The SPDE approach for spatio-temporal datasets with advection and diffusion

TL;DR

This paper introduces an advection-diffusion SPDE model for efficient prediction of large-scale spatio-temporal environmental data, combining physics-inspired statistical modeling with numerical methods.

Contribution

It develops a novel nonseparable spatio-temporal model based on SPDEs with a finite difference and finite element discretization, including stabilization techniques.

Findings

Effective modeling of solar radiation data

Enhanced computational efficiency for large datasets

Discussion of model advantages and limitations

Abstract

In the task of predicting spatio-temporal fields in environmental science using statistical methods, introducing statistical models inspired by the physics of the underlying phenomena that are numerically efficient is of growing interest. Large space-time datasets call for new numerical methods to efficiently process them. The Stochastic Partial Differential Equation (SPDE) approach has proven to be effective for the estimation and the prediction in a spatial context. We present here the advection-diffusion SPDE with first order derivative in time which defines a large class of nonseparable spatio-temporal models. A Gaussian Markov random field approximation of the solution to the SPDE is built by discretizing the temporal derivative with a finite difference method (implicit Euler) and by solving the spatial SPDE with a finite element method (continuous Galerkin) at each time step. The ''Streamline Diffusion'' stabilization technique is introduced when the advection term dominates the diffusion. Computationally efficient methods are proposed to estimate the parameters of the SPDE and to predict the spatio-temporal field by kriging, as well as to perform conditional simulations. The approach is applied to a solar radiation dataset. Its advantages and limitations are discussed.