Identifying the probabilistic structure of drained areas as a function of hypsometry in river networks

Classical morphological descriptors of river networks have been developed in 2D planar system. The third z-dimension (elevation), however, is necessary to model landscape evolution, particularly in a mountainous region when taking into account transport processes such as in glaciology and erosion. This paper aims at analyzing 3D morphometric properties of channel networks on the basis of probabilistic relationships relating the drained area and the elevation. The methodology is an extension of the fundamental planar power-law description proposed by Rodriguez-Iturbe et al. () representing the probability that the drained area A is higher than a threshold area a, P[A>a]similar to a(-beta), where beta is constant and represents an invariant property of natural river networks. A new scaling factor taking into account hypsometry is introduced in the previous power-law relationship, as P[A>a]similar to a(-tau)F(a/theta(z)), where F is a scaling function and theta is a measure of the system scale and a function of hypsometry. The new power law has one empirical parameter tau. A procedure was developed to calibrate this parameter and to calculate the corresponding uncertainty for applications on a single basin and then extended to a set of 70 basins in the Southern Rocky Mountains. The results show that the 3D power law exhibits scale-invariance properties with respect to theta, when applied to a single basin. Looking at the entire set of 70 basins, the criterion value associated with the uncertainty model shows a moderate loss in scaling quality compared with the basin-per-basin description but nevertheless remains acceptable. This 3D power law provides a geomorphologic basis for applications in mountainous regions where the relation between drained area and hypsometry has to be considered. Copyright (c) 2014 John Wiley & Sons, Ltd.