control the vertical scale ofcr

the simulated topography.Ž .iv Select points for erosion. Not all of the

unstable points are necessarily eroded at each itera-Ž .tion of the algorithm. Rinaldo et al. 1993 selected

Žonly the most unstable point for erosion the one.with the largest excess stress tyt , whereas wecr

use three different criteria that mimic the planarmodels. To generalize the Scheidegger model, aheadward algorithm is used. This algorithm begins

Ž .at the outlets and pits and progresses headwardthrough the current network configuration. In thisfashion, all unstable points with equal distance to theoutlet are eroded simultaneously, and every unstablepoint in the domain is eroded once before the flowdirections and contributing areas are updated. Togeneralize the Eden model, a random sequencingalgorithm is used in which a single unstable point israndomly selected for erosion. Invasion percolationis generalized using a ranked algorithm. This algo-rithm first determines the highest assigned randomnumber among the unstable points of the frontier. Itthen erodes all unstable points in the domain havingassigned numbers that are greater than or equal tothat value.

Ž .v Erode the selected points. Erosion reduces theelevation of the selected points until they are stable

Ž .with tst Rinaldo et al., 1993 . Thus, whencr

selected, a point is assigned a new elevation zszdŽ . yuq D lt rk A , where z is the elevation of thecr d

point immediately downstream and D l is the dis-tance separating the two points. The new elevationresults in a slope that is consistent with the slope–arearelationship SAAyu .

Ž . ( )vi Iterate the process by going back to step ii .The algorithms that we have just described assigndrainage directions in a way that is different from theplanar models and therefore develop different rivernetworks. However, if the initial surface is highabove the outlets and perfectly smooth, the algo-rithms reproduce the cluster growth rules of theplanar models. The height requirement ensures thatall of the cluster’s neighbors can be added to the


cluster in the present iteration. If the surface eleva-tion is too low, neighboring points may have tFtcr

and thus be immune from erosion. The surface isrequired to be smooth so that the headward algo-rithm grows the cluster by 1-pixel layers. The head-ward algorithm adds all points that currently draininto the cluster, and a smooth surface implies thatonly the adjacent neighbors drain into the cluster.The other variants have modes of growth that areinsensitive to surface roughness.

To understand the role of these two restrictions,three cases are analyzed for the headward and ran-dom models. In Case 1, the initial surface is smoothand high enough to ensure that t)t at all frontiercr

points. In Case 2, the initial surface is also high, buta small amount of white noise is added to the initialelevations. In Case 3, the initial surface has the sameroughness as Case 2, but its elevation is low enoughthat stable points appear along the frontier through-out the growth process.

6. Results for the models with elevation

Ensembles of five simulated topographies havebeen generated for each case using the headward,random, and ranked models. For statistical analysis,a 500=500 pixel domain was used for the headwardmodel and 200=200 pixel domains were used forthe other two models. For visual comparison below,200=200 pixel domains are used in all cases. Inorder to mimic their planar counterparts, the head-ward model uses a hexagonal grid and the randomand ranked models use square grids with eight neigh-bors.

Fig. 12 shows individual simulations of the threeŽ .headward cases. In Case 1 Fig. 12a , the headward

model has a frontier identical to the ScheideggerŽ .model Fig. 1a , but the dynamics within the grow-

ing basins develop different-looking networks. Thenew basins widen more rapidly, and the larger chan-nels have some long straight segments. The wideningresults from a dominance of large basins that is notobserved for Scheidegger’s model. For example,consider a basin that, by chance, has widened by asmall amount to increase its total area above theareas of neighboring basins. In the headward model,such a basin has lower slopes along its major streams

and thus lower elevations at its stream sources.Lower source points have a greater potential tocapture new points by being selected as their down-stream neighbors. The straightness of the largerchannels is related to the widening process and therestriction of the drainage directions. The dominantchannels are straight and angled so as to capturelateral area at the maximum allowable rate. At a laterstage of evolution, these dominant channels captureadditional lateral area by developing small side tribu-taries.

