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

We create virtual sources and receivers in a 3-D subsurface using the previously derived single-sided homogeneous Green's function representation. We employ Green's functions and focusing functions that are obtained using reflection data at the Earth's surface, a macrovelocity model, and the Marchenko method. The homogeneous Green's function is Green's function superposed with its time-reversal. Unlike the classical homogeneous Green's function representation, our approach requires no receivers on an enclosing boundary; however, it does require the source signal to be symmetric in time. We demonstrate that, in 3-D, the single-sided representation is an improvement over the classical representation by applying the representations to numerical data. We retrieve responses to virtual point sources with an isotropic and with a double-couple radiation pattern and compare the results to a directly modeled reference result. We also demonstrate the application of the single-sided representation for retrieving the response to a virtual rupture that consists of a superposition of double-couple point sources. This is achieved by obtaining the homogeneous Green's function for each source separately before they are transformed to the causal Green's function, time-shifted, and superposed. The single-sided representation is also used to monitor the complete wavefield that is caused by a numerically modeled rupture. However, the source signal of an actual rupture is not symmetric in time, and the single-sided representation can, therefore, only be used to obtain the causal Green's function. This approach leaves artifacts in the final result; however, these artifacts are limited in space and time.@en

Seismic interferometry is a method used to calculate wavefields for sources and receivers that are located where only sources or only receivers are available. There are correlation- or deconvolution-based interferometric methods that can be used to reposition the seismic array from the earth's surface to an arbitrary datum at depth. Based on the one-way reciprocity theorems of convolution and correlation type, we have determined that interferometric redatuming can be achieved in a deconvolution-only procedure in three steps. The first two steps consist of separately retrieving, for sources at the earth's surface, the downward- and upward-propagating Green's functions at receivers at the datum, which are then used in the third step to reposition the sources to the datum. For the involved deconvolutions, transmitted and backscattered wavefields need to be modeled with a velocity model between the acquisition and datum levels. Our numerical experiments demonstrate that the method can help to reduce nonphysical events and other artifacts that commonly arise in purely correlation-based procedures. If a high-quality overburden-velocity model is available, it correctly accounts for inhomogeneities in the overburden medium. Because the method's sensitivity to the velocity model is mainly introduced by backscattering at overburden heterogeneities, a smooth model is sufficient when overburden scattering is weak.

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When reflection images are studied, often only the zero-offset reflectivity is considered, however, taking into account the angle-dependent reflectivity can add additional information about the Earth's subsurface. This additional information can be used to extract the properties of the subsurface using the amplitude variation with offset (AVO) analysis techniques. However, the presence of a complex overburden can significantly deteriorate the AVO response, especially for deep targets. To overcome this problem, the overburden effects can be removed by redatuming the reflection response at a depth level below the overburden. The Marchenko method has the potential to correctly retrieve the angle-dependent reflectivity in acoustic media without distortions due to multiple scattering caused by the overburden. The method estimates the downgoing and upgoing Green's functions of a virtual source located in the subsurface from surface reflection data and an estimate of the direct arrival from the location of the virtual source. The estimated Green's functions represent accurate upgoing and downgoing wavefields as they contain all orders of internal multiple reflections of the subsurface. These internal multiple reflections contribute to retrieving the reflectivity accurately in the redatumed reflection response. By deconvolving the retrieved upgoing Green’s function with the downgoing Green’s function, a new reflection response is obtained, with virtual sources and virtual receivers in the subsurface. The resultant reflection response is free of spurious events related to internal multiples in the overburden and contains the correct amplitudes. The angle-dependent reflectivity of the redatumed response can be obtained by summing the reflection coefficients along lines of constant ray parameter or angle. Potentially, the retrieved angle-dependent reflection coefficients obtained by this method can be used as input in a subsequent inversion process to obtain the velocity and density of the subsurface.

When reflection images are studied, often only the zero-offset reflectivity is considered, however, taking into account the angle-dependent reflectivity can add additional information about the subsurface. This additional information can be used to extract the properties of the subsurface using amplitude variation with offset (AVO) analysis. However, the presence of a complex overburden can significantly deteriorate the AVO response, especially for deep targets. To overcome this problem, the overburden effects can be removed by redatuming the reflection response at a depth level below the overburden. In this paper, we show that the Marchenko redatuming method has the potential to correctly retrieve the angle-dependent reflectivity in an acoustic medium without distortions due to multiple scattering. The retrieved angle-dependent reflection coefficients can be used as input in a subsequent inversion process to obtain the velocity and density of the subsurface.@en

