A geostatistical inverse method for variably saturated flow in the vadose zone

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 the soil-water pressure head and each of the following: degree of saturation, saturated hydraulic conductivity, and a 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 on the basis of 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.