Reconstruction of a Spherically Symmetric Speed of Sound

Consider the inverse acoustic scattering problem for a spherically symmetric inhomogeneity of compact support that arises, among other places, in nondestructive testing. Define the corresponding homogeneous and inhomogeneous interior transmission problems, see, e.g., [D. Colton and P. Monk, Quart. J. Mech. Math., 41 (1988), pp. 97-125]. Here the authors study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem. It is shown in McLaughlin and Polyakov [J. Differential Equations, to appear] that these eigenvalues are the zeros of an average of the scattering amplitude, and a uniqueness theorem for the inverse acoustic scattering problem is presented where these eigenvalues are the given data. In the present paper an algorithm for finding the solution of the inverse acoustic scattering problem from this subset of transmission eigenvalues is developed and implemented. The method given here completely determines the sound speed when the size, measured by an integral, satisfies a particular bound. The algorithm is based on the Gel'fand-Levitan integral equation method [I. M. Gelfand and B. M. Levitan, Amer. Math. Soc. Trans., 1 (1951), pp. 253-304], [W. Rundell and P. E. Sacks, Inverse Problems, 8 (1992), pp. 457-482].