Towards a Complete Picture of Stationary Covariance Functions on Spheres Cross Time

With the advent of wide-spread global and continental-scale spatiotemporal datasets, increased attention has been given to covariance functions on spheres over time. This paper provides results for stationary covariance functions of random fields defined over $d$-dimensional spheres cross time. Specifically, we provide a bridge between the characterization in \cite{berg-porcu} for covariance functions on spheres cross time and Gneiting's lemma \citep{gneiting2002} that deals with planar surfaces. We then prove that there is a valid class of covariance functions similar in form to the Gneiting class of space-time covariance functions \citep{gneiting2002} that replaces the squared Euclidean distance with the great circle distance. Notably, the provided class is shown to be positive definite on every $d$-dimensional sphere cross time, while the Gneiting class is positive definite over $\R^d \times \R$ for fixed $d$ only. In this context, we illustrate the value of our adapted Gneiting class by comparing examples from this class to currently established nonseparable covariance classes using out-of-sample predictive criteria. These comparisons are carried out on two climate reanalysis datasets from the National Centers for Environmental Prediction and National Center for Atmospheric Research. For these datasets, we show that examples from our covariance class have better predictive performance than competing models.