Ž .In Case 2 Fig. 12b , new features are observed.During the growth, deviation from the uniformScheidegger frontier is introduced by the white noisein the initial surface elevations. The noise produces arandom pattern of internally draining networks onthe initial surface. As the cluster grows, these basinsare captured and eroded, thus irregularly advancingthe frontier. The final networks reflect the frontierform through an increased irregularity of the mainchannels.

Ž .In Case 3 Fig. 12c , a third mode of growth isobserved which is associated with a distinctive fron-tier. Because the initial surface has a low elevation,the shear stresses along the frontier are lower than inthe previous cases. If the elevations of the boundarypoints in the cluster are high, then the adjacentfrontier points may be stable. This stability wouldinhibit further growth at these points and alter thefrontier form. Since most of the elevation gains in anetwork occur along the channels with small areas,the length of such channels tends to be restricted.This causes the frontier to maintain a roughly con-stant distance from the larger channels, thus givingthe frontier a scalloped form.

Unlike the previous cases, the final networks inCase 3 tend to have basins that dominate far fromthe outlets. As the growing channels extend furtherfrom the open boundary, the elevations of the sourcesbecome higher. Because basins with smaller con-tributing areas have steeper trunk streams, theirgrowth is more restricted. At some distance from theoutlet, only a few basins are able to continue theirgrowth, and in the absence of competition, theycapture much lateral area.

The inclusion of elevation also affects the scalingcharacteristics. Fig. 3 shows the distribution of con-tributing areas for all three headward cases and the


Fig. 12. The growth of networks using the headward model under the three cases described in the text. The left panels show the frontier andnetwork configurations after 80 iterations, and the right panels show the final network configurations. Points outside the clusters are shadedgray, and points within the clusters are shaded according to their contributing areas.

planar Scheidegger model. The plots for the head-ward cases have significantly steeper slopes in log–log than the Scheidegger model. This includes Case

1, which has a mode of growth identical to Scheideg-ger’s model. Case 3 has bs0.47, which is slightlyhigher than the range of values observed in nature


Ž .Rigon et al., 1996 . Cases 1 and 2 show similardistributions to Case 3, but some significant devia-tions from power law scaling are observed in thecentral portion of the distribution where good scalingwould normally be seen for natural basins.

Hack’s law is shown in Fig. 6. The results parallelthose for the contributing area distribution. Again,some deviation from scaling is observed for Cases 1and 2. Case 3 clearly obeys a power law withhs0.55. This value is very similar to those seen in

Ž .nature Rigon et al., 1996 and is lower than the hvalue for Scheidegger.

Fig. 9 shows the dependence of the Euclideanbasin length on contributing area. For all three cases,the contributing area associated with a given Eu-clidean length is much larger than for Scheideggerbasins. This difference is due to the fact that thebasins have different means to capture area laterallyas described above. For Case 3, Hs0.86 is calcu-lated, which is much larger than H for Scheidegger’smodel. The new value is within the range observedfor natural basins but remains lower than the most

Ž .common values Rigon et al., 1996 . The sinuosityŽ .index h Hq1 s1.02 implies that the change in

Hack’s law from Scheidegger’s model is mainly dueŽto the difference in basin elongation rather than

. Ž .channel sinuosity . The sinuosity index h Hq1 isalso consistent with observations. Overall, basinsproduced by the headward model are more likenatural basins than any of the planar models previ-ously examined.

We now turn to the random model. Fig. 13 showsexamples of these basins along with snapshots takenduring their growth. The final configurations of allthree cases are visually similar. However, because

Žthese cases have clearly visible drainage divides like.the headward cases , their appearance is significantly

different from that of an Eden network at smallŽ .scales compare Figs. 1b and 13 . The frontiers

observed for Cases 1 and 2 are identical to Eden,although the drainage directions and the dynamicswithin the growing networks are different. The mostinteresting visual difference is the distinctive frontierin Case 3. This frontier is scalloped like the head-ward frontier in Case 3 and is less irregular than theprevious two random cases. This frontier form re-sults from the low elevations of the initial surface.

