Marchenko imaging can be seen as an internal multiple elimination scheme in imaging domain. By solving coupled Marchenko equations, the measured seismic data can be correctly redatumed to the subsurface with source positioned at subsurface and receivers positioned at acquisition surface. Then, the artefact-free image of the source point can be retrieved by multi-dimensionally deconvolving the redatumed up- and downgoing seismic data. The Marchenko Multiple Elimination scheme, derived from coupled Marchenko equations, is a data domain multiple elimination scheme. It can successfully predict and remove all orders of internal multiple reflections without model information or adaptive subtraction. The data domain multiple elimination scheme surgically removes internal multiple reflections without touching primary reflections. In this paper, we give a detailed comparison of both imaging domain and data domain Marchenko multiple elimination schemes. One synthetic model is used to illustrate the performance of both schemes and, the advantages and disadvantages of both schemes are discussed in detail.

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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.

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Marchenko methods can retrieve both Green’s functions and focusing functions from single-sided reflection data and a smooth velocity model, as essential components of a redatuming process. Recent studies also show that a modified Marchenko scheme can reconstruct primary-only reflection responses directly from reflection data without requiring a priori model information. To provide insight into the artifacts that arise when input data are not ideally sampled, we study the effects of subsampling in both types of Marchenko methods in 2D earth and data — by analyzing the behaviour of Marchenko-based results on synthetic data subsampled in sources or receivers. We show with a layered model that for Marchenko redatuming, subsampling effects jointly depend on the choice of integration variable and the subsampling dimension, originated from the integrand gather in the multidimensional convolution process. When reflection data are subsampled in a single dimension, integrating on the other yields spatial gaps together with artifacts while integrating on the subsampled dimension produces aliasing artifacts but without spatial gaps. Our complex subsalt model shows the subsampling may lead to very strong artifacts, which can be further complicated by having limited apertures. For Marchenko-based primary estimation (MPE), subsampling below a certain fraction of the fully-sampled data can cause MPE iterations to diverge, which can to some extent be mitigated by using more robust iterative solvers, such as LSQR. Our results, covering redatuming and primary estimation in a range of subsampling scenarios, provide insights that can inform acquisition sampling choices as well as processing parameterization and quality control, e.g., to set up appropriate data filters and scaling to accommodate the effects of dipole fields, or to help ensuring that the data interpolation achieves the desired levels of reconstruction quality that minimize subsampling artifacts in Marchenko-derived fields and images.@en

The Marchenko multiple elimination (MME) and transmission compensation schemes retrieve primary reflections in the two-way traveltime domain without model information or using adaptive subtraction. Both schemes are derived from projected Marchenko equations and are similar to each other, but they use different time-domain truncation operators. The MME scheme retrieves a new data set without internal multiple reflections. The transmission-compensated Marchenko multiple elimination scheme does the same and additionally compensates for transmission losses in the primary reflections. Both schemes can be solved with an iterative algorithm based on a Neumann series. At each iteration, a convolution or correlation between the projected focusing function and the measured reflection response is performed, and, after each convolution or correlation, a truncation in the time domain is applied. After convergence, the resulting projected focusing function is used for retrieving the transmission-compensated primary reflections and the projected Green's function is used for the physical primary reflections. We have determined that internal multiples are removed by using time-windowed input data that only contain primary reflections. We evaluate both schemes in detail and develop an iterative implementation that reproduces the presented numerical examples. The software is part of our open-source suite of programs and fits into the Seismic Unix software suite of the Colorado School of Mines.

