Inverse scattering problem in anisotropic media
Second-order elliptic equations in \(\mathbb{R}^n\) which carry no smallness assumptions on the coefficients are considered (and rewritten as an Laplace-Beltrami equation). From the scattering amplitudes the integrals are recovered. So it is possible to obtain a class of scattering invariants (constructed from the amplitude) by using results of geometric optics. These invariants are used to recover the metric tensor from the hodographic function (solving the hodographic problem). It is shown that for different operators from the considered class with the same scattering amplitude the metrics in semigeodesic coordinates can be reconstructed. If these metrics are close in terms of the Sobolev norm in semigeodesic coordinates, they are unique up to a diffeomorphism so one can pulling back the metrics into each other.
scientific article; zbMATH DE number 1330572