Application of stochastic time dependent parameters to improve the characterization of uncertainty in conceptual hydrological models

Abstract

The traditional description of a hydrological system with a deterministic, conceptual model and a lumped output error term does not explicitly consider the main mechanisms of uncertainty generation due to approximate process representation, unobserved variability in processes and influence factors, and input uncertainty. In this study, we test the description of such intrinsic uncertainty in conceptual models by making process rates stochastic through stochastic, time-dependent rate parameters. We analyze the advantages and challenges of this approach by using Bayesian inference to jointly estimate model parameters, parameters of the stochastic processes, and the time series of the stochastic parameters. Numerically, we use a particle filter to infer the stochastic time series and to approximate the marginal likelihood for Markov chain sampling of the constant parameters. Compared to the lumped error formulation, we achieve a more realistic description of uncertainty, as we obtain larger errors in intrinsic states, larger uncertainty during prediction than calibration periods (a feature missing for simple lumped error models), and autocorrelated model outputs. However, the additional degrees of freedom introduced with stochastic parameters can lead to an unintentional compensation for model deficits or input errors. This problem is symptomatic of model structure inadequacy or poor selection of the parameters that are made stochastic, and can be diagnosed through cross-validation and careful posterior analysis. The proposed approach is computationally more demanding and its implementation more challenging than the traditional description of hydrological systems using a deterministic model with a lumped error term. However, its advantages both in providing a more realistic representation of uncertainties and in diagnosing model deficiencies suggest its adoption and further development in future studies.