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Marchenko multiple elimination schemes are able to attenuate all internal multiple reflections in acoustic reflection data. These can be implemented with and without compensation for two-way transmission effects in the resulting primary reflection data set. The methods are fully automated and run without human intervention, but they require the data to be properly sampled and preprocessed. Even when several primary reflections are invisible in the data because they are masked by overlapping multiples, such as in the resonant wedge model, all missing primary reflections are restored and recovered with the proper amplitudes. Investigating the amplitudes in the primary reflections after multiple elimination with and without compensation for transmission effects shows that transmission effects are properly accounted for in a constant-velocity model. When the layer thickness is one quarter of the wavelength at the dominant frequency of the source wavelet, the methods cease to work properly. Full-wavefield migration relies on a velocity model and runs a nonlinear inversion to obtain a reflectivity model, which results in the migration image. The primary reflections that are masked by interference with multiples in the resonant wedge model are not recovered. In this case, minimizing the data misfit function leads to the incorrect reflector model even though the data fit is optimal. This method has much lower demands on data sampling than the multiple elimination schemes, but it is prone to getting stuck in a local minimum even when the correct velocity model is available. A hybrid method that exploits the strengths of each of these methods could be worth investigating.@en

Migration, velocity and amplitude analysis are the employed processing steps to find the desired subsurface information from seismic reflection data. The presence of free-surface and internal multiples can mask the primary reflections for which many processing methods are built. The ability to separate primary from multiple reflections is desirable. Connecting Marchenko theory with classical one-dimensional inversion methods allows to understand the process of multiple reflection elimination as a data-filtering process. The filter is a fundamental wave field, defined as a pressure and particle velocity that satisfy the wave equation. The fundamental wave field does not depend on the presence or absence of free-surface multiples in the data. The backbone of the filtering process is that the fundamental wave field is computed from the measured pressure and particle velocity without additional information. Two different multiples-free datasets are obtained: either directly from the fundamental wave field or by applying the fundamental wave field to the data. In addition, the known schemes for Marchenko multiple elimination follow from the main equation. Numerical examples show that source and receiver ghosts, free-surface and internal multiples can be removed simultaneously using a conjugate gradient scheme. The advantage of the main equation is that the source wavelet does not need to be known and no pre-processing is required. The fact that the reflection coefficients can be obtained is an interesting feature that could lead to improved amplitude analysis and inversion than would be possible with other processing methods.

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Green’s functions in an unknown medium can be retrieved from single-sided reflection data by solving a multidimensional Marchenko equation. This methodology requires knowledge of the direct wavefield throughout the medium, which should include forward-scattered waveforms. In practice, the direct field is often computed in a smooth background model, where such subtleties are not included. As a result, Marchenko-based Green’s function retrieval can be inaccurate, especially in severely complex media. In some cases, auxiliary transmission data may be available. In this extended abstract, we show how these data can be used to modify the Marchenko equation so that forward-scattered waveforms can be retrieved without additional knowledge of the medium.@en

With the Marchenko method it is possible to retrieve Green's functions between virtual sources in the subsurface and receivers at the surface from reflection data at the surface and focusing functions. A macro model of the subsurface is needed to estimate the first arrival; the internal multiples are retrieved entirely from the reflection data. The retrieved Green's functions form the input for redatuming by multidimensional deconvolution (MDD). The redatumed reflection response is free of internal multiples related to the overburden. Alternatively, the redatumed response can be obtained by applying a second focusing function to the retrieved Green's functions. This process is called Marchenko redatuming by double focusing. It is more stable and better suited for an adaptive implementation than Marchenko redatuming by MDD, but it does not eliminate the multiples between the target and the overburden. An attractive efficient alternative is plane-wave Marchenko redatuming, which retrieves the responses to a limited number of plane-wave sources at the redatuming level. In all cases, an image of the subsurface can be obtained from the redatumed data, free of artefacts caused by internal multiples. Another class of Marchenko methods aims at eliminating the internal multiples from the reflection data, while keeping the sources and receivers at the surface. A specific characteristic of this form of multiple elimination is that it predicts and subtracts all orders of internal multiples with the correct amplitude, without needing a macro subsurface model. Like Marchenko redatuming, Marchenko multiple elimination can be implemented as an MDD process, a double dereverberation process, or an efficient plane-wave oriented process. We systematically discuss the different approaches to Marchenko redatuming, imaging and multiple elimination, using a common mathematical framework.@en

