Hybrid Bayesian Smoothing on Surfaces

Abstract: Modeling spatial processes that exhibit both smooth and rough features poses a significant challenge. This is especially true in fields where complex physical variables are observed across spatial domains. Traditional spatial techniques, such as Gaussian processes (GPs), are ill-suited to capture sharp transitions and discontinuities in spatial fields. In this paper, we propose a new approach incorporating non-Gaussian processes (NGPs) into a hybrid model which identifies both smooth and rough components. Specifically, we model the rough process using scaled mixtures of Gaussian distributions in a Bayesian hierarchical model (BHM). Our motivation comes from the Community Earth System Model Large Ensemble (CESM-LE), where we seek to emulate climate sensitivity fields that exhibit complex spatial patterns, including abrupt transitions at ocean-land boundaries. We demonstrate that traditional GP models fail to capture such abrupt changes and that our proposed hybrid model, implemented through a full Gibbs sampler. This significantly improves model interpretability and accurate recovery of process parameters. Through a multi-factor simulation study, we evaluate the performance of several scaled mixtures designed to model the rough process. The results highlight the advantages of using these heavier tailed priors as a replacement to the Bayesian fused LASSO. One prior in particular, the normal Jeffrey's prior stands above the rest. We apply our model to the CESM-LE dataset, demonstrating its ability to better represent the mean function and its uncertainty in climate sensitivity fields. This work combines the strengths of GPs for smooth processes with the flexibility of NGPs for abrupt changes. We provide a computationally efficient Gibbs sampler and include additional strategies for accelerating Monte Carlo Markov Chain (MCMC) sampling.