Search

According to Huygens' principle, all points on a wave front act as secondary sources emitting spherical waves, and the superposition of these spherical waves forms a new wave front. In the mathematical formulation of Huygens' principle, the waves emitted by the secondary sources are represented by Green's functions. In many present-day applications of Huygens' principle, these Green's functions are replaced by their time-reversed versions, thus forming a basis for backpropagation, imaging, inversion, seismic interferometry, etc. However, when the input wave field is available only on a single open boundary, this approach has its limitations. In particular, it does not properly account for multiple reflections. This is remedied by a modified form of Huygens' principle, in which the Green's functions are replaced by focusing functions. The modified Huygens' principle forms a basis for imaging, inverse scattering, monitoring of induced sources, etc., thereby properly taking multiple reflections into account., Comment: 23 pages, 17 figures

High-resolution seismic reflections are essential for imaging and monitoring applications. In seismic land surveys using sources and receivers at the surface, surface waves often dominate, masking the reflections. In this study, we demonstrate the efficacy of a two-step procedure to suppress surface waves in an active-source reflection seismic dataset. First, we apply seismic interferometry (SI) by cross-correlation, turning receivers into virtual sources to estimate the dominant surface waves. Then, we perform adaptive subtraction to minimise the difference between the surface waves in the original data and the result of SI. We propose a new approach where the initial suppression results are used for further iterations, followed by adaptive subtraction. This technique aims to enhance the efficacy of data-driven surface-wave suppression through an iterative process. We use a 2D seismic reflection dataset from Scheemda, situated in the Groningen province of the Netherlands, to illustrate the technique's efficiency. A comparison between the data after recursive interferometric surface-wave suppression and the original data across time and frequency-wavenumber domains shows significant suppression of the surface waves, enhancing visualization of the reflections for following subsurface imaging and monitoring studies.

The study of wave propagation and scattering in time-dependent materials is a rapidly growing field of research. Whereas for 1D applications there is a simple relation between the wave equations for space-dependent and time-dependent materials, this relation is less straightforward for multi-dimensional materials. This paper discusses fundamental aspects of 2D electromagnetic and acoustic wave propagation and scattering in homogeneous, time-dependent materials. This encompasses a review of transmission and reflection at a single time boundary, a discussion of the Green's function and its symmetry properties in a piecewise continuous time-dependent material, a discussion of propagation invariants (including the net field-momentum density), general reciprocity theorems, and wave field representations. Analogous to the well-known expression for Green's function retrieval by time-correlation of passive measurements in a space-dependent material, an expression is derived for Green's function retrieval by space-correlation of passive measurements in a time-dependent material. The paper concludes with the discussion of the propagator matrix for a piecewise continuous time-dependent material, its symmetry properties and its relation with the Green's function., Comment: 46 pages, 4 figures, 4 tables

We derive time reflection and transmission coefficients for 1D acoustic waves encountering a time boundary at which the properties of the medium change instantaneously. The time reflection and transmission coefficients are shown to be identical to so-called reverse-space reflection and transmission coefficients which appear in the recursive computation of focusing wavefields used in seismology. We establish a bijectivity between the focusing wavefields and the wavefields produced by time scattering and show how this can be used to construct a space-time medium where the time scattering anticipates the space scattering and "computes" the exact inverse for the space scattering. The construction is shown to be independent of the boundary conditions chosen to compute the reflection and transmission coefficients. We demonstrate the construction with a simple numerical example of a single pulse encountering a series of time boundaries before reaching a spatial inhomogeneity. The time boundaries scatter the single pulse into a focusing wavefield that subsequently focuses through the spatial inhomogeneity. Under certain conditions, the transmitted wave has both the same wave shape and amplitude as the original pulse, yielding a transmission coefficient of unity. The reflection coefficient of the space-time medium is always non-zero however., Comment: 23 pages, 5 figures, 2 tables

Waves in space-dependent and time-dependent materials obey similar wave equations, with interchanged time- and space-coordinates. However, since the causality conditions are the same in both types of material (i.e., without interchangement of coordinates), the solutions are dissimilar. We present a systematic treatment of wave propagation and scattering in 1D space-dependent and time-dependent materials. After formulating unified equations, we discuss Green's functions and simple wave field representations for both types of material. Next we discuss propagation invariants, i.e., quantities that are independent of the space coordinate in a space-dependent material (such as the net power-flux density) or of the time coordinate in a time-dependent material (such as the net field-momentum density). A discussion of reciprocity theorems leads to the well-known source-receiver reciprocity relation for the Green's function of a space-dependent material and a new source-receiver reciprocity relation for the Green's function of a time-dependent material. A discussion of general wave field representations leads to the well-known expression for Green's function retrieval from the correlation of passive measurements in a space-dependent material and a new expression for Green's function retrieval in a time-dependent material. After an introduction of a matrix-vector wave equation, we discuss propagator matrices for both types of material. Since the initial condition for a propagator matrix in a time-dependent material follows from the boundary condition for a propagator matrix in a space-dependent material by interchanging the time- and space-coordinates, the propagator matrices for both types of material are interrelated in the same way. This also applies to representations and reciprocity theorems involving propagator matrices., Comment: 46 pages, 5 figures

