It is well known that the solution of an inverse problem is ill-posed and not unique. To avoid difficulties caused by this, when solving such a problem, Tikhonov's regularization terms are usually added to the norm quantifying the discrepancy between the model's predictions and experimental data. This regularization term however is often inadequate to perform the identification of a field of material properties that varies spatially. This is all the more difficult when dealing with the numerical solution of this inverse problem, for the sought field is spatially discretized and this discretization can influence the result of the identification. We will here examine an overall strategy using classical adaptive meshing methods used to circumvent these drawbacks. The first step consists of using two distinct meshes: one associated with the discretization of the sought spatial field and the other associated with the solution of the mechanical problems (forward and adjoint states). In the second step, we will introduce local error estimators that allow an oriented refinement of the mesh associated with the sought parameters. This general strategy is applied to a practical case study: the detection of underground cavities using experimental data obtained by an interferometric device on a satellite. We will then address the question of how the regularization terms and the error estimator driving the mesh refinement were selected.