ar X iv : m at h - ph / 0 50 60 48 v 1 1 8 Ju n 20 05 DYNAMIC INVERSE PROBLEM IN A WEAKLY LATERALLY INHOMOGENEOUS MEDIUM
نویسندگان
چکیده
An inverse problem of wave propagation into a weakly laterally inhomogeneous medium occupying a half-space is considered in the acoustic approximation. The half-space consists of an upper layer and a semi-infinite bottom separated with an interface. An assumption of a weak lateral inhomogeneity means that the velocity of wave propagation and the shape of the interface depend weakly on the horizontal coordinates, x = (x 1 , x 2), in comparison with the strong dependence on the vertical coordinate, z, giving rise to a small parameter ε << 1. Expanding the velocity in power series with respect to ε, we obtain a recurrent system of 1D inverse problems. We provide algorithms to solve these problems for the zero and first-order approximations. In the zero-order approximation , the corresponding 1D inverse problem is reduced to a system of non-linear Volterra-type integral equations. In the first-order approximation, the corresponding 1D inverse problem is reduced to a system of coupled linear Volterra integral equations. These equations are used for the numerical reconstruction of the velocity in both layers and the interface up to O(ε 2).
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