Computationally efficient inversion of steady-state stochastic moment equations of groundwater flow

The impact of various sources of uncertainty on predictions of groundwater flow is conveniently tackled by casting the governing equations in a stochastic framework. Different inverse stochastic approaches have been developed to condition hydrogeological models’ predictions not only on direct measurements of parameters but also on information on state variables. Here, we focus on the inversion of stochastic moment equations of groundwater flow, as originally proposed by Hernandez et al. [2003, 2006]. In their approach, hydraulic conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete pilot points. Prior estimates of pilot point values are obtained by generalized kriging. Posterior estimates at pilot points and (optionally) at measurement points are obtained by calibrating mean flow against measured values of head. The parameters are then projected onto a computational grid via kriging. Maximum likelihood calibration is employed to estimate not only hydraulic but also (optionally) unknown variogram parameters. The latter define the underlying geostatistical model. The approach yields covariance matrices for parameter estimation as well as head and flux prediction errors. The latter are obtained a posteriori by solving corresponding second-moment equations. Hernandez et al. [2003, 2006] implemented their inverse approach on a synthetic scenario, involving a general nonuniform flow condition in a bounded heterogeneous two-dimensional domain. Recently, Hendricks Franssen et al. [2009] performed a comparative synthetic study assessing the relative performance of this moment equations-based inverse method and several types of Monte Carlo inverse methodologies. Results of that study showed that that observed differences between the performances of the tested methods were not very large. However, Monte Carlo inversion of 100 realizations needed considerably more CPU time than geostatistical inversion of moment equations did. Bianchi-Janetti et al. [2009] applied the inverse moment-equations method to characterize the log-transmissivity distribution at a small scale field test site located in Montalto Uffugo (Italy). In their field application, information on hydraulic head is provided through self-potential signals recorded by a few surface electrodes during a pumping test, while only one transmissivity measurement was available. Notwithstanding the theoretical and conceptual advantages of the method, the inversion of moment equations is still based on an optimization process which requires the numerical calculation of the derivatives of the objective function with respect to model parameters. These are in turn calculated from the derivatives of nodal heads (i.e., the sensitivity matrix). This limits its applicability to situations where the number of parameters, e.g., hydraulic conductivity values at pilot points, is not large because of the associated relevant computational cost. Here, we embed exact equations satisfied by the sensitivity matrix of the (ensemble) mean hydraulic head, up to its second-order of approximation, within the inverse modeling process. This renders the nonlinear inversion of stochastic moment equations feasible for a large number of unknown hydraulic parameters. We illustrate our algorithm and procedure with a synthetic example.