An injective far-field pattern operator and inverse scattering problem in a finite depth ocean

The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R 2 and R 3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L 2 (∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.