Inverse problem for the reduced wave equation with fixed incident field

Abstract

The inverse problem for the reduced wave equation Δu+k2n2(x) u=0 is examined when n (x) is assumed to be real continuous and equal to unity outside some prescribed compact domain D0 in R3. The case where a finite set of measurements of the scattered or total field (generated by a single incident wave at a fixed frequency) at points outside D0 is considered. It is shown that the inverse problem can be expressed in terms of a system of linear functional equations, plus a quadratic nonlinear integral equation. By imposing an additional criterion for uniqueness of the functional equations and placing certain restrictions on the size of the measured data, a solution of the system can be generated by successive approximations. It is shown that the iterative scheme yields correction terms to the solution that are obtained by the common linearized approximation where n (x) is assumed to be a very small perturbation of a known quantity and only linear terms are retained.