On the retrieval of forward-scattered waveforms from acoustic reflection and transmission data with the Marchenko equation

TL;DR

This paper enhances the Marchenko equation method for retrieving Green's functions in acoustic media by incorporating transmission data to include forward-scattered waveforms, improving accuracy in reflection and transmission analysis.

Contribution

It introduces an auxiliary equation for forward-scattered components, enabling the inclusion of forward-scattered waveforms in Green's function retrieval using the Marchenko equation.

Findings

Incorporating transmission data improves Green's function accuracy.

The auxiliary equation can be solved in media with density contrast and constant velocity.

Numerical examples demonstrate successful inclusion of forward-scattered waveforms.

Abstract

A Green's function in an acoustic medium can be retrieved from reflection data by solving a multidimensional Marchenko equation. This procedure requires a-priori knowledge of the initial focusing function, which can be interpreted as the inverse of a transmitted wavefield as it would propagate through the medium, excluding (multiply) reflected waveforms. In practice, the initial focusing function is often replaced by a time-reversed direct wave, which is computed with help of a macro velocity model. Green's functions that are retrieved under this (direct-wave) approximation typically lack forward-scattered waveforms and their associated multiple reflections. We examine whether this problem can be mitigated by incorporating transmission data. Based on these transmission data, we derive an auxiliary equation for the forward-scattered components of the initial focusing function. We demonstrate that this equation can be solved in an acoustic medium with mass density contrast and constant propagation velocity. By solving the auxiliary and Marchenko equation successively, we can include forward-scattered waveforms in our Green's function estimates, as we demonstrate with a numerical example.