Generalisation of the solution of the inverse Richards' problem

In inverse problems defined by models that include partial differential equations, a part of the boundary conditions are unknown and are to be estimated from experimental measurements. We have shown in a previous contribution that the solution of the inverse Richards' problem can allow estimating percolation rates at the bottom of landfills through the use of measurements at the surface only. This can be a useful complement of the information furnished by the vadose measurement system, pointing to the possible presence of biases of in-situ equipment, and making it possible to use inexpensive mobile equipment to carry out surface measurements. In this article, we consider a generalisation which makes it possible to consider the presence of unknown nonlinear parameters, such as the effective hydraulic conductivity and the root uptake coefficients. This is accomplished using the method of separation of variables in the resulting estimation problem. Thanks to the linearity of the model, all these conditions can be expressed as linear functions of the unknown lower boundary condition. Otherwise, the relevant non-linear parameters are to be estimated from the data as well. Obviously, the correlation between the linear parameters contained in the unknown lower boundary conditions and the non-linear parameters can reduce the reliability of the monitoring procedure and hence the necessity of limiting the number of the latter.