Hortonian scaling and effect of cutoffs on statistics of self-affine river networks

The planar structure of river networks exhibits fractal properties. In particular, available data indicate a tendency for drainage areas A to be distributed according to a power law; their boundaries and main channels to form self-affine curves; the characteristic lengths of drainage areas to be independently self-affine in the one-dimensional space Z of total channel length, rendering the network elongated (and anisotropic); the inverse network density A/Z to be either constant or self-affine in Z, depending on whether or not the network fills the two-dimensional space of A; and (as found in a recent study) areas of given size, vegetative cover, and mean steady state soil moisture, weighted by their flow distance from the basin outlet, to be self-affine in the one-dimensional space of this distance. Various theoretical and semiempirical relationships have been proposed among exponents defining some of these and other scale dependencies. Expressions have been proposed for ensemble moments of A and some length measures associated with finite size river basins. We present a new model that views any self-affine basin property, Y(X), as belonging to an infinite hierarchy of mutually uncorrelated, statistically homogeneous random functions defined on elementary subbasins, each of which is characterized by a unique integral (spatial correlation) scale λ. We cite a mathematically rigorous hydrologic rationale for our model and use it in conjunction with published scaling relations to obtain the probability density function of λ; to derive analytical expressions for ensemble analogs of Horton's scaling laws; to deduce from them that streams of any Horton-Strahler order ω are associated with integral scales λ_ω ≤ λ ≤ λ_ω+1, where the ratio λ_ω+1/λ_ω is a constant independent of ω; to develop analytical expressions relating statistical moments of Y(X) to arbitrary lower and upper cutoff scales that may (but need not) be taken to represent data support and maximum watershed size; to describe ways of estimating the corresponding parameters; and to provide a theoretical basis for the heretofore unexplained observation that self-affine amplitude fluctuations of basin boundaries and main channels (as measured by their variance), having a common Hurst exponent, are larger in the former than in the latter.