Characterization of Spatially Heterogeneous Environmental Variables Through Multi-Modal Generalized Sub-Gaussian Distributions

Abstract

We provide a sound theoretical framework for the characterization of randomly heterogeneous spatial fields exhibiting multi-modal, long-tailed probability densities. Multi-modal distributions are at the core of conceptual models employed to represent heterogeneity of hydrogeological or geochemical systems across which one can otherwise distinguish diverse regions whose location is uncertain. Within each region, the quantity of interest shows a distinct heterogeneous pattern that can be described through a generally non-Gaussian distribution. Our analytical model embeds the joint formulation of the probability density of the target variable and its spatial increments. The distributions of the latter scale with separation distance between locations at which increments are evaluated. This feature is in line with documented experimental observations of a variety of Earth system quantities. Our stochastic modeling framework integrates approaches based on unimodal non-Gaussian fields described through a Generalized Sub-Gaussian model and (multi-modal) distributions resulting from mixtures of Gaussian fields. These are recovered as specific instances within our comprehensive formulation. We apply this framework to an experimental data set consisting of a collection of dissolution rate fields obtained from high-resolution nanoscale measurements acquired through Atomic Force Microscopy and documenting the dissolution behavior of a calcite sample under continuous flow conditions. Our findings demonstrate the capability of our stochastic approach to elucidate key statistical traits and scaling features inherent in the heterogeneous distributions of these types of environmental variables.