The inverse source problem

Abstract

The formulas Eq. (1.33) of Chapter 1 represent the solution to the radiation problem in a non-dispersive medium governed by the wave equation; i.e., they give the radiated field u +( r , t ) in terms of a known source q ( r , t ). These formulas were generalized to dispersive media in Chapter 2, where the radiation problem was solved directly in the frequency domain for a known source embedded in a uniform dispersive background medium. The inverse source problem (ISP), as its name indicates, is the inverse to the radiation problem, and in this problem one seeks the source q ( r , t ) from knowledge of its radiated field u +( r , t ). The question of what applications require a solution to an inverse source problem naturally arises. There are basically two such applications that consist of (i) imaging (reconstructing) the interior of a volume source from observations of the field radiated by the source and (ii) designing a volume source to act as a multi-dimensional antenna to radiate a prescribed field. In the first application actual field measurements are employed, thereby generating data that are then used to “solve” the ISP and thus “reconstruct” the interior of the source, whereas in the second application desired field data are used to “design” a source that will generate those data. Regarding the ISP, the two applications are essentially identical, differing only in emphasis; in application (i) we have to contend with measurement error and noisy data, whereas in application (ii) we have to contend with inconsistencies between the desired data and the constraints required of the source (antenna).