TABLE OF CONTENTS The inverse scattering problem for acoustic waves consists of recovering the shape of a scatterer from the scattered field. Inverse problems have inspired a wide variety of techniques in the engineering sciences, such as remote sensing, nondestructive testing, and imaging etc., and for this reason have been the object of study by scientists in a number of diverse disciplines. Rapid progress in this field has been made since the early 1970s, and a survey of these results can be found in [126], [121], and the references cited there. However, most of the activity in this field has been directed to the cases of 
and 
. It has been noticed that in some situations, for example, in a wave guide, remote sensing and imaging lead to more complicated problems. In a homogeneous, finite-depth ocean, Gilbert and Xu [211], [210] showed [6] that the "propagating" far-field pattern can carry only the information from the N + 1 propagating modes, where N is the largest integer less than (2kh ??)/2 ? and h is the depth of the ocean. This loss of information makes this problem different from inverse problems in 
since the far-field pattern operator is no longer injective. A particular example of this occurs for 0 < k < ?/2h; then N = ?1 and the far-field pattern is identically zero for any incoming waves. Even in the case of sufficiently large k
