🤖 AI Summary
This study addresses the challenge of modeling spatiotemporal convection–diffusion processes defined on two-dimensional manifolds, such as the sphere, by proposing a Gaussian random field framework grounded in convection–diffusion stochastic partial differential equations (SPDEs). The method constructs a covariance structure via Galerkin discretization on smooth, compact Riemannian manifolds and enables scalable spatiotemporal prediction through Bayesian inference. To the best of our knowledge, this work is the first to systematically apply the convection–diffusion SPDE framework to domains with complex geometry, thereby integrating physical interpretability with statistical flexibility. Empirical evaluations demonstrate that the model achieves both high predictive accuracy and computational efficiency on simulated spherical data as well as real-world global aerosol optical thickness observations.
📝 Abstract
The aim of this work is to propose a statistical model for spatio-temporal data on meshed surfaces based on the Stochastic Partial Differential Equation (SPDE) modeling approach. Specifically, we focus on a class of advection-diffusion SPDEs defined on smooth compact orientable closed Riemannian manifolds of dimension 2, and their discretization via a Galerkin approach. We demonstrate how this method enables the development of scalable algorithms for the simulation and prediction of Gaussian random fields that are solutions to the discretized SPDE. Additionally, we present recent developments in the inference of such models. The method is applied to a simulated spatio-temporal dataset exhibiting advective and diffusive behavior on the sphere, as well as to a real case study on aerosol optical depth in the atmosphere across the globe's surface.