The Marchenko algorithm can suppress the disturbing effects of internal multiples that are present in seismic reflection data. To achieve this, a set of coupled equations with four unknowns is solved. These coupled equations are separated into a set of two equations with two unknowns using a time window. The two unknown focusing functions can be resolved by an iterative or direct method. These focusing functions, when applied to reflection data, create virtual point-sources inside the medium. Combining individual virtual point-sources into a plane-wave leads to an efficient computation of images without internal multiples. In this study the internal multiples are eliminated in a redatuming step which is part of the imaging algorithm. To use the Marchenko algorithm with plane-wave focusing functions, the time window that separates the unknowns must be adapted. The design of the plane-wave Marchenko algorithm is explained and illustrated with numerically modeled and measured reflection data., Comment: 20 pages, 13 figures

Recently, the focus of reflection seismologists has shifted to applications where a high-resolution image of the subsurface is required. Least-Squares Reverse-Time Migration (LSRTM) is a common tool used to compute such images. Still, its high computational costs have led seismologists to use target-oriented LSRTM for imaging only a small target of interest within a larger subsurface block. Redatuming the data to the upper boundary of the target of interest is one approach to target-oriented LSRTM. Still, many redatuming methods cannot account for multiple scattering within the overburden. We apply a target-oriented least-squares reverse time migration algorithm that integrates Marchenko redatuming and double-focusing to a field dataset. This redatuming method accounts for all orders of multiple scattering in the overburden, thus improving the accuracy of target-oriented LSRTM. Moreover, we demonstrate the effectiveness of a double-focusing algorithm in reducing the data size by decreasing both spatial and temporal dimensions of the model and the data. The algorithm's performance is evaluated using field data acquired in the Norwegian Sea. The numerical results show that our target-oriented LSRTM algorithm can reduce the internal multiple effects and increase the resolution of the resulting image., Comment: This preprint has been published at Geophysics journal as: Aydin Shoja, Joost van der Neut, and Kees Wapenaar, (2024), "Target-oriented least-squares reverse-time migration with Marchenko redatuming and double-focusing: Field data application," GEOPHYSICS, vol 89, NO. 3, P 1-8

In this report we investigate whether and under what conditions virtual seismology via the acoustic Marchenko method can be applied to DAS data from a survey in the province of Groningen, The Netherlands. Virtual seismology allows to retrieve the band-limited Green's function between a virtual source at an arbitrary focal point in the subsurface, while accounting for all orders of multiples. The method requires the reflection response at the surface and an estimate of the traveltime between the surface and focal point. However, in order to successfully apply the method the reflection response needs to be free from surface waves and other direct waves, and properly scaled in order for the Marchenko scheme to converge. These limitations severely complicate the application of the Marchenko method to field data, especially seismic surveys on land. This report considers a full 2D geophone survey as well as a 1.5D approximation for a DAS survey, and compares the results of the virtual sources with an actual dynamite source. The results show that virtual seismology can be used to recreate the reflections recorded at the surface from the dynamite source using either geophone or DAS data., Comment: 7 pages, 6 figures

The overburden structures often can distort the responses of the target region in seismic data, especially in land datasets. Ideally, all effects of the overburden and underburden structures should be removed, leaving only the responses of the target region. This can be achieved using the Marchenko method. The Marchenko method is capable of estimating Green's functions between the surface of the Earth and arbitrary locations in the subsurface. These Green's functions can then be used to redatum wavefields to a level in the subsurface. As a result, the Marchenko method enables the isolation of the response of a specific layer or package of layers, free from the influence of the overburden and underburden. In this study, we apply the Marchenko-based isolation technique to land S-wave seismic data acquired in the Groningen province, the Netherlands. We apply the technique for combined removal of the overburden and underburden, which leaves the isolated response of the target region which is selected between 30 m and 270 m depth. Our results indicate that this approach enhances the resolution of reflection data. These enhanced reflections can be utilised for imaging and monitoring applications., Comment: 26 pages, 8 figures

