Covariance Models and Simulation Algorithm for Stationary Vector Random Fields on Spheres Crossed with Euclidean Spaces

This paper focuses on vector random fields defined on $\mathbb{S}^d\times \mathbb{R}^k$, $d \geq 2$ and $k \geq 1$, with covariance functions that depend on the geodesic distance in $\mathbb{S}^d$ and on the separation vector in $\mathbb{R}^k$. First, we propose parametric families of nonseparable covariance functions with closed-form expressions and explicit spectral representations. Then, we derive an algorithm for fast simulation of such random fields, which combines spectral simulation methods in $\mathbb{S}^d$ and $\mathbb{R}^k$ previously introduced in the literature and relies on importance sampling techniques. We provide computer codes and practical guidelines and describe the advantages of our proposal in comparison to other methods.