General statements of impossibility can be important in science and engineering. Ambiguities in inverse problems are cases of non-uniqueness where classes of different objects give the same response. A strong ambiguity is one which no additional data will remove the non- uniqueness, whereas a weak ambiguity is one which can be removed by additional data. In direct scattering theory, different potentials with one or more trapped modes may give the same R(k) or the e^i(alpha )R(k) where (alpha) is a real parameter at all wave-numbers k. In three-dimensional direct scattering theory, different material media and sources J, p give the same scattering matrix at all times (or wave numbers) at all scattering angles and all incident angles. Examples of strong ambiguities will be given including one where a temporal relaxation of a homogeneous body is equivalent to a totally different time-independent homogeneous body. Weak ambiguities will be presented including both examples of incident scatters. The conditions on the scatterers at spatial infinity and their trapped mode bound-state structure will be given.

Date Published: 29 December 1992
PDF: 10 pages
Proc. SPIE 1767, Inverse Problems in Scattering and Imaging, (29 December 1992); doi: 10.1117/12.139035

Published in SPIE Proceedings Vol. 1767:
Inverse Problems in Scattering and Imaging
Michael A. Fiddy, Editor(s)