Key words: Inverse problems, distributed parameter systems, infinite dimensional Kalman smoother, EM algorihm. In this presentation we are concerned with the estimation of the hydraulic conductivity profile on the basis of noisy observations of hydraulic heads. An augmented Kalman smoother is applied to the stochastic flow equation. The implementation of the smoother requires the solution of a Riccati type partial differential equation for the smoother covariance and a stochastic partial differential equation to obtain the...