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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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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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 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

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

We consider reflection data that have been subsampled by 70% and use Point-Spread-Functions to reconstruct the original data. The subsampled, original and reconstructed reflection data are used to image the medium of interest with the Marchenko method. The image obtained using the subsampled data shows artifacts caused by internal multiples, which are eliminated when the original and reconstructed data are used.@en

The southern boundary of Region IV of ancient Ostia coincides with the southern limit of the excavated area of the ancient city. The perceived expanse of the city is influenced by the extent of the excavation. It is not known if the unexcavated part lying south of Region IV also contains structures of antiquity which might have important historical significance. We have carried out high-resolution, shallow seismic reflection surveys along two profiles, using shear (transverse) waves. The goal of these pilot surveys was to see if any indication of ultra-shallow scatterers, indicating potential location of shallow-buried structures, can be found in the shear wave data. The results show very distinct back-scattered shear-wave arrivals from a mysterious tumulus, whose location along Line A was known. It has been possible to interpret with reasonable confidence the location of several conspicuous, shallow scatterers in the two seismic profiles. Use of shear waves and a high-frequency, electromagnetic shear-wave vibrator was crucial to achieve seismic a resolution of nearly 25 cm. The amplitude of the scattered energy is helpful to locate the relatively strong scatterers. Our results suggest that the unexcavated areas located south of Region IV most likely contain buried underground structures. 3-D shear-wave seismic reflections together with new seismic-imaging approaches will be promising to illuminate the unknown shallow subsurface of this important archeological site in a noninvasive manner.@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

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

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

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

Forecasting induced seismicity responses for field data is difficult if no detailed model of the subsurface is available, which generally is the case. As an alternative, reflection data of the subsurface and a non-detailed background model can be used in the Marchenko method to obtain virtual receivers in the subsurface. By employing homogeneous Green's function retrieval, the responses between the virtual receivers are obtained. This approach is applied to synthetic data and a double-couple signature is imposed on the response, to simulate a small-scale induced seismicity response. To simulate a large-scale response, several small-scale responses are superposed. These methods are applied to a field dataset as well to forecast a small-scale and large-scale induced seismicity response.@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.

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A virtual acoustic source inside a medium can be created by emitting a time-reversed point-source response from the enclosing boundary into the medium. However, in many practical situations the medium can be accessed from one side only. In those cases the time-reversal approach is not exact. Here, we demonstrate the experimental design and use of complex focusing functions to create virtual acoustic sources and virtual receivers inside an inhomogeneous medium with single-sided access. The retrieved virtual acoustic responses between those sources and receivers mimic the complex propagation and multiple scattering paths of waves that would be ignited by physical sources and recorded by physical receivers inside the medium. The possibility to predict complex virtual acoustic responses between any two points inside an inhomogeneous medium, without needing a detailed model of the medium, has large potential for holographic imaging and monitoring of objects with single-sided access, ranging from photoacoustic medical imaging to the monitoring of induced-earthquake waves all the way from the source to the earth's surface.

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The characterisation of complex overburden structures can be key for seismic imaging in many geologic settings. To this end, one of the main challenges can be the separation of overburden signals from those coming from deeper structures. Here, we present a data-driven, focusing-based approach to extract full-waveform overburden-only transmission responses from surface-acquired reflection seismic data. Our approach builds on the Marchenko redatuming framework, which is capable of retrieving subsurface fields at depth, containing all internal multiples, from surface reflection data. Intrinsic to the Marchenko scheme is the retrieval of focusing functions, which by definition are sensitive only to overburden properties. Here, we show that the overburden-only transmission can be obtained directly by combining the focusing coda, a by-product of Marchenko redatuming, with a model-based reference transmission. Our transmission estimate can be obtained by inversion of the focusing operator, as well as by a numerically stable, series-based iterative scheme. We present our transmission estimation approach for acoustic and elastic media, and validate both cases using numerical examples. Our examples show that, within its domain of validity, our series-based transmission estimates quickly converge to the desired transmission responses, with good accuracy both in terms of kinematics and absolute amplitudes.@en