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

We report the extension of the inverse acoustic scattering approach presented in (Kouri and Vijay, 2003) from the use of both reflection and transmission data (R_k/T_k) to the sole use of the reflection data (R_k). 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. We derive new inverse acoustic scattering series for reflection data which we evaluate for test cases both analytically and numerically (Dirac-δ interaction and the square well or barrier). Our results compare well to results obtained by (Weglein et al., 2001) for the square barrier and to previous results obtained in (Kouri and Vijay, 2003) using both transmission and reflection data.