Extending generalized Horton laws to test embedding algorithms for topologic river networks

River networks in the landscape can be described as topologic rooted trees embedded in a three-dimensional surface. We examine the problem of embedding topologic binary rooted trees (BRTs) by investigating two space-filling embedding procedures: Top-Down, previously developed in the context of random self-similar networks (RSNs), and Bottom-Up, a new procedure developed here. We extend the concept of generalized Horton laws to interior sub catchments and create a new set of scaling laws that are used to test the embedding algorithms. We compare the two embedding strategies with respect to the scaling properties of the distribution of accumulated areas Aω and network magnitude Mω for complete order streams ω. The Bottom-Up procedure preserves the equality of distributions A ω / E A ω = d M ω / E M ω ; a feature observed in real basins. The Top-Down embedded networks fail to preserve this equality because of strong correlations of tile areas in the final tessellation. We conclude that the presence or absence of this equality is a useful test to diagnose river network models that describe the topology/geometry of natural drainage systems. We present some examples of applying the embedding algorithms to self similar trees (SSTs) and to RSNs. Finally, a technique is presented to map the resulting tiled region into a three-dimensional surface that corresponds to a landscape drained by the chosen network. Our results are a significant first step toward the goal of creating realistic embedded topologic trees, which are also required for the study of peak flow scaling in river networks in the presence of spatially variable rainfall and flood-generating processes.