A physical explanation of the cumulative area distribution curve

Transcription

1 WATER RESOURCES RESEARCH, VOL. 34, NO. 5, PAGES , MAY 1998 A physical explanation of the cumulative area distribution curve Hemantha Perera and Garry Willgoose Department of Civil, Surveying and Environmental Engineering, The University of Newcastle, Callaghan, New South Wales, Australia Abstract. A physical explanation for the behavior of the cumulative area distribution (CAD) based on the Tokunaga channel network model is given. The CAD is divided into three regions. The first region, for small areas, is dependent on hillslope flow accumulation patterns and represents the catchment average of the hillslope flow accumulation in the diffusive erosion-dominated areas, upstream reaches, of the catchment. The second region represents that portion of the catchment dominated by fluvial erosion. This region is well described by a log-log linear power law, which results from the scaling properties of the channel network. The scale exponent, b, is highly sensitive to a parameter of the Tokunaga stream numbering scheme. The exponent b converges to -0.5 for higher order Tokunaga networks for parameters consistent with topological random networks. Small networks have lower values of b, which asymptotic converges to b = -0.5 as the catchment scale increase. The third region reflects the lowest reaches of the channel network, the scale of the catchment, and is a boundary effect. An explicit analytical solution to the scaling properties in the second region is derived on the basis of the Tokunaga network model. 1. Introduction The cumulative distribution of drainage area is the percentage of the catchment that has an area per unit width draining through a point which is greater than or equal to a given area. It is a geomorphologic measure that characterizes part of the catchment hydrology and is a measure commonly used by geomorphologists and hydrologists to characterize the scale invariant structure of channel networks [Tarboton et al., 1989; Rodriguez-Iturbe et al., 1992; Inaoka and Takayasu, 1993; La Barbera and Roth, 1994]. This cumulative area distribution (CAD) characterizes the scaling properties of drainage area. The CAD and its scaling properties are key components in the development of physically based models that link geomorphology and hydrology [Perera, 1997; Ibbitt et al., 1997]. Perera [1997] showed that slope of the CAD in both the hillslope and channel domains is important for describing the geomorphic impact on subsurface saturation excess runoff generation. Using a digital elevation model (DEM) of a catchment, the drainage path through, and the drainage area per unit width of each point in the catchment can be computed. For a grided DEM the cumulative area (A) is determined as the number of pixels within the catchment, which have drainage area, a (number of pixels), draining through greater than or equal to some specified drainage area (a *),.q(a >- a*) = N, i=1 N i = 0 if a i < a*, N, = 1 if a i -> a* this distribution as a means of characterizing the flow aggregation structure of channel networks. Other researchers [Moglen and Bras, 1994; Sun et al., 1994a, b, c; Inaoka and Takayasu, 1993] have used the CAD for the calibration of their geomorphology models. Figure la shows the CAD for the Middle Creek catchment, which is described elsewhere [Willgoose, 1994], from the Pokolbin region of Australia. The CAD has a characteristic type of behavior that is common for catchments in many different regions. These distributions can be divided into three regions [Moglen and Bras, 1994] as shown in Figure 1. Region 1 represents small drainage areas that we postulate to be the hillslope flow aggregation. Region 2 represents' that part of the catchment dominated by channelized flow. This region tends to follow a straight line in log-log scale, obeying the power law distribution as, A (a >- a *) cr (a*)* (2) where b is the scaling exponent. Region 3 consists of the few nodes on the main trunk of the channel network near to the catchment outlet. The large jumps and steps in region 3 result from the large tributaries that join the main trunk and contribute large percentages of area. In this region the CAD rapidly decreases with increasing drainage area. This paper deals with the characterization of the channel (region 2) and hillslope (region 1) regions of the CAD. The Tokunaga channel network model [Tokunaga, 1978] is used to simulate a range of channel networks to examine the channel behavior. The effect of Tokunaga parameters K and E on the scaling properties