Coupling empirical Bayes and Akaike's Bayesian Information Criterion to estimate Aquifer transmissivity fields

The correct management of groundwater and the prediction of solute transport in aquifers is based on numerical flow and transport models; for this reason, it is of main importance the knowledge of the spatial variability of the hydraulic parameters, such as the transmissivity. In this work we apply an empirical Bayes method combined with the Akaike’s Bayesian Information Criterion (ABIC) to estimate highly parameterized transmissivity fields. As observations for the inverse problem, we use both transmissivity and hydraulic head data. A numerical flow model is adopted to solve the forward problem. From a Bayesian point of view, ABIC uses prior information on the unknown function, such as the prior distribution (assumed Gaussian) and the covariance model of the unknowns. Due to the non-linearity of the forward problem, the approach is based on a successive linearization method; at each iteration, first we estimate the hyper-parameters (noise variance and the covariance model parameters) using ABIC, then we update the unknowns at each grid node (i.e. transmissivity values). When transmissivity observations are available, a first estimate of the unknowns can be performed using an empirical Bayes interpolation. The procedure allows to compute the posterior probability distribution of the target quantities and to quantify the uncertainty in the model prediction. The methodology was tested on a well-known literature case study that consists in a confined aquifer of 5000 m × 5000 m with 50 × 50 cells which contains a variety of boundary conditions (both Dirichelet and Neuman type) and considering different scales of heterogeneities. Results show the good performance of the inverse procedure; for this reason, we are working on implementing the methodology in a free and user friendly code. The software will be independent of any model. It will control the forward model and it will be able to read and write the necessary input and output files. This will allow the scientific community to apply the inverse approach to multiple case studies, not only related to groundwater.