Marchenko algorithms retrieve the wavefields excited by virtual sources in the subsurface, these are the Green's functions consisting of the primary and multiple reflected waves. The requirements for these algorithms are the same as for conventional imaging algorithms; they need an estimate of the velocity model and the recorded reflected waves. We investigate the dependence of the retrieved Green's functions using the Marchenko equation on the background velocity model and address the question: ``How well do we need to know the velocity model for accurate Marchenko focusing?". We present three different background velocity models and compare the Green's functions retrieved using these models. We show that these retrieved Green's functions using the Marchenko equation give correlation coefficients with the exact Green's function larger than 90% on average except near the edges of the receiver aperture. We also examine the presence of refracted waves in the retrieved Green's function. We show with a numerical example that the Marchenko focusing algorithm produces refracted waves only if the initial velocity model used for the iterative scheme is sufficiently detailed to model the refracted waves.

Reservoir simulations for subsurface processes play an important role in successful deployment of geoscience applications such as geothermal energy extraction and geo-storage of fluids. These simulators provide time-laps dynamics of the coupled poro-mechanical processes within the reservoir and its over-, under-, and side-burden environments. For more reliable operations, it is crucial to connect these reservoir simulation results with the seismic surveys (i.e., observation data). However, despite being crucial, such integration is challenging due to the fact that the reservoir dynamics alters the seismic parameters. In this work, a coupled reservoir simulation and time-lapse seismic methodology is developed for multiphase flow operations in subsurface reservoirs. To this end, a poromechanical simulator is designed for multiphase flow and connected to a forward seismic modeller. This simulator is then used to assess a novel methodology of seismic monitoring by isolating the reservoir signal from the entire reflection response. This methodology is shown to be able to track the development of the fluid front over time, even in the presence of a highly reflective overburden with strong time-lapse variations. These results suggest that the proposed methodology can contribute to a better understanding of fluid flow in the subsurface. Ultimately, this will lead to improved monitoring of reservoirs for underground energy storage or production., Comment: 11 pages + 2 pages supplementary material, 7 figures

We show the capability of the Marchenko method to retrieve not only propagating waves, but also evanescent waves, based on a recent derivation of the Marchenko method that does not depend on up-down decomposition inside the medium of interest. We show how these wave fields can be easily retrieved in the slowness-intercept-time domain and what the wave fields look like when they are transformed back to the space-time domain. It is vital for the retrieval of the coda of the wave field that the initial estimate of the focusing function is a direct arrival that contains both the up-going and down-going component of the evanescent wave field. This is because these events directly overlay each other in time.

The data-driven Marchenko method is able to redatum wavefields to arbitrary locations in the subsurface, and can, therefore, be used to isolate zones of specific interest. This creates a new reflection response of the target zone without interference from over- or underburden reflectors. Consequently, the method is well suited to obtain a clear response of a subsurface reservoir, which can be advantageous in time-lapse studies. The clean isolated responses of a baseline and monitor survey can be more effectively compared, hence the retrieval of time-lapse characteristics is improved. This research aims to apply Marchenko-based isolation to a time-lapse marine dataset of the Troll field in Norway in order to acquire an unobstructed image of the primary reflections, and retrieve small time-lapse traveltime difference in the reservoir. It is found that the method not only isolates the primary reflections but can also recover internal multiples outside the recording time. Both the primaries and the multiples can then be utilised to find time-lapse traveltime differences. More accurate ways of time-lapse monitoring will allow for a better understanding of dynamic processes in the subsurface, such as observing saturation and pressure changes in a reservoir or monitoring underground storage of hydrogen and CO2., Comment: 11 pages, 9 figures, accepted by Geophysical Prospecting

Least-Squares Reverse-Time Migration (LSRTM) is a method that seismologists utilize to compute a high-resolution subsurface image. Nevertheless, LSRTM is a computationally demanding problem. One way to reduce the computational costs of the LSRTM is to choose a small region of interest and compute the image of that region. However, finding representations that account for the wavefields entering the target region from the surrounding boundaries is necessary. This paper confines the region of interest between two boundaries above and below this region. The acoustic reciprocity theorem is employed to derive representations for the wavefields at the upper and lower boundaries of the target region. With the help of these representations, a target-enclosed LSRTM algorithm is developed to compute a high-resolution image of the region of interest. Moreover, the possibility of using virtual receivers created by Marchenko redatuming is investigated., Comment: 1558-0644 \c{opyright} 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information

