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.

Electromagnetic wave propagation and scattering in materials with time-variant parameters are subjects of an active field of research. The theory is usually restricted to transverse-electric or transverse-magnetic modes in one-or two-dimensional settings. Here we discuss the theory of three-dimensional, multi-component electromagnetic waves in a homogeneous, time-variant material. We present 3 × 3 causal and acausal Green’s matrices, and a 6 × 6 propagator matrix. We derive several fundamental properties such as the conservation of momentum density for a piecewise continuous, time-variant material, symmetry properties of the propagator matrix, and a relation between the causal and acausal Green’s matrices. These properties are used in derivation of a general wave-field representation (the counterpart of the Kirchhoff integral for a time-invariant material) and an expression for Green’s function retrieval with space correlations (the counterpart of Green’s function retrieval with time correlations in a time-invariant material).

Time materials are materials with acoustic, electromagnetic, or elastic properties that change while waves are propagating through them. Waves can scatter from such temporal inhomogeneities, but due to causality, they cannot reflect back in time. As a result, time-scattered wave fields do not resemble space-scattered wave fields—which are spatially mirrored—but instead look like copies of the original signal traveling back in time. Consequently, they can form an ingredient for physical solvers of inverse problems that rely on time-reversal transformations. Here, we investigate the use of time reflections for calculating so-called focusing wave fields. We establish a one-to-one mapping between focusing wave fields and the transmitted wave fields produced by time scattering. This mapping is then used to construct a space-time medium where the time scattering anticipates the space scattering and “computes” the exact inverse transmission response for the space scattering. 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 wave field that subsequently focuses through the spatial inhomogeneity. With certain mild assumptions, the transmitted wave has both the same wave shape and amplitude as the original pulse, yielding a transmission coefficient of unity.

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.

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.

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.

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.

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.