Marchenko inversion in a strong scattering regime including surface-related multiples

Marchenko redatuming is a powerful technique which uses all orders of scattering to construct an image of the subsurface. The algorithm is of great value when strong internal scattering is present, as it mitigates unwanted overburden artefacts when imaging a deeper subsurface target. The solution to the Marchenko equation, on which Marchenko methods are based, is often written in terms of an iterative substitution expansion (Neumann series) and conceptualized as a process of adding and removing seismic events, in an attempt to construct a focusing function. In this work we show that in some cases one may have to look beyond series expansion techniques to determine the correct solution. We prove that in the presence of surface-related multiples, the originally proposed Neumann series type algorithm may fail to converge, whereas in the absence of such multiples convergence is guaranteed. We study convergence properties of a class of such algorithms, which correspond to different choices of pre-conditioning of the Marchenko equation. We also propose and study the effectiveness of two other means of solving the Marchenko equation: LSQR (standard least squares) and a modified Levinson-type algorithm inspired by Merchant-Parks inversion, and discuss some of the challenges they pose.