Elastodynamic Marchenko inverse scattering: A multiple-elimination strategy for imaging of elastodynamic seismic reflection data

The Marchenko method offers a new perspective on eliminating internal multiples. Instead of predicting internal multiples based on events, the Marchenko method formulates an inverse problem that is solved for an inverse transmission response. This approach is particularly advantageous when internal multiples generate complicated interference patterns, such that individual events cannot be identified. Moreover, the retrieved inverse transmissions can be used for a wide range of applications. For instance, we present a numerical example of the single-sided homogeneous Green's function representation in elastic media. These applications require a generalization of the Marchenko method beyond the acoustic case. Formally these extensions are nearly straightforward, as can be seen in the chapter on plane-wave Marchenko redatuming in elastic media. Despite the formal ease of these generalizations, solving the aforementioned inverse problem becomes significantly more difficult in the elastodynamic case. We analyze fundamental challenges of the elastodynamic Marchenko method. Elastic media support coupled wave-modes with different propagation velocities. These velocity differences lead to fundamental limitations, which are due to differences between the temporal ordering of reflection events and the ordering of reflectors in depth. Other multiple-elimination methods such as the inverse scattering series encounter similar limitations, due to violating a so-called monotonicity assumption. Nevertheless, we show that the Marchenko method imposes a slightly weaker form of the monotonicity assumption because it does not rely on event-based multiple prediction. Another challenge arises from the initial estimate that is required by the Marchenko method. In the acoustic case, this initial estimate can be as simple as a direct transmission from the recording surface to the redatuming level. In the presence of several wave-modes, an acoustic direct transmission generalizes to a so-called forward-scattered transmission, which is not a single event but a wavefield with a finite temporal duration. Former formulations of the elastodynamic Marchenko method require this forward-scattered transmission as an initial estimate. However, in practice, this initial estimate is often unknown. We present an alternative formulation of the elastodynamic Marchenko method that simplifies the initial estimate to a trivial one. This approach replaces the inverse transmission, which is often referred to as a focusing function, by a so-called backpropagated focusing function. This strategy allows us to remove internal multiples, however, unwanted forward-scattered waves persist in the data. This insight suggests that forward-scattered waves cannot be predicted by the Marchenko method: either they are provided as prior knowledge, or they remain unaddressed. The remaining forward-scattered waves may be eliminated by exploiting minimum-phase behavior as additional constraint. This approach is inspired by recent developments of the acoustic Marchenko method that use a minimum-phase constraint to handle short-period multiples. Generalizing this strategy to the elastodynamic case is challenging because wavefields are no longer described by scalars but by matrices. Hence, we start by analyzing the meaning of minimum-phase in a multi-dimensional sense. This investigation illustrates that the aforementioned backpropagation turns the focusing function into a minimum-phase object. This insight suggests that, from a mathematical view point, the backpropagated focusing function can be seen as a more fundamental version of the focusing function. Moreover, we present attempts of using this property as additional constraint to remove unwanted forward-scattered waves. Given the remaining theoretical challenges of the elastodynamic Marchenko method, we analyze the performance of an acoustic approximation. We evaluate the effect of applying the acoustic Marchenko method to elastodynamic reflection data. For this analysis, we look for geological settings where an acoustic approximation could be impactful. The Middle East is a promising candidate because, due to its nearly horizontally-layered geology, elastic scattering effects are weaker for short-offsets, which are the main contributors to structural images. Therefore, we construct a synthetic Middle East model based on regional well-log data as well as knowledge about the regional geology. In contrast to field data examples, the synthetic study allows us to include or exclude elastic effects. Hence, we can inspect the artifacts caused by an acoustic approximation. The results indicate that the acoustic Marchenko method can be sufficient for multiple-free structural imaging in geological settings akin to the Middle East.Applied Geophysics and Petrophysic

