The adjoint method of optimal control for the acoustic monitoring of a shallow water environment

Despite the variety of applications in these research fields, adjoint methods have only very recently drawn attention from the ocean acoustics community. In ocean acoustic tomography and geoacoustic inversion, where the inverse problem is to recover unknown acoustic properties of the water column and the seabed from acoustic transmission data, the solution approaches are typically based on travel time inversion or standard matched-field processing in combination with metaheuristics for global optimization.

In order to complement the adjoint schemes already in use in meteorology and oceanography with an ocean acoustic component, this thesis is concerned with the development of the adjoint of a full-field acoustic propagation model for shallow water environments.

In view of the increasing importance of global ocean observing systems such as the European Seas Observatory Network, the Arctic Ocean Observing System and Maritime Rapid Environmental Assessment (MREA) systems for defence and security applications, the adjoint of an ocean acoustic propagation model can become an integral part of a coupled oceanographic and acoustic data assimilation scheme in the future.

Given the acoustic pressure field measured on a vertical hydrophone array and a modelled replica field that is calculated for a specific parametrization of the environment, the developed adjoint model backpropagates the mismatch (residual) between the measured and predicted field from the receiver array towards the source.

The backpropagated error field is then converted into an estimate of the exact gradient of the objective function with respect to any of the relevant physical parameters of the environment including the sound speed structure in the water column and densities, compressional/shear sound speeds, and attenuations of the sediment layers and the sub-bottom halfspace. The resulting environmental gradients can be used in combination with gradient descent methods such as conjugate gradient, or Newton-type optimization methods tolocate the error surface minimum via a series of iterations. This is particularly attractive for monitoring slowly varying environments, where the gradient information can be used to track the environmental parameters continuously over time and space.

In shallow water environments, where an accurate treatment of the acoustic interaction with the bottom is of outmost importance for a correct prediction of the sound field, and field data are often recorded on non-fully populated arrays, there is an inherent need for observation over a broad range of frequencies. For this purpose, the adjoint-based approach is generalized for a joint optimization across multiple frequencies and special attention is devoted to regularization methods that incorporate additional information about the desired solution in order to stabilize the optimization process.

Starting with an analytical formulation of the multiple-frequency adjoint approach for parabolic-type approximations, the adjoint method is progressively tailored in the course of the thesis towards a realistic wide-angle parabolic equation propagation model and the treatment of fully nonlocal impedance boundary conditions. A semi-automatic adjoint generation via modular graph approach enables the direct inversion of both the geoacoustic parameters embedded in the discrete nonlocal boundary condition and the acoustic properties of the water column. Several case studies based on environmental data obtained in Mediterranean shallow waters are used in the thesis to assess the capabilities of adjoint-based acoustic inversion for different experimental configurations, particularly taking into account sparse array geometries and partial depth coverage of the water column. The numerical implementation of the approach is found to be robust, provided that the initial guesses are not too far from the desired solution, and accurate, and converges in a small number of iterations. During the multi-frequency optimization process, the evolution of the control parameters displays a parameter hierarchy which clearly relates to the relative sensitivity of the acoustic pressure field to the physical parameters.

The actual validation of the adjoint-generated environmental gradients for acoustic monitoring of a shallow water environment is based on acoustic and oceanographic data from the Yellow Shark '94 and the MREA '07 sea trials, conducted in the Tyrrhenian Sea, south of the island of Elba.

Starting from an initial guess of the environmental control parameters, either obtained through acoustic inversion with global search or supported by archival in-situ data, the adjoint method provides an efficient means to adjust local changes with a couple of iterations and monitor the environmental properties over a series of inversions.

In this thesis the adjoint-based approach is used, e.g., to fine-tune up to eight bottom geoacoustic parameters of a shallow-water environment and to track the time-varying sound speed profile in the water column.

In the same way the approach can be extended to track the spatial water column and bottom structure using a mobile network of sparse arrays.

Work is currently being focused on the inclusion of the adjoint approach into hybrid optimization schemes or ensemble predictions, as an essential building block in a combined ocean acoustic data assimilation framework and the subsequent validation of the acoustic monitoring capabilities with long-term experimental data in shallow water environments.