Abstract
We study the regularity properties of Gaussian fields defined over spheres cross time. In particular, we consider two alternative spectral decompositions for a Gaussian field on $\mathbb{S}^{d}\times \mathbb{R}$. For each decomposition, we establish regularity properties through Sobolev and interpolation spaces. We then propose a simulation method and study its level of accuracy in the $L^{2}$ sense. The method turns to be both fast and efficient.
Highlights
Spatio-temporal variability is of major importance in many fields, in particular for anthropogenic and natural processes, such as earthquakes, geographic evolution of diseases, income distributions, mortality fields, atmospheric pollutant concentrations, hydrological basin characterization and precipitation fields, among others
A Gaussian random field (GRF) which is 2-weakly isotropic stationary on Sd ×R, is isotropic in the spatial variable and stationary in the time variable, it has an invariant distribution under rotations on the spatial variable, and under translations on the temporal variable
The present work has provided a deep look at the regularity properties of Gaussian fields evolving temporally over spheres
Summary
Spatio-temporal variability is of major importance in many fields, in particular for anthropogenic and natural processes, such as earthquakes, geographic evolution of diseases, income distributions, mortality fields, atmospheric pollutant concentrations, hydrological basin characterization and precipitation fields, among others. The tour de force in [17] characterizes isotropic Gaussian random fields on the sphere Sd through Karhunen–Loeve expansions with respect to the spherical harmonics functions and the angular power spectrum. The present paper extends part of the work of [17] to space-time Such extension is non-trivial and depends on two alternative spectral decompositions of a Gaussian field on spheres cross time. The Berg-Porcu representation in terms of Schoenberg functions inspires the proposal of alternative spectral decompositions for the temporal part, which become crucial to establish the regularity properties of the associated Gaussian field. Efficient simulation methods for random fields defined on the sphere cross time are, currently, almost unexplored. The manuscript is intended for complex-valued random fields over Sd × R with d ∈ N, except for Section 5 where the simulations considered are for real-valued random fields over S2 × R
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