Green’s function representations for Marchenko imaging without up/down decomposition

Abstract

Summary Marchenko methods are based on integral representations which express Green’s functions for virtual sources and/or receivers in the subsurface in terms of the reflection response at the surface. An underlying assumption is that inside the medium the wave field can be decomposed into downgoing and upgoing waves and that evanescent waves can be neglected. We present a new derivation of Green’s function representations which circumvents these assumptions, both for the acoustic and the elastodynamic situation. These representations form the basis for research into new Marchenko methods which have the potential to handle refracted and evanescent waves and to more accurately image steeply dipping reflectors.

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

“…In this section, we briefly review a recent derivation of the Marchenko-type representation by [25] for Green's function retrieval from reflection data. Consider the configuration in Fig.…”

“…We assume that the wavefield is recorded at ∂D U , where it can be decomposed into downgoing constituents p + and upgoing constituents p − , such that p = p + + p − . It has been shown that the wavefield at any location x ∈ D may then be expressed as [25] p (x, ω)…”

“…Further, F U is a so-called focusing function, which focuses at the upper boundary and obeys wave equation ( 2) with q = 0. This function is subject to the focusing condition [25]…”

“…In this section, we have derived a Marchenko equation for (vertical) dipole Green's functions, see (6). However, the theory can be modified for the retrieval of monopole Green's functions, see [25].…”

“…2(b). Based on (25), these data should match Θ L T LU F U a , resulting in a linear inverse problem that can be solved for F U a , for instance by LSQR [42]. Since F U i and F U d are only allowed to be non-zero at (−∞, −t U d + t ǫ ], so does F U a = F U i − F U d , which we enforce during the inversion by restricting the unknown quantity to this time interval.…”