Notions of support for far fields

In practical remote sensing, faraway sources radiate fields that, within measurement precision, are nearly those radiated by point sources. Algorithms like MUSIC (Devaney J. Acoust. Soc. Am. at press, Kirsch 2002 Inverse Problems 18 1025–40) correctly identify their number, their locations and their strengths based on observations of the near or far fields they radiate. Asymptotic perturbation formulae (Ammari et al 2005 SIAM J. Appl. Math. 65 2107–27, Brühl et al 2003 Numer. Math. 93 635–54) have been used to successfully locate small sparse inclusions based on remote measurements. The main motivation for this paper is to locate sources which are supported on sets that are larger and less sparse. Although the far field of a solution to the inhomogeneous Helmholtz equation does not determine the source, or its support, uniquely, we will show how to associate with any far field a unique union of well-separated-convex sets (UWSC sets) that is both big enough to support a source that can radiate that far field, and small enough that it must be contained in the UWSC-support of any source that radiates the same far field. This means that it makes theoretical sense to look for not only the number and the locations, but also the convex geometry of sources based on the far field they radiate. The only requirement is that sources be well separated—the diameter of each convex component is strictly smaller than the distance to the other components. We also give examples to illustrate the extent to which both the convexity and well-separated properties in UWSC are necessary, i.e. we will exhibit far fields with which it is not possible to associate a unique smallest compact set or, in R, a unique smallest disjoint union of convex sets.