of region 2 of the CAD are examined. We where a t is the total area of the catchment and a i is the will show that the parameter K is strongly related to the scaling drainage area of ith pixel. A log-log plot of A against a* is exponent, b, while E shows a less pronounced effect on b. called the CAD (Figure 1). Rodriguez-Iturbe et al. [1992] used The hillslope behavior of region 1 is studied using different hypothetical hillslope flow patterns and different sizes of the Copyright 1998 by the American Geophysical Union. hillslope area. The hillslope behavior of the CAD is a result of Paper number 98WR the average hillslope flow accumulation pattern of a catch /98/98WR ment. The behavior of region 3 of the CAD is shown to be a 1335

2 1336 PERERA AND WILLGOOSE: PHYSICAL EXPLANATION , < 102 > 10 ] = (a) i I iiii I i i i i i iii I i i i i 11 1 i i i : ' Observed _! region- 1 region-2. i region ] (b) -- Raw data Moving average of data points a network that does not suffer from those problems [Tarboton, 1996] and is scale invariant. Tokunaga used the Strahler ordering system [Strahler, 1952] to define orders of the channels but defined a different stream law. The Tokunaga model is built as follows. Analytical The average number,/,e,,, of streams of order n entering from sides into a stream of higher order k. Parameters E i (i = 1,..., rn - 1) and K are given by mem_l : m_lem_2 '- icei _ 1 -- nen_l '- 2E1 = E mem_ 2 -- m_lem_ El -- E2 mem_j: m_lem_l_j... +je = Ej E 2 E 3 Ej E1 E2 Ej_ =K where m is the highest order of the catchment. In the Tokunaga model E and K are assumed constants, so that E s = K i- E 1' Tokunaga [1978] showed that the parameters K = 2, E = 1 correspond to the mean properties of topologically distinct random networks [Shreve, 1966]. A drainage network with K = 2, E = 1 is shown in Figure 2 [Tokunaga, 1978]. 3. Channel Behavior The CAD generally obeys a log-log power law in region 2 (Figure la). This power law behavior has been observed in natural basins [Rodriguez-Iturbe et al., 1992; La Barbera and Roth, 1994] and simulated drainage basins [Sun et al., 1994a, b, -4,,,,,,111 i i i i iiill i,,,,,,, c; Inaoka and Takayasu, 1993] and appears to be a universal behavior of catchments. To study this universality, drainage Area (nodes) networks simulated with the Tokunaga model are examined, Figure 1. together with their dependence on the Tokunaga parameters (a) Observed and analytical cumulative area distribution of Middle Creak natural catchment in Pokolbin region K and E. of Australia. Data from 20 m DEM. (b) Slope of the observed Two drainage networks with parameters K = 2 and E = 1 CAD between each data point (raw data) and the moving are simulated assuming two different zeroth-order hillslope average of five data points. flow patterns, hill 1 and hill 2 (shown in Figures 3a and 3b and described in detail in the next section). K = 2 and E = 1 are typical of observed values [Tokunaga, 1978] and, as previously boundary effect. Finally, an analytical solution for the CAD in noted, are consistent with the topologically random network region 2 is presented based on the Tokunaga channel network model. Figure 4 displays the CAD of two 10th-order drainage model. Using the analytical solution, we demonstrate that scal- basins with the different hillslope flow patterns. In Figure 4, ing exponent, 4> converges to -0.5 for higher-order networks region 2 of both CADs can be approximated by a straight line for parameters K = 2 and E - 1, which are consistent with showing power law behavior with an exponent of approxitopologically random channel networks. This value for 4> of the topologically random channel network was also observed by De Vries et al. [1994]. The exponent for small networks generally deviates from this value but asymptotically converges to it with increasing area. (3) 2. The Tokunaga Channel Network Model Channel networks for this study are constructed on the basis of Tokunaga channel numbering system [Tokunaga, 1978], which is different from the Horton [1945] and Strahler [1952] laws of drainage composition. The Horton and Strahler drainage composition laws suffer from conceptual problems with respect to scale and similarity since their bifurcation ratios vary significantly with scale and are applicable only for scale invariant, and physically unrealistic, structurally Hortonian tworks [Scheidegger, 1968; Tarboton, 1996; Kirchner, 1993]. The Tokunaga cyclicity model is an alternative method of characterizing Figure 2. Hypothetical drainage network with K = 2 and E - 1 [Tokunaga, 1978]. The numbers represent the order.

