Multidimensional up-down deconvolution effectively eliminates surface-related multiples from ocean-bottom seismic data. Recently, several down-down deconvolution methods have been introduced as attractive alternatives. Whereas multidimensional up-down deconvolution fully accounts for lateral variations of the medium parameters, the underlying theory of some of the down-down deconvolution methods is essentially based on the assumption that the medium is horizontally layered. We derive representations for multidimensional down-down deconvolution for laterally varying media in a uniform manner and discuss the pros and cons of the various methods.

The methods of time-reversed acoustics and Green's function retrieval are traditionally deployed for classical inhomogeneous, time-invariant materials. The mutual relation between these methods is well established. Recently, similar methods have been proposed for homogeneous, time-variant materials. Here, we investigate their mutual relation and their relation with the corresponding methods in classical materials. For this analysis, we make use of the fact that the wave equations for both classes of material are similar, with the roles of time and space interchanged. However, the principle of causality holds for both classes of material; hence, here the roles of time and space are not interchanged. We find that (1) whereas classical time-reversed acoustics involves emission of a time-reversed single-component wave field from a (ideally closed) boundary into the inhomogeneous material, its idealized counterpart involves emission of a sign-reversed two-component wave field, recorded in a time-reversed material, from a single time instant into the actual time-variant material; and (2) whereas classical Green's function retrieval involves temporal cross-correlation of wave fields at two space locations in response to single-component sources on a (ideally closed) boundary, its counterpart involves spatial cross-correlation of wave fields at two time instants in response to two-component sources at a single time instant.

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.

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 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 match the exact Green’s function with a high accuracy. We also examine the presence of refracted waves in the retrieved Green’s function. 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. We show with numerical examples that the average slowness between the surface and the depth of the focal point is required for an accurate reflected wave retrieval. However, substantially more accurate velocity model knowledge is required in the presence of refracted waves.

According to Huygens' principle, all points on a wave front act as secondary sources emitting spherical waves, and the envelope 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 multiply reflected waves. 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 multiply reflected waves into account.

Recently, there has been an increasing interest in employing rotational motion measurements for seismic source inversion, structural imaging and ambient noise analysis. We derive reciprocity and representation theorems for rotational motion. The representations express the rotational motion inside an inhomogeneous anisotropic earth in terms of translational and rotational motion at the surface. The theorems contribute to the theoretical basis for rotational seismology methodology, such as determining the moment tensor of earthquake sources.

The study of wave propagation and scattering in time-dependent materials is a rapidly growing field of research. Whereas for one-dimensional 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 article discusses fundamental aspects of two-dimensional 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. This article 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.

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 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.

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.

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.

Summary Coda-wave interferometry employs the sensitivity of multiply scattered waves to detect minute changes of the propagation velocity. In most applications the underlying assumption is that the velocity changes take place approximately uniformly in a large region. Time-lapse seismic reservoir monitoring, on the other hand, aims to infer local time-lapse changes from seismic reflection measurements at the surface. Hence, coda-wave interferometry, applied directly to time-lapse seismic data for reservoir monitoring, cannot infer such local changes in a reservoir. Recently, we developed a Marchenko-based methodology to isolate the response of a target zone from the full reflection response. We apply this methodology to Troll-field data and successfully predict the internal multiples of the target zone. This confirms that in principle the Marchenko-based isolation method can be used to prepare time-lapse seismic data for coda-wave interferometry, which in turn can be used to infer time-lapse changes in a reservoir caused by production or storage of resources in the subsurface.

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.

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 a review of reflection and transmission coefficients, 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.

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 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 data set 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 estimate internal multiples outside the recording time. Both the primaries and the multiples can then be utilized 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.

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 (ROI) 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 article confines the ROI 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 ROI. Moreover, the possibility of using virtual receivers created by Marchenko redatuming is investigated.

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 simulations provide time-lapse dynamics of the coupled poromechanical 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.

SUMMARY Previous studies indicate that scattering may pose a trade-off for the performance of seismic interferometry (SI) applications for retrieving body-wave reflections of a target reflector. While it has been demonstrated that a higher scattering strength of the overburden improves the Green’s function estimated by cross-correlation SI, other theoretical and empirical studies showed that multiple scattering also gives rise to more artefacts. The implications of this trade-off are analysed in this numerical study for a lithospheric scenario with varying crustal scattering strength and passive illumination conditions. In this scenario, we apply SI by cross-correlation to elastodynamic responses to double-couple sources to reconstruct virtual Moho primary reflections. We include multidimensional deconvolution (MDD) methods in the analysis to investigate whether scattering-induced artefacts affect MDD methods in a similar way as was shown for the cross-correlation method. Our results show that there indeed exists a trade-off between the quality of the virtual primary reflection of the target that can be obtained by SI and the scattering strength of the overburden. Furthermore, we find that the full-field MDD method proves to be most resilient to the negative effects of multiple scattering for all illumination conditions and scattering strengths analysed.

In recent years, the Marchenko method has proven to be a viable tool to create virtual seismic sources and receivers in the subsurface from reflection measurements at the surface. Applications range from suppressing internal multiples in seismic imaging to forecasting responses to induced seismic sources. One of the attractive aspects of the Marchenko method is that no detailed subsurface model is needed; a smooth background model suffices. All information needed to treat the internal multiples correctly comes from the reflection measurements at the surface. One of the underlying assumptions of the Marchenko method is that the seismic wave field can be decomposed into downgoing and upgoing waves at any position in the subsurface where one wants to create a virtual source or receiver. Although in many situations this implies no significant restrictions, it may hamper the imaging of steeply dipping flanks and it prevents the treatment of refracted and evanescent waves.It can be shown that the Marchenko focusing function (the nucleus of the Marchenko method) can be expressed in terms of the so-called propagator matrix. The propagator matrix, which was introduced in geophysics in the nineteen-sixties for 1D systems and developed further in the nineteen-seventies for laterally varying 3D media, ‘propagates’ a wave field from one depth level to another. It does not rely on up-down decomposition and it accounts for propagating waves at all angles and for evanescent waves. By exploiting the link between the Marchenko focusing function and the propagator matrix, the applicability of the Marchenko method can be expanded. In the presentation we will review the underlying theory and discuss the potential application of the Marchenko method for refracted waves.

Marchenko-type integrals typically relate so-called focusing functions and Green’s functions via the reflection response measured on the open surface of a volume of interest. Originating from one dimensional inverse scattering theory, the extension to two and three dimensions set in motion various new developments regarding imaging in complex materials. This extension, however, is based on wavefield decomposition inside the volume and a truncated medium state, i.e. a version of the medium that is reflection-free underneath the focusing location, suggesting that evanescent, refracted and diving waves cannot be included in the representation. We elaborate on a new derivation for Marchenko-like integrals that (i) extends the concept of wavefield focusing by using a generalised homogeneous Green’s function, (ii) is based on partial differential equations and thereby allows for additional insights and a new physical intuition for Marchenko equations, (iii) unifies wavefield focusing for open and closed boundary systems, (iv) does not require wavefield decomposition or a truncated medium state, thus including the full wavefield Green’s function, (v) enables using forward modelling to obtain, e.g., Marchenko-type, time-compact focusing functions. We place a particular focus on the latter point, illustrating and investigating how to solve the underlying partial differential equations for various types of focusing functions. This paves the way for a deeper understanding of focusing functions as well as advanced full wavefield Marchenko schemes. While the derivations are generally presented for the 3D case, we show numerical examples in 1D.