To enhance monitoring of the subsurface, virtual sources and receivers inside the subsurface can be created from seismic reflection data at the surface of the Earth using the Marchenko method. The response between these virtual sources and receivers can be obtained through the use of homogeneous Green's function retrieval. A homogeneous Green's function is a superposition of a Green's function and its time reversal. The main aim of this paper is to obtain accurate homogeneous Green's functions from field data. Classical homogeneous Green's function retrieval requires an unrealistic enclosing recording surface; however, by using a recently proposed single-sided retrieval scheme, this requirement can be avoided. We first demonstrate the principles of using the single-sided representation on synthetic data and show that different source signatures can be taken into account. Because the Marchenko method is sensitive to recording limitations of the reflection data, we study five cases of recording limitations with synthetic data and demonstrate their effects on the final result. Finally, the method is demonstrated on a preprocessed field data set that fulfills the requirements for applying the single-sided Green's function retrieval scheme. The scheme has the potential to be used in future applications, such as source localization.@en

Time-reversal acoustics, seismic interferometry, back propagation, source-receiver redatuming and imaging by double focusing are all based in some way or another on Green's theorem. An implicit assumption for all these methods is that data are available on a closed boundary, a condition that is never met in geophysical practice. As a consequence, although direct and primary scattered waves are handled very well, most methods do not properly account for multiply scattered waves. This can be significantly improved by replacing the back-propagating Green's functions in any of the aforementioned approaches by Marchenko-based focusing functions. We show how this improves time-reversal acoustics, back propagation and source-receiver redatuming and we indicate how it enables the monitoring and forecasting of responses to induced seismic sources.@en

We aim to monitor and characterize signals in the subsurface by combining these passive signals with recorded reflection data at the surface of the Earth. To achieve this, we propose a method to create virtual receivers from reflection data using the Marchenko method. By applying homogeneous Green's function retrieval, these virtual receivers are then used to monitor the responses from subsurface sources. We consider monopole point sources with a symmetric source signal, for which the full wave field without artifacts in the subsurface can be obtained. Responses from more complex source mechanisms, such as double-couple sources, can also be used and provide results with comparable quality to the monopole responses. If the source signal is not symmetric in time, our technique based on homogeneous Green's function retrieval provides an incomplete signal, with additional artifacts. The duration of these artifacts is limited and they are only present when the source of the signal is located above the virtual receiver. For sources along a fault rupture, this limitation is also present and more severe due to the source activating over a longer period of time. Part of the correct signal is still retrieved, as is the source location of the signal. These artifacts do not occur in another method that creates virtual sources as well as receivers from reflection data at the surface. This second method can be used to forecast responses to possible future induced seismicity sources (monopoles, double-couple sources and fault ruptures). This method is applied to field data, and similar results to the ones on synthetic data are achieved, which shows the potential for application on real data signals.@en

In recent years, progress has been made in the field of virtual seismology. Using the novel data-driven Marchenko method, virtual sources and receivers can be created in the subsurface using only reflection data at the surface of the Earth and a background velocity model of the subsurface. Extensive studies have been performed on the application of the Marchenko method to 2D reflection data. This includes both synthetic reflection data and field reflection data. These studies have shown the potential improvements of the method for applications such as hydrocarbon imaging and wavefield monitoring. In the case of wavefield monitoring, a network of virtual receivers can be created in the subsurface to monitor the wavefield response of a subsurface source. This source response can be directly measured in the field, or it can be created from the reflection data, making it a virtual source response. Sensitivity studies have been performed to consider the acquisition limitations of the reflection data, as well as the influence of complex source signatures. This showed the potential of applying the Marchenko method for objectives such as monitoring complex induced seismicity signals and forecasting the responses of induced earthquakes. In this paper, we wish to present further studies for the creation of virtual sources and receivers from 3D reflection data. Similar studies for the application of the Marchenko method for hydrocarbon imaging have been performed, however, in these prior studies, both the source and receiver were virtual. In this paper, we also consider the case that the source is not created from the reflection data, but rather is recorded, and the influence of these types of sources on the monitoring. Furthermore, the limitations of the acquisition setup in the 3rd dimension are considered. @en

The earthquake seismology and seismic exploration communities have developed a variety of seismic imaging methods for passive- and active-source data. Despite the seemingly different approaches and underlying principles, many of those methods are based in some way or another on Green's theorem. The aim of this paper is to discuss a variety of imaging methods in a systematic way, using a specific form of Green's theorem (the homogeneous Green's function representation) as a common starting point. The imaging methods we cover are time-reversal acoustics, seismic interferometry, back propagation, source–receiver redatuming and imaging by double focusing. We review classical approaches and discuss recent developments that fully account for multiple scattering, using the Marchenko method. We briefly indicate new applications for monitoring and forecasting of responses to induced seismic sources, which are discussed in detail in a companion paper.@en

