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

TL;DR

This paper introduces a new family of covariance functions for modeling global spatial data on spheres, overcoming limitations of the Matérn family by allowing flexible smoothness and fractal dimension, with demonstrated improved prediction accuracy.

Contribution

It proposes a novel isotropic covariance family for spherical data that generalizes Matérn, enabling flexible smoothness and fractal modeling on the sphere.

Findings

The new covariance family allows for any admissible fractal dimension.

Simulation results support the model's consistency and flexibility.

Application to Earth data shows improved prediction accuracy.

Abstract

The Mat{\'e}rn 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 Mat{\'e}rn 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 Mat{\'e}rn model using chordal distances.