Some theory for anisotropic processes on the sphere

We investigate properties of an existing class of models for Gaussian processes on the sphere that are invariant to shifts in longitude. The class is obtained by applying first-order differential operators to an isotropic process and potentially adding an independent isotropic term. For a particular choice of the operators, we derive explicit forms for the spherical harmonic representation of these processes’ covariance functions. Because the spherical harmonic representation is a spectral one, these forms allow us to draw conclusions about the local properties of the processes. For one, the coefficients in the spherical harmonic representation relate to the equivalence and orthogonality of the measures induced by the models. It turns out that under certain conditions the models will lack consistent parameter estimability even when the process is observed everywhere on the sphere. We also consider the ability of the models to capture isotropic tendencies on the local level, a phenomenon observed in some data.