The incremental increase in contributing area along a main stem river, called here the incremental area function (IAF), has direct relevance to the spatial heterogeneity of environmental fluxes (water, sediment, nutrients, etc.) entering the stream from hillslopes and side tributaries. It also dictates, to a large extent, possible ecohydrologic discontinuities or transitions resulting from large tributary contributions. Mathematically, the IAF directly reflects the topological and geometrical structure of the river network and maps the two-dimensional landscape organization into a one-dimensional function. In this paper, we use two approaches to investigate the spatial heterogeneity of the IAF. First, we implement a multithreshold decomposition on IAF to study the distribution of distances between tributaries as a function of the imposed threshold contributing area and verify the presence of a simple power law scaling relationship between the threshold and the average distance between tributaries. Second, we use a wavelet-based multiscale approach and document the presence of statistical self-affinity (multifractality) in the IAF with a high intermittency coefficient, reflecting the complex arrangement of extreme contributions of different size tributaries. We propose a multiplicative cascade model, parameterized in terms of basin-specific properties, to statistically simulate the IAF along the main stem. Finally, we point out the relation between the IAF and the widely used width function of a basin and show how the latter can be constructed from the former via a convolution on the self-similar structure of a tree.
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