Fast and exact simulation of isotropic Gaussian random fields on $\mathbb{S}^{2}$ and $\mathbb{S}^{2}\times \mathbb{R}$

TL;DR

This paper introduces a fast, exact simulation method for isotropic Gaussian random fields on spheres and their extensions over space-time, utilizing block circulant matrices on regular grids for computational efficiency.

Contribution

It develops a novel simulation technique leveraging block circulant matrices for Gaussian fields on spheres and space-time, improving speed and accuracy over existing methods.

Findings

Enables fast and exact simulation of isotropic Gaussian fields on spheres.

Extends the method to space-time Gaussian fields with covariance depending on geodesic distance and time.

Uses regular grids and block circulant matrices for computational efficiency.

Abstract

We provide a method for fast and exact simulation of Gaussian random fields on spheres having isotropic covariance functions. The method proposed is then extended to Gaussian random fields defined over spheres cross time and having covariance functions that depend on geodesic distance in space and on temporal separation. The crux of the method is in the use of block circulant matrices obtained working on regular grids defined over longitude $\times$ latitude.