Huygens’ principle states that all points on a wavefront act as secondary sources which emit spherical waves. A new wavefront arises as the envelope of these waves. In the traditional form, the secondary waves in Huygens’ principle are described by dipole Green’s functions. This works well for forward wave field extrapolation. However, for inverse wave field extrapolation through inhomogeneous media (with time-reversed Green’s functions), internal multiple reflections are not properly handled. We discuss a modified version of Huygens’ principle, in which the Green’s functions are replaced by focusing functions. It appears that this modified version not only accounts for internal multiples but also for refracted waves. The focusing functions can be derived from the reflection response with the Marchenko method for full (i.e., non-decomposed) focusing functions. ... Read more

Geophysics education is often challenging, as it entails explaining complicated physical processes that take place inside the Earth. Because these processes happen below the surface of the Earth, it can be difficult for students to connect to the material and understand what is happening. As a result, it is hard for students to make the link between the abstract explanation of the processes to the physical measurements that are performed during fieldwork.

A novel way to close the gap between theory and fieldwork is the use of Virtual Reality or VR. VR allows a student to fully immerse themselves into a digital twin of reality and to experience and visualize processes that are invisible in real life. This is the purpose of Geoscience Processes Virtual Education or GeoProVE. In this application, we have developed a fully immersive and interactive scenario where a student can learn about Ground Penetrating Radar or GPR. The use performs a GPR measurement along a line and is guided with questions to understand how the data are acquired and why specific patterns arise. One of the major features is the ability to pull the subsurface out of the ground, to see how the waves propagate through the subsurface and interact with objects, such as pipes and the water table, in the subsurface. Several setups with increasing complexity are shown to the students, with a strong emphasis on challenge-based learning through a scoring system.

Aside from the GPR scenario, a scenario focused on offshore 3D seismics is also in development for GeoProVE, with the aim to create additional scenarios focused on ERT and geothermal applications. GeoProVE is intended to become fully open source so other developers can contribute to the knowledge base. The application has shown positive engagement from students for geophysics education. We will demonstrate the development of GeoProVE along with its main features. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

Monitoring seismic wavefields caused by induced seismicity in the subsurface is a difficult process. Ideally, it requires physical receivers in the subsurface, which is unpractical. Frequently, only measurements at the surface of the Earth are available, which give a limited amount of information about the subsurface. One way to improve the monitoring of the subsurface is through the use of virtual sources and receivers, which are not physically present but are created from the measured reflection data at the surface. This can be achieved through the use of the classical homogeneous Green's representation, however, this method requires two Green's functions measured on an enclosing boundary, which is an unrealistic requirement. Instead, a single-sided representation of the homogeneous Green's function can be used, where a focusing function, which is a wavefield that focuses from a single-sided boundary to a focal position in the subsurface without artifacts related to the internal multiples, is employed together with a Green's function. To obtain the Green's function and focusing function that are needed for this representation, the Marchenko method is used. This method employs reflection data, without free-surface multiples, at the surface of the Earth and an estimation of the first arrival, which can be modeled in a macro velocity model.

To test whether induced seismicity in the real subsurface can be monitored using the single-sided representation, synthetic data are first considered, which include a synthetic reflection response and macro velocity model. The Marchenko method is used in combination with these data to obtain the focusing functions and Green's functions that are required for the homogeneous Green's function representations. The classical representation and the single-sided representation of the homogeneous Green's function employ the Green's functions and focusing functions to obtain the homogeneous Green's function of the medium. The homogeneous Green's function is visualized by creating snapshots of the homogeneous Green's function and these snapshots are compared to a directly modeled reference wavefield. This demonstrates that the classical representation, when applied to data at an open acquisition boundary, yields significant artifacts in the results, while the single-sided representation obtains accurate results. It is also shown that the radiation pattern of a double-couple source can be included in the retrieval of the homogeneous Green's function. The synthetic reflection data are truncated by limiting the offsets and sampling distance and applying attenuation to simulate field conditions. These truncations show that the single-sided homogeneous Green's function contains artifacts and lacks physical events if the reflection data are not ideal. 2D field reflection data and a macro velocity model from the V\o ring basin are considered and pre-processed to account for these truncations. The classical and the single-sided homogeneous Green's function representation are both applied to the field data and the results show that the retrieval of the homogeneous Green's function is possible for 2D field data using point sources while employing the single-sided representation. The results of the classical representation contain a large amount of errors. It is also shown that a homogeneous Green's function can be retrieved that has a virtual source with a double-couple radiation pattern.

