Gibbs' Distribution on drainage networks

Abstract

We propose a spatial stochastic model for drainage networks defined on a square lattice of points. The probability of a particular spanning tree s draining a given basin represented by a set of lattice points is proportional to exp [−β H ( s )], where β is a parameter to be estimated and H ( s ) is defined to be the difference between total flow distance to the outlet through the drainage tree and total distance along shortest paths to the outlet. Thus H ( s ) is a global measure of sinuosity of the channels constituting the drainage tree. This probability distribution on trees is known as Gibbs' distribution and is well known in statistical mechanical contexts. The distribution with β = 0 (a uniform distribution) has been studied in a previous paper by the authors and was found to yield networks that are too sinuous. In this paper, β is allowed to be greater than zero, which produces more realistic networks that are not as sinuous. The question of spatial variability of β is addressed from an equilibrium point of view using ideas from statistical mechanics. To examine the question of spatial variability, the parameter β is defined in a Bayesian sense and allowed to vary randomly from basin to basin. In a case study we express β in terms of a log linear model with three independent variables: pixel size (grid spacing), drainage area, and average channel slope. We then estimate the parameters of this model with a set of 50 drainage trees obtained using digital elevation data for Willow Creek in Montana. Parameter estimates indicate that β tends to vary directly with slope and inversely with drainage area, which suggests heterogeneities within larger basins that are not adequately accounted for by Gibbs' distribution with a constant parameter value.