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WATER RESOURCES RESEARCH, VOL. 30, NO. 12, PAGES 3531-3539, DECEMBER 1994
Self-organized river basin landscapes:Fractal and multifractal characteristicsIgnacio Rodriguez-Iturbe,1 Marco Marani,2 Riccardo Rigon,3and Andrea Rinaldo2
Abstract. In recent years a new theory of the evolution of drainage networks andassociated landscapes has emerged, mainly in connection with the development offractal geometry and of self-organized criticality (SOC) concepts. This theory has muchimproved our understanding of the mechanisms which determine the structure ofnatural landscapes and their dynamics of evolution. In the first part of this paper themain ideas in the theory of landscape self-organization are outlined, and someremarkable features of the resulting structures are presented. In the second part weapply theoretical tools developed in the context of multifractal fields to the study of thescaling properties of the field of elevations of SOC landscapes. We observe that suchlandscapes appear to be more complex than simple self-affine fractals, although in somecases a simple fractal framework may be adequate for their description. We also showthat multiple scaling may emerge as a result of heterogeneity in the field propertiesreflecting climate and geology.
1. Introduct ion
The global frameworks of fractal analysis [Mandelbrot,1983] of fluvial structures [La Barbera and Rosso, 1987,1989; Tarboton et al., 1988; Rodriguez-Iturbe et al., 1992a,b, c; Rinaldo et al.f 1992; Turcotte, 1992; Chase, 1992;Die tier and Zhang, 1992; Ouchi and Matsushita, 1992;Lavallee et al, 1993; Rinaldo et al., 1993; Ijjasz-Vasquez etal., 1993; Rigon et al., 1993], and of self-organized criticality[Bak et al., 1987, 1988; Bak and Creutz, 1993] have been thefoundations of a new theory about the evolution of drainagenetworks and their associated landscapes. It has been shown[Rodriguez-Iturbe et al., 1992a; Rinaldo et al., 1992] thatoptimal channel networks (OCNs) obtained by minimizingthe local and global rates of energy expenditure evolveautomatically from arbitrary initial conditions to networkconfigurations exhibiting fractal and multifractal characteristics indistinguishable from those found in nature. Althoughdenditric structures obtained through optimality conditionshad been introduced before [Howard, 1971, 1990], we believe this was the first time fractality was observed in optimalstructures. Sun et al. [1994a] have studied the original modelof Rodriguez-Iturbe et al. [1992a] and Rinaldo et al. [1992],corroborating the fractal aspects of the planar structure ofnetworks developed under this framework.
Chance and the rules of optimum energy expenditure are
'Department of Civil Engineering, Texas A&M University, College Station.
2Istituto di Idraulica "G. Poleni," Universita di Padova, Padua,Italy.
3Dipartimento di Ingegneria Civile e Ambientale, Universita diTrento, Trent, Italy.Copyright 1994 by the American Geophysical Union.Paper number 94WR01493.0043-1397/94/94WR-01493505.00
the elements which control the evolution and final structureof the drainage network. Although the particular outcome ofan individual process is given by randomness, the interplayof such elements in the extended and dissipative systemwhich is the drainage network results in the same fundamental principles of evolution. The fractal characters of thesystems obtained from these principles are found not to bedependent on the initial conditions, thus suggesting thatnatural fractal structures like river networks may arise as ajoint consequence of optimality and randomness [Rodriguez-Iturbe et al., 1992c].
An important step forward was taken by Rinaldo et al.[1993], who observed that OCNs are a particular case ofself-organized critical structures. The theory of self-organized criUcality [Bak et al, 1987, 1988] provides in facta broad framework for the dynamics of fractal growth and anappealing scheme for the evolution of drainage networks[Rinaldo et al., 1993] which, unlike the theory of OCNs,does not explicitly state a global optimality principle as theguiding evolution concept. Again, the characteristics of thestate toward which the system evolves are independent ofthe initial conditions and are very similar to those observedin nature. In the mentioned scheme [Rinaldo et al., 1993;Rigon et al., 1994], landscapes evolve through the interplayof fluvial (erosive) and hiilslope (diffusive) processes. Theerosion process acts on the system, maintaining it in acritical state (i.e., a state in which the shear stress due towater flow cannot exceed a critical value), while a specialkind of diffusive process is considered as preserving thefractal nature of the elevation field. We believe that this useof the theory of self-organized criticality provides a clear linkbetween the fractal landforms observed in nature and thedynamics responsible for their growth.