The Marchenko method can be used to retrieve Green’s functions (including multiple scattering) between virtual sources in the subsurface and physical receivers at the surface or virtual receivers in the subsurface. Here we discuss a variant of the Marchenko method which retrieves the response between physical sources and virtual receivers in the subsurface. We discuss the theory and illustrate it with numerical examples. The main application of the proposed method is monitoring of induced seismicity with virtual receivers in the subsurface.@en

Marchenko Imaging is a new technology in geophysics, which enables us to retrieve Green's functions at any point in the subsurface having only reflection data. One of the assumptions of the Marchenko method is that the medium is lossless. One way to circumvent this assumption is to find a compensation parameter for the lossy reflection series so that the lossless Marchenko scheme can be applied. The main goals of this work are to: [1] use the Marchenko equation to estimate the attenuation in the subsurface, [2] find a compensation parameter for the lossy reflection series so that the lossless Marchenko scheme can be applied. We propose a novel approach which makes it possible to calculate an effective temporal Q‐factor of the medium between a virtual source in the subsurface and receivers at the surface. This method is based on the minimization of the artefacts produced by the lossless Marchenko scheme. Artefacts have a very specific behavior: if the input data to the Marchenko equation are over‐ or under‐ compensated, the resulting artefacts will have an opposite polarity. Thus, they can be recognized. This approach is supported by a synthetic example for a 1D acoustic medium without a free surface.@en

Marchenko Imaging is a new technology in geophysics which enables to retrieve Green's functions at any point in the subsurface having only reflection data. This method is based on the extension of the 1D Gelfand-Levitan-Marchenko equation to a 3D medium. One of the assumptions of the Marchenko method is that the medium is lossless. If the lossy reflection response is used in the Marchenko scheme, some artefacts in the Green's functions as well as in the seismic image are present. One way to circumvent this assumption is to find a compensation parameter for the lossy reflection series so that the lossless Marchenko scheme can be applied. The main tasks of this thesis are to: [1] use the Marchenko equation to estimate the attenuation in the subsurface, [2] find a compensation parameter for the lossy reflection series so that the lossless Marchenko scheme can be applied, and [3] to create an upscaling method for wave propagation. The Artefact Removal Method was created which makes it possible to calculate an effective temporal Q-factor of the medium between a virtual source in the subsurface and receivers at the surface. This method is based on the minimization of the artefacts produced by the lossless Marchenko scheme. The minimization was performed in three ways: [1] in the space-time domain, [2] in the frequency domain and [3] to the scales of the wavelet transform applied to the artefacts. This method can also be used to find the layers with high attenuation. The upscaling method which can be used to construct macro-scale homogenized viscoelastic properties of the medium from the micro-scale properties of a heterogeneous medium was developed. This is done through linking the macro- and micro- scale Lippmann-Schwinger equations which describe the wave field and the strain field scattering in an inhomogeneous medium, respectively. In this thesis, the macro-scale homogenized viscoelastic properties were calculated by using the T-matrix Approach and the Generalized Dvorkin-Mavko Attenuation Model. All theoretical results are supported by synthetic 1D modeling. The theoretical part of the thesis and the general work flow can be used for a very complex medium.

In wave theory, a Green’s function is defined as the response of a medium to an impulsive point source. The homogeneous Green’s function is the combination of the Green’s function and its time-reversal. Homogeneous Green’s functions can be retrieved if the medium is enclosed by a boundary where the full wavefield is recorded. In recent years, the Marchenko method has gained popularity, because unlike many conventional methods it does not require an enclosing boundary. Instead a single-sided boundary is all that is required. The method is sensitive to attenuation, which makes it difficult to apply to field data. We will show that by using corrections on the attenuated data, we can retrieve the Green’s functions in the subsurface. These results can be visualized in order to see how the wavefield propagates through the subsurface. @en

The Marchenko method makes it possible to compute subsurface-to-surface Green's functions from reflection measurements at the surface. Applications of the Marchenko method have already been discussed in many papers, but its implementation aspects have not yet been discussed in detail. Solving the Marchenko equation is an inverse problem. The Marchenko method computes a solution of the Marchenko equation by an (adaptive) iterative scheme or by a direct inversion. We have evaluated the iterative implementation based on a Neumann series, which is considered to be the conventional scheme. At each iteration of this scheme, a convolution in time and an integration in space are performed between a so-called focusing (update) function and the reflection response. In addition, by applying a time window, one obtains an update, which becomes the input for the next iteration. In each iteration, upgoing and downgoing focusing functions are updated with these terms. After convergence of the scheme, the resulting upgoing and downgoing focusing functions are used to compute the upgoing and downgoing Green's functions with a virtual-source position in the subsurface and receivers at the surface. We have evaluated this algorithm in detail and developed an implementation that reproduces our examples. The software fits into the Seismic Unix software suite of the Colorado School of Mines.

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