Stochastic analysis of field measured unsaturated hydraulic conductivity

Unsaturated hydraulic conductivity K values as a function of soil-water pressure head h were measured in the soil at 75 cm depth at 70 different sites separated from one another by a distance of l m along a horizontal transect. K field was viewed as a random function of spatial location x. Field data were analyzed (1) to examine the isotropy and stationarity of K, (2) to check the ergodicity of K in the mean and covariance functions, and (3) to characterize the distribution properties of K by estimating the higher-order correlations, that is, third and fourth cumulants. The mean functions were estimated by averaging over h and x. The covariance function was studied to investigate its spatial origin dependency. Logs and square roots of K were used for estimating the third and fourth cumulants. Results showed that spatial covariance functions are anisotropic and both lag and origin dependent, that is, spatially nonhomogeneous. Because the stationarity (statistical homogeneity) of K is scale dependent, which was indicated by the identification of locally stationary covariance regions, the ergodic properties of K are also scale dependent at smaller spatial scales. Results related to the distribution characteristics of K indicated that although ln K is marginally Gaussian distributed, in the context of spatial stochastic processes the random field of ln K is not Gaussian because the third and fourth cumulants of the field are still significantly different from zero and have the same order of magnitude as the first and second cumulants. The square root transformation, however, resulted in a random field that is approximately Gaussian although marginal distributions of K remained skewed. Analyses of ln K and K indicated that better transformations which would result in both marginal and joint Gaussian behavior for the random field of K are needed.