We implement the 3D Marchenko equations to retrieve responses to virtual sources inside the subsurface. For this, we require reflection data at the surface of the Earth that contain no free-surface multiples and are densely sampled in space. The required 3D reflection data volume is very large and solving the Marchenko equations requires a significant amount of computational cost. To limit the cost, we apply floating point compression to the reflection data to reduce their volume and the loading time from disk. We apply the Marchenko implementation to numerical reflection data to retrieve accurate Green's functions inside the medium and use these reflection data to apply imaging. This requires the simulation of many virtual source points, which we circumvent using virtual plane-wave sources instead of virtual point sources. Through this method, we retrieve the angle-dependent response of a source from a depth level rather than of a point. We use these responses to obtain angle-dependent structural images of the subsurface, free of contamination from wrongly imaged internal multiples. These images have less lateral resolution than those obtained using virtual point sources, but are more efficiently retrieved.

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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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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

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

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 Hessian matrix plays an important role in correct interpretation of the multiple scattered wave fields inside the FWI frame work. Due to the high computational costs, the computation of the Hessian matrix is not feasible. Consequently, FWI produces overburden related artifacts inside the target zone model, due to the lack of the exact Hessian matrix. We have shown here that Marchenko-based target-oriented Full Waveform Inversion can compensate the need of Hessian matrix inversion by reducing the nonlinearity due to overburden effects. This is achieved by exploiting Marchenko-based target replacement to remove the overburden response and its interactions with the target zone from residuals and inserting the response of the updated target zone into the response of the entire medium. We have also shown that this method is more robust with respect to prior information than the standard gradient FWI. Similarly to standard Marchenko imaging, the proposed method only requires knowledge of the direct arrival time from a focusing point to the surface and the reflection response of the medium.@en

Wavefield focusing can be achieved by Time-Reversal Mirrors, which involve in- and output signals that are infinite in time and waves propagating through the entire medium. Here, an alternative solution for wavefield focusing is presented. This solution is based on a new integral representation where in- and output signals are finite in time, and where the energy of the waves propagating in the layer embedding the focal point is reduced. We explore the potential of the proposed method with numerical experiments involving a 1D example and a cranium model consisting of a skull enclosing a brain.@en

Wavefield focusing is often achieved by time-reversal mirrors, where wavefields emitted by a source located at the focal point are evaluated at a closed boundary and sent back, after time-reversal, into the medium from that boundary. Mathematically, time-reversal mirrors are derived from closed-boundary integral representations of reciprocity theorems. In heterogeneous media, time-reversal focusing theoretically involves in- and output signals that are infinite in time and the resulting waves propagate through the entire medium. Recently, integral representations have been derived for single-sided wavefield focusing. Although the required input signals for this approach are finite in time, the output signals are not and, similar to time-reversal mirroring, the resulting waves propagate through the entire medium. Here, an alternative solution for double-sided wavefield focusing is derived. This solution is based on an integral representation where in- and output signals are finite in time, and where the energy of the waves propagating in the layer embedding the focal point is smaller than with time-reversal focusing. The potential of the proposed method is explored with numerical experiments involving a head model consisting of a skull enclosing a brain.@en

The Marchenko method is capable to create virtual sources inside a medium that is only accessible from an openboundary. The resulting virtual data can be used to retrieve images free of artefacts caused by internal multiples. Conventionally, the Marchenko method retrieves a so-called focusing wavefield that focuses the data from the recording surface to a point inside the medium. Recently, it was suggested to modify the focusing condition such that the new focusing wavefield creates a virtual plane wave source inside the medium, instead of a virtual point source. The virtual plane wave data can be used to image an entire surface inside the medium in a single step rather than imaging individual points on the surface. Consequently, the imaging process is accelerated significantly. We provide an extension of plane wave Marchenko redatuming for elastodynamic waves and demonstrate its performance numerically.@en

Marchenko imaging is a novel imaging technique that is capable to retrieve images from single-sided reflection measurements free of artefacts related to internal multiples (e.g. Behura et al., 2014; Broggini et al., 2012). An essential ingredient of Marchenko imaging is the so-called focusing function which can be retrieved from reflection data and a background model. Initially, the focusing function was defined such that it focuses inside the medium of interest as a point in time and in space (e.g. Wapenaar et al., 2014). The focusing property is used to retrieve the up- and downgoing Green’s functions associated to a virtual point source or receiver inside the medium. Subsequently, the retrieved Green’s functions are used to compute an image. Meles et al. (2017) introduced a new focusing function that focuses as a plane wave inside the medium. The new focusing function allows to retrieve medium responses associated to virtual plane wave sources or receivers inside the medium. Hence, imaging based on areal-sources as suggested by Rietveld et al. (1992) becomes possible including the benefits of the Marchenko method. In the following we compare Marchenko imaging using point and plane wave focusing.@en

