Corruption of accuracy and efficiency of Markov chain Monte Carlo simulation by inaccurate numerical implementation of conceptual hydrologic models

Abstract

Conceptual rainfall‐runoff models have traditionally been applied without paying much attention to numerical errors induced by temporal integration of water balance dynamics. Reliance on first‐order, explicit, fixed‐step integration methods leads to computationally cheap simulation models that are easy to implement. Computational speed is especially desirable for estimating parameter and predictive uncertainty using Markov chain Monte Carlo (MCMC) methods. Confirming earlier work of Kavetski et al. (2003), we show here that the computational speed of first‐order, explicit, fixed‐step integration methods comes at a cost: for a case study with a spatially lumped conceptual rainfall‐runoff model, it introduces artificial bimodality in the marginal posterior parameter distributions, which is not present in numerically accurate implementations of the same model. The resulting effects on MCMC simulation include (1) inconsistent estimates of posterior parameter and predictive distributions, (2) poor performance and slow convergence of the MCMC algorithm, and (3) unreliable convergence diagnosis using the Gelman‐Rubin statistic. We studied several alternative numerical implementations to remedy these problems, including various adaptive‐step finite difference schemes and an operator splitting method. Our results show that adaptive‐step, second‐order methods, based on either explicit finite differencing or operator splitting with analytical integration, provide the best alternative for accurate and efficient MCMC simulation. Fixed‐step or adaptive‐step implicit methods may also be used for increased accuracy, but they cannot match the efficiency of adaptive‐step explicit finite differencing or operator splitting. Of the latter two, explicit finite differencing is more generally applicable and is preferred if the individual hydrologic flux laws cannot be integrated analytically, as the splitting method then loses its advantage.