Many seismic imaging methods use wave field extrapolation operators to redatum sources and receivers from the surface into the subsurface. We discuss wave field extrapolation operators that account for internal multiple reflections, in particular propagator matrices, transfer matrices and Marchenko focusing functions. A propagator matrix is a square matrix that `propagates' a wave-field vector from one depth level to another. It accounts for primaries and multiples and holds for propagating and evanescent waves. A Marchenko focusing function is a wave field that focuses at a designated point in space at zero time. Marchenko focusing functions are useful for retrieving the wave field inside a heterogeneous medium from the reflection response at its surface. By expressing these focusing functions in terms of the propagator matrix, the usual approximations (such as ignoring evanescent waves) are avoided. While a propagator matrix acts on the full wave-field vector, a transfer matrix (according to the definition employed in this paper)`transfers' a decomposed wave-field vector (containing downgoing and upgoing waves) from one depth level to another. It can be expressed in terms of decomposed Marchenko focusing functions. We present propagator matrices, transfer matrices and Marchenko focusing functions in a consistent way and discuss their mutual relations. In the main text we consider the acoustic situation and in the appendices we discuss other wave phenomena. Understanding these mutual connections may lead to new developments of Marchenko theory and its applications in wave field focusing, Green's function retrieval and imaging., Comment: 43 pages, 4 figures

Geophysicists have widely used Least-squares reverse-time migration (LSRTM) to obtain high-resolution images of the subsurface. However, LSRTM needs an accurate velocity model similar to other migration methods. Otherwise, it suffers from depth estimation errors and out of focus images. Moreover, LSRTM is computationally expensive and it can suffer from multiple reflections. Recently, a target-oriented approach to LSRTM has been proposed, which focuses the wavefield above the target of interest. Remarkably, this approach can be helpful for imaging below complex overburdens and subsalt domains. Moreover, this approach can significantly reduce the computational burden of the problem by limiting the computational domain to a smaller area. Nevertheless, target-oriented LSRTM still needs an accurate velocity model of the overburden to focus the wavefield accurately and predict internal multiple reflections correctly. In recent years, Marchenko redatuming has emerged as a novel data-driven method that can predict Green's functions at any arbitrary depth, including all orders of multiples. The only requirement for this method is a smooth background velocity model of the overburden. Moreover, with Marchenko double-focusing, one can make virtual sources and receivers at a boundary above the target and bypass the overburden. This paper proposes a new algorithm for target-oriented LSRTM, which fits the double-focused data with modeled data at a boundary above the target of interest. Consequently, our target-oriented LSRTM algorithm correctly accounts for all orders of overburden-related multiples, resulting in a significant reduction of the artifacts caused by overburden internal multiple reflections in the target image compared to conventional LSRTM., Comment: This is a pre-copyedited, author-produced PDF of an article accepted for publication in Geophysical Journal International following peer review

Geophysical monitoring of subsurface reservoirs relies on detecting small changes in the seismic response between a baseline and monitor study. However, internal multiples, related to the over- and underburden, can obstruct the view of the target response, hence complicating the time-lapse analysis. In order to retrieve a response that is free from over- and underburden effects, the data-driven Marchenko method is used. This method effectively isolates the target response, which can then be used to extract more precise time-lapse changes. Additionally, the method also reveals target-related multiples that probe the reservoir more than once, which further define the changes in the reservoir. To verify the effectiveness of the method, a numerical example is constructed. This test shows that when using the isolated target response, the observed time differences resemble the expected time differences in the reservoir. Moreover, the results obtained with target-related multiples also benefit from the Marchenko-based isolation of the reservoir. It is, therefore, concluded that this method has potential to observe dynamic changes in the subsurface with increased accuracy., Comment: 34 pages, 7 figures, review copy

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.

Classical acoustic wave-field representations consist of volume and boundary integrals, of which the integrands contain specific combinations of Green's functions, source distributions and wave fields. Using a unified matrix-vector wave equation for different wave phenomena, these representations can be reformulated in terms of Green's matrices, source vectors and wave-field vectors. The matrix-vector formalism also allows the formulation of representations in which propagator matrices replace the Green's matrices. These propagator matrices, in turn, can be expressed in terms of Marchenko-type focusing functions. An advantage of the representations with propagator matrices and focusing functions is that the boundary integrals in these representations are limited to a single open boundary. This makes these representations a suitable basis for developing advanced inverse scattering, imaging and monitoring methods for wave fields acquired on a single boundary., Comment: 71 pages, 7 figures, 1 table

