Abstract. Data scarcity and model over-parameterisation, leading to model equifinality and large prediction uncertainty, are common barriers to effective hydrological modelling. The problem can be alleviated by constraining the prior parameter space using parameter regionalisation. A common basis for regionalisation in the UK is the HOST database which provides estimates of hydrological indices for different soil classifications. In our study, Base Flow Index is estimated from the HOST database and the power of this index for constraining the parameter space is explored. The method is applied to a highly discretised distributed model of a 12.5 km 2 upland catchment in Wales. To assess probabilistic predictions against flow observations, a probabilistic version of the Nash-Sutcliffe efficiency is derived. For six flow gauges with reliable data, this efficiency ranged between 0.70 and 0.81, and inspection of the results shows that the model explains the data well. Knowledge of how Base Flow Index and interception losses may change under future land use management interventions was then used to further condition the model. Two interventions are considered: afforestation of grazed areas, and soil degradation associated with increased grazing intensity. Afforestation leads to median reduction in modelled runoff volume of 24% over the simulated 3 month period; and a median peak flow reduction ranging from 12 to 15% over the six gauges for the largest simulated event. Uncertainty in all results is low compared to prior uncertainty and it is concluded that using Base Flow Index estimated from HOST is a simple and potentially powerful method of conditioning the parameter space under current and future land management.

[1] When constructing a hydrological model at the macroscale (e.g., watershed scale), the structure of this model will inherently be uncertain because of many factors, including the lack of a robust hydrological theory at that scale. In this work, we assume that a suitable conceptual model structure for the hydrologic system has already been determined; that is, the system boundaries have been specified, the important state variables and input and output fluxes to be included have been selected, the major hydrological processes and geometries of their interconnections have been identified, and the continuity equation (mass balance) has been assumed to hold. The remaining structural identification problem that remains, then, is to select the mathematical form of the dependence of the output on the inputs and state variables, so that a computational model can be constructed for making simulations and/or predictions of the system input-state-output behavior. The conventional approach to this problem is to preassume some fixed (and possibly erroneous) mathematical forms for the model output equations. We show instead how Bayesian data assimilation can be used to directly estimate (construct) the form of these mathematical relationships such that they are statistically consistent with macroscale measurements of the system inputs, outputs, and (if available) state variables. The resulting model has a stochastic rather than deterministic form and thereby properly represents both what we know (our certainty) and what we do not know (our uncertainty) about the underlying structure and behavior of the system. Further, the Bayesian approach enables us to merge prior beliefs in the form of preassumed model equations with information derived from the data to construct a posterior model. As a consequence, in regions of the model space for which observational data are available, the errors in preassumed mathematical form of the model can be corrected, improving model performance. For regions where no such data are available the ''prior'' theoretical assumptions about the model structure and behavior will dominate. The approach, entitled Bayesian estimation of structure, is used to estimate water balance models for the Leaf River Basin, Mississippi, at annual, monthly, and weekly time scales, conditioned on the assumption of a simple single-state-variable conceptual model structure. Inputs to the system are uncertain observed precipitation and potential evapotranspiration, and outputs are estimated probability distributions of actual evapotranspiration and streamflow discharge. Results show that the models estimated for the annual and monthly time scales perform quite well. However, model performance deteriorates for the weekly time scale, suggesting limitations in the assumed form of the conceptual model.

[1] 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.Citation: Bulygina, N., and H. Gupta (2011), Correcting the mathematical structure of a hydrological model via Bayesian data assimilation, Water Resour. Res., 47, W05514,

[1] In hydrological modeling, two areas of application present particular challenges, first the modeling of ungauged catchments, and second the modeling of catchment nonstationarity; for example due to effects of land use change. The ungauged catchment problem requires that prior knowledge of the catchment is combined with evidence of behavior; for example from a regionalization exercise and/or spot flow measurements. Simulation of the effects of land use change requires that prior knowledge of the catchment is combined with information on the effects of that change on model parameters, generally in the absence of direct observations with which to condition the parameters. In both cases, ideally, all available sources of information about the behavior should be considered, and integrated in a way that maximizes the value of the information for model identification and uncertainty estimation. Using a formal Bayesian procedure, we combine three different sources of knowledge into a catchment scale conceptual model: (1) small-scale physical properties; (2) regionalized signatures of flow; and (3) available flow measurements. Applying the methodology to a distributed model for the Hodder catchment, UK, the physics-based information source contributed most to improving model performance, followed by peak flow times, and lastly the regionalized signatures. The flood frequency curve was evaluated under scenarios of land use change, and those changes that were significant relative to model uncertainty were identified.