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The ability to separate primary reflections from multiples is important for making subsurface images. Many existing methods need some form of model information and adaptive subtraction. Marchenko methods have been modified to operate at the acquisition surface. The associated filters can be computed from the reflection response without any model information. They are a function of a freely chosen time instant that defines the time window of the filter. The scheme can be implemented without adaptive subtraction. Applying the filter to the reflection response removes all multiples from the overburden that would arrive within the time window in which the filter is defined. For this reason, the first event in the result is a primary reflection event that can be taken and stored in a new dataset containing only primary reflections. From data of a resonant wedge model with thin layers, the images after MME show that destructive interference effects are removed by MME. Reflectors are imaged that are missing in the images of the reflection response. Thin layer effects cause incomplete prediction and removal of multiples. When MME treats the combined reflections from thin layers as a single complicated event, their combined multiples from other reflectors are properly removed.@en

We have seen many developments in Marchenko equation-based methods for internal multiple attenuation in the past years. Starting from a wave-equation based method that required a smooth velocity model, there are now Marchenko equation-based methods that do not require any model information or user-input. In principle, these methods accurately predict internal multiples. Therefore, the role of the adaptive filter has changed for these methods. Rather than needing an aggressive adaptive filter to compensate for inaccurate internal multiple predictions, only a conservative adaptive filter is needed to compensate for minor amplitude and/or phase errors in the internal multiple predictions caused by imperfect acquisition and preprocessing of the input data. We demonstate that a conservative adaptive filter can be used to improve the attenuation of internal multiples when applying a Marchenko multiple elimination (MME) method to a 2D line of streamer data. In addition, we suggest that an adaptive filter can be used as a feedback mechanism to improve the preprocessing of the input data.@en

Seismic images provided by reverse time migration can be contaminated by artefacts associated with the migration of multiples. Multiples can corrupt seismic images, producing both false positives, i.e. by focusing energy at unphysical interfaces, and false negatives, i.e. by destructively interfering with primaries. Multiple-related artefacts can be dealt with via Marchenko methods, either via Green’s functions redatuming or data domain schemes (i.e., multiple prediction / primary synthesis algorithms). Data domain Marchenko methods were originally designed to operate on point source gathers, and can therefore be computationally demanding when large problems are considered. However, computationally attractive schemes operating on plane-wave datasets were also derived, by adapting Marchenko point source gathers methods to include plane-wave concepts. As a result, current Marchenko algorithms allow fully data-driven synthesis of primary reflections associated with point and plane-wave source responses. Numerical tests show that while the best images are obtained when well sampled point source gathers are processed, using few multiple-free plane-wave gathers can be used as an unexpensive and effective processing step.@en

Since the introduction of the Marchenko method in geophysics, many variants have been developed. Using a compact unified notation, we review redatuming by multidimensional deconvolution and by double focusing, virtual seismology, double dereverberation and transmission-compensated Marchenko multiple elimination, and discuss the underlying assumptions, merits and limitations of these methods.@en

Seismic images provided by reverse time migration can be contaminated by artefacts associated with the migration of multiples. Multiples can corrupt seismic images, producing both false positives, that is by focusing energy at unphysical interfaces, and false negatives, that is by destructively interfering with primaries. Multiple prediction/primary synthesis methods are usually designed to operate on point source gathers and can therefore be computationally demanding when large problems are considered. A computationally attractive scheme that operates on plane-wave datasets is derived by adapting a data-driven point source gathers method, based on convolutions and cross-correlations of the reflection response with itself, to include plane-wave concepts. As a result, the presented algorithm allows fully data-driven synthesis of primary reflections associated with plane-wave source responses. Once primary plane-wave responses are estimated, they are used for multiple-free imaging via plane-wave reverse time migration. Numerical tests of increasing complexity demonstrate the potential of the proposed algorithm to produce multiple-free images from only a small number of plane-wave datasets.

