Monte Carlo assessment of parameter uncertainty in conceptual catchment models: the Metropolis algorithm George Kuczera a , *, Eric Parent b a Department of Civil, Surveying and Environmental Engineering, University of Newcastle, NSW, 2308,Callaghan, Australia b Laboratoire de Gestion du Risque En Sciences de l’Eau, ENGREF, 19 av du Maine, 75732, Cedex 15, Paris, France Received 6 February 1998; accepted 13 August 1998 Abstract Two Monte Carlo-based approaches for assessing parameter uncertainty in complex hydrologic models are considered. The first, known as importance sampling, has been implemented in the generalised likelihood uncertainty estimation (GLUE) framework of Beven and Binley. The second, known as the Metropolis algorithm, differs from importance sampling in that it uses a random walk that adapts to the true probability distribution describing parameter uncertainty. Three case studies are used to investigate and illustrate these Monte Carlo approaches. The first considers a simple water balance model for which exact results are known. It is shown that importance sampling is inferior to Metropolis sampling. Unless a large number of random samples are drawn, importance sampling can produce seriously misleading results. The second and third case studies consider more complex catchment models to illustrate the insights the Metropolis algorithm can offer. They demonstrate assessment of parameter uncertainty in the presence of bimodality, evaluation of the significance of split-sample tests, use of prior information and the assessment of confidence limits on hydrologic responses not used in calibration. When compared with the capabilities of traditional inference based on first-order approximation, the Metropolis algorithm provides a quantum advance in our capability to deal with parameter uncertainty in hydrologic models.  1998 Elsevier Science B.V. All rights reserved. Keywords: Bayesian inference; Conceptual catchment models; Importance sampling; Markov Chain Monte Carlo sampling; Parameter uncertainty; Rainfall-runoff models 1. Introduction This study is concerned with the calibration of conceptual catchment models and, in particular, the realistic quantification of parameter uncertainty and its effect on model predictions made using data independent of the calibration data. A conceptual catchment model lies somewhere between physically- based reductionist models and black box models. Physically-based reductionist models (Grayson et al., 1992) attempt to scale up to the catchment scale the known physics of the hydrologic phenomena at the laboratory scale. Black box models such as neural networks (Chen et al., 1990) and ARMA models (Box and Jenkins, 1976) make no explicit attempt to employ the known physics of the hydrologic phenom- enon. Conceptual models attempt to avoid the scaling problems encountered in reductionist models by focussing on the processes believed by the hydrologist to be dominant and using control volumes over which state variables and fluxes are temporally and spatially averaged (Nash and Sutcliffe, 1970). Although a mass balance is enforced for each control volume, the flux Journal of Hydrology 211 (1998) 69–85 0022-1694/98/$ - see front matter  1998 Elsevier Science B.V. All rights reserved. PII: S0022-1694(98)00198-X * Corresponding author. Fax: 0061 2492 16991; E-mail: cegak @cc.newcastle.edu.au