Feasible optimality implies Hack's lapdodds/files/papers/others/1998/rigon1998a.pdfFeasible...
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WATER RESOURCES RESEARCH, VOL. 34, NO. 11, PAGES 3181-3189, NOVEMBER 1998 Feasible optimality implies Hack's law Riccardo Rigon Dipartimcnto di Ingcgneria Civile c Ambientale, Universita di Trento, Trento, Italy Ignacio Rodriguez-Iturbe Department of Civil Engineering, Texas A&M University, College Station Andrea Rinaldo1 Centra Intcrnazionale di Idrologia "Dino Tonini," Universita di Padova. Padova, Italy Abstract. We analyze the elongation (the scaling properties of drainage area with mainstream length) in optimal channel networks (OCNs) obtained through different algorithms searching for the minimum of a functional computing the total energy dissipation of the drainage system. The algorithms have different capabilities to overcome the imprinting of initial and boundary conditions, and thus they have different chances of attaining the global optimum. We find that suboptimal shapes, i.e., dynamically accessible states characterized by locally stationary total potential energy, show the robust type of elongation that is consistently observed in nature. This suggestive and directly measurable property is not found in the so-called ground state, i.e., the global minimum, whose features, including elongation, are known exactly. The global minimum is shown to be too regular and symmetric to be dynamically accessible in nature, owing to features and constraints of erosional processes. Thus Hack's law is seen as a signature of feasible optimality thus yielding further support to the suggestion that optimality of the system as a whole explains the dynamic origin of fractal forms in nature. 1. Introduction Spanning, loopless network configurations characterized by minimum total energy dissipation (optimal channel networks (OCNs)) show many interesting properties including robust scale-invariant properties and a striking resemblance to fluvial landforms (see, for a review, Rodriguez-Iturbe and Rinaldo [1997]). The search for OCNs entails a complex optimization problem related to the evolution of fluvial networks [Ro driguez-Iturbe et al., 1992a, b; Rinaldo et al., 1992] for which key statistical features of the global optima are exactly known [Maritan et al., 1996b]. The interest for this problem has also widened because extensive experimental observations on the features of fluvial landforms can be derived through accurate and objectively analyzable digital terrain maps (see, for a re view, e.g., Rodriguez-Iturbe and Rinaldo [1997]). Whether by the attitude of singling out differences rather than common characters when comparing landforms or by the lack of perception of the importance of the issue, geomor- phologists have so far eschewed the issue of distinctive statis tics of channel networks. We argue, following Rigon et al. [1996] and Rinaldo et al. [1998], that the matching of five directly measurable scaling exponents (defining topological ag gregation, structure and length of drainage paths, elongation, and general shape of the drainage basin) is a stringent com parison test which OCNs meet beautifully [e.g., Rodriguez- 'Also at Dipartimcnto di Ingcgneria Idraulica, Marittima e Gcotec- nica, Universita di Padova, Padova, Italy. Copyright 1998 by the American Geophysical Union. Paper number 98WR02287. 0043-1397/98/98 WR-02287S09.00 Iturbe and Rinaldo, 1997]. Furthermore, the optimality as sumed by OCNs has been validated theoretically under a broad theoretical framework [Sinclair and Ball, 1996; Rinaldo et al., 1998]. OCNs are obtained by selecting the spanning network con figuration, say s, that minimizes the energy dissipation of the system, defined as H(s) = 2 AJ (1) where i spans all sites of an arbitrary lattice and.4, is a measure of the number of sites upstream to i connected by the network. H{s) is a nonlocal quantity (At = X 'jenn(i) Aj + R. where nn(i) are the nearest neighbors of/' in the lattice draining into i and /?, is a distributed injection term, i.e., a measure of landscape-forming rainfall events, conventionally taken as unity, R, = 1 [see, e.g., Rodriguez-Iturbe and Rinaldo, 1997]) which represents the flow collected at each site in the open, injected system. It is usually termed the total contributing area at i and embeds long-range interactions; the y power of At is an exponent capturing the physics of the erosional process, and it usually takes on the value y = 0.5 [Rodriguez-Iturbe et al., 1992b; Howard, 1990, 1994] (for a review of the linkage of y to the dominant sediment transport process in landscape evolu tion models see Rodriguez-Iturbe and Rinaldo [1997, chapter IV]. The optimization problem defined by (1) is parameter- free (one only needs to choose the network whose energy dissipation function is lowest) but entails an involved selection procedure among the critically large number of spanning net works in a given domain. Interestingly, Rinaldo et al. [1997] found that for a given set of dynamical rules the energy landscape is riddled with a large number of local minima characterized by a range of similar 3181
Transcript of Feasible optimality implies Hack's lapdodds/files/papers/others/1998/rigon1998a.pdfFeasible...
