A comparative study of Markov chain Monte Carlo methods for conceptual rainfall‐runoff modeling

Abstract

One challenge that faces hydrologists in water resources planning is to predict the catchment's response to a given rainfall. Estimation of parameter uncertainty (and model uncertainty) allows assessment of the risk in likely applications of hydrological models. Bayesian statistical inference provides an ideal means of assessing parameter uncertainty, whereby prior knowledge about the parameter is combined with information from the available data to produce a probability distribution (the posterior distribution) that describes uncertainty about the parameter and serves as a basis for selecting appropriate values for use in modeling applications. Widespread use of Bayesian techniques in hydrology has been hindered by difficulties in summarizing and exploring the posterior distribution. These difficulties have been largely overcome by recent advances in Markov chain Monte Carlo (MCMC) methods that involve Monte Carlo sampling of the posterior distribution. This study compares four MCMC sampling algorithms in the context of rainfall‐runoff modeling. The algorithms compared include a conventional Metropolis‐Hastings algorithm used previously in hydrological applications which uses a combination of block and single‐site updating and an adaptive Metropolis algorithm that has characteristics that are well suited to model parameters with a high degree of correlation and interdependence, as is often evident in hydrological models. In addition to these, two other algorithms are evaluated to clarify the relative importance of updating all model parameters as a block versus updating each parameter one at a time. The MCMC techniques are compared for simplicity, ease of use, statistical efficiency in exploration of the parameter space, and speed of implementation, using 11 years of daily rainfall‐runoff data from the Bass river catchment in Australia. The results show that the adaptive Metropolis algorithm is superior in many respects and can offer a relatively simple basis for assessing parameter uncertainty in hydrological modeling studies and that the efficiency of the adaptive algorithm is not solely attributed to the block‐updating element of the algorithm.