Shape retrieval of an obstacle immersed in shallow water from single-frequency farfields using a complete family method

The cross sectional contour of a sound-soft closed cylindrical obstacle placed in an acoustic planar waveguide modelling a shallow water configuration is retrieved from a limited knowledge of scattered farfield patterns in the water layer at a single frequency. A complete Dirichlet family of fundamental solutions of the corresponding boundary value problem is introduced (Green's functions of the waveguide). Iterative construction of the contour is carried out by minimizing a two-term cost functional. The first term measures how well the data are fitted, the second term how well the boundary condition is satisfied. In practice, a star-shaped contour is sought while the scattered field is taken as a finite weighted sum of Green's functions whose source locations evolve with the retrieved contour. Reconstructions from independently generated synthetic data for both convex and non-convex shapes at a low and a high frequency are shown. Influence of numerical parameters (initial shape, number of Green's functions and sampling nodes of the contour, relative weight of each term in the cost functional) and physical ones (location of sources, location and positioning accuracy of the receivers, measurement noise) is investigated. The good efficiency of this complete family method is confirmed in a demanding situation where, in addition to filtering out of high-spatial-frequency wavefields with range, only finitely many modes are propagated; and where lack of information due to aspect-limited data is not alleviated by frequency diversity.