WATER RESOURCES RESEARCH, VOL. 34, NO. 11, PAGES 3181-3189, NOVEMBER 1998
Feasible optimality implies Hack's lawRiccardo RigonDipartimcnto di Ingcgneria Civile c Ambientale, Universita di Trento, Trento, Italy
Ignacio Rodriguez-IturbeDepartment of Civil Engineering, Texas A&M University, College Station
Andrea Rinaldo1Centra Intcrnazionale di Idrologia "Dino Tonini," Universita di Padova. Padova, Italy
Abstract. We analyze the elongation (the scaling properties of drainage area withmainstream length) in optimal channel networks (OCNs) obtained through differentalgorithms searching for the minimum of a functional computing the total energydissipation of the drainage system. The algorithms have different capabilities to overcomethe imprinting of initial and boundary conditions, and thus they have different chances ofattaining the global optimum. We find that suboptimal shapes, i.e., dynamically accessiblestates characterized by locally stationary total potential energy, show the robust type ofelongation that is consistently observed in nature. This suggestive and directly measurableproperty is not found in the so-called ground state, i.e., the global minimum, whosefeatures, including elongation, are known exactly. The global minimum is shown to be tooregular and symmetric to be dynamically accessible in nature, owing to features andconstraints of erosional processes. Thus Hack's law is seen as a signature of feasibleoptimality thus yielding further support to the suggestion that optimality of the system asa whole explains the dynamic origin of fractal forms in nature.
1. Introduction
Spanning, loopless network configurations characterized byminimum total energy dissipation (optimal channel networks(OCNs)) show many interesting properties including robustscale-invariant properties and a striking resemblance to fluviallandforms (see, for a review, Rodriguez-Iturbe and Rinaldo[1997]). The search for OCNs entails a complex optimizationproblem related to the evolution of fluvial networks [Rodriguez-Iturbe et al., 1992a, b; Rinaldo et al., 1992] for which keystatistical features of the global optima are exactly known[Maritan et al., 1996b]. The interest for this problem has alsowidened because extensive experimental observations on thefeatures of fluvial landforms can be derived through accurateand objectively analyzable digital terrain maps (see, for a review, e.g., Rodriguez-Iturbe and Rinaldo [1997]).
Whether by the attitude of singling out differences ratherthan common characters when comparing landforms or by thelack of perception of the importance of the issue, geomor-phologists have so far eschewed the issue of distinctive statistics of channel networks. We argue, following Rigon et al.[1996] and Rinaldo et al. [1998], that the matching of fivedirectly measurable scaling exponents (defining topological aggregation, structure and length of drainage paths, elongation,and general shape of the drainage basin) is a stringent comparison test which OCNs meet beautifully [e.g., Rodriguez-
'Also at Dipartimcnto di Ingcgneria Idraulica, Marittima e Gcotec-nica, Universita di Padova, Padova, Italy.Copyright 1998 by the American Geophysical Union.Paper number 98WR02287.0043-1397/98/98 WR-02287S09.00
Iturbe and Rinaldo, 1997]. Furthermore, the optimality assumed by OCNs has been validated theoretically under a broadtheoretical framework [Sinclair and Ball, 1996; Rinaldo et al.,1998].
OCNs are obtained by selecting the spanning network configuration, say s, that minimizes the energy dissipation of thesystem, defined as
H(s) = 2 AJ (1)
where i spans all sites of an arbitrary lattice and.4, is a measureof the number of sites upstream to i connected by the network.H{s) is a nonlocal quantity (At = X'jenn(i) Aj + R. wherenn(i) are the nearest neighbors of/' in the lattice draining intoi and /?, is a distributed injection term, i.e., a measure oflandscape-forming rainfall events, conventionally taken asunity, R, = 1 [see, e.g., Rodriguez-Iturbe and Rinaldo, 1997])which represents the flow collected at each site in the open,injected system. It is usually termed the total contributing areaat i and embeds long-range interactions; the y power of At isan exponent capturing the physics of the erosional process, andit usually takes on the value y = 0.5 [Rodriguez-Iturbe et al.,1992b; Howard, 1990, 1994] (for a review of the linkage of y tothe dominant sediment transport process in landscape evolution models see Rodriguez-Iturbe and Rinaldo [1997, chapterIV]. The optimization problem defined by (1) is parameter-free (one only needs to choose the network whose energydissipation function is lowest) but entails an involved selectionprocedure among the critically large number of spanning networks in a given domain.
