International audienceUse of a deterministic fractal-multifractal (FM) geometric method to model width functions of natural river networks, as derived distributions of simple multifractal measures via fractal interpolating functions, is reported. It is first demonstrated that the FM procedure may be used to simulate natural width functions, preserving their most relevant features like their overall shape and texture and their observed power-law scaling on their power spectra. It is then shown, via two natural river networks (Racoon and Brushy creeks in the United States), that the FM approach may also be used to closely approximate existing width functions

Puente, C. E.

Sivakumar, B.

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A deterministic width function modelC. E. Puente, B. SivakumarTo cite this version:C. E. Puente, B. Sivakumar. A deterministic width function model. Nonlinear Processes inGeophysics, European Geosciences Union (EGU), 2003, 10 (6), pp.525-529. <hal-00302262>HAL Id: hal-00302262https://hal.archives-ouvertes.fr/hal-00302262Submitted on 1 Jan 2003HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Nonlinear Processes in Geophysics (2003) 10: 525–529Nonlinear Processesin Geophysics© European Geosciences Union 2003A deterministic width function modelC. E. Puente and B. SivakumarDepartment of Land, Air & Water Resources, University of California, Davis, USAReceived: 11 September 2002 – Revised: 13 March 2003 – Accepted: 15 May 2003Abstract.Use of a deterministic fractal-multifractal (FM) geometricmethod to model width functions of natural river networks,as derived distributions of simple multifractal measures viafractal interpolating functions, is reported. It is first demon-strated that the FM procedure may be used to simulate naturalwidth functions, preserving their most relevant features liketheir overall shape and texture and their observed power-lawscaling on their power spectra. It is then shown, via two nat-ural river networks (Racoon and Brushy creeks in the UnitedStates), that the FM approach may also be used to closelyapproximate existing width functions.1 IntroductionThe width function, w(r), defined such that w(r)dr is thearea of a basin whose flow distance to the outlet is betweenr and r + dr , is an important geomorphological qualifier ofa river network. As such a function captures the essentialfeatures of the basin’s response as implied by the instanta-neous unit hydrograph (Rodriguez-Iturbe and Rinaldo, 1997;Gupta and Waymire, 1998), its proper modeling has becomeincreasingly relevant.Recently, a variety of sophisticated stochastic models forthe width function have been introduced (Marani et al., 1994;Veneziano et al., 2000). While the work of Marani etal. (1994) relied on fractional Brownian motion and on ran-dom multiplicative cascades in order to capture, respectively,the power spectra and the multifractal characteristics of nat-ural width functions; Veneziano et al. (2000) introduced aniterated random pulse (IRP) model that, while accounting ex-plicitly for the self-similarity of natural river networks, prop-erly captures their non-negativity, non-stationarity, and thepower-law decay of their spectral density function.The purpose of this article is to illustrate by means of fewexamples that a deterministic fractal-multifractal (FM) rep-Correspondence to: C. E. Puente (cepuente@ucdavis.edu)resentation (Puente, 1992, 1994), successfully employed tomodel data sets corresponding to a host of geophysical pro-cesses, such as rainfall (Puente and Obrego´n, 1996; Obrego´net al., 2002a, b), turbulence (Puente and Obrego´n, 1999), andgroundwater contamination transport (Puente et al., 2001a,b), may also be used to represent the overall structure of thewidth function of natural catchments.2 The Fractal-Multifractal approachFigure 1 illustrates the construction of a derived distribu-tion via the FM approach (Puente, 1992, 1994). As is seen,such a set named dy is found transforming a (determinis-tic) binomial multifractal measure, dx, (bottom left), viaa (deterministic) fractal interpolating function f . The setdy is obtained considering all relevant events in x corre-sponding to a level y and adding their contributions, i.e.dy(B) = dx{x : f (x) ∈ B}, for all Borel subsets B on y.For a given set of N + 1 points in the plane{(x0, y0), (x1, y1), . . . , (xN , yN ); x0 < x1 < . . . < xN },measures dx and dy may be calculated iterating a suitableset of N contractile affine mappings (according to an appro-priate set of N weights pn that dictate the intermittencies ofdx) and counting over a desired