The proposed research looks at a new class of spatial models derived from the convolution representation of Gaussian process models. By expanding the class of distributions for the underlying process being convolved, a range of flexible spatial models results. These models are especially useful for inverse problems and spatial processes over time.

The research is motivated by an inverse problem from hydrology and a space-time problem from meteorology. The research will address both theoretical and methodological aspects within a Bayesian framework. In terms of flexible convolution models, this research will specifically examine convolutions of Markov random fields and convolutions of temporally evolving processes, and will put these into a general framework of convolutions of normally-distributed processes.

In the areas of both hydrology and meteorology, physical processes and available data exist at multiple scales, so multi-resolution versions of these models are needed. From an implementation standpoint, new computational methods are necessary. Particularly for inverse problems such as the hydrology example, evaluating the likelihood is computationally expensive, thus requiring efficient methods. The proposed research also addresses these computational needs, exploring the use of coupled chains, parallel computing, and surrogate modeling.