Numerical methods for estimating the degree of practical stability of inverse problems in geoelectrics

The practical inverse problem of geoelectrics in the finite-parametric classes of media is considered on a closed bounded subset of a finite-dimensional space, which creates a favorable background for the correct solution of the problem. In this case, with certain additional constraints, the inverse problem can be regarded as conditionally correct and can be theoretically stable, although it may, at the same time, remain ill-conditioned if the degree of detail in the description (and parameterization) of the medium is not reconciled with the noise level, the volume, and spatial structure of the input data. This leads to the practical instability (ambiguity) of the solution of this problem (Berdichevsky and Dmitriev, 2008). By analyzing the properties of the forward operator of the problem in the finite-parametric classes of the media, one can derive quantitative a priori and a posteriori estimates for the degree of practical stability for the solution of the inverse problem (Shimelevich et al., 2011). On the basis of these estimates, the requirements can be formulated for developing the parameterization models of the medium and designing the stable algorithms for solving the inverse problem. These estimates are also useful for objectively estimating (verifying) previous interpretations.