Extreme value statistics of scalable data exemplified by neutron porosities in deep boreholes

Spatial statistics of earth and environmental (as well as many other) data tend to vary with scale. Common manifestations of scale-dependent statistics include a tendency of increments to have symmetric, non-Gaussian frequency distributions characterized by heavy tails that decay with separation distance or lag; power-law scaling of sample 5 structure functions (statistical moments of absolute increments) in midranges of lags; linear relationships between log structure functions of successive orders at all lags, known as extended self-similarity or ESS; and nonlinear scaling of structure function power-law exponents with function order, a phenomenon commonly attributed in the literature to multifractals. Elsewhere we proposed, explored and demonstrated a new 10 method of geostatistical inference that captures all of these phenomena within a unified theoretical framework. The framework views data as samples from random fields constituting scale-mixtures of truncated (monofractal) fractional Brownian motion (tfBm) or fractional Gaussian noise (tfGn). Important questions not addressed in previous studies concern the distribution and statistical scaling of extreme incremental values. Of spe-15 cial interest in hydrology (and many other areas) are statistics of absolute increments exceeding given thresholds, known as peaks over thresholds or POTs. In this paper we explore for the first time the statistical behavior of POTs associated with samples from scale-mixtures of tfBm or tfGn. We are fortunate to have at our disposal thousands of neutron porosity values from six deep boreholes, in three diverse depositional envi-20 ronments, which we show possess the properties of such samples thus following the theory we proposed. The porosity data are of additional value in revealing a remarkable transition from one scaling regime to another at certain lags. The phenomena we uncover are of fundamental importance for the analysis of fluid flow and solute as well as particulate transport in complex hydrogeologic environments.