Marchenko redatuming is a novel scheme used to retrieve up- and downgoing Green's functions in an unknown medium.Marchenko equations are based on reciprocity theorems and are derived on the assumption of the existence of functions exhibiting space-time focusing properties once injected in the subsurface. In contrast to interferometry but similarly to standard migration methods, Marchenko redatuming only requires an estimate of the direct wave from the virtual source (or to the virtual receiver), illumination from only one side of the medium and no physical sources (or receivers) inside the medium. In this contribution we consider a different time-focusing condition within the frame of Marchenko redatuming that leads to the retrieval of virtual plane-wave responses. As a result, it allows multiple-free imaging using only a 1-D sampling of the targeted model at a fraction of the computational cost of standard Marchenko schemes. The potential of the new method is demonstrated on 2-D synthetic models.

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Marchenko redatuming is a novel scheme used to retrieve up- and down-going Green's functions in an unknown medium. Marchenko equations are based on reciprocity theorems and are derived on the assumption of the existence of so called focusing functions, i.e. functions which exhibit time-space focusing properties once injected in the subsurface. In contrast to interferometry but similarly to standard migration methods, Marchenko redatuming only requires an estimate of the direct wave from the virtual source (or to the virtual receiver), illumination from only one side of the medium, and no physical sources (or receivers) inside the medium. In this contribution we consider a different time-focusing condition within the frame of Marchenko redatuming and show how this can lead to the retrieval of virtual plane-wave-responses, thus allowing multiple-free imaging using only a 1 dimensional sampling of the targeted model. The potential of the new method is demonstrated on a 2D synthetic model.@en

Disregarding their possible use in imaging, surface multiples are primarily still dealt with by 3D SRME (convolutional and/or wave equation modelling in different ratios) despite its well-documented shortcomings related to the need for adaptive subtraction and its well-known limitations in shallow water regimes. However, where necessary, this single prediction-subtraction process can easily be augmented by other methods generating additional predictions and then used in a simultaneous multi-model subtraction method, or the process can be completely replaced by an inversion algorithm (e.g. EPSI and its variants).@en

Disregarding their possible use in imaging, surface multiples are primarily still dealt with by 3D SRME (convolutional and/or wave equation modelling in different ratios) despite its well-documented shortcomings related to the need for adaptive subtraction and its well-known limitations in shallow water regimes. However, where necessary, this single prediction-subtraction process can easily be augmented by other methods generating additional predictions and then used in a simultaneous multi-model subtraction method, or the process can be completely replaced by an inversion algorithm (e.g. EPSI and its variants).@en

A number of seismic processing methods, including velocity analysis (Sheriff and Geldart, 1999), make the assumption that recorded waves are primaries - that they have scattered only once (the Born approximation). Multiples then represent a source of coherent noise and must be suppressed to avoid artefacts. There are different approaches to mitigate free surface multiples (see Dragoset et al. (2010) for an overview), but internal multiples still pose a problem and usually cannot be removed without high computational cost or knowledge of the medium. Recently, Marchenko redatuming has been developed to image a medium in the presence of internal multiples (Wapenaar et al., 2014). Using Marchenko redatuming in combination with convolutional interferometry, Meles et al. (2016) have developed a method which allows the construction of a primaries-only data set from existing seismic reflection data and an initial velocity model. The method was proposed for the acoustic case and appears to be robust with respect to even huge inaccuracies in the employed velocity model. In this paper we investigate the impact of such primaries-only data on a simple velocity analysis workflow, as opposed to using the full data set with multiples. We use semblance analysis (Sheriff and Geldart, 1999) and compare the results obtained with three different data sets: the full reflection data with multiples, primaries data calculated with prior knowledge of the subsurface, and primaries data calculated with an entirely incorrect constant velocity model. We then use the velocity models that we construct to perform reverse time migration (RTM) of each of the data sets. We find that the velocities found are robust with respect to errors in the initial model used for Marchenko redatuming, and the method produces good results if non-hyperbolic moveout effects are avoided. @en