Vector subspace methods in inverse acoustic wave scattering

Abstract

The problems of detecting, locating, and identifying acoustic scatterers embedded in a known inhomogeneous background from the multistatic data matrix collected from an arbitrary unstructured, mixed sensor array are addressed using the distorted wave Born approximation. Based on this formulation two classes of problems are considered: (i) detecting and locating a finite set of point scatterers, (ii) imaging an arbitrary distribution of point or extended scatterers. Problems of the first class are shown to admit solutions based on the singular value decomposition (SVD) of the multistatic data matrix considered as a linear mapping from the finite vector space of transmitter inputs to the finite vector space of receiver outputs. In this case the SVD of the data matrix is shown to lead directly to generalized time-reversal algorithms that allow super-resolution location estimation. Problems of the second class are also addressed using the SVD but this time it is based on the multistatic data matrix considered as a linear mapping from the Hilbert space of scatterer distributions to the finite vector space of receiver outputs. In this case a generalized form of the filtered backpropagation algorithm is derived and shown to lead to a minimum norm image of the scatterer distribution.