Investigating the application of random fields on manifolds in geostatistical modeling

Abstract

The twofold goal of this manuscript is first to briefly review some of the frameworks that allow the study and modeling of non-stationary random fields, and second, to notice that accounting for high-order dependencies in geological processes can be achieved by generating stationary and non-stationary random fields in a high dimensional ambient space and then taking the projection into lower dimensions of the field restricted to an immersed submanifold. First, a brief review of the most common methods for non-stationary modeling is provided. Then we demonstrate that it is possible to obtain higher-order characteristics in the random fields, under the spatial deformation framework. The analytical framework for the geometry of excursion sets is introduced, in order to make consistent the analysis of connectivity patterns indirectly, by looking at the topological characteristics of the excursion sets. Finally, we introduce some ideas for future research.