Generalized Sub-Gaussian Processes: Theory and Application to Hydrogeological and Geochemical Data

We start from the well-documented scale dependence displayed by the probability distribution and associated statistical moments of a variety of hydrogeological and soil science variables and their spatial or temporal increments. These features can be captured by a Generalized Sub-Gaussian (GSG) model, according to which a given variable, Y, is subordinated to a (typically spatially correlated) Gaussian random field, G, through a subordinator, U. This study extends the theoretical framework originally proposed by Riva et al. (2015, 10.1002/2015WR016998) to include the possibility of selecting a general form of the subordinator, thus enhancing the flexibility of the GSG framework for data interpretation and modeling. Analytical expressions for the GSG process associated with lognormal, Pareto, and Gamma subordinator distributions are then derived. We demonstrate the ability of the GSG modeling framework to capture the way key features of the statistics associated with two data sets transition across scales. The latter correspond to variables, which are typical of a geochemical and a hydrogeological setting, that is, (i) data characterizing the micrometer-scale surface roughness of a crystal of calcite, collected within a laboratory-scale setting, resulting from induced mineral dissolution, and (ii) a vertical distribution of decimeter-scale porosity data, collected along a deep kilometer-scale borehole within a sandstone formation and typically used in hydrogeological and geophysical characterization of aquifer systems. The theoretical developments and the successful applications of the approach we propose provide a unique framework within which one can interpret a broad range of scaling behaviors displayed by a variety of Earth and environmental variables in various scenarios. Plain Language Summary Characterization of hydrogeological and geochemical systems aims at assessing the heterogeneity and scale dependency exhibited by their attributes and the associated key statistics. It has been shown that complex scaling features documented for the statistics of a wide range of Earth, environmental (and several other) variables, and their spatial/temporal increments can be captured through a Generalized Sub-Gaussian (GSG) model. The latter relies on the subordination of a Gaussian random field through a subordinator. This study extends the theoretical framework originally proposed for the GSG model to include multiple choices of the subordinator distribution. We provide the theoretical formulation and discuss the main features of the GSG model resulting from (i) a general form of the subordinator and (ii) three selected distributional forms. We show the effectiveness of the GSG modeling framework for the interpretation of real data encompassing a considerably wide range of scales by analyzing (i) a set of surface topography (roughness) data collected on a calcite sample in a laboratory-scale geochemical setting and (ii) a field-scale distribution of porosity data, collected along a deep borehole within a sandstone formation.