Stochastic hydrological process models have two conceptual advantages over deterministic models. First, even though water flow in a well-defined environment is governed by deterministic differential equations, a hydrological system, at the level we can observe it, does not behave deterministically. Reasons for this behavior are unobserved spatial heterogeneity and fluctuations of input, unobserved influence factors, heterogeneity and variability in soil and aquifer properties, and an imprecisely known initial state. A stochastic model provides thus a more realistic description of the system than a deterministic model. Second, hydrological models simplify real processes. The resulting structural deficits can better be accounted for by stochastic than by deterministic models because they, even for given parameters and input, allow for a probability distribution of different system evolutions rather than a single trajectory. On the other hand, stochastic process models are more susceptible to identifiability problems and Bayesian inference is computationally much more demanding. In this paper, we review the use of stochastic, time-dependent parameters to make deterministic models stochastic, discuss options for the numerical implementation of Bayesian inference, and investigate the potential and challenges of this approach with a case study. We demonstrate how model deficits can be identified and reduced, and how the suggested approach leads to a more realistic description of the uncertainty of internal and output variables of the model compared to a deterministic model. In addition, multiple stochastic parameters with different correlation times could explain the variability in the time scale of output error fluctuations identified in an earlier study.
