Scaling relations for watersheds

We study the morphology of watersheds in two and three dimensional systems subjected to different degrees of spatial correlations. The response of these objects to small, local perturbations is also investigated with extensive numerical simulations. We find the fractal dimension of the watersheds to generally decrease with the Hurst exponent, which quantifies the degree of spatial correlations. Moreover, in two dimensions, our results match the range of fractal dimensions $1.10\le d_f\le 1.15$ observed for natural landscapes. We report that the watershed is strongly affected by local perturbations. For perturbed two and three dimensional systems, we observe a power-law scaling behavior for the distribution of areas (volumes) enclosed by the original and the displaced watershed and for the distribution of distances between outlets. Finite-size effects are analyzed and the resulting scaling exponents are shown to depend significantly on the Hurst exponent. The intrinsic relation between watershed and invasion percolation, as well as relations between exponents conjectured in previous studies with two dimensional systems, are now confirmed by our results in three dimensions.

Figure 1

Example of a watershed for uncorrelated systems with linear size $L=129$ lattice sites in (a) two and (b) three dimensions. The upper and lower catchments drain to the top and bottom boundary, respectively.

Figure 2

The fractal dimension $d_f$ of the watershed as a function of the Hurst exponent $H$ of the system in two (red stars) and three dimensions (green open circles) according to left and right hand axes, respectively. In two dimensions, each point corresponds to an estimate of the fractal dimension by fitting the power law in Eq. (1) to the watershed masses for system sizes $L=\{5,17,65,257,1025,4097\}$. For each system size, the mass was averaged over ${10}^4$ landscape realizations. In three dimensions, the system sizes $L=\{5,9,17,33,65,129,257\}$ were used with the same number of realizations. The lines show the fractal dimension of the watershed obtained for the Alps (solid) and close to the Big Lakes (dotted), characterizing, according to the left hand axes, the range of estimates for natural landscapes as summarized in Table . In the typical range of natural landscapes, $0.3<H<0.5$, the simulation results are in agreement with the natural ones.

Figure 3

(a) Example of a displacement (top, green) of the original watershed [bottom, red, the same as in Fig. 1a] after a perturbation at the spot marked by a cross for an uncorrelated artificial landscape. The dotted line connects the two outlets of the area before (cross) and after (dot) perturbation and defines the distance $R$. (b) The watershed close to the big lakes in the United States [light (red) line]. A perturbation of 2 m at a spot (cross) near Thunder Bay caused a change in the watershed [dark (blue) line]. The watershed displacement encloses an area of about 3730 km${}^2$ [29]. The dot marks the new outlet of the area after the perturbation.

Figure 4

(a) Data collapse for the distribution $P\left(N_s\right|R)$ with $R=1$ (filled) and $R=10$ (open) and system sizes $L=\{129,257,513\}$ (triangles, circles, and squares, respectively), using the scaling $P\left(N_s\right|R)=L^{2\alpha }f\left[N_s{L}^2\right]$. The solid lines represent fits to the data of a power law with exponent $\alpha =2.23\pm 0.03$. The inset shows the collapse of the same data for $L=513$, when the $x$ axis is rescaled by $R^d$, visualizing the lower cutoff from the power-law behavior. (b) The average number of sites (area) $〈N_s〉$ enclosed by the original and the perturbed watershed at an outlet distance $R$ for the same landscapes and system sizes as in (a). The lines show the expression given by Eq. (4). Each data point, in both (a) and (b), corresponds to an average over 2000 realizations of linear system size $L=513$ and over 4000 for $L=129$ and 257. The error bars are smaller than the size of the symbols.

Figure 5

The average mass $〈M〉$ of the invasion percolation cluster connecting the two outlets with a distance $R$ for uncorrelated two dimensional landscapes of sizes $L=\{129,257,513\}$ (triangles, circles, and squares, respectively). The lines show the predictions according to Eq. (6) for different system sizes. The inset shows the average mass varying with the number of enclosed sites. The line is a power-law fit to the data yielding an exponent $0.95\pm 0.01\approx d_f^{*}/2$. Each data point corresponds to an average over 2000 realizations of systems with size $L=513$ and over 4000 for $L=129$ and 257. The error bars are smaller than the size of the symbols.

Figure 6

Data collapse of the size distribution $P\left(M\right)$ of the mass $M$ of the invasion percolation cluster connecting the two outlets for uncorrelated landscapes of three different system sizes $L=\{129,257,513\}$ (triangles, circles, and squares, respectively). The line represents a power-law fit to the data for the largest landscape (squares) yielding an exponent $1+{\beta }^{*}=1.21\pm 0.02$. Each data point is an average over 2000 realizations of systems with size $L=513$ and over 4000 for $L=129$ and 257. The error bars are smaller than the size of the symbols.

Figure 7

(a) The largest of all observed changed volumes (dark, blue) attached to the perturbed watershed (light, red) for the same uncorrelated three dimensional system of linear size $L=129$, as used in Fig. 1b. The upper and lower catchments drain to the top and bottom borders, respectively. (b) The same changed volume as in (a) without the watershed for better visibility. The lines mark the sites where the original watershed intersects the system boundaries.

Figure 8

Data collapse of the distribution $P\left(N_s\right)$ of the number of sites (volume) enclosed by the original and the perturbed watershed for uncorrelated three dimensional systems of size $L=\{33,65,129\}$, (squares, circles, triangles, respectively). The line represents a power-law fit to the data yielding an exponent $\beta =1.31\pm 0.05$. Each data point is an average over 2000 realizations. The error bars are smaller than the size of the symbols.

Figure 9

Number of cubic boxes $C$ of size $\varepsilon$ covering the changed volume, shown in Fig. 7, as obtained from a box counting method. Each data point consists of a single measurement. The line corresponds to ${\varepsilon }^{2.48}$.

Figure 10

The exponents $\alpha ,\beta ,\rho ,\sigma$ (squares, circles, triangles up, and triangles down, respectively) as a function of the Hurst exponent $H$ for perturbations in 3D. Each data point consists of a similar measurement as performed in Fig. 8 to obtain $\beta$ for uncorrelated systems.