Marchenko-type focusing functions: Generalisation, modelling and imaging
Authors:   , Diekmann Leon Daniel 
Published: May 2023

Imaging is a field of mathematics and physics that aims to retrieve information about the internal structure of an object that can only be accessed on its boundary. Many imaging methods are based on the following principle: a source outside of the object emits a wave. The wave propagates through the object. Wherever the physical structure of the object changes, scattered waves are induced. These scattered waves are measured by receivers outside of the object, and these scattered data are used to invert for the interior composition of the medium under investigation.
The Marchenko integral was originally introduced for one-dimensional inverse scattering problems in the context of quantum mechanics. It can be related to Green's functions and so-called focusing functions - fields that produce a focus when injected into a medium from a single side. About ten years ago, the Marchenko integral was extended to two and three dimensions. This paved the way for, e.g., the elimination of imaging artefacts due to multiple scattering and Green's function retrieval for virtual source locations. However, many questions about the full potential as well as the accuracy of the Marchenko equation in two and three dimensions remain unanswered.
In this thesis we present a new derivation for the multidimensional Marchenko integral. Our derivation is based on a generalised framework for wavefield focusing and circumvents the limiting assumptions of the previous extension. As we use partial differential equations rather than integral equations to define focusing functions, it allows for new physical insights. For instance, our approach indicates that it is possible to model Marchenko-type focusing functions with a conventional wave equation. Ultimately, this enables us to study Marchenko-type focusing in different 2D and 3D media and learn about the accuracy of the concept. We present a straightforward modelling approach for 1D as well as a least-squares modelling approach for 2D and 3D. The latter suggests that the Marchenko integral might be inherently approximative in multiple dimensions. We also discuss Green's function retrieval with our newly derived Marchenko integral, i.e. without wavefield decomposition. This method allows for estimating Green's functions for virtual sources inside of the medium. While it requires single-sided scattering data and an estimate of the first arrival of the desired Green's function there is no need to have an actual source or receiver inside of the medium. Our results demonstrate that we can retrieve good estimates of the full-spectrum Green's functions, involving evanescent and refracted waves, which were believed to not be retrievable with the previously derived Marchenko integral. Ultimately, we discuss imaging with these Marchenko-based Green's functions. Being able to include measurements for virtual sources inside of the medium allows for a natural linearisation of the imaging problem. Thus, we use the Marchenko integral to linearise state-of-the-art imaging approaches, similar to full waveform inversion or least-squares reverse time migration, and estimate the scattering potential. Our Marchenko-based linearisation accounts for all orders of scattering and performs slightly better than a single-scattering approximation.