A stochastic Tokunaga model for stream networks

The Tokunaga cyclic model describes average network topology. A stochastic generalization is proposed. The stochastic model assumes that actual tribut ary numbers are random realizations from a negative binomial distribution w hose mean is defined by the Tokunaga parameters epsilon(1) and K. These par ameters can be interpreted as representing the effects of regional controls . Upon these regional controls is superimposed an inherent spatial variabil ity in network topology. A third parameter alpha characterizes this spatial variability. When alpha becomes large, the negative binomial model approac hes a Poisson model. A goodness-of-fit test based on a chi(2) test statisti c is developed, and an inference framework for estimation of parameters and stream-related statistics is described. This methodology is illustrated on tributary data from three catchments, one of the order of 5 and two of the order of 8. It is shown that the stochastic Tokunaga model using the negat ive binomial distribution is not inconsistent with the tributary data, wher eas the Poisson model is unambiguously rejected by the data. Monte Carlo Ba yesian methods are used to evaluate the uncertainty in the Tokunaga paramet ers and in stream number related statistics such as the bifurcation ratio. It is shown that tributary data from the order-5 network provide little pow er for discriminating between model hypotheses. The tributary data for the two order-8 basins are significantly different from the asymptotic stream n umber statistics predicted by Shreve's random network model. Finally, the p roblem of space filling or preservation of nontopological properties is con sidered in the context of the stochastic Tokunaga model.