number of bins (the resolu-tion of the records to be fitted) over x and y (Puente, 1994).Such mappings have the specific formwn(xy)=(an 0cn dn)(xy)+(enfn), (1)where an, cn, en, and fn are related to the interpolating pointsand the free scaling parameters dn, 0 ≤ |dn| < 1, via theconditionswn(x0y0)=(xn−1yn−1)(2)andwn(xNyN)=(xnyn)(3)526 C. E. Puente and B. Sivakumar: A deterministic width function modelxydyfdxFigure 1:Fig. 1. The fractal-multifractal framework in two dimensions. Amultifractal measure dx is transformed via a fractal interpolatingfunction f into a derived measure dy.for n = 1, 2, . . . , N , and yield a fractal interpolating func-tion whose graph’s dimension Df is: (a) 1, if ∑ |dn| ≤ 1,and (b) ≥ 1, from ∑ |dn|aDf−1n = 1, if ∑ |dn| > 1 (Barns-ley, 1988).A vast class of deterministic derived measures dy maybe obtained by varying the parameters of f and dx, i.e.(xn, yn), dn and pn, and by allowing projections to be foundat directions other than y, i.e. at angles θ different from 0 de-grees. At the end, the FM methodology results in a largeuniverse of “data sets”, that, depending on the fractal di-mension Df , include a wide variety of multifractal records,even having “chaotic” or “stochastic” properties, and Gaus-sian distributions, in the limit when Df tends to two (Puente,2004). The geometric approach may easily be tailored togenerate sets of arbitrary resolutions (e.g. Obrego´n et al.,2002b) which turn out to have the desirable feature of be-ing neither self-similar nor self-affine (as seen in Fig. 1), asfound in many practical applications.It so happens that the set dy in Fig. 1 (and many others)has a similar shape and texture as found in width functionsof natural river networks and such observation motivates us-ing the FM approach to represent natural width functions.In what follows, it is shown how the FM approach may in-deed be employed to simulate general width functions and tomodel specific width functions as observed in nature.Figure 2:Fig. 2. Examples of (normalized) width functions, c1 to c4 are gen-erated via the FM procedure, while c5 and c6 correspond to Racoonand Brushy creeks.3 Simulated and natural width functionsAs explained in Veneziano et al. (2000), natural widthfunctions possess three important characteristics: (a) non-negativity, (b) non-stationarity, and (c) power-law decay ofthe spectral density function, S(f ) ∼ f−β , with values ofβ typically ranging between 1.6 and 2.0. To illustrate suchfeatures, Fig. 2 includes six such plausible graphs (normal-ized so that their areas equal one unit), cases c1 to c6, madeof 1 721 data points for c1 to c5 and 1 719 points for c6, andFig. 3 shows their corresponding power spectra.As seen in Fig. 2, all such graphs are indeed non-negative,they possess similar high-frequency textures, and they arealso non-stationary as they begin and end at zero. As shownin Fig. 3, the power spectra of these sets have similar decayand their exponents β are bounded by the value of 2.0, asshown by the straight lines on all graphs.It happens that while the first four sets in Fig. 2 were ob-tained via the aforementioned deterministic FM methodol-ogy and are given by fractal interpolating functions whosegraphs have dimensions of, in order, 1.42, 1.48, 1.52 and1.62 (see Table 1), the last two (c5 and c6) correspond totwo natural catchments: Racoon creek of 384 km2 located inC. E. Puente and B. Sivakumar: A deterministic width function model 527Figure 3:Fig. 3. Power spectra of the width functions shown in Fig. 2.Table 1. FM parameters for simulated width functions c1 to c4 asshown in Fig. 2. Values not presented below are set as follows:x0 = 0, y0 = 0, and x2 = 1.“Basin” Coordinates Angle θx1 x3 y1 y2 y3c1 0.700 1.800 0.800 0.0c2 0.700 1.500 1.800 0.800 0.000 8.0c3 0.600 2.000 1.000 0.0c4 0.600 2.000 1.300 12.0“Basin ” Scalings Intermittenciesd1 d2 d3 p1 p2 p3c1 0.600 -0.800 0.500 0.500c2 0.540 -0.730 -0.500 0.440 0.330 0.230c3 -0.800 -0.630 0.510 0.490c4 -0.890 -0.630 0.470 0.530Table 2. FM parameters for approximated width functions ofRacoon and Brushy creeks as depicted in Fig. 4. Values not pre-sented below are set as follows: x0 = 0, y0 = 0, and x2 = 1Basin Coordinates Angle θx1 y1 y2Racoon 0.607 2.015 1.095 -0.67Brushy 0.550 2.494 2.077 9.90Basin Scalings Intermittenciesd1 d2 p1 p2Racoon -0.881 -0.615 0.530 0.470Brushy -0.917 -0.653 0.492 0.508Pennsylvania and Brushy creek of 322 km2 situated in Al-abama, both extending for about 50 km from their respectiveoutlets (Veneziano et al., 2000). These results show that theFM approach provides, by varying its key parameters, greatmany