Articles | Volume 19, issue 2
Hydrol. Earth Syst. Sci., 19, 729–745, 2015
https://doi.org/10.5194/hess-19-729-2015
Hydrol. Earth Syst. Sci., 19, 729–745, 2015
https://doi.org/10.5194/hess-19-729-2015

Research article 04 Feb 2015

Research article | 04 Feb 2015

Scalable statistics of correlated random variables and extremes applied to deep borehole porosities

A. Guadagnini1,2, S. P. Neuman1, T. Nan1, M. Riva1,2, and C. L. Winter1 A. Guadagnini et al.
  • 1Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona 85721, USA
  • 2Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano, Piazza L. Da Vinci 32, 20133 Milan, Italy

Abstract. We analyze scale-dependent statistics of correlated random hydrogeological variables and their extremes using neutron porosity data from six deep boreholes, in three diverse depositional environments, as example. We show that key statistics of porosity increments behave and scale in manners typical of many earth and environmental (as well as other) variables. These scaling behaviors 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 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 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 special interest in hydrology (and many other areas) are statistics of absolute increments exceeding given thresholds, known as peaks over threshold or POTs. In this paper we explore the statistical scaling of data and, for the first time, corresponding POTs associated with samples from scale mixtures of tfBm or tfGn. We demonstrate that porosity data we analyze possess properties of such samples and thus follow the theory we proposed. The porosity data are of additional value in revealing a remarkable cross-over from one scaling regime to another at certain lags. The phenomena we uncover are of key importance for the analysis of fluid flow and solute as well as particulate transport in complex hydrogeologic environments.

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Short summary
Previously we have shown that many earth-system and other variables can be viewed as samples from scale mixtures of truncated fractional Brownian motion or fractional Gaussian noise. Here we study statistical scaling of extreme absolute increments associated with such samples. As a real example we analyze neutron porosities from deep boreholes in diverse depositional units. Phenomena we uncover are relevant to the analysis of fluid flow and solute transport in complex hydrogeologic environments.