Hierarchical Bayesian Approach to Boundary Value Problems with Stochastic Boundary Conditions

Abstract

Boundary value problems are ubiquitous in the atmospheric and ocean sciences. Typical settings include bounded, partially bounded, global, and limited area domains, discretized for applications of numerical models of the relevant fluid equations. Often, limited area models are constructed to interpret intensive datasets collected over a specific region, from a variety of observational platforms. These data are noisy and they typically do not span the domain of interest uniformly in space and time. Traditional numerical procedures cannot easily account for these uncertainties. A hierarchical Bayesian modeling framework is developed for solving boundary value problems in such settings. By allowing the boundary process to be stochastic, and conditioning the interior process on this boundary, one can account for the uncertainties in the boundary process in a reasonable fashion. In the presence of data and all its uncertainties, this idea can be related through Bayes' theorem to produce distributions of the interior process given the observational data. The method is illustrated with an example of obtaining atmospheric streamfunction fields in the Labrador Sea region, given scatterometer-derived observations of the surface wind field.