Exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow

We present novel equations for the exact sensitivity matrix of the (ensemble) mean hydraulic head under steady state groundwater flow conditions. These equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads. Our formulation allows considerable improvement of the methodology proposed by Hernandez et al. (2003, 2006) and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters. The spatial distribution of the natural logarithm, Y, of conductivity is parameterized within the pilot points framework. Whereas prior values of Y at pilot points are obtained by a variant of kriging, posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads. The maximum likelihood function also includes a regularization term. By means of a synthetic example and upon adopting formal model information criteria we explore the influence of (1) the number of pilot points and (2) the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying Y variogram. We find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the Kashyap's Bayesian measure.