3 PERERA AND WILLGOOSE: PHYSICAL EXPLANATION 1337 mately rb = That rb is independent of the zeroth-order hillslope flow pattern suggests that the hillslope drainage pattern does not influence the scaling properties of the CAD in region 2, only the intercept in the vertical axis in Figure 4. The effect of changes in the values of K and E are examined next. Figure 5 shows the CAD of the seventh-order simulated basins using the hill 2 zeroth-order hillslope. Results for parameter values ofk = 4, E = 1 andk -- 2, E = 2 are compared with the results for nominal parameter values of K = 2 and E Figure 5 shows that when K increases from 2 to 4, with constant E, the slope of region 2, 4 increases from approximately -0.5 to -0.25, suggesting an inverse relationship between K and rb. There is no observable effect on the slope with the changes in E. This dependence on K of the CAD is because K characterizes the scaling properties of the network. A more highly branched network will have a higher value of K so that the CAD declines more slowly. 4. Hillslope Behavior In this section we examine how the zeroth-order basin hill- slope flow path patterns influence the form of the CAD, in particular, region 1. Two hypothetical hillslope drainage patterns are shown in Figures 3a and 3b, named hill 1 and hill 2. I I I I I I I I I I I,I I I I I I I I I I I I I I 1 I I I I I 111 I I I I I I I I I I IIIIIIIIII IIIIIIIIII I I I I I I I I I I I'1 I I I I I I I I II (a) II,,tillIll, II,[ll[[,,]lllHq I,,t,,,t I ' '''''"l [ [1'1["1 [ lit[,,l[ 10 8, -- Hill- 1, -... Hl11-2 '-'1 O _., _ -..,, < region-3 g 104 ß. --_ I... I... I... I... I... I Figure 4. Cumulative area distributions of 10th-order simulated catchments using Tokunaga K = 2 and E -- 1 for the two different hillslope flow patterns from Figure 3. The main difference between these patterns is in the form of the lateral inflow into the first order channel. Tokunaga channel networks up to order 10 have been simulated using these different zeroth-order networks. The CAD of orders 1 through order 10 are shown in Figures 6 and 7. The CAD of the first-order networks in Figures 6 and 7 are different because of the differences in the zeroth-order hill- slope and flow path patterns. In region 1 of Figures 6 and 7, the slope of the CAD initially increases with increasing area, with the slope then decreasing with further increasing area. This behavior has been observed in natural catchments (e.g., Figure lb). In hillslope regions a high percentage of the area of the catchment has a very small drainage area, leading to fast decline in cumulative area with increasing drainage area and thus increasing the slope of CAD initially on the left-hand side of region 1. As the drainage area increases, tributaries merge, resulting in a faster increase in the drainage area leading the upward concave region of the curve on the right-hand side of region 1 (Figure lb). Despite the differences in the flow path patterns used in the ' 10 5 o 104 < D - > 103 r E =1 K=4 ' -- 1 ' 'E E =2, K=2 = E 1 =1, K=2 (b) Figure 3. Hypothetical first-order basins with zeroth-order hillslope flow patterns. Arrows indicate the flow directions. (a) Hill 1 and (b) hill ? Figure 5. Cumulative area distribution of seventh-order catchment for hill 2 hillslope flow patterns from Figure 3b using different values of Tokunaga K and E.

4 _ 1338 PERERA AND WILLGOOSE: PHYSICAL EXPLANATION randomly selecting various sizes of first-order areas with the, 108 hill 2 drainage pattern (with 145, 181, 225, 277, 337, and 405 nodes, covering the range of first-order areas within the Pokolbin catchment in Figure 1). Figure 8 shows that the effect of 106 variable areas (with the same flow path patterns) does not significantly affect the shape of the CAD in region 1. In a natural drainage basin, where the area of the hillslopes vary, 104 the shape of the region 1 of the CAD thus represents the average hillslope flow accumulation pattern. Figures l a, 6, and 7 show that the hillslope behavior of the 102 CAD (region 1) are different from the channel network behavior in region 2, reflecting the different flow accumulation 1 patterns on channel and hillslope. If the convergence properties of the hillslope and channel were the same then region and 2 would not be distinguishable. Thus the difference in regions 1 and 2 appears to be a useful indicator of changing Figure 6. Cumulative area distribution of order-1 through flow aggregation, and potentially of erosion process, as exhiborder-10 networks for hill 1 drainage pattern from Figure 3a ited by geomorphology. using Tokunaga K - 2 and E1 = 1. Numbers representhe orders. Broken lines show the boundary of the different regions. 