The quantitative characterization of self-organized critical(SOC) landscapes and their comparison with natural terrainproperties (including power laws in the probability distribution of cumulated areas [Rodriguez-Iturbe et al., 1992b] andof stream lengths [Tarboton et al., 1988; Rodriguez-Iturbe et
3531
3532 RODRIGUEZ-ITURBE ET AL.: SELF-ORGANIZED RIVER BASIN LANDSCAPES
al., 1992b], multifractal spectra of new growth site measures, i.e., width functions [Rinaldo et al., 1992, 1993], andscaling relationship of slopes with cumulated area [Tarbotonet al., 1989; Rigon et al., 1993]) was the subject of previouspapers [Rinaldo et al., 1993; Rigon et al., 1994]. This paperdescribes a systematic analysis of the fractal properties ofthe resulting topographies and a comparison with measuresof the roughness of natural terrains.
2 . A Conceptua l Framework forLandscape Self-Organization
We will now briefly recall a general scheme for theevolution of drainage networks in which the structure of thefield of elevations organizes itself, evolving through a sequence of critical states [Rinaldo et al., 1993] under theaction of fluvial and hillslope processes.
Fluvial erosion may be dynamically described by anequation like dz/dt = af(r - rc), with/(:c) = 0 for x < 0(where t is the local shear stress, tc is the critical thresholdfor erosion, and a provides the timescale for the erosionprocess). The shear stress t is proportional to yVz, where yis the flow depth and Vz is the local slope. From localoptimality principles and empirical evidence we have that y rc erosion exists. The localoptimality principle also leads to the scaling of slopes withdischarge as Vz,- = Qi~0'5, or, using drainage area as asurrogate of discharge, as Vz,- ~ A/~ .
Rinaldo et al. [1993] studied the dynamic evolution resulting from local exceedance of an erosion threshold. Briefly,the evolution rules are as follows. Each site / in a two-dimensional lattice has two associated variables, the elevation z; and the total flow Q{ surrogated by the draining areaAi. An arbitrary initial planimetric network is obtained bychoosing the elevations of the landscape randomly or in asystematic way. For each site the shear stress is thencomputed as r,- « A°-5Az,-, Az being the drop along thedrainage direction fixed by the steepest descent (since latticelengths are equal to one, Az is equal to the slope). Thecomputed t,- are compared with a given threshold value tc,and the exceedances are identified throughout the lattice.The maximum exceedance is placed, say, at they'th site, andthe elevation of this site is reduced to the value which yieldst = tc. Drainage directions are then recomputed because ofthe modified elevation at the site./. The process is repeateduntil the system evolves to a critical state in which there areno exceedances of t above tc. This critical state is thenperturbed at random by adding elevation to a node. Theperturbation may lead to a readjustment of the structure, andthis is repeated until further perturbations do not inducevariations in the configuration of the system.
It is important to note that the total energy dissipation ofthe system decreases during the evolution and stabilizes atvalues very close to those obtained in OCNs, whose dynamics are controlled by the explicit minimization of the totalrate of energy expenditure. An explanation of the equivalence of OCNs and self-organized critical networks (SOCs)relies on the equivalence of minimizing the total rate ofenergy expenditure and the total potential energy of thesystem [Ijjasz-Vasquez et al., 1993]. In fact, the evolution ofself-organized networks leads the system to the state ofminimum possible potential energy compatible with the
restriction that slopes scale with area as Vz,- = A/ . Therole of randomness is crucial as the perturbations are thedriving mechanism that maintains the evolution of the system.
As described above, the evolutionary strategy from anyinitial condition used by Rinaldo et al. [1993] is one in whichthe maximum exceedance of the threshold shear stress isidentified and the elevation of the site is reduced so that thelocal shear is at the critical value. Two different evolutionarystrategies have also been implemented. The first one is arandom strategy in which a site is chosen at random amongall those t exceeding rc and the elevation is then reduced tomake t = tc. The dynamics then continues as in the originalstrategy described before. The second strategy is a parallelprocessing one in which all sites with r > tc have theirelevations simultaneously updated to make r = rc. Although conceptually different, the three evolutionary strategies produce self-organized networks with identical fractaland multifractal characteristics. Furthermore, in all casesthe total energy expenditure is very close to that obtained forOCNs in similar domains [Rinaldo et al., 1993].
