Abstract. Geostatistical theory has shown promise in dealing with issues of stability, uniqueness, and identity of estimates inherent in inverse problems of subsurface flow. Here the geostatistical method is extended to three-dimensional, unsteady flow in variably saturated porous geological media (the vadose zone) that are modeled using the Richards equation and the van Genuchten-Mualem constitutive equations. The saturated hydraulic conductivity, c•, and n parameters of this relationship are treated as spatially correlated, statistically independent, stochastic processes for representing heterogeneity of porous media. For given covariance functions of the parameters the adjoint-state sensitivity method is used to calculate first-order approximations for covariances of capillary pressure and moisture content and cross covariances between capillary pressure, moisture content, and the hydraulic properties. These covariances and cross covariances are then used in a successive linear estimator (SLE) to estimate the conditional means of the heterogeneous hydraulic property fields based on measurements of pressure and moisture content data. A sequential conditioning approach for our SLE was also applied to data sets collected at different sampling times during a transient infiltration event. This approach has the benefit of reducing the size of the matrices and so helps avoid numerical stability problems. On the basis of our study, pressure and moisture content data sets collected at later times of an infiltration event or during steady state flow were found to provide better estimates (smaller mean-square error compared to the true field) of the hydrological parameters of the vadose zone than data from very early times.

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Beginning in the early 1900s, poly-factorial, poly-microbial pneumonia was identified as a disease affecting bighorn sheep (Ovis canadensis) and it continues to threaten bighorn populations, posing an ongoing management challenge. In May

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We present a geostatistically based inverse model for characterizing heterogeneity in parameters of unsaturated hydraulic conductivity for threedimensional¯ow. Pressure and moisture content are related to perturbations in hydraulic parameters through cross-covariances, which are calculated to ®rst-order. Sensitivities needed for covariance calculations are derived using the adjoint state sensitivity method. Approximations of the conditional mean parameter ®elds are then obtained from the cokriging estimator. Correlation between parameters and pressure ± moisture content perturbations is seen to be strongly dependent on mean pressure or moisture content. High correlation between parameters and pressure data was obtained under saturated or near saturated¯ow conditions, providing accurate estimation of saturated hydraulic conductivity, while moisture content measurements provided accurate estimation of the pore size distribution parameter under unsaturated¯ow conditions. IntroductionDif®culty in detailed characterization of spatially varying hydraulic properties of ®eld sites remains one of the main challenges for successful prediction of¯ow and transport in the vadose zone. Characterizing the vadose zone in detail using core samples or small-scale hydraulic tests is prohibitively time-consuming and costly and, consequently, is beyond the means of most studies. For these reasons there has been much interest and research into using the more easily measured states of the¯ow system, i.e. pressure head and moisture content, to estimate soil hydraulic properties. In the following discussion we refer to the soil hydraulic properties as primary information and measured states of the¯ow system as secondary information. This use of secondary information for the purpose of determining hydraulic properties of the porous medium is the essence of the socalled`inverse problem' in subsurface hydrology. While the ®eld scale inverse problem is inherently ill-posed and the solution non-unique, the dif®culties en-