outcomes that may be considered suitable represen-tations (simulations) of natural width functions, simulationswhich, by varying smoothly with respect to their parameters,provide an opportunity to represent natural width functions,at any given resolution, in their entirety.In what follows, it shall be shown how the FM approachmay be used to find suitable approximations of natural widthfunctions, considering the aforementioned creeks.4 FM approximations to Racoon and Brushy creeksThis section presents the results obtained while solving theinherent inverse problem to approximate the width functionsfor the Racoon and Brushy creeks.Based on previous experience with other data sets(Obrego´n et al., 2002a, b), an objective function account-ing for the cumulative distributions of the records and theirderivatives was minimized, performing the search via a ge-netic algorithm (Duan et al., 1992). Specifically, an objec-tive function 1 was used adding: (a) the sum of square dif-ferences between cumulative distributions of real (normal-ized) width functions and FM sets (∑12); and (b) the sumof square differences between cumulative distributions ofnormalized absolute one-lag derivatives of real (normalized)width functions and FM sets (∑12d ).While Table 2 includes the obtained parameters from theoptimization exercise for both catchments, Fig. 4 summa-rizes pictorially their results, as it contains: (a) the normal-ized width functions (so that their areas are equal to one unit);(b) their cumulative distributions; and (c) their power spectra.528 C. E. Puente and B. Sivakumar: A deterministic width function modelFigure 4:Fig. 4. Actual and FM fitted widthfunctions, cumulative distributions, andpower spectra for Racoon and Brushycreeks. For the actual sets, the widthfunction and the power spectra areshown in white, whereas the cumulativedistribution is highlighted in gray. Thehorizontal and vertical scales for thethree graphs corresponding to Racooncreek are: [1, 1721], [0, 0.0014]; [0, 1],[0, 1]; and [0.01, 1.0], [0.001, 10.0] asin c5 in Fig. 3. Those scales for Brushycreek are: [1, 1719], [0, 0.0019]; [0, 1],[0, 1]; and [0.01, 1.0], [0.001, 10.0], asin case c6 in Fig. 3.As may be inferred, the FM representations for the Racoonand Brushy creek catchments are based on binomial multi-fractal measures and on fractal interpolating functions pass-ing by three points (leading to seven free parameters, seeTable 2, for x0, y0 and x2 were kept fixed), whose graphshave dimensions of 1.60 and 1.66, respectively. These resultsstem from very faithful descriptions of the records’ cumula-tive distributions and of their derivatives (not shown), as im-plied by very low values of the attributes∑12,∑12d , andhence 1 (all less than 1), that started, as the search began, atvalues greater than 20. The closeness of such fits may also beappreciated in the close agreement in the quantities L1 andL2 (real values in parenthesis), that give the lengths of cu-mulative distributions of the records and of their derivatives,when calculated from (0, 0) to (1, 1), such that the records’domain is set to one unit.As may be seen in Fig. 4, the obtained FM width func-tions capture the overall trends present in the actual func-tions, including good approximations of the peaks, and pos-sess similar textures that translate into good agreement of thepower-law decay in the power spectra for both catchments.Notice that this happens despite small deviations from thescaling lines at high frequencies, as the FM approximationsshown, that do not optimize the desired scaling law, turnedout to produce smoother width functions compared to the ac-tual sets.As typical width functions share similar features to theones included here (see for example Rodriguez-Iturbe andRinaldo, 1997), the results in Fig. 4 further illustrate that theFM approach may be used to model them, using fractal in-terpolating functions whose graphs have fractal dimensionsranging between 1.4 and 1.7. As specific fits of other widthfunctions may certainly be obtained in a similar fashion, itis envisioned that a “physical” meaning may be assignedto their FM geometric parameters (or groups of them) interms of actual characteristics of the underlying watersheds,as it has been found in other applications (e.g. Puente et al.,2001a, b). Such an exercise may ultimately lead to a regionalparameterization of a host of width functions.5 ConclusionsThis article has shown that the FM methodology may beemployed to obtain suitable approximations of width func-tions of natural catchments as derived distributions of sim-ple multifractal