Are river basins optimal channel networks?

Advances in Water Resources 16 (1993) 69-79

Are river basins optimal channel networks? E.J. ljjdsz-Vdsquez, a R.L. Bras, a I. Rodriguez-lturbe, a'b R. R i g o n c & A. Rinaldo ~ aRalph M. Parsons Laboratory, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA b Instituto Internacional de Estudios Avanzados, PO Box 17606, Parque Central, Caracas, Venezuela Clstituto di Idraulica "G. Poleni', Universita di Padova, via Loredan 20, 1-35131, Padova, Italy

Drainage networks generated with a simulation model based on the scaling relationship between slopes and areas were found to have total energy expenditure values very near the minimum value of optimal channel networks. Using this model to grow networks inside the boundaries of real basins, it is shown that the drainage networks identified with the aid of digital elevation maps in such basins tend to organize themselves in configurations that minimize total energy expenditure. The role of perturbations in the search for configurations with lower energy and the existence of unstable equilibrium landscapes are also examined. Key words: Geomorphology, river networks, optimal networks, landscape

simulation, digital elevation maps, unstable landscapes. ate step in the comparison of total energy expenditure in real basins obtained from DEMs and the value predicted by the OCN formalism. Section 2 describes the behavior of the model. Section 3 reviews the theory of optimal channel networks and their properties. Section 4 proceeds to show that the values of total energy expenditure of networks simulated by the model are very near those predicted by the OCN formalism and therefore the model may be used to determine near-optimal values of energy expenditure of drainage networks in large domains. In Section 5 networks are grown with the slope-area model using boundaries of real basins identified with DEMs. The values of total energy expenditure in the simulated and the real basins are shown to be very similar, suggesting that river networks tend to organize themselves to minimize energy expenditure. Section 6 shows the equivalence of total energy expenditure and the constrained minimization of potential energy and provides an explanation for the mechanism through which networks grow and organize to minimize total energy expenditure. Section 7 studies the relationship between the concepts of minimum total energy and stability of landscapes.

1 INTRODUCTION Based on three principles of minimum energy expenditure, Rodriguez-Iturbe et al. 13 have recently proposed a theory that links energy dissipation, the threedimensional structure of river basins and the spatial organization of drainage networks. The first two principles are able to explain many empirical facts which have been observed in the hydraulic characteristics of channels. The third principle requires minimum total energy expenditure in the network as a whole and leads to an organization of drainage structures that shares many important characteristics with river networks, for example Horton's laws, the distribution of link lengths and fractal and multifractal properties.lO, ~, t3 Networks that drain a given area and have minimum total energy expenditure are called optimal channel networks (OCN). The method that has been used to construct OCNs is analogous to random search procedures for the traveling salesman problem. However, the task of finding an O C N is not an easy one and the largest domains that can be studied are much smaller than real basins at the resolution available with digital elevation maps (DEM). The purpose of this paper is to use a model of drainage network simulation, based on the scaling relationship between slopes and areas as an intermedi-

2 THE S L O P E - A R E A M O D E L The slope-area model is a simple model that simulates river networks and the three-dimensional structure of

Advances in Water Resources 0309-1708/93/$06.00

© 1993 Elsevier Science Publishers Ltd. 69

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E.J. Ijj{zsz-V{zsquez, R.L. Bras, I. Rodriguez-Iturbe, R. Rigon & A. Rinaldo

river basins using the scaling relationship between slopes and flows observed in river networks: S ~ Q-O

