A catalogue of nonseparable positive semidefinite kernels on the product of two spheres

We present a catalogue of 26 parametric families of matrix-valued positive semidefinite kernels for modeling the spatial correlation structure of vector random fields defined over the product of two hyperspheres. Such a geometry has been proved successful to account for complex sources of seasonality and direction-dependence of phenomena regionalized on a large portion of planet Earth. All the kernels are nonseparable, as they cannot be written as the product of positive semidefinite kernels defined on hyperspheres, and sufficient validity conditions on their parameters are identified. Their analytical spectral representations and a spectral simulation algorithm are also provided. A side product is the derivation of new matrix-valued isotropic covariance kernels on hyperspheres, together with their analytical spectral representations.