Local Tomographic Methods in SONAR

Abstract

. Tomographic methods are described that will reconstruct object boundaries in shallow water using SONAR data. The basic ideas involve microlocal analysis, and they are valid under weak assumptions even if the data do not correspond exactly to our model. 1 Introduction Integrals over spheres are important in pure mathematics [12], [20], [22] and in applications in partial dierential equations [15] and for physical problems including SONAR [10] [21], seismic testing [21], and RADAR [4]. In this article, we will describe the application to SONAR and geophysical testing and prove a general uniqueness theorem for local data. We will give a singularity detection method for the linear problem that requires only local data. We will explain why this method is valid for data that do not t our model as long as certain fairly weak assumptions hold. Our results are all valid in any dimension, in particular, n = 2 and n = 3. In each of these applied problems, after a linearization, the original ..