On Green’s function retrieval by iterative substitution of the coupled Marchenko equations
01 Nov 2015-Geophysical Journal International (Oxford University Press)-Vol. 203, Iss: 2, pp 792-813
TL;DR: In this article, the authors propose an iterative substitution of the coupled Marchenko equations to retrieve the Green's functions from a source or receiver array at an acquisition surface to an arbitrary location in an acoustic medium.
Abstract: Iterative 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.
Citations
More filters
TL;DR: In this article, a filter is computed from the measured reflection response and does not require a background model, and the filter is a focusing wavefield that focuses inside a layered medium and removes all internal multiples between the surface and the focus depth.
Abstract: We present an imaging method that creates a map of reflection coefficients in correct one-way time with no contamination from internal multiples using purely a filtering approach. The filter is computed from the measured reflection response and does not require a background model. We demonstrate that the filter is a focusing wavefield that focuses inside a layered medium and removes all internal multiples between the surface and the focus depth. The reflection response and the focusing wavefield can then be used for retrieving virtual vertical seismic profile data, thereby redatuming the source to the focus depth. Deconvolving the upgoing by the downgoing vertical seismic profile data redatums the receiver to the focus depth and gives the desired image. We then show that, for oblique angles of incidence in horizontally layered media, the image of the same quality as for 1D waves can be constructed. This step can be followed by a linear operation to determine velocity and density as a function of depth. Numerical simulations show the method can handle finite frequency bandwidth data and the effect of tunneling through thin layers.
187 citations
TL;DR: In this article, a new method of wavefield extrapolation based on inverse scattering theory produces accurate estimates of these subsurface scattered wavefields, while still using relatively little information about the Earth's properties.
Abstract: S U M M A R Y Seismic imaging provides much of our information about the Earth’s crustal structure. The principal source of imaging errors derives from simplicistically modeled predictions of the complex, scattered wavefields that interact with each subsurface point to be imaged. A new method of wavefield extrapolation based on inverse scattering theory produces accurate estimates of these subsurface scattered wavefields, while still using relatively little information about the Earth’s properties. We use it for the first time to create real target-oriented seismic images of a North Sea field. We synthesize underside illumination from surface reflection data, and use it to reveal subsurface features that are not present in an image from conventional migration of surface data. To reconstruct underside reflections, we rely on the so-called downgoing focusing function, whose coda consists entirely of transmission-bornmultiple scattering. As such, we provide the first field data example of reconstructing underside reflections with contributions from transmitted multiples, without the need to first locate or image any reflectors in order to reconstruct multiple scattering effects.
122 citations
Cites methods from "On Green’s function retrieval by it..."
...As noted by van der Neut et al. (2015) and Vasconcelos et al. (2015), the solution of the focusing functions at iteration K can be written in a compact form by means of a Neumann series expansion: f+(K ) = K∑ k=0 ( R∗ R)kf+d f−(K ) = R K∑ k=0 ( R∗ R)kf+d ....
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TL;DR: In this article, a data-driven wavefield focusing approach was proposed to decompose the Green's function at a virtual receiver at depth in its downgoing and upgoing components, which were then used to create a ghost-free image of the medium with either cross-correlation or multidimensional deconvolution.
Abstract: Standard imaging techniques rely on the single scattering assumption. This requires that the recorded data do not include internal multiples, i.e., waves that have bounced multiple times between reflectors before reaching the receivers at the acquisition surface. When multiple reflections are present in the data, standard imaging algorithms incorrectly image them as ghost reflectors. These artifacts can mislead interpreters in locating potential hydrocarbon reservoirs. Recently, we introduced a new approach for retrieving the Green’s function recorded at the acquisition surface due to a virtual source located at depth. We refer to this approach as data-driven wavefield focusing. Additionally, after applying source-receiver reciprocity, this approach allowed us to decompose the Green’s function at a virtual receiver at depth in its downgoing and upgoing components. These wavefields were then used to create a ghost-free image of the medium with either crosscorrelation or multidimensional deconvolution, presenting an advantage over standard prestack migration. We tested the robustness of our approach when an erroneous background velocity model is used to estimate the first-arriving waves, which are a required input for the datadriven wavefield focusing process. We tested the new method with a numerical example based on a modification of the Amoco model.
113 citations
TL;DR: In this article, an adaptive substitution of the multidimensional Marchenko equation has been introduced to integrate internal multiple reflections in the seismic imaging process, without the need of a macro velocity model of the subsurface.
Abstract: Iterative substitution of the multidimensional Marchenko equation has been introduced recently to integrate internal multiple reflections in the seismic imaging process. In so-called Marchenko imaging, a macro velocity model of the subsurface is required to meet this objective. The model is used to back-propagate the data during the first iteration and to truncate integrals in time during all successive iterations. In case of an erroneous model, the image will be blurred (akin to conventional imaging) and artifacts may arise from inaccurate integral truncations. However, the scheme is still successful in removing artifacts from internal multiple reflections. Inspired by these observations, we rewrote the Marchenko equation, such that it can be applied early in a processing flow, without the need of a macro velocity model. Instead, we have required an estimate of the two-way traveltime surface of a selected horizon in the subsurface. We have introduced an approximation, such that adaptive subtracti...
105 citations
TL;DR: In this paper, the up-and down-going Green's functions from virtual sources in the subsurface are reconstructed in convolutional interferometry by combining purely reflected, up and downgoing Green’s functions.
