A physical explanation of the cumulative area distribution curve

A physical explanation for the behavior of the cumulative area distribution (CAD) based on the Tokunaga channel network model is given. The CAD is divided into three regions. The first region, for small areas, is dependent on hillslope flow accumulation patterns and represents the catchment average of the hillslope flow accumulation in the diffusive erosion-dominated areas, upstream reaches, of the catchment. The second region represents that portion of the catchment dominated by fluvial erosion. This region is well described by a log-log linear power law, which results from the scaling properties of the channel network. The scale exponent, phi, is highly sensitive to a parameter of the Tokunaga stream numbering scheme. The exponent phi converges to -0.5 for higher order Tokunaga networks for parameters consistent with topological random networks. Small networks have lower values of phi, which asymptotic converges to phi = -0.5 as the catchment scale increase. The third region reflects the lowest reaches of the channel network, the scale of the catchment, and is a boundary effect. An explicit analytical solution to the scaling properties in the second region is derived on the basis of the Tokunaga network model.