The propagator matrix "propagates" a full wave field from one depth level to another, accounting for all propagation angles and evanescent waves. The Marchenko focusing function forms the nucleus of data-driven Marchenko redatuming and imaging schemes, accounting for internal multiples. These seemingly different concepts appear to be closely related to each other. With this insight, the strong aspects of the propagator matrix (such as the handling of evanescent waves) can be transferred to the focusing function. Vice-versa, the propagator matrix inherits from the focusing function that it can be retrieved from the reflection response, which reduces its sensitivity to the subsurface model., Comment: 19 pages, 2 figures

Marchenko methods are based on integral representations which express Green's functions for virtual sources and/or receivers in the subsurface in terms of the reflection response at the surface. An underlying assumption is that inside the medium the wave field can be decomposed into downgoing and upgoing waves and that evanescent waves can be neglected. We present a new derivation of Green's function representations which circumvents these assumptions, both for the acoustic and the elastodynamic situation. These representations form the basis for research into new Marchenko methods which have the potential to handle refracted and evanescent waves and to more accurately image steeply dipping reflectors., Comment: 42 pages, 5 figures

We create virtual sources and receivers in a 3D 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 macro velocity model and the Marchenko method. The homogeneous Green's function is a 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 3D, 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 represenation 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., Comment: \{copyright} 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

With the Marchenko method, Green's functions in the subsurface can be retrieved from seismic reflection data at the surface. State-of-the-art Marchenko methods work well for propagating waves but break down for evanescent waves. This paper discusses a first step towards extending the Marchenko method for evanescent waves and analyses its possibilities and limitations. In theory both the downward and upward decaying components can be retrieved. The retrieval of the upward decaying component appears to be very sensitive to model errors, but the downward decaying component, including multiple reflections, can be retrieved in a reasonably stable and accurate way. The reported research opens the way to develop new Marchenko methods that can handle refracted waves in wide-angle reflection data., Comment: 15 pages, 6 figures

The homogeneous Green's function is the difference between an impulse response and its time-reversal. According to existing representation theorems, the homogeneous Green's function associated with source-receiver pairs inside a medium can be computed from measurements at a boundary enclosing the medium. However, in many applications such as seismic imaging, time-lapse monitoring, medical imaging, non-destructive testing, etc., media are only accessible from one side. A recent development of wave theory has provided a representation of the homogeneous Green's function in an elastic medium in terms of wavefield recordings at a single (open) boundary. Despite its single-sidedness, the elastodynamic homogeneous Green's function representation accounts for all orders of scattering inside the medium. We present the theory of the elastodynamic single-sided homogeneous Green's function representation and illustrate it with numerical examples for 2D laterally-invariant media. For propagating waves, the resulting homogeneous Green's functions match the exact ones within numerical precision, demonstrating the accuracy of the theory. In addition, we analyse the accuracy of the single-sided representation of the homogeneous Green's function for evanescent wave tunnelling.

With the Marchenko method it is possible to retrieve the wave field inside a medium from its reflection response at the surface. To date, this method has predominantly been applied to naturally occurring materials. In this paper we extend the Marchenko method for applications in metamaterials with, in the low-frequency limit, effective negative constitutive parameters. We illustrate the method with a numerical example, which shows that the method works well for vertically propagating plane waves and for dipping plane waves with small horizontal slownesses. The proposed method has potential applications for example in non-destructive testing of layered materials., Comment: 37 pages, 11 figures, 2 tables

The Marchenko method retrieves the responses to virtual sources in the Earth's subsurface from reflection data at the surface, accounting for all orders of multiple reflections. The method is based on two integral representations for focusing- and Green's functions. In discretized form, these integrals are represented by finite summations over the acquisition geometry. Consequently, the method requires ideal geometries of regularly sampled and co-located sources and receivers. Recently new representations were derived, which handle imperfectly sampled data. These new representations use point-spread functions (PSFs) that reconstruct results as if they were acquired using a perfect geometry. Here, the iterative Marchenko scheme is adapted, using these new representations, to account for imperfect sampling. This new methodology is tested on a 2D numerical example. The results show clear improvement between the proposed scheme and the standard iterative scheme. By removing the requirement for perfect geometries, the Marchenko method can be more widely applied to field data., Comment: arXiv admin note: text overlap with arXiv:2003.10802

In recent years, a variety of Marchenko methods for the attenuation of internal multiples has been developed. These methods have been extensively tested on 2D synthetic data and applied to 2D field data, but only little is known about their behaviour on 3D synthetic data and 3D field data. Particularly, it is not known whether Marchenko methods are sufficiently robust for sparse acquisition geometries that are found in practice. Therefore, we start by performing a series of synthetic tests to identify the key acquisition parameters and limitations that affect the result of 3D Marchenko internal multiple prediction and subtraction using an adaptive double-focusing method. Based on these tests, we define an interpolation strategy and use it for the field data application.