Remarks on the Inverse Scattering Problem for Acoustic Waves

A fact that is often overlooked in inverse scattering theory is that there are many inverse scattering problems and the ones chosen for study are frequently done so on the basis of mathematical convenience rather than physical reality. Of course in some cases mathematical convenience and physical reality coincide and there is no problem. However in many other situations this is not the case. For example the two most commonly studied inverse scattering problems in acoustics are, given the far-field pattern of the scattered wave at a fixed frequency, 1) to determine the shape of a sound-soft or sound-hard obstacle and 2) to determine the “index of refraction” of a penetrable obstacle under the assumption that the pressure is smoothly varying across the boundary of the obstacle [3]. These problems only make sense if it is known a priori that the object is in fact sound-soft or sound-hard or, if the object is penetrable, that the pressure is smoothly varying, i.e. that certain physical properties of the object are known as apriori. If multi-frequency far-field data is assumed known then further assumptions are needed on the dispersion properties of the scattering object. From this point of view, the first problem in inverse scattering theory is to find out what can be determined about the scattering object if nothing at all is known apriori about its physical composition or dispersive properties. This is the problem we will address ourselves to in this paper.