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The Mat{\'e}rn family of isotropic covariance functions has been central tothe theoretical development and application of statistical models forgeospatial data. For global data defined over the whole sphere representingplanet Earth, the natural distance between any two locations is the greatcircle distance. In this setting, the Mat{\'e}rn family of covariance functionshas a restriction on the smoothness parameter, making it an unappealing choiceto model smooth data. Finding a suitable analogue for modelling data on thesphere is still an open problem. This paper proposes a new family of isotropiccovariance functions for random fields defined over the sphere. The proposedfamily has a parameter that indexes the mean square differentiability of thecorresponding Gaussian field, and allows for any admissible range of fractaldimension. Our simulation study mimics the fixed domain asymptotic setting,which is the most natural regime for sampling on a closed and bounded set. Asexpected, our results support the analogous results (under the same asymptoticscheme) for planar processes that not all parameters can be estimatedconsistently. We apply the proposed model to a dataset of precipitable watercontent over a large portion of the Earth, and show that the model gives moreprecise predictions of the underlying process at unsampled locations than doesthe Mat{\'e}rn model using chordal distances.

The $\mathcal{F}$-family of covariance functions: A Mat\'ern analogue for modeling random fields on spheres