5. Erosion Process-Based Interpretation of the Cumulative Area Distribution Willgoose et al. [1991b] argue that the area-slope relationship two set of networks in Figures 6 and 7, the channel aggregation of a catchment obeys a power law with a change in the expopatterns for higher-order networks are the same, as K and E nent as the catchment switches from diffusive erosion domiare same. Region 1 has no effect on the shape with increasing nance (i.e., sediment flux rate constant x f(slope)) for order. We interprethis to mean that region 1 represents the smaller areas to fluvial erosion dominance (i.e., sediment hillslope component of the catchment where scale invariant flux rate constant x f(area, slope)) for large areas. The channel network aggregation has no influence. Points (pixels) area-slope relationship of the Middle Creek catchment is on hillslopes have smaller drainage areas relative to points shown in Figure 9 [Willgoose, 1994]. The change in the expo- (pixels) on channel network. The hillslope flow patterns dom- nent for larger drainage areas occurs at an area of approxiinate for small drainage areas, while with increasing area the mately 10 pixels (200 m2/m). Drainage areas less than 10 pixels influence of the hillslope decreases and the channel accumu- are dominated by diffusion, and areas more than 10 pixels are lation pattern dominates. Thus the CAD for small drainage dominated by fluvial erosion. This switch from diffusion to areas depends on hillslope flow path patterns. fluvial dominance at an area of 10 pixels coincides with the In the discussion above the areas of all of the first-order boundary between regions 1 and 2 in the CAD of Figure l a. basins have been assumed to be equal. In a natural drainage This coincidence of area between the area-slope relationship basin the first-order areas will vary randomly. To test this and CAD can be observed in most of the natural catchments used effect, drainage networks up to order 6 were simulated by by Moglen and Bras [1994], suggesting that region 1 of the CAD is diffusion dominated and that region 2 is fluvially dominated _ glon- -! 6 10 s :, _, region-2-1 :',,, 1 ''.,,,,,6...,7, ,,, , 104 < : 101 i I i I tilt I I i i i i i i i I i i I I I I I! x.. order 6 order. order4 i_. orders3 order Figure 7. Cumulative area distribution of order-1 through order-10 networks of hill 2 drainage pattern from Figure 3b Figure 8. Region 1 of the cumulative area distributions of using Tokunaga K - 2 and E = t, Numbers represent the order-through order-6 networks simulated randomly selectorders. Broken lines show the boundary of the different ing variousizes 145, 225, 277, 237, and 405 nodes of first-order regions. areas of hill 2 drainage pattern.

5 _. _ PERERA AND WILLGOOSE: PHYSICAL EXPLANATION 1339 We thus asser that the boundary between regions 1 and 2 of the CAD will move to the left for catchments with less diffu- sion transport. This hypothesis is examined by simulating a drainage network assuming a unit area (one pixel) for firstorder areas (exterior link areas) and interior subbasin areas (interior link areas). This means that the catchment accumulation pattern is dominated by channel flow. CAD of order 10 with K - 2 and E 1-1 are shown in Figure 10, and the log-log linearity extends up to the leftmost region. This also indicates that if the hillslope drainage pattern is the same as channel flow pattern, both hillslope and channel regions (regions 1 and 2) of the CAD will exhibit a similar behavior without break in slope. This factor can be observed in the CAD (Figure 11) of a synthetic catchment simulated by the SIBERIA basin evolution model [Willgooset al., 1991a]. This catchment is dominated by fluvial erosion since the diffusion component of SI- BERIA has been turned off. Figure 11 shows that the log-log linearity is continuous almost up to the area of two pixels, at which numerical diffusion in the SIBERIA solver became im- portant [Willgoose et al., 1989]. Thus region 1 is small for catchments with little diffusive erosion and vice versa. In natural drainage basins, hillslope areas (small areas) are dominated by diffusive erosion and as area increases moving downstream erosion is increasingly dominated by fluvial processes. Channel flow composition patterns are determined by fluvial erosion so that region 2 is dominated by fluvial erosion and region 1 is dominated by diffusion. Moglen and Bras [1994] noted similar behavior with a variant of the SIBERIA model. They showed that an increase in diffusivity moved the transition from region 1 to region 2 to the right which is consistent with our hypothesis. 