How Bayesian data assimilation can be used to estimate the mathematical structure of a model

Abstract

In previous work, we presented a method for estimation and correction of non-linear mathematical model structures, within a Bayesian framework, by merging uncertain knowledge about process physics with uncertain and incomplete observations of dynamical input-state-output behavior. The resulting uncertainty in the model input-state-output mapping is expressed as a weighted combination of an uncertain conceptual model prior and a data-derived probability density function, with weights depending on the conditional data density. Our algorithm is based on the use of iterative data assimilation to update a conceptual model prior using observed system data, and thereby construct a posterior estimate of the model structure (the mathematical form of the equation itself, not just its parameters) that is consistent with both physically based prior knowledge and with the information in the data. An important aspect of the approach is that it facilitates a clear differentiation between the influences of different types of uncertainties (initial condition, input, and mapping structure) on the model prediction. Further, if some prior assumptions regarding the structural (mathematical) forms of the model equations exist, the procedure can help reveal errors in those forms and how they should be corrected. This paper examines the properties of the approach by investigating two case studies in considerable detail. The results show how, and to what degree, the structure of a dynamical hydrological model can be estimated without little or no prior knowledge (or under conditions of incorrect prior information) regarding the functional forms of the storage–streamflow and storage–evapotranspiration relationships. The importance and implications of careful specification of the model prior are illustrated and discussed.

Appendix

Bias and Nash–Sutcliffe coefficient (NS) performance measures for the output predictions Q, when either the ensembles, expected value estimate, maximum likelihood estimate, or 25, 50, 75 and 95% probability mass regions are used as predictors: $Bias = {\frac{{\sum\nolimits_{t} {Q_{t}^{pred} - Q_{t}^{true} } }}{{\sum\nolimits_{t} {Q_{t}^{true} } }}}*100\% ,\,NS = 1 - {\frac{{\sum\nolimits_{t} {(Q_{t}^{pred} - Q_{t}^{true} )^{2} } }}{{\sum\nolimits_{t} {(Q_{t}^{true} - Q_{0}^{true} )^{2} } }}}.$ Where the ‘pred’ and ‘true’ super-indices denote predicted and true values, respectively, and Q ^true ₀ denotes the mean of the sequence of true values. For the probability mass regions, the measures are computed using pdf-weighted averages.