Covariance functions for mean square differentiable processes on spheres

Many applications in spatial statistics involve data observed over large regions on the Earth’s surface. There is a large statistical literature devoted to covariance functions capable of modeling the degree of smoothness in data on Euclidean spaces. We adapt some of this work to covariance functions for processes on spheres, where the natural distance is great circle distance. In doing so, we define the notion of mean square differentiable processes on spheres and give necessary and sufficient conditions for an isotropic covariance function on a sphere to correspond to an m times mean square differentiable process. These conditions imply that if a process on a Euclidean space is restricted to a sphere of lower dimension, the process will retain its mean square differentiability properties. The restriction requires the covariance function to take Euclidean distance as its argument. To address the issue of whether covariance functions using Euclidean distance result in poorly fitting models, we introduce an analog to the Matérn covariance function that is valid on spheres with great circle distance metric, as the usual Matérn that is only generally valid with Euclidean distance. These covariance functions are compared with several others in applications involving satellite and climate model data.