Random Walk and Random Roughness Models of Drainage Networks

Abstract

A random walk model of a drainage network is generated on an underlying matrix by selecting at random the drainage direction out of the elementary areas. In a random roughness model, roughness heights of the elementary areas rather than drainage directions are assigned at random. The resulting topography of that model then uniquely determines the drainage pattern. An equivalence of the two models is suggested by the similarities, found in the mode of construction and by the equality of the probabilities of occurrence of some simple configurations. The random roughness model can yield values of primary probabilities, i.e., probabilities which can be used in the construction of a random walk model. The various types of drainage patterns are classified by the sets of primary probabilities which would generate them. Thus outcomes of homogeneous and isotropic matrices are classified as pure dendritic patterns. Outcomes of homogeneous matrices with probability sets derived from a sloping plane roughness model are classified as general dendritic (including parallel) patterns. Trellis, annular, and other patterns are considered as subgroups of the class of patterns generated on all other possible matrices.