6. Analytical Solution for the Channel Network Domain It is possible to derive an explicit analytical solution to the scaling properties of the cumulative area distribution in the channel network domain, when drainage networks are described by the Tokunaga model. This result yields the log-log linearity observed. The first-order exterior link areas (ae) are 10 6 '7' 0 s 3 4 < 10 3 ', Figure 10. Simulated and analytical cumulative area distribution of an order-10 Tokunaga network with unit interior and exterior link areas for Tokunaga K = 2 and E1 = 1. assumed constant and interior link areas (ai) proportional to ae as ai = ca, where c is a constant. To simplify the derivation below, we initially assume a = 1, and c = 1, so that exterior and interior link areas are unity. A more general analytical solution is given later in this section. From (3) the Tokunaga cyclic system gives the number of n th-order channels directly draining laterally into an rnthorder channel,,,nn, as rnnn = E1 K(m-1)-n (4) Therefore the total number of channels directly draining laterally into the rnth-order channel is rn-1 Z mnn = E1 q- E1K q-... q- E1Km-3 q- E1 Km-2 n=l =El K-1 (5) ,,,, [ [ [ I [,][j ] I ] [][][J t I [ [][[IJ ] [ [ [ Observed Analytical 0.1 < 10 2 ;> 101 ', : Area/unit width (m) Figure 9. The area-slope data for the Middle Creek catchment in the Pokolbin region of Australia. The dots are the raw data from 20-m DTM, and the circles are average of 20 points with the same area to more clearly show the mean trend [ Willgoose, 1994] i i i i i iii 10 4 Figure 11. Observed and analytical cumulative area distribution of a synthetic catchment simulated by SIBERIA model using only fluvial erosion since no diffusive erosion modeled [1451lgoose, 1994].

6 , 1340 PERERA AND WILLGOOSE: PHYSICAL EXPLANATION _ g V, El= 1... x... K=2, E =O ! Figure 12. Analytical cumulative area distributions of order-10 Tokunaga networks with unit exterior and interior link areas for different Tokunaga K and E 1, so that the number of links in an m th-order channel L m can be written as K m-1-1 Lm = 1 + E1 K- 1 (6) With all links having unit area, the total area of an m th-order basin (am) equals the total number of links in that basin. This is equal to the cumulative area, m A 1 (number of pixels), which has the drainage area (a) greater than or equal to the firstorder area a 1 (where a 1 = a e = 1 node), mal(a --> al = 1) = am = 2m l- i (7) where mldb i (i = 1... m) is the number of ith-order channels in an order rn basin. In general, the cumulative area (man) of an m th-order drainage basin, which has a drainage area greater than or equal to the area of nth-order subbasin (an) within an mth-order basin, can then be written as (see Appendix A) man(a an)= 2m l- 1 + ml&n 2 + E 1 q- K + [(2 + E1 q- K) 2-8K] 1/2 Q = 2 (13) Knowing K and E 1, the drainage area for any given order basin and the cumulative area for that basin area can then be calcu- lated. This analytical solution for the values K - 2 and E is compared with the previously simulated 10th-order Tokunaga network with unit exterior and interior link areas (Figure 10). The fit of the analytical solution to the simulations is good, with the calculations for the analytical solution given in Appendix B. The effect of changes in the values of parameters K and E on the analytical solution has also been examined. Figure 12 compares the analytical CAD of 10th-order basin for K -- 4, E1 = 1;K= 2, E 1 = 1;K= 1.5, E 1 = 1;K= 2, E1 = 2; and K - 2, E1 = 0.5. From Figure 12, when K increases for constant E 1, the slope qb of the CAD decreases, and when K decreases, qb increases. There is no apparent change in when E 1 changes with constant K. This results are consistent with the previously discussed simulation results. Figure 12 shows that the slope of the CAD gradually changes, asymptotically reaching a constant with increasing area. This behavior was examined by computing the analytical CAD by increasing the order. The slopes of the distributions between area of order i and order i + 1 (i = 1, 2, 3,..- ) for catchments of orders 10, 20, and 30 are computed and shown in Figure 13. As area increases the scaling coefficient, qb converges to -0.5 for K = 2 and E 1 = 1. De Vries et al. [1994] derived a relationship of qb with Horton's constants and topological dimension. They computed qb for an infinite topologically random channel network, which is in agreement with our analytical results for Tokunaga parameters K - 2 and E1-1. The order-10 basin in Figure 13 did not attain this constant value because of the slow convergence. Middle Creek behaved similarly (Figure lb), not attaining a constant 4> because of small size. Figure 13 suggests that slow convergence and the finite size of natural catchments control variation of The slope of the CAD diverges from the constant value for higher drainage areas near to the outlet (Figure 13) because of the boundary effect of the networks. Hancock [1997] observed i 1 m /..L i 1 + E 1 K- 1 and the area of the n th-order subbasin is an = 2nt where, from Tokunaga [1978], mp'n = P(m-n-1)(2 + mp'm = 1 (Q(m-n-1)_ p(m-n-1))q(2 q_ E1_ p) + (Q_p) P= 2 2 1/2 ß "8" (8) _ -0,6 (9) ca (11) Order (i) Figuee 13, Slope of the cumulative area distributions for (12) varying order of Tokunaga networks with unit exterior and interior link areas using Tokunaga K -- 2 and E1 = 1.

