Discrete Gaussian Vector Fields On Meshes

TL;DR

This paper develops discrete intrinsic Gaussian processes for vector-valued data on meshes, capturing geometry and curvature, and demonstrates their application to climate data downscaling and ocean current inference.

Contribution

It introduces a novel framework for discrete Gaussian vector fields on meshes that incorporates geometric properties and is applicable to environmental data modeling.

Findings

Successfully modeled wind data downscaling on a global scale

Captured harmonic flows and boundary conditions effectively

Inferred ocean currents from sparse observations

Abstract

Though the underlying fields associated with vector-valued environmental data are continuous, observations themselves are discrete. For example, climate models typically output grid-based representations of wind fields or ocean currents, and these are often downscaled to a discrete set of points. By treating the area of interest as a two-dimensional manifold that can be represented as a triangular mesh and embedded in Euclidean space, this work shows that discrete intrinsic Gaussian processes for vector-valued data can be developed from discrete differential operators defined with respect to a mesh. These Gaussian processes account for the geometry and curvature of the manifold whilst also providing a flexible and practical formulation that can be readily applied to any two-dimensional mesh. We show that these models can capture harmonic flows, incorporate boundary conditions, and model non-stationary data. Finally, we apply these models to downscaling stationary and non-stationary gridded wind data on the globe, and to inference of ocean currents from sparse observations in bounded domains.