measures via fractal interpolating functions,in a rather parsimonious way. As such, it has been shownthat such a deterministic procedure may be used to not onlysimulate sets that properly preserve all essential features, ascaptured by stochastic models (Veneziano et al., 2000), butalso fit specific width function sets, which random models,due to their specific random structure, cannot easily do.The FM approach, while providing parsimonious descrip-tions of whole width functions leading to vast compressionratios (1721 to 7 or about 246 to 1 in the examples consid-ered), represents a new approach for understanding the com-plexities of natural width functions such that their inherentgeometries are fully captured and at any given resolution.Due to its scope, the FM method cannot be safely comparedwith other existing (stochastic) procedures that by definitionare able to simulate plausible realizations of width functions(even if with fewer parameters than a wholistic FM repre-sentation may have). In a sense, the FM methodology, byrepresenting specific width functions, just parameterizes re-alizations as given by stochastic representations.Given that space-filling Peano river basins yield simplemultifractal width functions that correspond to the parentmeasure of the FM procedure (Marani et al., 1991), it is envi-sioned that further analysis of other width functions using theFM approach may provide, via the parameters of the inherentfractal interpolating functions, a plausible physical interpre-C. E. Puente and B. Sivakumar: A deterministic width function model 529tation of the geometric parameters and a subsequent regionalclassification of river networks.Acknowledgements. The work presented in this article was sup-ported in part by the USEPA via Grant GAD # R82 4780 and byNASA under Grant NAG5–7441.The data sets used in this workwere kindly provided to us by Glenn E. Moglen. Comments byanonymous reviewers are gratefully acknowledged.ReferencesBarnsley, M. F.: Fractals Everywhere, Academic, San Diego, Calif.,1988.Duan, Q., Sorooshian, S., and Gupta, V.: Effective and efficientglobal optimization for conceptual rainfall-runoff models, WaterResour. Res., 28(4), 1015–1031, 1992.Gupta, V. K. and Waymire, E. C.: Spatial variability and scale in-variance in hydrologic regionalization, in Scale Dependence andInvariance in Hydrology, edited by G. Sposito, Cambridge Univ.Press, New York, 88–135, 1998.Marani, A., Rigon, R., and Rinaldo, A.: A note on fractal channelnetworks, Water Resour. Res., 27, 3041–3049, 1991.Marani, M., Rinaldo, A., Rigon, R., and Rodrı´guez-Iturbe, I.: Ge-omorphological width functions and the random cascade, Geo-phys. Res. Lett., 21(19), 2123–2126, 1994.Obrego´n, N., Sivakumar, B., and Puente, C. E.: A deterministic ge-ometric representation of temporal rainfall: Sensitivity analysisfor a storm in Boston, J. Hydrol., 269(3-4), 224–235, 2002a.Obrego´n, N., Puente, C. E., and Sivakumar, B.: Modeling high reso-lution rain rates via a deterministic fractal-multifractal approach,Fractals, 10(3), 387–394, 2002b.Puente, C. E.: Multinomial multifractals, fractal interpolators, andthe Gaussian distribution, Phys. Lett. A, 161, 441–447, 1992.Puente, C. E.: Deterministic fractal geometry and probability, Int.J. Bifurcations Chaos, 4(6), 1613–1629, 1994.Puente, C. E.: A universe of projections: May Plato be right?,Chaos, Solitons and Fractals, 19/2, 241–253, 2004.Puente, C. E. and Obrego´n, N.: A deterministic geometric represen-tation of temporal rainfall: Results for a storm in Boston, WaterResour. Res., 32(9), 2825–2839, 1996.Puente, C. E. and Obrego´n, N.: A geometric platonic approach tomultifractality and turbulence, Fractals, 7(4), 403–420, 1999.Puente, C. E., Robayo, O., Dı´az, M. C., and Sivakumar, B.: Afractal-multifractal approach to groundwater contamination, 1.Modeling conservative tracers at the Borden site, Stoch. Env.Res. Risk Assess., 15(5), 357–371, 2001a.Puente, C. E., Robayo, O., and Sivakumar, B.: A fractal-multifractalapproach to groundwater contamination, 2. Predicting conserva-tive tracers at the Borden site, Stoch. Env. Res. Risk Assess.,15(5), 372–383, 2001b.Rodrı´guez-Iturbe, I. and Rinaldo, A.: Fractal River Basins: Chanceand Self-organization, Cambridge Univ. Press, New York, 1997.Veneziano, D., Moglen, G. E., Furcolo, P., and Iacobellis, V.:Stochastic model of the width function, Water Resour. Res.,36(4), 1143–1157, 2000.

A deterministic width function model

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A deterministic width function model

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