Threshold dependent (TD) processes as in the case ofcritical shear stress are most important in the carving andevolution of river channels. Nevertheless, often thresholdindependent (TI) processes act on the formation of thelandscape of a river basin, especially at the hillslope scale.The lowering of elevations throughout the basin may in thiscase be described by a field equation like
dz— =a/(r-Tc) + /3F(Vz).at
0)
The first term of the right-hand side describes TD fluvialerosion which is zero when t < tc. The second term modelsthe TI processes which are essentially related to the gradientof the elevation field, Vz. The TI processes act everywhere,but since a » B [Rigon et al., 1994] the first term plays thecentral role whenever erosion forces are acting.
Rigon et al. [1994] present a scheme for landscape self-organization which accounts for the presence of TI processes and is an extension of the one described in theprevious section. After the readjustments following a number of perturbations leading to a SOC landscape a diffusioncomponent (TI) is introduced in the scheme. The site i withthe largest drop to one of its neighbors/, Azy, is identifiedand its elevation z, decreased by a quantity proportional toAz,y. The mass removed from z, is redistributed among theneighbors j with elevation zt lower than z,-. Mass redistribution is done according to
Zj ~ Zj
kSnn
0 < v < \ , (2)
where k G nn indicates that the sum is over the nearestneighbors lower than /. The points draining into i changetheir Vz while the points./ change both z and Vz, and thus awhole set of adjustments occurs following a TI activity atany point. The above procedure produces a lowering of theslopes which results in smoother hillslopes and gentlerlandscapes when the iterations continue in time. Indeed,without the TI processes the hillslopes of the resultinglandscapes look unrealistically rough or at least highly
RODRIGUEZ-ITURBE ET AL.: SELF-ORGANIZED RIVER BASIN LANDSCAPES 3533
Plate 1. Self-organized landscape resulting from both TD and TI processes [after Rigon et al., 1994].
uncommonly so. One of the relevant effects of TI processesis the lowering of the fractal dimension of the landscape[Rigon et al., 1994]. An example of the effect of repeatedcycles of TD and TI processes is shown in Plate 1. Both vand the length of the iterative process are important, andthrough their effects a large variety of landscapes can beobtained.
3. Fractal and Multifractal Descriptorsof Landscapes
Let us consider the asymptotic field of elevations z(x, t =«>) = z(x) produced by the above processes, defined at everypoint x in a subset of U2. If z(x) has the property
dz(x, + Ae) - z(x,) = Aw[z(x2 + e) - z(x2)], (3)
where e is a unit vector, A is a separation distance, X|, x2 aretwo arbitrary points, and the d above the equals signindicates equality in distribution (the exponent H is calledthe Hurst exponent [Mandelbrot, 1983; Feder, 1988]), it issaid to be simple scaling [Lavallee et al., 1993]. The unique
fractal dimension D characterizing the simple scaling field isrelated to Hurst exponent H via the relationship D = 3 - H[Feder, 1988]. It is important to note that a simple scalingfield is postulated to have constant statistical characteristicswith respect to position and orientation. Equation (3) holds,in fact, for any choice of x12 and e.
Many examples of fractal analyses of topography andbathymetry measured along a linear track can be found in theliterature [e.g., Malinverno, 1989; Turcotte, 1992] documenting the fractal nature of natural landscapes. Most of thesestudies are based on monofractal (or, simple scaling) modelsthat do not seem entirely consistent with the properties ofmeasured field data. Recent contributions [Schertzer andLovejoy, 1989; Lavallee et al., 1993] give a different interpretation of the fractal characters observed in real topographies, arguing that geographical fields are generally multi-fractal and that inconsistencies are inevitable when theseentities are forced into narrower geometric frameworksinvolving a single fractal dimension. In view of these considerations the characteristics of the SOC landscapes introduced in the previous sections will be analyzed with the
3534 RODRIGUEZ-ITURBE ET AL.: SELF-ORGANIZED RIVER BASIN LANDSCAPES
* ■ ■ ' ' ' " " ' ' ' ' ' " ' ^ ^ ^ ^
120 ■ j y & $ - v - - ; N
100 • m80 H p .60 _ . , . " , •
40 g*sn*?> . .
2 0
ScIp -0
20 4 0 60 80 100 120
Figure 1. Plot of the set of points 5 that belong to predetermined ranges of elevations. Isolines are identified atdifferent heights for an elevation field resulting from TDprocesses only. The solid area refers to the set 5s2, i.e., theset of points whose elevation is lower than 2 (z(x) ^ 2).Different levels of shading indicate sets S where 2 < z(x) <3 and so on. /
purpose of linking the dynamics of fractal growth throughSOC processes and the resulting fractal or multifractalstructures.
