Dynamic inverse problem in a weakly laterally inhomogeneous medium: theory and numerical experiment

Abstract

An inverse problem of wave propagation into a weakly laterally inhomogeneous medium occupying a half-space is considered in the acoustic approximation. The halfspace 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 ).