(1)

where S is slope, Q is water flow and 0 is a scaling exponent with a value usually around 0.5. Among others, Leopold et al. 7 report a mean observed value of 0 in streams of the US of 0"49. Tarboton et al. 18 found a value of 0.5 in the analysis of drainage networks obtained from digital elevation maps (DEMs). They used contributing area as a surrogate for discharge. Flint 3 relates (1) to Horton's slope and Schumm's area laws. Theoretically, relationship (1) has been the object of numerous studies. It can be obtained formally in onedimensional profiles subject to uplift and erosion with a sediment transport function of the form Qmsn. In this case the equilibrium profiles has slopes that scale as in eqn (1) with O= ( m - 1 ) / n . 2'5'15' 19 Another way of looking at relationship (1) is as the result of the minimization of energy expenditure. TM 23, 24 In general, eqn (1) is at the heart of every study of the threedimensional structure of river basins. The slope-area model simulates the elevation field over a gridded domain. Each grid box is termed a pixel following the notation in DEMs. At every iteration the model assigns flow directions along the steepest slope downhill. Following these flow directions, the model calculates the drainage a r e a A i of each pixel i. Then, the slope at pixel i at the next iteration is set to 5"; = kA~ where k is a constant and 0 is the scaling parameter. Again, A; has been taken as a surrogate for Qi. The model keeps the elevation of the outlets (and any lakes) constant and uses them to recalculate elevations of the pixels draining to the outlet using the slopes Si at each pixel. If a pixel has a lower elevation than its neighbors then it captures more drainage area which implies that its slope (and therefore its elevation) will be lower at the next timestep. This reinforcing feedback is the basic mechanism for the growth of the drainage network. The process is iterated until an equilibrium landscape is reached. The initial condition is usually a flat plateau with the same elevation but very small random perturbations in order to properly define drainage directions. This initial elevation field has been used in other models of basin evolution. 2°2j The elevation of the outlet (or outlets) is kept at a lower level. Boundaries are closed except for the outlet. Figure 1 shows an example of how a landscape and its network develop in a square basin draining through one of its corners. This simulation uses 0 = 0.5. The drainage network is presented by defining pixels that have a contributing area of at least five pixels as channels. This threshold area concept has been used to infer networks from DEMs. 19 The last figure is the equilibrium landscape. At this point the entire area is drained by the network and the slope of each pixel is exactly equal to kAi °5.

3 OPTIMAL CHANNEL NETWORKS (OCN) The concept of minimization of energy expenditure has been applied to different aspects of river basins. 16~24 Recently, Rodriguez-Iturbe et al. 13 presented three principles of energy expenditure in drainage networks that explain many of the existing empirical relationships for individual channel characteristics, as well as global network properties. The three principles of optimal energy expenditure are: 13 (1) the energy expenditure in any link of the network is minimum, (2) the energy expenditure per unit area of channel is the same everywhere in the network, and (3) the energy expenditure in the network as a whole is minimum. Principles (1) and (2) explain relationships observed in the field, for example: velocity tends to be constant throughout the network, ~6'22 depth and width of channels scale with flow as d ~ Q0.5 and w ~ Q0.5,7 and slopes scale with flow as S ~ Q o.5. The third optimality principle has implications on the overall structure of the network and the way different elements are arranged and connected to deliver water and sediment out of the basin. Analytically, this third principle can be written 13 as the search of the drainage network for a configuration where the value of

i

is minimum. Qi are flows, L i a r e link lengths and k is a constant. Usually Qi is replaced by its surrogate variable Ai, the contributing area. Rodriguez-Iturbe et al. 13 implemented an optimization procedure similar to the one used by Lin 8 for the traveling salesman problem. Starting from a network that drains a given area, random changes are applied to the flow directions at different pixels and the new configuration is accepted if the total energy expenditure (eqn (2)) is smaller. In this way the procedure finds networks with minimum global energy dissipation and these are called optimal channel networks (OCN). It has been shown that different properties of natural river networks like Horton's laws, 4 distribution of link lengths, Melton's law, and fractal properties like the power-law cumulative distributions of streams lengths 17 and contributing areas j2 are well reproduced by OCNs. 1° l J, J3, 14 Furthermore, the properties of OCNs are not affected by the search procedure, the initial network used in the search, or quenched randomness. ~ All this evidence points to the possibility that river networks may indeed minimize their total energy dissipation. However, random search strategies are computer intensive procedures and only domains of up tO 10 4 pixels have been analyzed. This size is much smaller than the size of basins in DEMs at the scale of resolution available.

Are river basins optimal channel networks?

