Application of the Gelfand–Levitan method to geoacoustic inversion in shallow water

The Gelfand–Levitan inverse technique is an exact inversion method originating in quantum scattering theory [I. M. Gelfand and B. M. Levitan, Am. Math. Soc. Transl. 1, 253–304 (1955)]. At a single frequency, the required input data are the Fourier transform of the plane wave reflection coefficient as a function of incident vertical wave number. The method was originally employed in underwater acoustics for a deep-water environment where a simple relationship exists between the reflection coefficient and the depth-dependent Green’s function, which is obtained by Hankel transforming measurements of the point source pressure field [A. A. Merab, Sc.D. thesis, MIT/WHOI Joint Program, Woods Hole, MA (January 1987)]. In shallow water, the Green’s function is characterized by poles corresponding to the eigenvalues of the perfectly trapped modes in the waveguide. These poles complicate the relationship between the Green’s function and the reflection coefficient, and therefore the application of the Gelfand–Levitan technique. However, through application of the Darboux transform to the governing wave equation and the reflection coefficient, the Gelfand–Levitan method can be employed for shallow-water geoacoustic inversion [J. R. McLaughlin and S. Wang, in Mathematical and Numerical Aspects of Wave Propagation, edited by J. A. DeSanto (SIAM, Philadelphia, 1998), pp. 232–236]. In this talk, the Darboux transform will be illustrated and examples of geoacoustic inversion for synthetic shallow-water acoustic data will be presented. [Work supported by ONR.]