Regularity properties and simulation of Gaussian random fields on the sphere across time

In many geosciences applications, the phenomena of interest are observed on large portion
of the Planet Earth. When such phenomena evolves over time, a valid model for the study of these
observations is to consider them as partial realization of a spatio-temporal random field, where the
spatial component is defined on a sphere, considering this last as a more realistic representation of the
globe. We Introduce a family of Gaussian random fields constructed by spectral methods via a double
Karhunen-Loéve type representation. Based on later result on the literature we claim that this family
represent the class of Isotropic and stationary Gaussian random fields over Sd × R. In the formulation,
we exploit the theory about positive definite functions on the sphere and its extension to the spatio-
temporal context. We then study regularity properties of the proposed family via Sobolev spaces and
interpolation techniques. Then we approximate the elements of this family with a double truncation
and we analyse the accuracy of this technique evaluating the error in L2 -norm and through simulation
experiments.

     Joint work with Alfredo Alegría and milio Porcu