Bayesian parameter inference in hydrological modelling using a Hamiltonian Monte Carlo approach with a stochastic rain model

Abstract. Conceptual models of the rainfall-runoff behaviour of hydrological catchments have proven to be useful tools for making probabilistic predictions. However, model parameters need to be calibrated to measured data and their uncertainty quantified. Bayesian statistics is a consistent framework for learning from observed data, in which knowledge about model parameters is described through probability distributions. One of the dominant sources of uncertainty in rainfall-runoff modelling is the true rainfall over the catchment, which often needs to be inferred from a few rain-gauge and runoff measurements. modelling this uncertainty naturally leads to stochastic differential equation models, which render traditional inference algorithms such as the Metropolis algorithm infeasible due to their expensive likelihood functions. Therefore, in hydrology and other applied fields of research, error models are traditionally oversimplified for ease of inference as additive errors on the output, leading to biased parameter estimates and unreliable predictions. However, thanks to recent advancements in algorithms and computing power, full-fledged Bayesian inference with stochastic models is no longer off-limits for hydrological applications. We demonstrate this with a case study from urban hydrology, for which we employ a highly efficient Hamiltonian Monte Carlo inference algorithm with a time-scale separation.