The F-family of covariance functions: A Matern analogue for modeling random fields on spheres

The Matern 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 distance between any two locations is the great circle distance. In this setting, the Matern family of covariance functions has a restriction on the smoothness parameter, making it an unappealing choice to model smooth data. Finding a suitable analogue for modelling data on the sphere is still an open problem. This paper proposes a new family of isotropic covariance functions for random fields defined over the sphere. The proposed family has a parameter that indexes the mean square differentiability of the corresponding Gaussian field, and allows for any admissible range of fractal dimension. Our simulation study mimics the fixed domain asymptotic setting, which is the most natural regime for sampling on a closed and bounded set. As expected, our results support the analogous results (under the same asymptotic scheme) for planar processes that not all parameters can be estimated consistently. We apply the proposed model to a dataset 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 the Matern model using chordal distances. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

The Matern family of isotropic covariance functions plays a central role in the development and application of statistical models for geospatial data, but has limitations when modeling data on the sphere. This paper proposes a new family of isotropic covariance functions for random fields defined over the sphere, with a parameter to index the mean square differentiability and allow for a range of fractal dimensions.