Probabilistic structure of the distance between tributaries of given size in river networks

We analyze the distribution of the distances between tributaries of a given size (or of sizes larger than a given area) draining along either an open boundary or the mainstream of a river network. By proposing a description of the distance separating prescribed merging contributing areas, we also address related variables, like mean (or bankfull) flow rates and channel and riparian area widths, which are derived under a set of reasonable hydrologie assumptions. The importance of such distributions lies in their ecological, hydrologic, and geomorphic implications on the spreading of species along the ecological corridor defined by the river network and on the propagation of infections due to water-borne diseases, particularly in view of exact theoretical predictions explicitly using the alongstream distribution of confluences carrying a given flow. Use is made here of real river networks, suitably extracted from digital elevation models, optimal channel networks, and exactly solved tree-like constructs like the Peano and the Scheidegger networks. The results obtained redefine theoretically in a coherent and general manner and verify observationally the distribution function of the above distances and thus provide the general probabilistic structure of tributaries in river networks. Specifically, we find that the probability of exceedence of the alongstream distance d of tributaries of size larger than a has the explicit form P(≥d) = exp (-Cd/a^H/(1+H)), where C is a constant that depends on the choice of boundary conditions and H ≤ 1 is the Hurst exponent.