A supCBI process with application to streamflow discharge and a model reduction

We propose a new stochastic model for streamflow discharge timeseries as a jump-driven process, called a superposition of continuous-state branching processes with immigration (a supCBI process). It is a non-Markovian model having the capability of reproducing the subexponential autocorrelation found in the hydrological data. The Markovian embedding as a version of matrix analytic methods is applied to the supCBI process, successfully yielding analytical formulae of statistical moments and autocorrelation. The supCBI process is identified at study sites, where hourly streamflow discharge data are available. We also consider another Markovian embedding as a model reduction of the supCBI process to a continuous-time binary semi-Markov chain of high- and low-flow regimes. We show that waiting times can be modeled using a mixture of exponential distributions, suggesting that semi-Markov chains serve as effectively reduced models of the supCBI process.