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

Abstract

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.

Statistical Characterization of Heterogeneous Dissolution Rates of Calcite from In situ and Real-Time AFM Imaging

“…The adopted GSG model relies on a log-normal subordinator, which has been already tested for the interpretation of the spatial statistics of a wide range of data (e.g., Riva et al, 2015;Siena et al 2019). We note that our theoretical framework includes the possibility of selecting a general form of the subordinator (Siena et al 2020). The application of alternative formulations of the GSG model on dissolution rates will be the subject of future investigations.…”

“…Brand et al (2017) highlight a dependency between the distribution parameters and the temporal window across which dissolution rates are evaluated. Pollet-Villard et al (2016) and Siena et al (2020) show examples of the application of geostatistics for the study of the spatial heterogeneity of reactive processes data at the microscopic level. Pollet-Villard et al (2016) rely on common variogram analyses of topography data measured experimentally through AFM imaging of a dissolving orthoclase surface in far-from-equilibrium conditions.…”

“…Pollet-Villard et al (2016) rely on common variogram analyses of topography data measured experimentally through AFM imaging of a dissolving orthoclase surface in far-from-equilibrium conditions. Siena et al (2020) interpret the statistics of VSI-based measurements of calcite surface topography subject to dissolution in close-to-equilibrium conditions within the recently developed statistical framework of analysis relying on a generalized sub-Gaussian (GSG) formulation (Riva et al 2015). The latter constitutes a modeling approach which enables one to characterize jointly the statistical behavior of a given (spatially heterogeneous) quantity, Y( ) (x denoting a vector of spatial coordinates), and its spatial (or temporal) increments, ΔY( ) = Y( + ) − Y( ) evaluated at multiple separation distances (or lags), s. This approach has been shown to fully embed statistical features which have been documented by observations of a broad spectrum of hydrogeological variables [e.g., porosity (Guadagnini et al 2014;Painter 1996;Riva et al 2015), permeability (Painter 1996;Riva et al 2013b, a) or hydraulic conductivity (Guadagnini et al 2013;Liu and Molz 1997;Meerschaert 2004)] and resulting in a clear scale-dependent behavior of the sample distributions (and associated statistical moments) of spatial increments (see also Sect.…”

Solute transport in bounded porous media characterized by generalized sub-Gaussian log-conductivity distributions

“…Key documented manifestations of such a behavior include the observation that the distribution of spatial increments of such variables taken between two points separated by a given spatial distance (lag) tend to be symmetric and to develop heavier tails and sharper peaks as lag decreases. Such a behavior has been displayed (among others) by log-hydraulic conductivity and permeability (Painter, 1996(Painter, , 2001Liu and Moltz, 1997;Meerschaert et al, 2004;Siena et al, 2012Siena et al, , 2019Riva et al, 2013aRiva et al, , 2013bGuadagnini et al, 2018), electrical resistivity (Painter, 2001), vadose zone hydraulic properties (Guadagnini et al, 2012(Guadagnini et al, , 2013(Guadagnini et al, , 2014, neutron porosity (Riva et al, 2015a), sediment transport (e.g., Ganti et al, 2009), fully developed turbulence (Boffetta et al, 2008), and micro-scale geochemical data (Siena et al, 2020).…”

“…According to the GSG model, the departure of the distribution of a variable and its two-point increments from the Gaussian one is given by the action of a (spatially uncorrelated) subordinator on an otherwise spatially correlated Gaussian random field. This modeling strategy allows representing jointly within a unique framework the above-documented behavior (as described by probability distributions and/or moments) of a quantity and its incremental values and has been successfully applied to the interpretation of main features displayed by various subsurface attributes (Riva et al 2015a;Guadagnini et al, 2018;Siena et al, 2020, and references therein). These concepts have already been employed in preliminary analytical and numerical studies of flow and transport in porous media whose log-conductivity is characterized through a GSG model.…”