A Method for Defining Gaussian Probability Densities for Forward Modeling in Finite Dimensions Using the Method of Tarantola

The inverse problem of recovering an unknown boundary function given some information about the (interior) solution is considered. It is assumed that the forward problem is governed by a linear partial differential equation (PDE) and that homogeneous boundary conditions exist on the rest of the boundary. A method developed by Tarantola (Tar87) is used which is based upon statistical inference on finite dimensional random vectors. Within this framework, a method for constructing Gaussian probability density functions which model the forward problem is proposed.