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Fig. 1. Isometric view and drainage network (identified with a threshold contributing area of five pixels) at different iterations of the simulation. Fig. 1 continues on opposite page. 4 TOTAL ENERGY EXPENDITURE IN OCNs AND NETWORKS GENERATED BY THE S L O P E - A R E A MODEL Given that the slope area model is able to simulate basins with a large number of pixels in their domains, it is of interest to compare the total energy expenditure of OCNs and networks simulated by the slope-area model in identical domains. If the random search algorithm of OCNs and the slope-area model produce networks of similar total energy then the efficient slope-area model

can be used to test whether real basins obtained from DEMs optimize total energy expenditure. In order to make a fair comparison, 100 repetitions of the search procedure of OCNs in a 24 x 24 domain were performed. The experiments start from different random initial networks that drain the domain under study. An example of such random initial networks is presented in Fig. 2. The search procedure rearranges the elements of the network into configurations with smaller total energy dissipation E (measured using eqn (2)). Figure 3 presents the configuration with the lowest

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Fig. 1. --contd. value of E among the networks obtained. Figure 4 shows the histogram of the final values of total energy expenditure E. Even though there is some variation in these values (between 1620 and 1680 where k = 1, Qi -~ Ai and the pixels are considered of unit area in eqn (2)), it is very small compared to the amount by which E has been reduced from the initial condition (from a mean value of 2250 to a mean value of 1640). Furthermore, it has been shown that the statistical properties of the different networks obtained with the random search procedure are identical.ll Now, using the same 24 × 24 domain and as initial landscape a plateau with the same mean elevation but different random perturbations, 100 repetitions of the slope-area model were carried out similar to the

simulation shown in Fig. 1. The histogram of final total energies is presented in Fig. 5. Figure 6 presents together the histograms of energies for the initial random networks used in the OCN procedure (at the right-hand side), the final OCNs and the networks simulated with the slopearea model (at the left-hand side). The overlap between the histograms of OCNs and networks generated by the model and the small difference between these two histograms compared to the distance between them and the histogram for random networks supports the idea of using the slopearea model to generate networks with near-optimal energy expenditures. The computational demands of the random search procedure of OCNs prevent the construction of histograms with enough points for large domains. However, a few OCNs were constructed in a 64 × 64 domain (each of which requires approximately 20 h of CPU time) and were compared against networks of the slope-area model. The difference in total energy between them remained within a few percentage points.

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Fig. 4. Histogram of values of total energy expenditure for 100 repetitions of the OCN random search procedure.

Are river basins optimal channel networks?

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Fig. 5. Histogram of values of total energy expenditure for 100 networks simulated with the slope area model.

In the search for optimality, the river network may find itself trapped in local minima. It is therefore of interest to examine the role of perturbations as a mechanism to move the system out of these minima. Physically, these perturbations may represent the local inhomogeneities found by the landscape during its evolution. For this purpose, the elevation field of an equilibrium landscape obtained with the slope-area model was perturbed and given back as an initial condition to the model. Figure 7 presents the final networks of the equilibrium landscape before the perturbations. The total energy expenditure E was measured each time a new equilibrium was reached. Then, a new perturbation was applied. Figure 8 presents the decrease of the total energy expenditure E resulting from this process. The network rearranges into configurations with smaller values of E until it stabilizes at a

lower energy state. Figure 9 shows the final network after the perturbation process, when the lower energy state is reached. Even though the changes between the networks in Figs 7 and 9 are small, they are noticeable. The perturbations in the elevation field allow the system to arrange some defects in its configuration and find states with smaller values of E. Notice, however, that the reduction in E is small. Figure 10 presents the histogram of total energy expenditure of 100 different networks obtained with the slope-area model after perturbations were applied and a new stable equilibrium with lower energy was found. The histogram shifts to the left from its position in Fig. 5 (which corresponds to the equilibrium networks before the perturbations) and the overlap with the histogram of OCNs is even larger. Nevertheless, the change is small and the perturbation process requires repeated runs of the model, making it very difficult to be implemented in large domains. Given that the improvement in E is small, it seems reasonable

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to use the results of the model without perturbations to compare the energy expenditure level of DEMs and the value for OCNs.

