Focusing and Green's function retrieval in three-dimensional inverse scattering revisited: A single-sided Marchenko integral for the full wave field

Abstract

The Marchenko integral, key to inverse scattering problems across many disciplines, is a long-standing equation that relates single-sided reflection data and Green's functions for virtual source locations inside of an inaccessible, one-dimensional volume. The concept was later expanded to two and three dimensions, yielding important advances in imaging complex media, particularly in the context of geophysics. However, this expansion is based on a set of coupled Marchenko equations which requires up and down decomposition of the wave fields at both the level of the measurement surface and the level of the virtual source of the desired Green's function. The underlying theory implies that the recently developed Marchenko relations, while enabling novel applications, carry intrinsic limitations. For example, this scheme cannot incorporate evanescent or refracted waves, and in turn practical implementations must discard data to meet such requirements. We present a derivation that circumvents these limitations, thereby yielding a Marchenko integral akin to those in recent advances, but that is more general than previously assumed. We set up a wave equation based framework to describe the physical concept of focusing functions by introducing homogeneous Green's functions of the second kind. Based on this, we derive integral representations for both closed and open boundary volumes. Owing to our perspective on the integral formalism, we present an inverse scattering approach for retrieving Green's functions from single-sided reflection data—with the same practical applicability of recent methods but without any limitations due to one-way decomposition. Finally, we illustrate the capability of the Marchenko method to obtain the full wave field, including evanescent and refracted waves, within an unknown scattering medium by means of a numerical example.