The Matérn family of isotropic covariance functions has been central to the theoretical development and application of statistical models for geospatial data. For global data defined over the whole sphere representing planet Earth, the natural definition of the distance between two locations is the great-circle distance. In this setting, the Matérn family is no longer valid, and finding a suitable analogue for modelling data on the sphere has for some time been an open problem.

This paper proposes a new family of isotropic covariance functions for random fields defined on the sphere. The family has four parameters, one of which indexes the mean square differentiability of the corresponding Gaussian field. The new family also allows for any admissible range of fractal dimension.

We describe a simulation to show the behaviour of the maximum likelihood parameter estimation under fixed domain asymptotics, this being the relevant asymptotic regime for sampling a closed set. As expected, the results support the analogous result for planar processes that not all parameters can be estimated consistently under fixed domain asymptotics.

We apply the proposed model to a data-set of precipitable water content over a large portion of the Earth and show that the model gives more precise predictions of the underlying process at unsampled locations than does Matérn model using chordal distances. Technical details are given in an Appendix.

Keywords: Great-circle distance; Fractal Dimensions; Matérn covariance; Mean Square Differentiability