Covariance Functions for Multivariate Gaussian Fields evolving temporally over Planet Earth

The construction of valid and flexible cross-covariance functions is a fundamental task for modelling multivariate space-time data arising from climatological and oceanographical phenomena. Indeed, a suitable specification of the covariance structure allows to capture both the space-time dependencies between the observations and the development of accurate predictions. For data observed over large portions of planet Earth it is necessary to take into account the curvature of the planet. Hence the need for random field models defined over spheres across time. In particular, the associated covariance function should depend on the geodesic distance, which is the most natural metric over the spherical surface. In this work, we provide a complete characterisation for the space-time cross-covariances, that are continuous, geodesically isotropic in the spatial variable and stationary in the temporal one, and discuss some fundaments and construction principles of matrix-valued covariances on spheres. We propose a flexible parametric family of matrix-valued covariance functions, with both marginal and cross structure being of the Gneiting type. We additionally introduce a different multivariate Gneiting model based on the adaptation of the latent dimension approach to the spherical context. Finally, we assess the performance of our models through simulation experiments, and we analyse a bivariate space-time data set of surface air temperatures and atmospheric pressures.