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.

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.

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 data set. 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 minimize 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 2-D seismic reflection data set 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 subsequent subsurface imaging and monitoring studies.

Waves in space-dependent and in 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 time- and space-coordinates), the solutions are dissimilar. We present a systematic treatment of wave propagation and scattering in 1D space-dependent and in 1D 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 general 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, and to Marchenko-type focusing functions.

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.

Many seismic imaging methods use wavefield extrapolation operators to redatum sources and receivers from the surface into the subsurface. We discuss wavefield 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 wavefield 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 wavefield that focuses at a designated point in space at zero time. Marchenko focusing functions are useful for retrieving the wavefield 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 wavefield vector, a transfer matrix (according to the definition used in this paper) 'transfers' a decomposed wavefield 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 wavefield focusing, Green's function retrieval and imaging.

Minimum-phase properties are well-understood for scalar functions where they can be used as physical constraint for phase reconstruction. Existing scalar applications of the latter in geophysics include, for example the reconstruction of transmission from acoustic reflection data, or multiple elimination via the augmented acoustic Marchenko method. We review scalar minimum-phase reconstruction via the conventional Kolmogorov relation, as well as a less-known factorization method. Motivated to solve practice-relevant problems beyond the scalar case, we investigate (1) the properties and (2) the reconstruction of minimum-phase matrix functions. We consider a simple but non-trivial case of 2 × 2 matrix response functions associated with elastodynamic wavefields. Compared to the scalar acoustic case, matrix functions possess additional freedoms. Nonetheless, the minimum-phase property is still defined via a scalar function, that is a matrix possesses a minimum-phase property if its determinant does. We review and modify a matrix factorization method such that it can accurately reconstruct a 2 × 2 minimum-phase matrix function related to the elastodynamic Marchenko method. However, the reconstruction is limited to cases with sufficiently small differences between P- and S-wave traveltimes, which we illustrate with a synthetic example. Moreover, we show that the minimum-phase reconstruction method by factorization shares similarities with the Marchenko method in terms of the algorithm and its limitations. Our results reveal so-far unexplored matrix properties of geophysical responses that open the door towards novel data processing tools. Last but not least, it appears that minimum-phase matrix functions possess additional, still-hidden properties that remain to be exploited, for example for phase reconstruction.

For the elastodynamic wave equation discussed in Appendix A.4 in Ref. 1, the expressions for matrices ∼L 6 1 and ∼L 6 2 in Eqs. (A32) and (A33) must be multiplied by 61. In other words, the signs of ∼L _ ₁and ∼L _ ₂have to be changed, whereas the signs of ∼L 1 and ∼L 2 remain unchanged. With these corrections, matrix ∼L , defined in Eq. (A5), fulfills the symmetry property formulated by Eq. (A6). The corrections have no consequences for Eqs. (A36)-(A38).

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.

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.

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. To retrieve a response that is free from the 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. In addition, the method also reveals target-related multiples that probe the reservoir more than once, which further defines the changes in the reservoir. To verify the effectiveness of the method, a numerical example is constructed. This test finds 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 the potential to observe dynamic changes in the subsurface with increased accuracy.