Intrinsic Random Functions on the sphere

TL;DR

This paper extends intrinsic random functions to spherical domains, develops a universal kriging formula, and explores the connection to splines, highlighting the inapplicability of thin-plate splines on the sphere.

Contribution

It introduces a novel framework for modeling non-stationary spatial processes on the sphere using IRFs and connects these to reproducing kernel Hilbert spaces.

Findings

Low-frequency truncation is crucial for IRFs on the sphere.

Universal kriging formula for the sphere is derived.

Thin-plate splines are unsuitable for surface fitting on the sphere.

Abstract

Spatial stochastic processes that are modeled over the entire Earth's surface require statistical approaches that directly consider the spherical domain. Here, we extend the notion of intrinsic random functions (IRF) to model non-stationary processes on the sphere and show that low-frequency truncation plays an essential role. Then, the universal kriging formula on the sphere is derived. We show that all of these developments can be presented through the theory of reproducing kernel Hilbert space. In addition, the link between universal kriging and splines is carefully investigated, whereby we show that thin-plate splines are non-applicable for surface fitting on the sphere.