Acoustical Holography is, at Best, Only a Partial Solution to the Inverse Scattering Problem

Abstract

Coherent acoustical imaging and acoustical holography are rarely employed solely for the purpose of recording and reconstructing a coherent acoustical field. Usually, the ultimate information desired concerns the characteristics of an acoustic source, a scatterer, or the inhomogeneity in a medium. The image or holographic reconstruction is an intermediate step in extracting this information (often by human visual recognition). The determination of such characteristics from measurements at a distance of emitted or scattered acoustic fields is, in fact, an inverse scattering problem. In a companion paper by the author in this volume, (hereafter referred to as “Stone-C”), the Exact Inverse Scattering Theory is reviewed as it applies to the acoustic probing problem. The purpose of this paper is to quantify the relationships among imaging, holography, and inverse scattering. It is shown that a holographic reconstruction is only a partial solution to the inverse scattering problem at best. Furthermore, it is shown that holographic data is sufficient to provide an exact solution to the inverse scattering problem, using the Exact Theory. Finally, a common but rarely acknowledged approximation associated with the interpretation of a holographic reconstruction is clarified. It is often assumed and/or stated that the field reconstructed from a hologram is the field which existed in the region of the scatterer (in optical terms, in “object space”) when the hologram was recorded. The nature of this approximation is quantified, and the conditions under which it becomes a good approximation are derived.