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.

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

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.

/pdf/data-driven-wavefield-focusing-and-imaging-with-6rg8yzr1jz.pdf

Data-driven wavefield focusing and imaging with multidimensional deconvolution: Numerical examples for reflection data with internal multiples

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...

Adaptive overburden elimination with the multidimensional Marchenko equation

[1] We present results from the tomographic analysis of seismic data from the Parkfield area using three different inversion codes. The models provide a consistent view of the complex velocity structure in the vicinity of the San Andreas, including a sharp velocity contrast across the fault. We use the inversion results to assess our confidence in the absolute location accuracy of a potential target earthquake. We derive two types of accuracy estimates, one based on a consideration of the location differences from the three inversion methods, and the other based on the absolute location accuracy of “virtual earthquakes.” Location differences are on the order of 100–200 m horizontally and up to 500 m vertically. Bounds on the absolute location errors based on the “virtual earthquake” relocations are ∼50 m horizontally and vertically. The average of our locations places the target event epicenter within about 100 m of the SAF surface trace.

Fine-scale structure of the San Andreas fault zone and location of the SAFOD target earthquakes : Preparing for the San Andreas fault observatory at depth: Earthquakes and crustal structure

Marchenko redatuming is a revolutionary technique to estimate Green’s functions from virtual sources in the subsurface using only data measured at the earth’s surface, without having to place either sources or receivers in the subsurface. This goal is achieved by crafting special wavefields (so-called focusing functions) that can focus energy at a chosen point in the subsurface. Despite its great potential, strict requirements on the reflection response such as knowledge and accurate deconvolution of the source wavelet (including absolute scaling factor) and co-location of sources and receivers have so far challenged the application of Marchenko redatuming to real-world scenarios. I combine the coupled Marchenko equations with a one-way version of the Rayleigh integral representation to obtain a new redatuming scheme that handles internal as well as free-surface multiples using dual-sensor, band-limited seismic data (with an unknown source signature) from any acquisition system that presents arbit...

Rayleigh-Marchenko redatuming for target-oriented, true-amplitude imaging

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.

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Internal multiple prediction and removal using marchenko autofocusing and seismic interferometry

Katrin Löer,1 Nima Riahi2 and Erik H. Saenger1,3 1International Geothermal Centre, Bochum University of Applied Sciences, Lennershofstr. 140, D-44801 Bochum, Germany. E-mail: katrin.loeer@hs-bochum.de 2Schweizerische Bundesbahnen SBB (Swiss federal railways SFR), Hilfikerstr. 1, CH-3000 Bern 65, Switzerland 3Faculty of Geosciences, Institute of Geology, Mineralogy und Geophysics, Ruhr-Universität Bochum, Universitätsstr. 150, D-44780 Bochum, Germany

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Three-component ambient noise beamforming in the Parkfield area

We have evaluated an explicit relationship between the representations of internal multiples by source-receiver interferometry and an inverse-scattering series. This provides a new insight into the interaction of different terms in each of these internal multiple prediction equations and explains why amplitudes of estimated multiples are typically incorrect. A downside of the existing representations is that their computational cost is extremely high, which can be a precluding factor especially in 3D applications. Using our insight from source-receiver interferometry, we have developed an alternative, computationally more efficient way to predict internal multiples. The new formula is based on crosscorrelation and convolution: two operations that are computationally cheap and routinely used in interferometric methods. We have compared the results of the standard and the alternative formulas qualitatively in terms of the constructed wavefields and quantitatively in terms of the computational cost using examples from a synthetic data set.

/pdf/relating-source-receiver-interferometry-to-an-inverse-7t2149juxo.pdf

Relating source-receiver interferometry to an inverse-scattering series to derive a new method to estimate internal multiples

/pdf/diffracted-and-pseudo-physical-waves-from-spatially-limited-2n8b1bqlme.pdf

Diffracted and pseudo-physical waves from spatially limited arrays using source–receiver interferometry (SRI)

An automated algorithm uses recordings of acoustic energy across a spatially-distributed array to derive information about multiply scattered acoustic waves in heterogeneous media. The arrival time and scattering-order of each recorded diffracted acoustic wave, and the exact sequence of diffractors encountered by that wave, are estimated without requiring an explicit model of the medium through which the wave propagated. Individual diffractors are identified on the basis of their unique single-scattering relative travel-time curves (move-outs) across the array, and secondary (twice-scattered) waves are detected using semblance analysis along temporally offset primary move-outs. This information is sufficient to estimate travel times and scattering paths of all multiply diffracted waves of any order, and for these events to be identified in recorded data. The algorithm is applied to synthetic acoustic data sets from a variety of media, including different numbers of point-diffractors and a medium with strong heterogeneity and non-hyperbolic move-outs.

Katrin Löer

Papers

Internal multiple prediction and removal using marchenko autofocusing and seismic interferometry

Three-component ambient noise beamforming in the Parkfield area

Relating source-receiver interferometry to an inverse-scattering series to derive a new method to estimate internal multiples

Diffracted and pseudo-physical waves from spatially limited arrays using source–receiver interferometry (SRI)

Automatic identification of multiply diffracted waves and their ordered scattering paths