Inverse problem for the reduced wave equation with fixed incident wave. II

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Abstract
The reduced wave equation Δu+k2n2(x)u = 0 for x in R3 is considered where [n2−1] is a real function of compact support given by the domain D. Sources outside D produce an incident field ui (satisfying the equation with n≡1 for all x). This generates a scattered field us(x,n) satisfying the reduced wave equation and the radiation condition. The scattered field is measured at a set of points {xj}Nj = 1 outside D. If n* is a known quantity for which us(x,n*) can be computed, the inverse problem considered here, consists of solving the nonlinear system of functional equations us(xj,n) −us(xj,n*) = bj, j = 1,2,...,N for an n(x) such that ℱD (n2−n2*)2W dx is minimized. With restrictions on the data set {bj}Nj = 1, a unique solution can be generated in a certain ball by iterating a system of nonlinear integral equations. It is shown that n(x), with less stringent conditions than that of continuity, may be treated.

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