There are many ways to characterize rough surfaces. Oneway to study the fractal properties of a field is to define,through the introduction of a threshold value of elevation T,sets that have theoretically determined characteristics withrespect to fractal geometry. One such set is the exceedanceset S>.7, introduced as the set of all the points x satisfyingz(x) > T. Likewise, one can define the perimeter set PT asthe subset of the points of 5^T having at least one nearestneighbor belonging to the set ST). In the case of asimple scaling structure the sets S>r are characterized by asingle fractal dimension D(5>r) = D while D(PT) = 2 - H,both values being independent of T [Lavallee et al., 1993].These properties provide a means of checking whethersimple scaling arises for z(x).
Figure 1 shows the sets of points which belong to differentranges in elevation, from which we have obtained the abovesets 5 defining the elevation field z(x, ta) for a SOClandscape obtained when only TD actions are operating, i.e.,whose field equation is
dz— = af(r- tc).a t
(4)
The initial condition z(x, 0) was obtained by Eden [1961]growth [Rinaldo et al., 1993]. Here rc = 1 on a 128 x 128lattice. Elsewhere [Rigon et al., 1994], we have shown thatthe statistical features of the landscape are independent ofthe threshold value tc. Figure 2 shows contour lines of the
120
100
Figure 2. As in Figure 1, for the elevation field in Plate 1,where both TD and TI processes were allowed to occur.
landscape z(x, tB = °°) shown in Plate 1, obtained by a cycleof TI processes defined by the field equation
dz— =/3F(Vz),at
(5)
acting from the initial condition z(x, ta). The resulting fieldis described in Figure 1.
Box counting has been applied to the graphs in Figures 1and 2 to determine the fractal dimension of each contourline. For each contour, box-counting [Mandelbrot, 1983;Feder, 1988] data are represented reasonably well (Figure 3)by a straight line whose slope is the fractal dimension D(PT)in a log-log plot of the number of boxes N(S) that cover thewhole contour versus the box size 5. This indicates that anycontour of a self-organized landscape may be considered aself-similar fractal. The main results of the analyses areillustrated in Table 1. We note that the fractal dimensions ofthe isolevel lines obtained for different contour elevations
o
6.5
6
5 i
5
4.5 DX
4
3.5
3
1 \
- 4 . 5 - 3 . 5 - 3
log 6-2.5
Figure 3. Typical dependence of A/(5) on 5 found in thebox-counting procedure applied to one of the isolines ofFigure 1 (TD processes only).
Table 1. Elevations, Fractal Dimensions (Box Counting)of the Border Sets, and Correlation Coefficients of Box-Counting Scaling for TD and TI SOC LandscapesContour TD Actions TI Actions
z(x) = T D(PT)TD r D(PT)Tl
1 1.38 0.99 1 . 3 0 0 . 9 91.5 1.42 0.96 1 . 3 0 0 . 9 82 1.43 0.98 1 . 3 0 0 . 9 82.5 1.43 0.94 1 . 2 9 0 . 9 73 1.39 0.96 1 . 2 6 0 . 9 73.5 1.37 0.99 1 . 2 2 0 . 9 64 1.35 0.92 1 . 2 1 0 . 9 4
are slightly different. Similar results were observed for theoptimal topographies obtained by Sun et al. [1994b] byenforcing an artificial perfect scaling of slopes to an optimalchannel network [Rodriguez-Iturbe et al., 1992b, c; Rinaldoet al., 1992]. The contours near the average height have verysimilar fractal dimensions, and, overall, the surfaces produced seem to show the signs of a relatively complexself-affine fractal. However, in the above analysis, samplesize effects are important both at the higher and at the lowerelevations, thus partially impairing the related results. In therange of elevations where the sample size is constant weobserve a consistent behavior analogous to the topographiesobtained by Sun et al. [1994b]. Moreover, we will show laterthat other analyses suggest that multiple scaling does notoccur in the topographic field obtained via SOC procedureswith space-constant field properties. Fast Fourier transformanalyses [e.g., Cochran et al., 1967; Feder, 1988; Malin-verno, 1989], roughening measurements [e.g., Mandelbrot,1983; Feder, 1988; Matsushita and Ouchi, 1989; Dubuc etal., 1989a, b] and height-height correlation measurements[e.g., Mandelbrot, 1983; Feder, 1988; Sun et al., 1994a, b]also agree with the results shown by the contours near theaverage elevation.
