An inverse scattering problem for the two-dimensional Helmholtz equation

Type Thesis

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Summary

A two dimensional inverse scattering problem in which an acoustic plane wave is incident on a cylindrical object with the index of refraction varying only in two spatial directions is discussed. First, a modified method of lines procedure is developed to and the scattering amplitudes of the wave scattered from the object and compared with exact scattering amplitudes for some exactly solvable examples to verify its performance. Then, an inversion procedure is developed that uses the scattering amplitude for finding the object's profile. For that, Reese T. Prosser's procedure for the inversion in three dimensions is modified to our two dimensional case. Then, the back-scattered data is used to find both the Born approximation and Reese T. Prosser's first order approximation to find inversion results. Since the procedure requires the data to be available both for on-shell and off-shell values of the wave number, an interpolation method is developed to recover the needed data and inversion results are compared with exact values for a solid cylinder, a hollow cylinder and a shifted cylinder with constant index of refraction.