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The transmission compensated primary reflections can be obtained from the single-sided acoustic reflection response in the two-way traveltime domain. This is achieved by eliminating free-surface and internal multiple reflections and compensating for transmission losses in primary reflections without model information. The substantial computational cost of the proposed scheme can be reduced by an order of magnitude with a fast implementation version. This is achieved by using the previously computed filter functions as initial estimate for every new truncation time value. We evaluate the success of the scheme with simple and complex 2-D numerical examples. We find that the scheme has excellent performance in most cases, except for the case where strong reflectors are present. In such case, the current scheme suffers from lack of convergence.@en

Marchenko methods compute a focusing function for a receiver at the acquisition surface and a virtual source in the subsurface. Computing the focusing function requires subsurface information. The method has been modified to operate at the acquisition surface. The focusing function becomes a fundamental wave field as known since many decades. These can be computed from the up- and down-going parts of the data without any subsurface information. The up- and down-going parts can be obtained from up-down decomposition, or from up-down decomposition of the data followed by free surface multiple removal and wavelet deconvolution. The primary reflection dataset is obtained from applying the fundamental wave field to the data, or directly from the up-going part of the fundamental wave field. In the first option, the obtained primary reflections are the same as in the data, with all transmission effects and possibly the source ghost and source wavelet. In the second option, the obtained dataset is a primary reflection impulse response where the amplitudes have been compensated for transmission effects.@en

The Marchenko multiple elimination (MME) scheme is derived from the coupled Marchenko equations. It is proposed for filtering primary reflections with two-way traveltime from the measured acoustic data. The measured acoustic reflection data are used as its own filter and no model information or adaptive subtraction is required to apply the method. The data obtained after MME are better suited for velocity model construction and artefact-free migration than the measured data. We apply the MME scheme to a measured laboratory data set to evaluate the success of the method. The results suggest that the MME scheme can be the appropriate choice when high-quality pre-processing is performed successfully.@en

Internal multiple reflections have been widely considered as coherent noise in measured seismic data, and many approaches have been developed for their attenuation. The Marchenko multiple elimination (MME) scheme eliminates internal multiple reflections without model information or adaptive subtraction. This scheme was originally derived from coupled Marchenko equations, but it was modified to make it model independent. It filters primary reflections with their two-way traveltimes and physical amplitudes from measured seismic data. The MME scheme is applied to a deepwater field data set from the Norwegian North Sea to evaluate its success in removing internal multiple reflections. The result indicates that most internal multiple reflections are successfully removed and primary reflections masked by overlapping internal multiple reflections are recovered.@en

We have derived a scheme for retrieving the primary reflections from the acoustic surface-reflection response by eliminating the free-surface and internal multiple reflections in one step. This scheme does not require model information and adaptive subtraction. It consists only of the reflection response as a correlation and convolution operator that acts on an intermediate wavefield from which we compute and capture the primary reflections. For each time instant, we keep one value for each source-receiver pair and store it in the new data set. The resulting data set contains only primary reflections, and from this data set, a better velocity model can be built than from the original data set. A conventional migration scheme can then be used to compute an artifact-free image of the medium. We evaluated the success of the method with a 2D numerical example. The method can have a wide range of applications in 3D strongly scattering media that are accessible from one side only.@en

The Marchenko methods can retrieve both Green's functions and focusing functions from single-sided reflection data and an initial focusing function. Recent studies also show that they can be applied to eliminating multiples directly from reflection data without requiring model information as the input. In this work, we study the effect of limited acquisition sampling on both the output focusing functions and the primary estimation results obtained from Marchenko methods. For focusing functions output from the coupled Marchenko system, downsampling in either source or receiver leads to absolute amplitude errors, with the addition of spatial aliasing brought by receiver subsampling. For the projected Marchenko system used in primary estimation, downsampling in these two dimensions can completely contaminate output primaries with multiples, with similar spatial gaps brought by receiver downsampling. Therefore, the acquisition geometry has a significant impact on results of the Marchenko methods, which must be taken into account in realistic acquisition scenarios.@en