Averaging Properties of Channel Networks Using Methods in Stochastic Branching Theory

Abstract

Methods in branching theory are used to average properties of channel networks, resulting in expressions for the instantaneous unit hydrograph (IUH) in terms of fundamental network characteristics (Z, α, β), where α parameterizes the link (channel segment) length distribution and β is a vector of hydraulic parameters. Several possibilities for Z are considered, including N, (N, D), (N, M), $\tilde{D}$ , and (N, $\tilde{D}$), where N is magnitude (number of first-order streams), D is topological diameter, M is order, and $\tilde{D}$ is mainstream length. Linear routing schemes, including translation, diffusion, and general linear routing, are used, and it is demonstrated that translation routing leads to an IUH identical to that obtained by use of the width function, where, for a given distance x, the width of a network is defined to be the number of links some point of which lies at channel distance x from the outlet (analogous to population size in branching theory).

The IUH is taken to be the conditional expectation of actual basin response given Z, and it is derived based on assumptions that the links are independent and identically distributed random variables and that the network is a member of a topologically random population. Uncertainty in use of this expectation to approximate actual basin response is given by the corresponding conditional variance.

Asymptotic (for large N,D, and $\tilde{D}$) results are available for several cases. For example, when Z = N asymptotic considerations lead to a Weibull probability density function for the IUH for all linear routing schemes, with only a single parameter depending on the particular routing method.

A simulation study compares different possibilities for Z in terms of ability to predict actual IUH characteristics. These characteristics include peak and time to peak.