Scaling Relations for Watersheds

Abstract

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.

References in corpus (5)

• Explosive percolation via control of the largest cluster
• Optimal path cracks in correlated and uncorrelated lattices
• New efficient methods to calculate watersheds
• Impact of Perturbations on Watersheds
• Fractality of eroded coastlines of correlated landscapes

Cited by in corpus (17)

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• Bohman-Frieze-Wormald model on the lattice, yielding a discontinuous percolation transition
• Retention capacity of random surfaces
• A universal approach for drainage basins
• Corrections to Scaling for Watersheds, Optimal Path Cracks, and Bridge Lines
• Stacked triangular lattice: Percolation properties
• Schramm-Loewner evolution and perimeter of percolation clusters of correlated random landscapes
• How to share underground reservoirs
• The influence of statistical properties of Fourier coefficients on random surfaces
• Exact evaluation of the cutting path length in a percolation model on a hierarchical network
• Retention capacity of correlated surfaces
• Minimal spanning trees at the percolation threshold: a numerical calculation
• Loop erased random walk on percolation cluster: Crossover from Euclidean to fractal geometry
• Loop erased random walk on a percolation cluster is compatible with Schramm-Loewner evolution