Inverse acoustic scattering series using the volterra renormalization of the Lippman-Schwinger equation in one dimension

The inverse scattering problem has enormous importance both for practical and theoretical applications, such as seismic exploration, nondestructive testing, and medical imaging. Based on the early work of Jost and Kohn \cite{jost52}, Moses \cite{Moses56}, Razavy \cite{Razavy75} and Prosser \cite{prosser1969}, Weglein and co-workers have pioneered inverse scattering series methods that require no assumed propagation velocity model. Kouri and Vijay formulated the 1-D acoustic scattering series in terms of a Volterra kernel with reflection and transmission data\cite{Kouri03}. It can be further proved that the Born-Neumann series solution of the Volterra equation converges absolutely, irrespective of the strength of the velocity interaction. Following this previous work of Kouri, higher orders of the Volterra Inverse Scattering Series (VISS) with reflection and transmission data ($Rk/Tk$) are analyzed here. In addition, for the seismic exploration applications, we also extended the VISS approach to the case where only the reflection data is available. The cases of single square barriers or wells and Gauss barriers and wells are studied to illustrate how well the Volterra Inverse Scattering Series performs the inversion. The results demonstrate that the Volterra inverse scattering series method is an effective tool in inverse scattering.