7 PERERA AND WILLGOOSE: PHYSICAL EXPLANATION 1341 a similar scale-dependent variation of tk for his experimental laboratory catchments as well as for the synthetic catchments simulated by the SIBERIA model. We can derive an expression for the asymptotic exponent, tkoo, for the CAD with Tokunaga parameters. Peckham [1995] derived a relationship for tkoo with Horton ratios, log Rc log R (14) where R c = limm oo {Lm+l/Lm} is the link ratio. For equal link lengths (14) is similar to that of De Vdes et al. [1994]. Combining (6) and (14) yields K- i+e (K m-f- 1) } Rc = m-.->oo lira K- 1 + Ei(K Rc = m--->oo lim 1/- -:r2: From Tokunaga [1978] 1/K m- - / - 1/K m / ;-( / - + E ( /Kin m) )} (15) (16) Rc = K K > 1 (17) RB =--= lim {Q} (18) rnl&n (rn-n)-->oo Substituting for RB and Rc in (14) yields log (Q/K) tkoo = log Q (19) Exponent tkoo is computed for different values of K and E1 using in Table 1. These results show that tk is very sensitive to the parameter K but less so to the parameter El, which is compatible with our previous results. It can be observed from Figure 12 that all the CADs display equal slopes between areas of one pixel and five pixels (area of order-2 network) on x axis irrespective of the values of K or El. The reason for this behavior is that K effects only the number of branches in networks of order 2 and above. The assumption of unit area of interior and exterior links in the above derivation is not necessary. If it is assumed that the area of exterior first order links, a e (where a e = a 1), and area of all interior links a i are related by a i = ca, then the total area of an nth-order drainage basin given in (9) can be rewrit- ten as an-- al[n/ l + C(nl&l- 1)] (20) and the cumulative area in (8) for n - 1 can be rewritten as and forn -> 2,, 4 (a -> ao = a [ml q- C(ml - 1)] (21) man(a >an)--cal{ml& rnld n - m/ i 1 +El K- 1 i=2 so that for known average values of a l, c,, and E l, CAD, of a drainage basin can be analytically derived. The sole difference between (8) and (22) is the appearance of a factor reflecting the ratio of interior to exterior link areas. Table 1. Asymptotic Values of the Exponent tk for Different K and E 1 Computed Using Equation (20) K E 1 q[ We evaluated this general analytical solution on the two catchment used before, the Middle Creek catchment and synthetic catchment simulated by SIBERIA model. The analytical and observed CADs are compared in Figures la and 11. These figures show that the analytical solutions are in good agreement with observed CADs. 7. Discussion The cumulative area distribution (CAD) can be divided into three regions (Figure la). Region 1 represents the hillslope flow aggregation pattern of diffusive erosion transport dominated, upstream, drainage areas. Different hillslope flow aggregation patterns lead to different shapes for region 1. In general, the relationship between log area and log cumulative area observed in the field is nonlinear. In natural basins the hillslope flow paths will vary from subcatchmento subcatchment, so that region 1 is the average flow accumulation pattern of all first-order basins within the catchment. Region 2 is that part of the cumulative area distribution dominated by fluvial erosion and the channel network. The cumulative area distribution in region 2 is well described by a log-log linear power law and results from the scaling properties of the channel network. The scale exponent, k, for the Middle Creek basin is about Rodriguez-Iturbet al. [1992] studied the cumulative area distribution of five drainage basins in North America. They found the exponent k to be approximately equal to for all their basins. Inaoka and Takayasu [1993] found = for the river patterns simulated by their erosion model for landforms. La Barbara and Roth [1994] derived an equation for cumulative area distribution based on Horton's laws, related to the fractal properties of river networks, and found an exponent of = Sun e! al. [1994a, b, c] showed that their minimum energy dissipation drainage network model had an exponent of k This study concludes that this scale exponent ( ) is sensitive to the scaling properties of the channel network. In the Tokunaga network numbering model the scale behavior is controlled by the parameter K. We find that the value of exponent is highly sensitive to this parameter. The exponent k converges to a constant for higher-order catchments with the topologically random network (K = 2 and E 1 = 1), giving = Increasing K from 2 to 4 decreases the exponent k from approximately -0.5 to -0.25, inversely proportional to K. The parameter E1 does not appear to significantly affect scaling behavior of the network and thus. Region 3 represents those large drainage areas along the main trunk of the channel network. The boundary of regions 2 and 3 of the CAD move to the right when the order of the catchment is increased (Figures 6 and 7), indicating a boundary effect. Figure 13 shows that the slope of the CAD decreases rapidly when the area is close to the total basin area, indicating that the behavior of region 3 is a boundary effect. Ibbitt et al. [1997] found a similar trend in the slope of the CAD across the

