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

Abstract

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 distance between any two locations is the great circle distance. In this setting, the Matérn 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 Matérn model using chordal distances.

Surface finite element approximation of spherical Whittle--Matérn Gaussian random fields

“…An example of a spherical Gaussian random field is the Matérn-Whittle field, which is defined as the solution u to the stochastic partial differential equation (SPDE) (1) (κ 2 − ∆ S 2 ) β u = W, where β, κ > 0 are regularity parameters and W denotes white noise on the sphere. Matérn-Whittle fields are the spherical analogue to Matérn fields on R d [21,17].…”

“…This approach may not be possible in the case of surfaces, since the Matérn covariance function adapted to a surface may cease to be positive definite. One way to address this issue in the particular case of the sphere is to define a new family of admissible covariance functions that capture the desired covariance behavior [1]. Another possibility is to use finite element techniques in order to approximate solutions to SPDE (1).…”

“…One way to address this issue in the particular case of the sphere is to define a new family of admissible covariance functions that capture the desired covariance behavior [1]. Another possibility is to use finite element techniques in order to approximate solutions to SPDE (1). Finite element approaches have been recently studied in the case of Euclidean domains, see for instance [4,3,6].…”