When any point is first added to the cluster byerosion, it has a small contributing area and thereforeis assigned a steep slope according to the slope–arearelationship. The slope remains high until the point israndomly selected for another erosion event, despiteany increases in contributing area as the clustercontinues to grow. This delay in the updating ofslopes causes the channels throughout the basin tohave steeper slopes than one would expect from theircurrent areas. When the headwater points have simi-lar elevations to the initial surface, t falls below t ,cr

limiting the ability of the cluster to capture newpoints until sufficient updating within the basin hasoccurred.

Unlike the dramatic changes observed betweenthe Scheidegger and headward models, the randommodel has scaling properties similar to those ofEden. Fig. 4 shows the distribution of contributingareas for the three cases. All three distributions aresimilar to that for Eden. As with the headwardmodel, deviation from scaling is observed for Cases1 and 2, but for Case 3, we observe bs0.42. Thisb value is larger than Eden’s and is very close to the

Ž .average of those reported in Rigon et al. 1996 . Fig.7 shows Hack’s law for basins simulated with therandom model. The exponent hs0.61 for Case 3 isidentical to that for the planar Eden model. Likewise,in Fig. 10, one finds Hs0.69, which is identical tothe H for Eden’s model. The addition of elevation tothis model did not produce exponents substantiallycloser to those of natural basins. Basins simulatedwith the random model continue to be too elongatedat large scales.

Finally, we turn to the ranked model. Fig. 14shows basins developed by the ranked model for thethree cases described above. Cases 1 and 2 both havecluster frontiers that are identical to invasion percola-

Ž .tion compare Figs. 1c and 14 , but their final net-works are substantially different. The drainage di-vides are more visible and the large channels haveless sinuosity. During early stages of growth, thenetworks develop highly sinuous trunk streams likethose of invasion percolation, but as more pointsbecome part of the basin, smaller tributaries begin toform. Such tributaries initially have small areas andtherefore steep slopes, but they sometimes providemuch shorter paths to the basin headwaters. This


Fig. 13. The growth of networks using the random model under the three cases described in the text. For Cases 1 and 2, the left panels showthe frontier and network configurations after 1.6 million iterations. For Case 3, the left panel shows the growth after 6.4 million iterations.The right panels show the final network configurations.

characteristic causes their source elevations to below enough to capture the headwater territory andthus reduces the sinuosity of the largest channels.

Less reorganization occurs in Case 3 because ofthe low elevation of the initial surface. The channelsthat extend early capture very little lateral territory


Fig. 14. The growth of networks using the ranked model under the three cases described in the text. The left panels show the frontier andnetwork configurations after 8000 iterations, and the right panels show the final network configurations.

and therefore have small areas along their lengths.Because this leads to steep slopes, the sources havetFt . These channels are thus unable to growcr

further until sufficient elaboration and readjustment

have occurred near the outlet to increase areas andreduce source elevations. This restriction leads to amore uniform mode of growth and less reorganiza-tion during the late stages of development. Although


not observed in the example shown in Fig. 14, thefinal network configuration can be quite similar tothose in Cases 1 and 2.

The scaling properties developed by the rankedmodel are different from those of invasion percola-tion networks. Fig. 5 shows the distribution of con-tributing areas for the ranked model, and all threecases adhere well to power laws at intermediateareas. For Case 3, bs0.44 is estimated, which islarger than the b value for invasion percolation andsimilar to the values observed in nature. Fig. 8 showshow the main stream length scales with contributingarea for the three cases. A value hs0.58 is esti-mated for Case 3, which is also in good agreementwith the values observed in nature. Given the visualdifferences described above, one suspects that thereduction in h is due to a reduction in channelsinuosity at large scales. Fig. 11 gives Hs0.97,indicating that the basins remain very close to self-similar. Thus, the reduction in h is due to a decrease

Ž .in sinuosity which is reflected in h Hq1 s1.14.This sinuosity measure is lower than for invasionpercolation but remains above the values observedfor natural basins.