5 TOTAL ENERGY E X P E N D I T U R E IN DEMs AND N E T W O R K S G E N E R A T E D BY THE S L O P E - A R E A MODEL

The next step is then to use actual basins identified from digital elevation maps and use their boundaries and outlet location as boundary conditions for the slopearea model. Table 1 compares the total energy expenditure of slope-area networks simulated using the actual domains of four different basins across the USA and the total energy expenditure of the actual basins. In every case, the difference between the total energy dissipation E (measured using eqn (2)) in the simulated and the real basin is less than 5%. As an example, Fig. 11 presents three networks in the same domain taken from North Fork Cour d'Alene river basin in Idaho, USA. Figure l l(a) is the network extracted from the DEM using a threshold contributing area of 50 pixels. Figure l l(b) shows a network simulated with the slope area model. Figure l l(c) shows a random network. All three networks are presented using the same threshold. While the values of E in the first and second cases are 4.2 × 105 and

4.0 x 105 (in pixel units), E for the third network has the much larger value of 6.1 x 105. The similarity of values of energy expenditure between the real and the simulated network suggest that river networks tend towards a state of minimum energy expenditure. It has been found, for some basins, that the scaling relationship Scx A 0 between slopes and contributing areas does not hold for all the values of areas but instead, there are two scaling regimes] 8 Figure 12 illustrates such behavior in one of the basins studied (Big Creek, Idaho, USA). In this figure, the slopes of the pixels in the basin are grouped into bins according to their contributing area. The circles represent the mean value of the slopes for the pixels in each bin. The break observed in scaling has been used by Tarboton et al. 8 to identify the threshold value of contributing area that separates hillslopes from channels. The form of the scaling relationship with two different scaling regimes can be used in the slope-area model. This change affects the flow pattern below the areas at which the break occurs. To illustrate this behavior, Fig. 13 shows the network of a basin grown with the break in scaling. While the basin in Fig. 7 was simulated with S oc A -°5 for every value of A, the basin in Fig.13 has Scx A -°5 if A is larger than 20 pixels and Sex A 00 (i.e. S constant) otherwise. In the original model the

Table 1. Comparison of the total energy expenditure of four actual basins and four slope-area networks simulated using the actual domains of the basins

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Are river basins optimal channel networks? pattern is aggregated at all scales while in the modified version of the model it tends to be parallel below the threshold value. Using the values of the two scaling regimes between slopes and areas observed in Fig. 12 (S cx A -0.12 if A < 210 pixels and S cx A -0.5 otherwise) and the real boundary of the Big Creek basin, the slopearea model was used to simulate the drainage network. The total energy expenditure E (calculated using those pixels identified as channels above the threshold area) is 4.54 x 105 (in pixel units) for the real basin and 4.51 x 105 for the simulated network. Summarizing, this section has shown that the difference in total energy expenditure of networks simulated with the slope-area model using actual basin boundaries from DEMs and the total energy expenditure of the actual river network is small. This suggests that drainage networks tend to organize themselves so as to minimize energy dissipation while delivering water and sediment out of the basin.

6 POTENTIAL ENERGY AND TOTAL ENERGY EXPENDITURE In this section, the third principle of optimal energy expenditure 13 is shown to be equivalent to the minimization of the total sum of elevations, constrained by the first two principles. This new interpretation hints to the mechanism through which networks grow and organize to minimize total energy expenditure. Using the first two principles of optimal energy expenditure 13 and the scaling relationship that can be derived from them:

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which is the same as expression (2) for the total energy expenditure E. By minimizing E, the network is also minimizing the sum of elevations Ep (which can be seen as a measure of the total potential energy of the basin). Notice that the state of minimum potential energy for the basin is not the flat plane because of the constraint (3) on the slopes. With a flat plane, the network is not able to deliver water and sediment out of the basin. In the slope-area model, each pixel is set to drain into the steepest direction downhill. Given that the slope of each pixel comes from the preceding iteration, by choosing the lowest neighbor the pixel is setting its elevation to the lowest possible value. As the network connects all the points in the basin, information is transmitted across the entire domain. On one hand, a change in a pixel's elevation affects the elevation of all the pixels uphill. On the other, the capture of additional area by pixels uphill changes the contributing area and elevation of pixels downhill. This interaction and communication may be the mechanism through which the principle of global optimality is embedded in the network growth process.

