Anisotropic statistical scaling of vadose zone hydraulic property estimates near Maricopa, Arizona

Abstract

[1] Fluid flow and mass transport in the vadose zone are strongly influenced by spatial variability in soil hydraulic properties. It has become common to characterize spatial variability geostatistically and to solve corresponding flow and/or transport problems stochastically. The typical (though not only) approach is to treat log saturated hydraulic conductivity, Y 5 log 10 K s , and perhaps other medium properties as statistically homogeneous, isotropic, or anisotropic multivariate-Gaussian random fields with unique variance and autocorrelation scale. A growing body of literature suggests that Y as well as many other variables may possess heavy-tailed, non-Gaussian distributions, and/or scaledependent statistical parameters representing a multiscale, hierarchical structure. Elsewhere, we have demonstrated that these important phenomena are difficult to detect, and are not fully tractable, with standard geostatistical methods. They are however detectable and tractable with a novel geostatistical method of analysis which treats such variables as samples from sub-Gaussian random fields subordinated to truncated fractional Brownian motion (tfBm) or truncated fractional Gaussian noise (tfGn). Each such field is a mixture of Gaussian components having random variances. The purpose of this paper is to explore the extent to which hydraulic parameters of unsaturated soils exhibit heavy-tailed distributions and statistical scaling of the above type. As a test bed we have selected an experimental site near Maricopa, Arizona, at which soil texture data have been collected in boreholes and trenches to a depth of 15 m over an area of 3600 m 2 . Since hydraulic parameters are difficult to measure to such depths we estimate them using a neural network pedotransfer model. Input data include percent sand, silt, and clay. Outputs include estimates of saturated and residual volumetric water content, saturated hydraulic conductivity, and (due to their wide applicability) parameters of the van Genuchten-Mualem constitutive model of unsaturated soil property variations with capillary pressure head. We find indeed that vertical and horizontal increments of our hydraulic parameter estimates exhibit heavy-tailed frequency distributions and statistical scaling which conform closely to our novel geostatistical framework. Its generality and applicability to the Maricopa site has far reaching implications vis-a-vis the geostatistical characterization of vadose zone properties and the stochastic modeling of flow and transport in unsaturated media at other sites.

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

“…Accordingly, we set λ l = 0 and λ u to a sufficiently large number to ensure that the TPV γ G (s; λ l , λ u ) reduces, within both working lag ranges, to the PV γ (s) = Bs 2H . Then, in a manner analogous to that outlined most recently by Guadagnini et al (2013Guadagnini et al ( , 2014, we obtain ML estimates of A in two ways: once by adopting corresponding method-of-moment estimatesĤ w andĤ b from Table 2 and once by estimating the latter jointly with A. Both sets of estimates are obtained upon fitting the theoretical PV γ (s) = Bs 2H to sample scale parametersσ (s n ) such as those plotted versus s n in Fig.…”

“…Statistical scaling of hydrogeological data such as permeability or hydraulic conductivity has been studied amongst others by Painter (2001), Meerschaert et al (2004), Kozubowski et al (2006), Siena et al (2012Siena et al ( , 2014, Riva et al (2013bRiva et al ( , 2013c, and Guadagnini et al (2012Guadagnini et al ( , 2013Guadagnini et al ( , 2014. Whereas research in the subsurface hydrology literature has not addressed specifically the distribution and statistical scaling of extreme incremental values, spatial correlations between values significantly in excess of the mean have been studied vis-à-vis variables such as transmissivity and their relevance to transport processes has been highlighted.…”

“…Work by our group has demonstrated theoretically (Neuman, 2010(Neuman, , 2011Guadagnini and Neuman, 2011;Siena et al, 2012;Neuman et al, 2013), computationally Neuman et al, 2013) and on the basis of varied pedological, hydrological and hydrogeological data (Siena et al, 2012(Siena et al, , 2014Riva et al, 2013b, c;Guadagnini et al, 2012Guadagnini et al, , 2013Guadagnini et al, , 2014) that statistical scaling behaviors of the kind traditionally attributed to multifractals can be interpreted more simply and consistently by viewing the data as samples from stationary sub-Gaussian random fields subordinated to truncated fBm (tfBm) or fGn (tfGn). Such sub-Gaussian fields are scale mixtures of stationary Gaussian fields with random variances (Andrews and Mallows, 1974;West, 1987) that we model as being lognormal or Lévy-stable (Samorodnitsky and Taqqu, 1994).…”

New scaling model for variables and increments with heavy‐tailed distributions

“…Whereas distributions of Y (or its logarithm) are at times slightly asymmetric with relatively mild peaks and tails, those of DY tend to be symmetric with peaks that grow sharper, and tails that become heavier, as the separation distance (lag) between pairs of Y values decreases. Documented examples include porosity [Painter, 1996;Guadagnini et al, 2014Guadagnini et al, , 2015, permeability [Painter, 1996;Riva et al, 2013aRiva et al, , 2013b and hydraulic conductivity [Liu and Molz, 1997;Meerschaert et al, 2004;Guadagnini et al, 2013], electrical resistivity [Painter, 2001;Yang et al, 2009], soil and sediment texture [Guadagnini et al, 2014], sediment transport rate [Ganti et al, 2009], rainfall [Kumar and Foufoula-Georgiou, 1993], measured and simulated turbulent fluid velocity [Castaing et al, 1990;Boffetta et al, 2008], and magnetic fluctuation [von Papen et al, 2014] data. No statistical model known to us captures these behaviors of Y and DY in a unified and consistent manner.…”

Simulation and analysis of scalable non-Gaussian statistically anisotropic random functions

“…Our own experience shows this to be specifically true for quantities controlling subsurface flow and transport such as soil textural composition, log permeability, porosity and unsaturated flow parameters. Previously we (Siena et al, 2012(Siena et al, , 2014Riva et al, 2013a, b;Guadagnini et al, 2012Guadagnini et al, , 2013Guadagnini et al, , 2014Guadagnini et al, , 2015 were able to capture some key aspects of such scaling by treating Y or DY as standard sub-Gaussian random functions Y(x) = UG(x) (e.g., Samorodnitsky and Taqqu, 1994) in which x is a (spatial or temporal) coordinate, G(x) is a scalable zero-mean stationary Gaussian random function and U is a non-negative random variable independent of G(x). A major attraction of this model is that all multivariate moments of Y and DY (that exist) are fully defined by the distribution of U and the first two (one-and two-point) moments of G. The model, however, did not allow us to reconcile two seemingly contradictory observations, namely that whereas sample frequency distributions of Y (or its logarithm) exhibit relatively mild non-Gaussian peaks and tails, those of DY display peaks that grow sharper and tails that become heavier with decreasing separation distance or lag.…”