Impedance profile recovery from transmission data

Abstract

We consider the problem of reconstructing the impedance profile from transmitted data. The problem (which arises for example in one‐dimensional tomography) is quite different from the problem in reflection seismology where the excitation and measurements are made at the surface. In our previous investigations, we used transmutation techniques to relate the transmitted data for time 0<t<∞ to the reflection data. This allows the use of the Gelfand–Levitan theory to recover the profile. In the present work, we consider the more difficult situation in which the transmitted data are given only up to a finite time. We study various maps and show that we can pose a fixed point problem to solve for the unknown profile. A new way of characterizing the Gelfand–Levitan theory for this problem based on minimization procedures is also presented.