8 1342 PERERA AND WILLGOOSE: PHYSICAL EXPLANATION range of areas shown in Figure 13 for the low Strahler order network. This paper has shown that the scale properties for drainage area can be simply parameterized with a scale invariant network ordering model. We believe on the basis of observed evidence [Rodriguez-Iturbet al., 1992; Inaoka and Takayasu, 1993; Sun et al., 1994a, b, c] and the physical explanation based on Tokunaga network that this log-log linear behavior is a universal property of observed landforms and results from the natural self-organizing property of channel networks during their development. The slope of the line appears to be about -0.5 for topologically random channel networks. Appendix A A detailed derivation of the analytical solution for the CAD based on the Tokunaga stream numbering scheme is given below. From (7), mal(a > al = 1) = a m = 2m/_l, (A1) The number of links in a second-order channel is given by (6) as K-1 L2 = I + ElK_ 1 (A2) Then the number of links or pixels which have the drainage area (a) less than second-order subbasin area (a2) within m th-order basins can be given as m/-l1 q- m/-l2 1 q- E K - m/.l2 (A3) The last term, m/,r2, is the number of links (or pixels) at the outlets of the second-order networks which have the drainage area equal to a 2. Therefore the cumulative area which has drainage area (a) greater than or equal to a 2 is ma 2(a --> a 2) = 2mp, mill, 1 -- mill'2 1 q- E1 K 1 + mill'2 For a third-order basin (a 3), ma3(a->a3) =2m/.rl- 1--m/.rl--m/.r2 1 +ELK_ 1 (A4) Table 2. Calculation of the Analytical Solution for an Order-10 Tokunaga Network With Unit Exterior and Interior Link Areas Appendix B Sample calculations of the analytical solution for an order-10 basin is given below. Tokunaga parameters K = 2 and E = 1 are used for calculations. Exterior and interior link areas are assumed as a unit (one pixel). From (12) and (13), [( ) 2-8 x 2] 1/2 P= 2 = [( ) 2-8 x 2] 1/2 Q= 2 =4 Substituting m = 10, P = 1, Q = 4, and E = 1 in (11) m/-rn can be calculated for n - 1 to 10 as in Table 2. Now, loan(a -> an) for n = 1 to 10 can be calculated as in Table 2 by substituting E = 1, K = 2, and m/,rn (n = 1 to 10) in (8). According to the Tokunaga model, n/-rl (for n = 1 to 9) = 10/-rl0-n+l (B1) and the calculated values are given in Table 2. Now, as all n/,r for n - 1 to 10 are known, a n values can be calculated from (9), as shown in Table 2. Acknowledgment. The research work presented in this paper was supported by the Australian Research Council (ARC). References De Vries, H., T. Becker, and B. Eckhardt, Power law distribution of --m 3 l+e1 K- 1 +m 3 discharge in ideal networks, Water Resour. Res., 30, , (AS) Hancock, G., Experimental testing of the SIBERIA landscape evolution model, Ph.D. thesis, Dep. of Civ., Surv. and Environ. Eng., In general, for an n th-order subbasin, Univ. of Newcastle, Callaghan, Australia, Horton, R. E., Erosional development of streams and their drainage basins: Hydrological approach to quantitative geomorphology, Bull. Geol. Soc. Am., 56, , man(a --> an) = 2mp, 1- I--rap, l--rap, 2 1 +ELK_ Ibbitt, R. P., G. R. Willgoose, and M. J. Duncan, The Ashley River channel network study: Channel network simulation models com-... mp'n 1 +El K- 1 + mp'n pared with data from the Ashley River, New Zealand, Misc. Rep. 319, Natl. Inst. for Water and Atmos. Res., Christchurch, New Zealand, Inaoka, H., and H. Takayasu, Water erosion as a fractal growth pro- = 2mla, 1-1 +mld, n- mld, i 1 + E1 K- 1 ' cess, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., 47(2), , Kirchner, J. W., Statistical inevitability of Horton's laws and the ap- ' The area of an n th-order subbasin is the total number of links parent randomness of stream channel network, Geology, 21, , in the subbasin; La Barbera, P2; and G2 R0th, invariance and SCaling properties i n the distribution of contributing area and energy in drainage basin, Hyan = 2n/-rl - 1 (A7) drol. Processes, 8, , 1994.