Figure 4 shows the log-log plot of the power spectrum(S(k)) (ensemble-)averaged over all south-north (S-N) andeast-west (E-W) topographic profiles obtained from verticalcuts (or transects) through a SOC landscape produced by TDactions (equation (4)). The data can be fitted by a straight line
V0.1
0.001
0.01
Figure 4. Power spectrum averaged over all S-N and E-Wtransects of the TD 128 x 128 SOC landscape considered.The slope in the log-log plot is -B = -1.6, yielding D - 1.7for the (mean) fractal dimension of the transects.
2 y
> Variation Q\ CO.s i / 2-D=0.5
o-1 y / 11.4
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2
Log box size
Figure 5. Results of the variation method analysis appliedto a transect of the TD 128 x 128 SOC landscape considered.Averaging over all transects gives D = 1.65 for the meanfractal dimension [after Rigon et al., 1994].
(i.e., S(k) « k~p) with slope -B(B = 1.6) giving, as shownlater, D = (5 - B)I2 = 1.7 ± 0.1 [Feder, 1988].
Figure 5 shows the results of the variation method analysis[Dubuc et al., 1989a, b] on a single vertical cut of the surfaceof a TD 128 x 128 SOC landscape with rc = 1. The rangevariation of the height z within a box of size L scales asproportional to LH (D = 2 - H) if the object is simplescaling. Once ensemble averaging is performed, the resultingfractal dimension is D = 1.65 ± 0.05.
The analysis of mean deviation, first introduced for self-affine fractals by Matsushita and Ouchi [1989], studies thedeviation a2(L) as a function of the horizontal length scale Las
°-2(LH^2 2 (^-(zhr (6)i j = \
where tq is the elevation at the lattice sites x = (/, j) and{z)l is the average height for the surface within a grid of sizeLxL For a self-affine fractal, a is related to L by a powerlaw a oc LH. Figure 6 shows the log dependence of o- on Lfrom which we derive D - 1.66 ± 0.05. We have repeatedall the above analyses for the TD + TI processes describedin (1) leading to the landscape in Plate 1 and have obtained
1 •-0.25
S-
-0.3 . S .I-0"35COa_J -0.4
• S >.-< î2-D=0.34i
-0.45
. ^ "1.1 1.2 1.3 1.4
LogN
1.5 1.6 1.7
Figure 6. Dependence of mean deviation o\L) on the horizontal length scale L obtained applying the Matsushita andOuchi [1989] analysis to the TD 128 x 128 SOC landscapeconsidered. This procedure gives the value D = 1.66 for themean fractal dimension [after Rigon et al., 1994].
3536 RODRIGUEZ-ITURBE ET AL.: SELF-ORGANIZED RIVER BASIN LANDSCAPES
1.7 dzat=- «/(r - rc) r^^.ryT^^
1.6 , V//vwAq 1.5
1.4
AAA V % = «/(r - re) + 0F(Q, Vz)
1.3
0 1 0 2 0 3 0 4 0 5 0 6 0E-W transect number
Figure 7. Values of the fractal dimension for different E-Wtransects of TD and TD + TI topographies.
consistent results for the mean fractal dimensions. Nevertheless, we observed a nonnegligible range in the individualfractal dimensions of the vertical cuts of the topography andin the contour lines at different elevations. Figure 7 shows aplot of the fractal dimensions for different E-W one-dimensional tracks of the topographies of TD and TD + TItopographies. One notices that the self-affine characteristicsof the evolving surface rendered by the TI type of self-organization are less rough. The above discussion suggestedthat the complexity of the self-affine fractal surface neededfurther analysis.
There are other ways to determine the scaling nature of afield, namely tho£e related to the definition of generalizedvariograms [Lavallee et al., 1993], which are essentially ageneralization of height-height correlation analyses. The qthmoment of a range, or generalized variogram, Ce/(X), isdefined as
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RODRIGUEZ-ITURBE ET AL.: SELF-ORGANIZED RIVER BASIN LANDSCAPES3539
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R. Rigon, Dipartimento di Ingegneria Civile e Ambientale, Universita di Trento, Mesiano di Povo, 1-38050 Trento, Italy.
I. Rodriguez-Iturbe, Department of Civil Engineering, TexasA&M University, College Station, TX 77843.
(Received November 22, 1993; revised April 25, 1994;accepted June 8, 1994.)
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