Extension of PE-adjoint optimization to multiple frequencies

Recently the concept of optimal boundary control by adjoint modelling has been introduced in shallow water acoustics for solving inverse problems. Analytic adjoint models have been derived for both the standard parabolic equation (SPE) and for Claerbout´s wide-angle approximation (WAPE), and the use of penalization methods has been investigated. This paper discusses the extension of the adjoint optimization formalism to multiple frequencies and shows how a broad-band approach can enhance the performance of the inversion process. For a sparse array geometry in particular, the generalization of the adjoint-based approach to a joint optimization across multiple frequencies is necessary to compensate for the lack of vertical sampling of the propagation modes.