Marchenko Green's Function Retrieval in Layered Elastic Media from Two-Sided Reflection and Transmission Data

Abstract

By solving a Marchenko equation, Green's functions at an arbitrary (inner) depth level inside an unknown elastic layered medium can be retrieved from single-sided reflection data, which are to be collected at the top of the medium. So far, an exact solution could only be obtained if the medium obeys stringent monotonicity conditions and if all forward-scattered (non-converted and converted) transmissions between the acquisition level and the inner depth level are known a-priori. We introduce an alternative Marchenko equation by revising the window operators that are applied in its derivation. We also introduce an auxiliary equation for transmission data, which are to be collected at the bottom of the medium, and a coupled equation, which is based on both reflection and transmission data. We show that the joint system of the Marchenko equation, the auxiliary equation and the coupled equation can be succesfully inverted when broadband reflection and transmission data are available. This results in a novel methodology for elastodynamic Green's function retrieval from two-sided data. Apart from these data, our approach requires P- and S-wave transmission times between the inner depth level and the top of the medium, as well as two angle-dependent amplitude scaling factors, which can be estimated from the data by enforcing energy conservation.

Marchenko-type focusing functions: Generalisation, modelling and imaging

“…Hence, the extension of the Marchenko method to two and three dimensions paved the way for various new methodologies and applications, like Marchenko imaging (Wapenaar et al, 2014b), target-oriented imaging (Ravasi et al, 2016), multiple elimination (Zhang and Slob, 2019; and monitoring (Brackenhoff et al, 2019a;van IJsseldijk and Wapenaar, 2021). While most of these applications are for acoustic waves, the Marchenko scheme was recently also extended to elastodynamic waves (da Costa Filho et al, 2014;Reinicke and Wapenaar, 2019;Reinicke et al, 2020;van der Neut et al, 2022). The extension of the Marchenko method to two and three dimensions also has some drawbacks.…”