A Markov Chain Monte Carlo Scheme for parameter estimation and inference in conceptual rainfall‐runoff modeling

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Abstract
A fully Bayesian approach to parameter estimation and inference in conceptual rainfall‐runoff models (CRRMs) is presented. Computations are performed using a Markov chain Monte Carlo (MCMC) method based on the Metropolis‐Hastings algorithm. Single‐site and block updating schemes are used for model parameters subject to nonnegativity restrictions as well as interval, equality, and order constraints. Diagnostics for the convergence of the Markov chain and CRRM assessment are also considered. The MCMC approach produces samples from the joint posterior distribution of the model parameters. This provides more information than single‐point estimates and avoids the need to use a normal approximation to the posterior as the basis for inference. The methodology is demonstrated using an eight‐parameter conceptual rainfall‐runoff model and two case studies from southeastern Australia. The first case study considers a watershed with high runoff yield over a 12‐year period. The second case study considers a watershed with low yield over a 17‐year period. The results indicate that (1) Bayesian methods provide an objective framework for model criticism and choice, (2) the proposed strategies for handling constraints on model parameters are effective, (3) the model parameters are sensitive to likelihood function selection, (4) the conventional approach of using a power transformation and an autoregressive process to stabilize error variance and model dependence in the residuals may have limited success, and (5) some care is required in the implementation of the MCMC approach and reliable results will be difficult to obtain when CRRM complexity exceeds the limitations of the rainfall‐runoff data at hand. A key finding is that the MCMC scheme presented herein provides a powerful means of identifying specific inadequacies in the structure of CRRMs.