Multi-dimensional Inverse acoustic scattering series using the Volterra renormalization of the Lippmann-Schwinger equation

Abstract

SUMMARY We report the extension of the Volterra inverse acoustic scattering series (VISS) approach presented in (Lesage et al., 2013) using reflection data (Rk) to multi-dimensions. The approach consists in combining two ideas: the renormalization of the Lippmann-Schwinger equation to obtain a Volterra equation framework (Kouri and Vijay, 2003) and the formal series expansion using reflection coefficients (Moses, 1956). The benefit of formulating acoustic scattering in terms of a Volterra kernel is substantial. Indeed the corresponding Born-Neumann series solution is absolutely convergent independent of the strength of the coupling characterizing the interaction. While treating the depth variation in the same manner as in the onedimensional case, additional lateral and longitudinal variations are addressed through Fourier expansions of the pressure wave, the reflection data and the velocity perturbation. We derive new multi-dimensional inverse acoustic scattering series for reflection data which we evaluate numerically for 2dimensional velocity models presenting depth and lateral variations. Our results compare well to results obtained by (Liu et al., 2005).