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

We derive and solve novel equations satisfied by 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 zero- and second-order (in terms of the standard deviation of the natural logarithm of hydraulic conductivity, Y) approximations of stochastic Moment Equations of flow on measurements of hydraulic conductivity and heads. Relying on direct calculation of the derivatives of (mean) hydraulic heads with respect to model parameters allows considerable improvement of the methodology originally proposed by Hernandez et al. [2003, 2006], not only in terms of accuracy of the solution, but also in terms of the CPU time required for the parameter estimation. An advantage of this methodology is that the system matrices are identical for all the parameters and coincide with those used to solve the equations of the forward problem, thus rendering the nonlinear inversion of stochastic Moment Equations feasible for a large number of unknown hydraulic parameters. We parameterize Y geostatistically from (uncertain) Y measurements and unknown values at discrete “pilot point” locations. 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/plausibility term penalizing large departures of the calculated Y values from their prior estimates. The variance and covariance of head estimates are then obtained a posteriori by solving corresponding Moment Equations. By means of a synthetic example and upon adopting formal model information/discrimination criteria we explore the influence of (a) the number of pilot points and (b) the order of approximation of the governing mean flow equation on our ability to properly reconstruct the log-conductivity and head fields and identify the relative weight of the regularization term together with the statistical parameters of the underlying Y variogram. We find that, whereas none of the adopted information criteria can identify the optimum number of pilot points, the plausibility weight and variogram parameters values can be determined by the Kashyap’s Bayesian measure (KIC). Our results suggest that the geostatistical inversion of groundwater flow Moment Equations can benefit from successive inversions of zero- and second-order equations to provide a robust and computationally affordable estimate of hydraulic and statistical parameters of the problem