Inverse geophysical and potential scattering on a small body

摘要

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.