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Inverse geophysical and potential scattering on a small body
Simple and numerically stable approaches to approximate solution of inverse geophysical and potential scattering problems are described. The method we propose consists of two steps. Let {nu}(z) be the inhomogeneity (potential), and let D be its support, First, we find approximations to the zeroth moment (total intensity){integral}{sub D}{nu}(z)dz and the first moment (center of gravity) {integral}{sub D}z{nu}(z)/{integral}{sub D}{nu}(z)dz of the function {nu}(z). We call this step ``inhomogeneity localization``, because in many cases the center of gravity lies inside D or is located close to it. Second, we refine the above moments and find the tensor of the second central moments of {nu}(z). Using this information, we find an ellipsoid D and a real constant {nu}, such that the inhomogeneity (potential){nu}(z) = {nu}, z {epsilon} D, and {nu}(z) = 0, z {epsilon} D, fits best the scattering data and has the same zeroth, first, and second moments. We call this step ``approximate inversion``. The proposed method does not require any intensive computations, it is very simple to implement and it is relatively stable towards noise in the data