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=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 scale-dependent 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.