Quantifying input and model errors in conceptual rainfall-runoff models using Bayesian total error analysis

Calibration and prediction in conceptual rainfall-runoff (CRR) modelling is affected by the sampling and measurement uncertainty in the forcing/response data and by the structural error of the model conceptualisation. This study presents a robust Bayesian Total Error Analysis methodology (BATEA) for dealing with these multiple sources of uncertainty. The core idea is to pose the CRR model calibration as a Bayesian hierarchical model with latent variables describing uncertainties in the data and the CRR model. This provides the opportunity to directly and comprehensively address all sources of uncertainty. A critical challenge is to characterize model error. In the past this has been thwarted by the convenient but indefensible treatment of CRR models as deterministic descriptions of catchment dynamics. Here it is argued that CRR models are fundamentally stochastic because sub-grid variability of catchment processes in time and space cannot be uniquely described by models operating at hillslope or larger scales. Acceptance that CRR models are intrinsically stochastic paves the way for a more rational description of model error. We advance the hypothesis that CRR model error can be characterized by storm-dependent random variation of one or more CRR model parameters. In addition to model error, we describe the treatment of data uncertainty in the observed input and output data using the hierarchical BATEA methodology. A practical implementation of BATEA requires the estimation of a large number of latent variables (realisations of storm-dependent model parameters). This yields high dimensional optimisation and sampling problems to estimate the most likely parameter sets and to characterise their uncertainty. Optimisation strategies using Newton-type methods are presented, along with practical implementations of Metropolis and Gibbs samplers to elicit the full posterior distribution of model parameters. Diagnostic tests guiding the selection of the distributions of latent variables are discussed. Case studies are presented illustrating the BATEA approach and the insights it generates about predictive uncertainty in CRR models operating under conditions of data and model uncertainty.