It is interesting to compare the headward, random,and ranked models for Case 3. The models all en-force a slope–area relationship with exponent us0.5, but they do so using different mechanisms.Therefore, the models have not only different modes

Ž .of cluster growth as described above but also dif-ferent reorganization dynamics within the growingbasins. We find that the planar scaling properties ofthe drainage networks depend on these differences.For example, Hs0.86 for the headward model,Hs0.69 for the random model, and Hs0.97 forthe ranked model. It should be noted that the differ-ences in boundary conditions do not affect the expo-nents. We have simulated the headward model with asingle outlet as was done for the random and rankedmodels and verified that the scaling characteristicsdo not change significantly.

7. Conclusions

Ž .In the first part of the paper Sections 3 and 4 ,we analyzed the topological and geometrical scalingcharacteristics of Scheidegger, Eden, and invasionpercolation networks and compared them to natural

river networks. Even though the networks generatedby the three algorithms are visually distinct, the Edenand invasion percolation networks have similar b

and h values. Small differences are observed in theirtopological characteristics, but the most obvious dif-ferences are in their geometrical scaling properties.Scheidegger and Eden basins become more elon-gated at larger scales, whereas invasion percolationbasins are essentially self-similar. The channels ofScheidegger and Eden are nearly straight, whereasinvasion percolation channels have a very high sinu-osity.

The geometrical differences between the Schei-degger, Eden, and invasion percolation networks canbe understood in terms of their modes of networkgrowth. Uniform headward growth, as in Scheideg-ger’s model, develops self-affine basins and non-fractal channels. As one allows the main channels to

Žgrow more rapidly i.e., shifting from Howard’s.conceptual model to Glock’s , the basins become

increasingly self-similar with channels that are moresinuous. Hence, for these planar models, the mode ofnetwork growth has an influence on the scalingcharacteristics of the basin.

When compared with published statistics for natu-ral basins, our simulations show that all the planarmodels are inadequate for river networks. The low Hvalues of Scheidegger and Eden basins are not ob-served in nature, and the sinuosity of invasion perco-lation channels is unrealistically high.

Ž .In the second half of the paper Sections 5 and 6 ,we extended the planar models to include the effectsof surface elevation on the network properties. In-cluding elevation in the Scheidegger and invasionpercolation models significantly affects the horizon-tal scaling properties without necessarily changingthe mode of cluster growth. In the Scheideggerextension, the large basins have a greater ability tocapture lateral area because of how the drainagedirections are assigned and updated as the clustergrows. In the invasion percolation extension, signifi-cant network reorganization occurs after the initialdevelopment, which reduces the sinuosity of thelargest channels. Much less pronounced effects areobserved when extending Eden’s model.

The simulations with elevation also show that themode of growth depends on certain elevation charac-teristics of the initial surface. For the headward


model, the mode of growth depends on the rough-ness of the initial surface since the pre-existingdrainage divides are inherited by the growing clusterfrontier. For all three models with elevation, themode of growth is affected by the height of theinitial surface because frontier points must have suf-ficient shear stress to be eroded. The mode of growth

Žaffects the quality of the observed power laws espe-.cially for the headward model with the most exact

power laws occurring when the initial surface is lowenough to halt the erosion at some frontier points. Insuch cases, the scaling exponents for the headwardand ranked models are found to be different fromthose of the Scheidegger and invasion percolationmodels. The new exponent values are in better agree-ment with those of natural basins.

We conclude that the inclusion of elevation instandard planar models of river network formationmay in some cases produce important changes in thescaling properties of such networks, making themmore similar to natural river networks. The effects ofelevation vary with the planar growth rule and, forexample, are larger for Scheidegger than for Eden.The present results are based on simple assumptionson the sequencing of erosion events, which controlthe points added to the cluster, and the updating ofelevations within the cluster. In reality, channel net-works may grow in more complex and gradual waysand display a mixture of the features observed in oursimulations.

Acknowledgements

The support of U.S. Army Research Office grantsDAAH-04-95-1-0181 and DAAH-04-96-1-0099 isgratefully acknowledged.

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