Interestingly, Rinaldo et al. [1997] found that for a given setof dynamical rules the energy landscape is riddled with a largenumber of local minima characterized by a range of similar
3181
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Figure 8. Scaling of ratios of moments of total mainstream length, i.e., (L")/(L"~1) versus A, whichdenned, through their slope and the finite-size argument [Rigon et al., 1996], the Hack exponent h for (a) theEden-like networks of Figure 4a, (b) the t = 0 networks of Figure 4b, and (c) the optimal networks obtainedby simulated annealing shown in Figure 4c.
exponents sought and/ is an arbitrary scaling function [Maritanet al., 1996a]), has been analyzed in detail to compute theexponents B by direct inspection and to compute H by thederived relationship {A")l(An~l) « L\+H (where n is anyinteger). Notice that H also relates to self-affine properties ofthe basin boundaries. The probability distribution of totalmainstream length to the divide L (the length L is computed,at each site, by measuring the flow path obtained by choosingthe upstream site with largest contributing area) is also analyzed for finite-size scaling effects, i.e., P[L S: /] « l~*f(l/A'')(where £ is a scaling exponent and h is Hack's coefficient[Rigon et al., 1996]). We have evaluated the exponent £ bydirect inspection, and we have evaluated h by the derivedrelationship (L")l{L"~l) « Ah, for a set of different momentsn. The internal consistency of the exponents is discussed, forexample, by Rodriguez-Iturbe and Rinaldo [1997].
Figures 5a-5c show the area distributions for the networks inFigures 4a-4c. Analogously, Figures 6a-6c show the relatedlength distributions. Figures 7a-7c show, for the same sequence, the scaling analyses of the ratios of five consecutivemoments of the area distribution, plotted against Lp whichallows for an accurate estimation of the Hurst coefficient H. Itshould be noted that a slight multiscaling effect seems to affect,in particular, the plots of the optimal networks of Figure 4c.Whether or not this is an artifact remains to be seen. Nevertheless, truly accurate statistics would have required significantly larger sizes of the computational lattice, which in turnwould have implied prohibitive computational burden (evenwith up-to-date computational technology) for simulated annealing. Finally, Figures 8a-8c show the evaluation of Hack'sexponents h through the analysis of the moments of the length
distribution, i.e., log-log plots of <Ln>/<Ln_1) <x Ah. In fact,Rigon et al. [1996] have shown that Hack's law is best evaluated(and indeed validated and understood) in the above framework.
The reference observational values have been widely discussed [Rodriguez-Iturbe and Rinaldo, 1997]. Real networksconsistently show linked exponents B, £, H, and h and welldefined ranges. These ranges are [Rinaldo et al., 1998] B =0.41 - 0.46, £ = 0.65 - 0.90, H = 0.70 - 1.00, and h =0.53 - 0.60. We thus accept that a network is realistic whenall scaling exponents fall within the above ranges.
Eden networks, under the choice of the boundary conditionsof Figure 4a, have B = 0.55, £ = 0.91, H = 0.70, and h =0.58. We observe that two exponents fall outside the feasiblerange, in particular, that describing aggregation structurewhich unfavors large contributing areas, while the others arereasonable. This suggested, in our view erroneously, thatchance-dominated networks could capture the essence of natural patterns (this conclusion is more evident if one considersa few other exponents, chiefly describing the wandering of themain course and boundaries, which we have not shown here forbrevity). Feasible optima have perfect scores, as B = 0.45,£ = 0.85, H = 0.75, and h = 0.56. Global optima (or ourbest approximation to it) tend to the analytically known valuesB = 0.50 and £ = 1 [Maritan et al, 1996a], as B = 0.50 and £ =1.01. The Hurst exponent is somewhat low (H = 0.94 ratherthan H = 1 as expected) for the problems of accuracy discussed above, and Hack's exponent h is 0.5. Global optima aretherefore far from realistic.
The statistical results (Figure 8) confirm that the globallyoptimal arrangement does not show elongation (h «* 0.5) as
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3188 RIGON ET AL.: FEASIBLE OPTIMALITY IMPLIES HACK'S LAW
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opposed to feasible optima (h ~ 0.57). Repeating the computation with even more degrees of freedom, that is, by allowing multiple outlets along all sides, further strengthens ourconclusion (see, e.g., Sun et al. [1994a, b, c] for the particularcase of simulated annealing and multiple outlets).
Thus the results shown in Figures 5 and 6 reinforce thesuggestion that the absolute minimum does not show Hack'slaw or any other significant and stringent statistics of real rivernetworks. Feasible optimal networks instead show strong resemblance to real networks.
4. ConclusionsWe have shown that Hack's law, one of the most common
empirical relationships found in the river basin, is naturallyderived from optimality principles only in connection to thenature of the stationary states achieved. Dynamically accessible stationary states, found by imperfect search proceduresseemingly relatable to nature's myopic search for a stable nichewithin a complex fitness landscape, show a striking resemblance to real basins. On the contrary, globally optimal configurations, obtained by screening procedures that allow un-physical freedom to the search process, do not elongate. Wethus conclude that feasible optimality is likely to be the productof the self-organized dynamics of fluvial networks and thatHack's law, whose ubiquity is widely acknowledged, reinforcesthe conclusions of previous work, suggesting that frustratedoptimality of complex dissipative systems with many degrees offreedom might be the dynamic reason for scale-free growthand form in nature.
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