Correcting the mathematical structure of a hydrological model via Bayesian data assimilation

The goal of model identification is to improve our understanding of the structure and behavior of a system so the model can be used to make inferences about its input-state-output response. It is conventional to preselect some model form and evaluate its "suitability" against historical data. If deemed unsuitable, ways must be found to "correct" the model through some intuitive process. Here, we discuss a Bayesian data assimilation process by which historical observations can be used to diagnose what might be wrong with the presumed mathematical structure of the model and to provide guidance toward fixing the problem. In previous work we showed how, given a suitable conceptual model for the system, the Bayesian estimation of structure (BESt) method can estimate the stochastic form for structural equations of a model that are consistent with historical observations at the spatiotemporal scale of the data while explicitly estimating model structural contributions to prediction uncertainty. However, a prior assumption regarding the form of the equations (an existing model) is often available. Here, we extend BESt to show how the mathematical form of the prior model equations can be corrected/improved to be more consistent with available data while remaining consistent with the presumed physics of the system. Conditions under which convergence will occur are stated. The potential of the extended BESt approach is demonstrated in the context of basin-scale hydrological modeling by correcting the equations of the HyMod model applied to the Leaf River catchment and thereby improving its representation of system input-state-output response.