Next, the application of the single-sided representation is considered in greater detail. The representation is used to forecast a wavefield in the subsurface as well as to monitor a wavefield in the subsurface. For the monitoring of the wavefield, it is assumed that a physical source in the subsurface causes a wavefield which is measured at the surface of the Earth. The Marchenko method is used to create virtual receivers inside the subsurface, which are used in combination with the physical measurement in the single-sided representation. This is a one-step process, because the Marchenko method is only used to create the virtual receivers. The single-sided representation of the homogeneous Green's function requires the source wavelet to be symmetric in time, which is unlikely for physical sources. Hence, a different single-sided representation can be used, which retrieves the causal Green's function and does not require a symmetric source wavelet. The single-sided representation of the causal Green's function can retrieve a majority of the correct events, however, the results contain anti-symmetric artifacts when the physical source is located above the virtual receiver. To forecast a wavefield in the subsurface, given a specific source configuration, the single-sided representation of the homogeneous Green's function can be used. In this case, a two-step process is applied, where both the source and the receiver in the subsurface are created by the Marchenko method and are therefore both virtual. After the homogeneous Green's function is obtained, it can be convolved with a non-symmetric wavelet. To demonstrate the difference between the one-step monitoring process and the two-step forecasting process, 2D synthetic reflection data are utilized. For the source configuration, a rupture plane is considered, which is modeled by superposing and time-shifting point sources, which contain a double-couple radiation pattern and are all scaled differently to simulate the heterogeneity of the rupture plane. The total wavefield created by this rupture plane is monitored using the single-sided representation of the causal Green's function. There are anti-symmetric artifacts present in the result, related to each point source, however, the correct wavefield is retrieved above the shallowest source location and below this source location after the first arrivals of all sources. The single-sided representation of the homogeneous Green's function is applied to forecast a virtual rupture plane, by retrieving the homogeneous Green's function for each source separately. The retrieved homogeneous Green's functions are transformed to causal Green's functions, shifted in time and superposed to forecast the total wavefield, which is free of the anti-symmetric artifacts at any depth. Both the monitoring approach and the forecasting approach are tested on 2D field data and the retrieved wavefields show similar results as were seen when the synthetic data were used. When the total wavefield is forecasted, there are no anti-symmetric artifacts present and when the wavefield is monitored, there are artifacts, however, they are only present in part of the result, below the sources before and during the first arrival of each source.

To test the application of the single-sided representation in 3D, a 3D implementation of the Marchenko method is required. The implementation is straightforward from a theoretical standpoint, as the surface integrals are performed over two dimensions instead of just one. The practical implementation is more difficult, however. The Marchenko method requires that the reflection data are well sampled in both space and time for sources and receivers, hence, the 3D reflection data are of a large size. As a result, not only a large amount of storage space is required, but the loading time of the reflection data is high, both of which are unpractical for efficient computation. We limit these problems by pre-transforming the reflection data to the frequency domain and compressing the data using floating point arrays, which reduces the storage space and loading time. Two datasets are considered, one modeled in a simple four layer model and the other in a subsection of the complex 3D Overthrust model. For both models, a Green's function inside the medium is retrieved, using a first arrival in the Marchenko method that was modeled in the exact medium, and compared to a reference Green's function that was directly modeled. The results for both models are accurate for the single Green's function. Next, imaging is performed for the models, however, instead of modeling the first arrivals, they are estimated using an Eikonal solver, because the modeling time of all the first arrivals is too high. The results of the imaging using the Marchenko method are compared to the results of conventional imaging, which demonstrates that artifacts, related to the internal multiples, are attenuated.

The 3D implementation of the Marchenko method is used to retrieve the Green's functions and focusing functions in 3D using 3D synthetic reflection data modeled in the Overhtrust model. The classical homogeneous Green's function representation and the single-sided representation of the causal Green's function and the homogeneous Green's function are all applied using these data, for three different combinations of a virtual source and a virtual receiver. The results are compared to a directly modeled wavefield, which shows that the result obtained by using the classical representation is contaminated by artifacts and lacks physical events. The result of the single-sided representation of the causal Green's function contains anti-symmetric artifacts related to the focusing function when the virtual receiver is located below the virtual source. The result of the single-sided representation of the homogeneous Green's function shows a good match to the reference result. The single-sided representation of the homogeneous Green's function is also applied using an Eikonal solver to obtain the first arrival that is required for the Marchenko method. The homogeneous Green's function that is obtained in this way shows a small decrease in quality for the result, however, this approach is more computationally feasible. The single-sided representation is used in combination with the Eikonal solver to retrieve a large amount of virtual receivers, so that the propagation of the wavefield in the subsurface can be visualized in time through the use of snapshots. This reveals that the part of the wavefield that is traveling at angles that are close to the normal of the surface is retrieved properly, while the part of the wavefield that is traveling at greater angles to the normal is reconstructed with less accuracy. This lack of proper retrieval is caused by the limited aperture of the reflection data. A rupture plane in 3D is considered and constructed in a similar way as is done for the 2D synthetic data. Point sources are used to model wavefields, which are time-shifted and superposed, however, to further represent the heterogeneity of the rupture plane, each wavefield is modeled using an unique causal wavelet. Both monitoring, using the single-sided causal Green's function representation, and forecasting, using the single-sided homogeneous Green's function representation, are performed on the rupture plane configuration. The two-step forecasting approach yields accurate results, for a given distribution of sources. The one-step monitoring approach retrieves accurate results above the shallowest source location, however, the result contains artifacts at the locations below the shallowest source, before and during the first arrival of each source. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more