Abstract: Standard seismic processing steps such as velocity analysis and reverse time migration (imaging) usually assume that all reflections are primaries: Multiples represent a source of coherent noise and must be suppressed to avoid imaging artifacts. Many suppression methods are relatively ineffective for internal multiples. We show how to predict and remove internal multiples using Marchenko autofocusing and seismic interferometry. We first show how internal multiples can theoretically be reconstructed in convolutional interferometry by combining purely reflected, up- and downgoing Green’s functions from virtual sources in the subsurface. We then generate the relevant up- and downgoing wavefields at virtual sources along discrete subsurface boundaries using autofocusing. Then, we convolve purely scattered components of up- and downgoing Green’s functions to reconstruct only the internal multiple field, which is adaptively subtracted from the measured data. Crucially, this is all possible without detailed modeled information about the earth’s subsurface. The method only requires surface reflection data and estimates of direct (nonreflected) arrivals between subsurface virtual sources and the acquisition surface. The method is demostrated on a stratified synclinal model and shown to be particularly robust against errors in the reference velocity model used.
86 citations
References
More filters
TL;DR: In this paper, a method for the elimination of all surface-related multiples by means of a process that removes the influence of the surface reflectivity from the data is proposed.
Abstract: The major amount of multiple energy in seismic data is related to the large reflectivity of the surface. A method is proposed for the elimination of all surface-related multiples by means of a process that removes the influence of the surface reflectivity from the data. An important property of the proposed multiple elimination process is that no knowledge of the subsurface is required. On the other hand, the source signature and the surface reflectivity do need to be provided. As a consequence, the proposed process has been implemented adaptively, meaning that multiple elimination is designed as an inversion process where the source and surface reflectivity properties are estimated and where the multiple-free data equals the inversion residue. Results on simulated data and field data show that the proposed multiple elimination process should be considered as one of the key inversion steps in stepwise seismic inversion.
740 citations
"On Green’s function retrieval by it..." refers methods in this paper
...Unlike in the conventional surface-related multiple elimination methodology (Verschuur et al. 1992), the source signature is deconvolved by these methods and the appropriate scaling of the output reflection response (as required by our scheme) is guaranteed....
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TL;DR: In this paper, a multidimensional multiple-attenuation method is presented that does not require any subsurface information for either surface or internal multiples. But it does not consider the relationship between forward and inverse scattering.
Abstract: We present a multidimensional multiple‐attenuation method that does not require any subsurface information for either surface or internal multiples. To derive these algorithms, we start with a scattering theory description of seismic data. We then introduce and develop several new theoretical concepts concerning the fundamental nature of and the relationship between forward and inverse scattering. These include (1) the idea that the inversion process can be viewed as a series of steps, each with a specific task; (2) the realization that the inverse‐scattering series provides an opportunity for separating out subseries with specific and useful tasks; (3) the recognition that these task‐specific subseries can have different (and more favorable) data requirements, convergence, and stability conditions than does the original complete inverse series; and, most importantly, (4) the development of the first method for physically interpreting the contribution that individual terms (and pieces of terms) in the inv...
497 citations
TL;DR: This work uses time-reversal logic to create a new downward-continued data set with virtual sources (VS's) at the geophone locations, and focuses energy that passes through the overburden into useful primary energy for the VS.
Abstract: We present a way to image through complex overburden. The method uses surface shots with downhole receivers placed below the most complex part of the troublesome overburden. No knowledge of the velocity model between shots and receivers is required. The method uses time-reversal logic to create a new downward-continued data set with virtual sources (VS's) at the geophone locations. Time reversal focuses energy that passes through the overburden into useful primary energy for the VS. In contrast to physical acoustics, our time reversal is done on a computer, utilizing conventional acquisition with surface shots and downhole geophones. With this approach, we can image below extremely complex (realistic) overburden — in fact, the more complex the better. We recast the data to those with sources where we actually know and can control the waveform that has a downward-radiation pattern that may also be controlled, and is reproducible for 4D even if the near-surface changes or the shooting geometry is altered sl...
414 citations
Book•
01 Jan 1993
TL;DR: In this paper, the reciprocity theorem was chosen as the central theme of the seismic wave theory, and the seismic experiment was formulated as terme of a geological system response to a known source function.
Abstract: Progress in seismic data processing requires the knowledge of all the theoretical aspects of the acoustic wave theory. We choose the reciprocity theorem as the central theme, because it constitutes the fundaments of the seismic wave theory (Fokkema and van den Berg, 1993). In essence, two states are distinguished in this theorem. These can be completely different, although they share the same time -invariant domgin of application and they are related via an interaction quantity. The particular choice of the two states determines the acoustic application. This makes it possible to formulate the seismic experiment in terme of a geological system response to a known source function.
334 citations
TL;DR: The results show that the surface-related multiple-elimination process is very effective in time gates where the moveout properties of primaries and multiples are very similar (generally deep data), as well as for situations with a complex multiple-generating system.
Abstract: A surface-related multiple-elimination method can be formulated as an iterative procedure: the output of one iteration step is used as input for the next iteration step (part I of this paper). In this paper (part II) it is shown that the procedure can be made very efficient if a good initial estimate of the multiple-free data set can be provided in the first iteration, and in many situations, the Radon-based multiple-elimination method may provide such an estimate. It is also shown that for each iteration, the inverse source wavelet can be accurately estimated by a linear (least-squares) inversion process. Optionally, source and detector variations and directivity effects can be included, although the examples are given without these options. The iterative multiple elimination process, together with the source wavelet estimation, are illustrated with numerical experiments as well as with field data examples. The results show that the surface-related multiple-elimination process is very effective in time gates where the moveout properties of primaries and multiples are very similar (generally deep data), as well as for situations with a complex multiple-generating system.
302 citations
"On Green’s function retrieval by it..." refers background in this paper
...In free-surface multiple elimination, similar adaptive filters have proven to be of great help in matching predicted events (multiples) to field observations (Verschuur & Berkhout 1997)....
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