The inverse backscattering problem and the inverse fixed angle scattering problem

University Of Delaware, Newark DE

Investigators

Abstract

The main goal of this project is to study the recovery of acoustic properties of a medium, such as the earth, by probing the medium with sound waves mechanically generated on the boundary of the medium and measuring the medium response on the boundary. This question will be studied theoretically with a focus on the recovery of the acoustic property from the data coming from a small number of experiments. This results of this study will provide insight in a variety of settings, including the exploration of the subsurface of other planets as well as medical imaging settings, where it may not be possible to generate acoustic waves at many different locations in the medium or measure the medium response at many different locations. The project will focus on two specific problems - the potential recovery backscattering problem for the constant velocity wave equation and the fixed angle velocity recovery problem for the non-constant velocity wave equation. The backscattering problem measurements are equivalent to certain measurements for the ultra-hyperbolic equation with potential, so the corresponding coefficient recovery problem for the ultra-hyperbolic equation will be studied by an adaptation of the Bukhgeim-Klibanov method. The fixed angle velocity recovery problem will be studied by an adaptation of the Bukhgeim-Klibanov method used to study the fixed angle potential recovery problem, with new techniques devised to overcome the difficulties associated with the unknown velocity affecting the structure of the solution of the differential equation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.