A new semi-Bayesian robust estimation method for spatially distributed models

Models with spatially distributed parameters are commonly used in geophysical sciences. An important problem is to estimate the model's driving conditions using incomplete observations of the state variables,  obtained in-situ or remotely. This problem could be ill-posed when the information content of observations is not sufficient. The situation is further aggravated if the parameters of the model are not well known. A usual approach is to consider the extended formulation where the driving conditions and parameters are estimated simultaneously. The extended problem, however, often suffers from equifinality. For the high-dimensional distributed models (including those in hydraulics and hydrology applications) the variational approach is often used due to its computational feasibility. In this approach one looks for the mode of the posterior pdf using minimization methods. However, if the uniqueness of the solution is not guarantied, the posterior pdf could be multimodal or even infinite-modal, i.e. the mode may no longer be considered as a useful representation of the posterior pdf. In contrast, the mean of any (bounded) pdf is unique. The posterior mean is computed in full Bayesian methods, such as MCMC for example. However, these algorithms are not useful in high-dimensional estimation.

We present a hybrid semi-Bayesian method which combines the elements of Bayesian and variational estimation. Here, each distributed coefficient is represented as a product of its mean value and the 'shape' function. The latter are estimated using the variational estimation, whereas the means - using the direct Bayesian approach. The two steps constitute an estimation cycle, which is repeated after the information exchange. In a broad sense, the method can be considered as a variant of the variational Expectation-Maximization method. The method is illustrated in application to the convection-diffusion transport model (generalized Burgers equation). Next, the results of solving the river discharge estimation problem in the SWOT data processing context are presented. In the latter case the full Saint-Venant equations based hydraulic model SIC is used.