An acoustic Lippmann-Schwinger inversion method: applications and comparison with the linear sampling method

Abstract

We introduce an imaging method based on solving the Lippmann-Schwinger equation of acoustic scattering theory. We compare and contrast the proposed Lippmann-Schwinger inversion with the well-established linear sampling method using numerical examples. We demonstrate that the two imaging methods are physically grounded in different but related wave propagation problems: Lippmann-Schwinger inversion seeks to reconstruct the space and time dependence of a scatterer based on the observed scattered field in a performed physical experiment, whereas the linear sampling method seeks to focus wave fields in a simulated virtual experiment by estimating the space and time dependence of an inverse source function that cancels the effects of the scatterer at a specified focusing point. In both cases, the medium in which the waves propagate is the same; however, neither method requires prior knowledge or assumptions on the physical properties of the unknown scatterer—only knowledge of the background medium is needed. We demonstrate that the linear sampling method is preferable to Lippmann-Schwinger inversion for target-oriented imaging applications, as Lippmann-Schwinger inversion gives nonphysical results when the chosen imaging domain does not contain the scatterer.