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it is possible to show that minimizing the total energy expenditure E is equivalent to the minimization of the sum of elevations:

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where hi(i) is the drop from pixel j(i) to its neighbor downstream andj(i) indexes the pixels along the flowing path from i to the outlet. The summation in eqn (5) can be reorganized by counting how many times a certain drop hn appears. This number is equal to the number of times a flowing path goes through pixel n, and this is equal to the number An of pixels draining through n.

7 M I N I M U M TOTAL ENERGY EXPENDITURE AND THE STABILITY OF LANDSCAPES It is possible to think of landscapes that, if given to the slope area model, would not be altered. One such landscape is the classic equilibrium form used by Smith and Bretherton 15 and later by Loewenherz 9 in their analysis of stability and channelization of surfaces. Figure 14(a) shows the landscape and Fig. 14(b) the parallel flow directions of the configuration. At every point the slope S is exactly equal to A-o.5 and therefore the landscape is at equilibrium and remains unaltered if given to the slope-area model. However, if a small random perturbation in elevation is applied as was done in Section 4, then the configuration changes radically. Figure 14(c) shows the equilibrium after only one perturbation and Fig. 14(d) after I0 perturbations. Figure 15 shows the dramatic drop in the value of the total energy expenditure even after a single perturbation.

76

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Are river basins optimal channel networks?

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Fig. 11. Three different networks in the same boundary domain of the North Folk Cour d'Alene river, Idaho. (a) Drainage network identified from a digital elevation map. (b) Drainage network simulated with the slope area model. (c) Random drainage network.

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E.J. Ijj[tsz-Vasquez, R.L. Bras, I. Rodriguez-Iturbe, R. Rigon & A. Rinaldo

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The system quickly reaches a state of low E after three or four perturbations are applied and then remains at that level. This experiment illustrates the nature of the search for an optimal network configuration.As presented schematically in Fig. 16, this is a problem with many local minima and the system is able to move between them if its elevation field is perturbed. Each of these local minima (which have very similar values of total energy expenditure) is a different configuration and their large

Are river basins optimal channel networks?

number is consistent with the enormous variety of channel networks found in nature. Even though the details of their structures are different, these configurations have comm o n statistical properties and near-optimal values of E. Furthermore, there are also unstable equilibrium states with high values of E which, when perturbed, move quickly to configuration with low total energy expenditure. An example of these unstable equilibrium landscapes is the one shown in Fig. 14(a). 8 CONCLUSIONS A slope-area model based on the scaling relationship between slopes and areas was shown to generate networks with total energy expenditure very near the minimum value of optimal channel networks. This permitted the use of efficient slope-area simulations to test whether real basins are in minimum energy states. Direct optimization at the scale of real basins is numerically impossible. The slope-area model was then used to simulate networks using boundaries of real basins. The values of total energy expenditure of the real and the simulated networks were very similar, suggesting that indeed river basins organize in configurations of minimum energy dissipation. A plausible explanation for this tendency was given based on the equivalence of minimum total energy dissipation and constrained minimum potential energy. Finally, the effect of perturbations in the landscape on the behavior of the total energy expenditure was examined, showing how inhomogeneities allow the network to rearrange and correct expensive defects in terms of energy expenditure and how unstable equilibrium landscapes move quickly towards lower levels of energy if perturbed. ACKNOWLEDGMENTS

This work was supported at M I T by the US Army Research Office (Agreement DAAL03-89-K-0151). The views, opinions and/or findings contained in this report are those of the authors and should not be construed as an official Department of the A r m y position, policy or decision, unless so designated by other documentation. REFERENCES

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