Inverse elastic scattering in three dimensions

Abstract

It is shown how to find the density and the elastic constants in the half-space x3≥0 from the value of the scattered displacement field us on the plane x3=0. The source of the field is a time-harmonic force applied at a source point xs on x3=0, and the field is measured at a geophone position xg on this plane. The medium is assumed to differ slightly from a uniform medium so that us(xg, xs, ω) can be calculated by the single scattering or Born approximation. As is to be expected, us contains more than enough information for the desired reconstruction. Therefore, a subset of the data, involving only longitudinal waves, is used. From it the density and the elastic constants are determined explicitly. A similar problem in which the elastic medium occupies all of space, with perturbations confined to a half-space, is treated also.