  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.
