• A Geostatistical Inverse Method for Variably Saturated Flow in the Vadose Zone

      Yeh, T.-C. Jim; Zhang, Jinqi; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1995-10-11)
      A geostatistical inverse technique utilizing both primary and secondary information is developed to estimate conditional means of unsaturated hydraulic conductivity parameters (saturated hydraulic conductivity and pore -size distribution parameters) in the vadose zone. Measurements of saturated hydraulic conductivity and pore -size distribution parameters are considered as the primary information, while measurements of steady -state flow processes (soil -water pressure head and degree of saturation) are regarded as the secondary information. This inverse approach relies on the classical linear predictor (cokriging) theory and takes the advantage of the spatial cross- correlation between soil -water pressure head, degree of saturation, saturated hydraulic conductivity, and pore -size distribution parameter. Using an approximate perturbation solution for steady, variably saturated flow under general boundary conditions, the cross- covariances between the primary and secondary information are derived. The approximate solution is formulated based on a first -order Taylor series expansion of a discretized finite element equation. The sensitivity matrix in the solution is evaluated by an adjoint state sensitivity approach for flow in heterogeneous media under variably saturated conditions. Through several numerical examples, the inverse model demonstrates its ability to improve the estimates of the spatial distribution of saturated hydraulic conductivity and pore -size distribution parameters using the secondary information.
    • An Iterative Geostatistical Inverse Method For Steady-Flow In The Vadose Zone

      Zhang, Jinqi; Yeh, T.-C. Jim; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1996-02-01)
      An iterative stochastic inverse technique utilizing both primary and secondary information is developed to estimate conditional means of unsaturated hydraulic conductivity parameters (saturated hydraulic conductivity and pore -size distribution parameters) in the vadose zone. Measurements of saturated hydraulic conductivity and pore -size distribution parameter are considered as the primary information, while measurements of steady -state flow processes (soil -water pressure head and degree of saturation) are regarded as the secondary information. This inverse approach is similar to the classical geostatistical approach, which utilizing a linear estimator that depends on the cross- covariance and covariance functions of unsaturated hydraulic conductivity parameters and flow processes. The linear estimator is, however, improved successively by solving the governing flow equation and by updating the residual covariance and cross- covariance functions, in an iterative manner. Using an approximate perturbation solution for steady, variably saturated flow under general boundary conditions, the covariances of secondary information and the cross -covariance between the primary and secondary information are derived. The approximate solution is formulated based on a first -order Taylor series expansion of a discretized finite element equation. The sensitivity matrices in the solution are evaluated by an adjoint state sensitivity approach for flow in heterogeneous media under variably saturated conditions. As a result, the nonlinear relationships between unsaturated hydraulic conductivity parameters and flow processes are incorporated in the estimation. Through some numerical examples, the iterative inverse model demonstrates its ability to improve the estimates of the spatial distribution of saturated hydraulic conductivity and pore -size distribution parameters compared to the classical geostatistical inverse approach. In addition, the inconsistency problem existing in classical geostatistical inverse approach is alleviated. The estimated fields of unsaturated hydraulic conductivity parameters and flow fields not only retain their observed values at sample locations, but satisfy the governing flow equation as well.