Abstract
The fractal scaling of river networks has been described in the context of both thin and fat fractals. Whereas the thin-fractal characterization is presented as a fractal dimension, D, derived from the length properties of the river channels, a fat-fractal characterization is given as a scaling exponent, β, derived from the behavior of river-channel area. Several authors have related D to the bifurcation ratio, Rb, and length ratio, Rl, of idealized Hortonian-network trees. Here, for these types of trees, we asymptotically relate β to Rb, Rl, and to a diameter exponent, Δ, which governs the downstream channel-widening process. Using this result, we present a linkage of D to β and discuss the implications of the relative values of Rb, Rl, and Δ on this linkage and on network channel behavior. Finally, we illustrate bias in the estimation of β from preasymptotic trees.
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