Inverse problems of plane wave scattering by 1D inhomogeneous layers



Inverse problems of the plane waves scattering of inclined incidence of the SH type by an inhomogeneous half-space (in particular, by a transition layer) or by an inhomogeneous layer with a free boundary are considered. The characteristics of an inhomogeneous elastic medium, i.e., the wave propagation velocity υ(z) and the density ρ(z), are functions of depth z and should be determined in the inverse problems. The following are considered: the inverse problems with data for a fixed angle of incidence θ0 of a plane wave (the angle between the vector of the normal to the plane wave front and z-axis) when the shape φ0(ξ,θ0) of an incident wave is known; the inverse problems with data for a family of angles θ0 both for known and unknown shapes of the incident wave. The functions φ0(ξ,θ0) and φ1(ξ,θ0) (the shapes of incident and reflected waves), the functions φ0(ξ,θ0) and u(H,ξ,θ0) (the shapes of the incident wave and the free boundary oscillation field), only the function u(H,ξ,θ0) (in the inverse problem with the unknown function φ0(ξ,θ0), or other data are given as data corresponding to any fixed value of θ0. Possible application areas of the inverse problems under consideration are specified.

This paper is essentially a review. A new result presented here has been obtained for inverse problems with the data {φ0(ξ,θ0), φ1(ξ,θ0)} or {φ(ξ,θ0), u(H,ξ,θ0)} for a set of angles θ0. The results, received earlier, (the uniqueness theorem, the solution method) are extended to the case when the limiting point Ō0 of this set is zero. Also, a new explicit formula, which makes it possible to find the functions υ(z) and ρ(z) using the data of these inverse problems for Ō0 = 0, has been obtained.