Parametric Bayesian estimation of point-like pollution sources of groundwater layers

This paper considers the problem of estimating point-like pollution sources of groundwater layers. To cope with the ill-posed character of this problem, a parametric Bayesian framework has been recently established. In this framework, where the priors for the source parameters are either uniform or Gaussian and the observation noise is homogeneous, a stochastic Markov Chain Monte Carlo (MCMC) algorithm has been proposed to compute the posterior distribution of both source parameters and noise variance. Here, we consider a more general model with truncated-Gaussian priors for pollution quantity and spreading time parameter, which gathers advantages of uniform and Gaussian choices, and an inhomogeneous noise, which accounts for the spatial diversity among sensors. For this model, we extend the existing stochastic algorithm, then propose a concurrent deterministic algorithm based on the variational Bayesian (VB) approach. This approach designs an approximation of the posterior law based on a separable from. The proposed MCMC and VB algorithms target the exact posterior and the approximated posterior, respectively. It is further shown that the former is more accurate, while the latter is computationally more efficient. Results of numerical experiments conducted using an experimental platform to compare the performances of the proposed schemes are presented.