9 PERERA AND WILLGOOSE: PHYSICAL EXPLANATION 1343 Moglen, G. E., and R. L. Bras, Simulation of observed topography using a physically-based basin evolution model, Tech. Rep. 340, Ralph M. Parsons Lab., Dep. of Civ. Eng., Mass. Inst. of Technol., Cambridge, Peckham, S. D., Self-similarity in the three-dimensional geomorphology and dynamics of large river basins, Ph.D. thesis, Fac. of the Grad. School, Univ. of Colo., Boulder, Perera, H. J., The hydro-geomorphic modelling of sub-surface saturation excess runoff generation, Ph.D. thesis, Dep. of Civ., Surv. and Environ. Eng., Univ. of Newcastle, Callaghan, Australia, Rodriguez-Iturbe, I., E. J. Ijjasz-Vasquez, R. L. Bras, and D. G. Tarboton, Power law distributions of discharge, mass, and energy in river basins, Water Resour. Res., 28, , Scheidegger, A. E., Horton's law of stream numbers, Water Resour. Res., 4, , Shreve, R. L., Statistical law of stream numbers, J. Geol., 74, 17-37, Strahler, A. N., Hypsometric (area-altitude) analysis of erosional topography, Geol. $oc. Am. Bull., 63, , Sun, T., P. Meakin, and T. Jossang, The topography of optimal drainage basin, Water Resour. Res., 30, , 1994a. Sun, T., P. Meakin, and T. Jossang, A minimum energy dissipation model for drainage basins that explicitly differentiates between channel networks and hillslopes, Phys. A, 210, 24-47, 1994b. Sun, T., P. Meakin, and T. Jossang, Minimum energy dissipation model for river basin geometry, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., 49, , 1994c. Tarboton, D. G., Fractal river networks, Hortons law and Tokunaga cyclicity, J. Hydrol., 187, , Tarboton, D. G., R. L. Bras, and I. Rodriguez-Iturbe, The analysis of river basins and channel networks using digital terrain data, Tech. Rep. 326, Ralph M. Parsons Lab., Mass. Inst. of Technol., Cambridge, Tarboton, D. G., R. L. Bras, and I. Rodriguez-Iturbe, A physical basin for drainage density, Geomorphology, 5, 59-79, Tokunaga, E., Consideration on the composition of drainage networks and their evolution, Geogr. Rep. 13, Tokyo Metrop. Univ., Willgoose, G. R., A physical explanation for an observed area-slopeelevation relationship for catchments with declining relief, Water Resour. Res., 30, , Willgoose, G. R., R. L. Bras, and I. Rodriguez-Iturbe, A physically based channel network and catchment evolution model, Tech. Rep. 322, Ralph M. Parsons Lab., Mass. Inst. of Technol., Cambridge, Willgoose, G. R., R. L. Bras, and I. Rodriguez-Iturbe, A physically based coupled network growth and hillslope evolution model, 1, Theory, Water Resour. Res., 27, , 1991a. Willgoose, G. R., R. L. Bras, and I. Rodriguez-Iturbe, A physical explanation of an observed link-area slope relationship, Water Resour. Res., 27, , 1991b. H. Perera and G. Willgoose, Department of Civil, Surveying and Environmental Engineering, The University of Newcastle, University Drive, Callaghan, NSW 2308, Australia. ( cegrw@cc. newcastle.edu.au) (Received September 19, 1997; revised January 14, 1998; accepted January 23, 1998.)