Intrinsic random functions 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. In practice, such processes rarely are second-order stationary — that is, they do not have constant mean values or the covariance function that depends only on their angular distance. Here, in order to model non-stationary processes on the sphere, we extend the notion of intrinsic random functions and show that low-frequency truncation plays an essential role. We show that 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.