A numerical scheme to solve the Lippmann-Schwinger equation for a linearly varying background

We consider the problem of solving the Helmholtz equation in 2D and 3D, where the velocity can be decomposed into a sum of two components: (i) the background velocity component that varies linearly with depth, and (ii) a smooth, compactly supported velocity perturbation. The integral equation equivalent of this problem yields the Lippmann-Schwinger equation (LSE), and in this paper we present a new numerical scheme to solve it. Iterative methods to solve the resulting linear system involves repeated computation of the integral appearing in the LSE. Using the truncated kernel approach introduced by Vico et al. (2016), we are able to exploit the translation invariance of the background in the horizontal direction and the compact support of the perturbation, to design a scheme that handles the singularity of the integrand and efficiently computes this integral. We present 3D numerical examples where we compute the solution to the LSE using our method for different velocity models.