On Green's function retrieval by iterative substitution of the coupled Marchenko equations

Abstract

S U M M A R YIterative substitution of the coupled Marchenko equations is a novel methodology to retrieve the Green's functions from a source or receiver array at an acquisition surface to an arbitrary location in an acoustic medium. The methodology requires as input the single-sided reflection response at the acquisition surface and an initial focusing function, being the time-reversed direct wavefield from the acquisition surface to a specified location in the subsurface. We express the iterative scheme that is applied by this methodology explicitly as the successive actions of various linear operators, acting on an initial focusing function. These operators involve multidimensional crosscorrelations with the reflection data and truncations in time. We offer physical interpretations of the multidimensional crosscorrelations by subtracting traveltimes along common ray paths at the stationary points of the underlying integrals. This provides a clear understanding of how individual events are retrieved by the scheme. Our interpretation also exposes some of the scheme's limitations in terms of what can be retrieved in case of a finite recording aperture. Green's function retrieval is only successful if the relevant stationary points are sampled. As a consequence, internal multiples can only be retrieved at a subsurface location with a particular ray parameter if this location is illuminated by the direct wavefield with this specific ray parameter. Several assumptions are required to solve the Marchenko equations. We show that these assumptions are not always satisfied in arbitrary heterogeneous media, which can result in incomplete Green's function retrieval and the emergence of artefacts. Despite these limitations, accurate Green's functions can often be retrieved by the iterative scheme, which is highly relevant for seismic imaging and inversion of internal multiple reflections.

Target-enclosed seismic imaging

“…It can be shown that linear imaging of the redatumed data is equivalent to the imaging of primaries (Wapenaar et al, 2017). Hence, Marchenko imaging can also be interpreted as an internal multiple elimination process (Meles et al, 2015;van der Neut and Wapenaar, 2016;da Costa Filho et al, 2017). However, it has also been recently shown that novel imaging conditions can be derived for multiply reflected waves, given that a detailed model of the subsurface is available (Halliday and Curtis, 2010;Fleury and Vasconcelos, 2012;Ravasi et al, 2014Ravasi et al, , 2015b.…”

“…These Green's functions can be computed in a detailed subsurface model or they can be directly measured in boreholes. In the latter case, the recorded waveforms should be decomposed into their upgoing and downgoing components (Mehta et al, 2007;van der Neut et al, 2016), as illustrated in Figure 1a. The required Green's functions can also be computed by solving a multidimensional Marchenko equa- Figure 1.…”

“…This strategy has also been referred to as multidimensional deconvolution (Wapenaar and van der Neut, 2010;Ravasi et al, 2015a;van der Neut et al, 2016). In this paper, least-squares inversion is implemented by the LSQR algorithm (Paige and Saunders, 1982).…”

Inversion of the Multidimensional Marchenko Equation

“…In this framework, that is discussed extensively by Van der Neut et al (2015), Green's functions in the subsurface are represented as vectors, in which all relevant traces are concatenated in the time-space domain. Vector g − contains the upgoing Green's functions with multiple sources located at the surface and a receiver positioned at a specified focal point.…”

“…The direct part of the focusing function is also removed by the window: Θf + 1d = 0. The coda and the upgoing part, however, contain data only before the direct wavefield (and after the time-reversed direct wavefield) Van der Neut et al, 2015). Hence they are preserved by the window function: Θf …”