Self-Affinity of Scheidegger’s River Patterns

Abstract

We investigated the scaling property of the drainage basin patterns generated by computer-simulation of the Scheidegger’s river network model, which is a type of directed percolation model. Growth exponents are obtained from the inclination of the log-log plot of the length L (parallel direction) and the width W (perpendicular direction) of the drainage basins as a function of the drainage area A, i.e., L∼A^ν⁄⁄ and W∼A^ν⊥. The values are found to be ν_⁄⁄=2⁄3 and ν_⊥=1⁄3, and the patterns are not self-similar, but self-affine. The relation ν_⁄⁄+ν_⊥=1 indicates that the inner structure is nonfractal or compact. Theoretical considerations are also given to these results.