  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.
