Simple statistical models for river networks

Abstract

The geometric scaling properties associated with simple river network models have been studied using computer simulations. In these models the river networks are comprised of random trajectories which start at randomly selected positions on a square lattice and continue until they either reach an edge of the lattice or join a previous trajectory to form a branched network. For those cases where the trajectories are either self-avoiding random walks (SAWs) or indefinitely growing self-avoiding walks (IGSAWs) the river basins are compact but have fractal basin boundaries with a dimensionally near to 5/4. For the IGSAW model the longest river is a self-similar fractal with a fractal dimensionality close to or equal to that of the IGSAW itself. If the trajectory is a directed walk, then the model is very similar to the Scheidegger river network model. In this case the basin boundary and the longest channel are self-affine Brownian processes. The results obtained from the IGSAW and directed walk models can be described in terms of a simple scaling model. For the SAW models this scaling picture does not apply and it appears that practical simulations are far from the asymptotic (large system size) limit.