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Subsurface inverse scattering problems: quantifying, qualifying, and achieving the available information

In inverse scattering problems, only a limited amount of independent data is actually available whenever the finite accuracy of the measurement set up is taken into account. In this paper, we deal with the problem of quantifying such an amount in the subsurface sensing case. In particular, an alternative formulation of the problem is given which also allows to understand how to dimensionate the measurement setup in an optimal fashion. Analytical results are reported for the case of a lossless soil, while a numerical study is carried out in the general case. By relying on the same formulation and tools, we also discuss the kind of unknown profiles that can actually be retrieved. In particular, it is shown that the class of retrievable functions exhibits intrinsic multiresolution features. This suggests that adoption of wavelet expansions to represent the unknown function may enhance the reconstruction capabilities. Numerical examples support this conclusion.