Scaling Relations for Watersheds

TL;DR

This study investigates the morphology and response of watersheds to perturbations in 2D and 3D systems, revealing how spatial correlations affect fractal dimensions and scaling behaviors, with implications for natural landscapes.

Contribution

It provides new insights into how watershed morphology depends on spatial correlations and confirms theoretical relations between exponents in 3D systems.

Findings

Fractal dimension decreases with Hurst exponent.

Watersheds are highly sensitive to local perturbations.

Scaling exponents depend on the Hurst exponent.

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 $1.10 \leq d_f \leq 1.15$ 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.