Seismic blending and deblending of crossline sources

Blending is a recent seismic acquisition design, which allows seismic shots to interfere. Current processing techniques are not capable to deal with blended data. Consequently, the blended data must be deblended (separated) as if they were acquired in a conventional way. I propose a new acquisition design based on blended crossline sources. In contrast to existing blended- acquisition designs that only blend in 2D (inline direction and time), this design blends sources in 3D (inline direction, crossline direction and time). Blended crossline sources allow to increase the data quality and/or to reduce the acquisition costs. While most blended- acquisition designs blend two sources, the proposed acquisition design blends up to seven sources. In order to realize this increase in number of blended sources without degrading the data quality, both the blended-acquisition design and the deblending method must be improved. To enhance the blending, I introduce a new incoherency measure of the blended-acquisition design, and propose three incoherent blending patterns. A 2D synthetic data example il- lustrates that the deblending quality indeed is optimized by maximizing the incoherency of the blended acquisition. To enhance the deblending, I derive a 3D deblending method. In contrast to 2D deblending methods, this method exploits both the crossline and inline direc- tion to deblend sources. The 3D deblending method significantly increases the deblending quality as illustrated by a 3D synthetic data example. The feasibility of blended crossline sources is proven on a 3D complex synthetic data example. Two acquisition configurations are examined: The Wide Crossline Source Array that aims to reduce the acquisition costs, and the Dense Crossline Source Array that increases the data quality. Both of them provide excellent deblending results with quality factors of 14.2 dB and 20.8 dB respectively.Applied Geophysics and PetrophysicsGeoscience & EngineeringCivil Engineering and Geoscience

An acoustic imaging method for layered non-reciprocal media

Given the increasing interest for non-reciprocal materials, we propose a novel acoustic imaging method for layered non-reciprocal media. The method we propose is a modification of the Marchenko imaging method, which handles multiple scattering between the layer interfaces in a data-driven way. We start by reviewing the basic equations for wave propagation in a nonreciprocal medium. Next, we discuss Green’s functions, focusing functions, and their mutual relations, for a non-reciprocal horizontally layered medium. These relations form the basis for deriving the modified Marchenko method, which retrieves the wave field inside the non-reciprocal medium from reflection measurements at the boundary of the medium. With a numerical example we show that the proposed method is capable of imaging the layer interfaces at their correct positions, without artefacts caused by multiple scattering.Accepted Author ManuscriptImPhys/Acoustical Wavefield ImagingApplied Geophysics and Petrophysic

Elastodynamic single-sided homogeneous Green's function representation: Theory and examples

The homogeneous Green’s function is the Green’s function minus its timereversal. Many wavefield imaging applications make use of the homogeneous Green’s function in form of a closed boundary integral. Wapenaar et al. (2016a) derived an accurate single-sided homogeneous Green’s function representation that only requires sources/receivers on an open boundary. In this abstract we will present a numerical example of elastodynamic singlesided homogeneous Green’s function representation using a 2D laterally invariant medium. First, we will outline the theory of the single-sided homogeneous Green’s function representation. Second, we will show numerical results for the elastodynamic case.Applied Geophysics and Petrophysic

Unified wave field retrieval and imaging method for inhomogeneous non-reciprocal media

Acoustic imaging methods often ignore multiple scattering. This leads to false images in cases where multiple scattering is strong. Marchenko imaging has recently been introduced as a data-driven way to deal with internal multiple scattering. Given the increasing interest in non-reciprocal materials, both for acoustic and electromagnetic applications, a modification to the Marchenko method is proposed for imaging such materials. A unified wave equation is formulated for non-reciprocal materials, exploiting the similarity between acoustic and electromagnetic wave phenomena. This unified wave equation forms the basis for deriving reciprocity theorems that interrelate wave fields in a non-reciprocal medium and its complementary version. Next, these theorems are reformulated for downgoing and upgoing wave fields. From these decomposed reciprocity theorems, representations of the Green's function inside the non-reciprocal medium are derived in terms of the reflection response at the surface and focusing functions inside the medium and its complementary version. These representations form the basis for deriving a modified version of the Marchenko method to retrieve the wave field inside a non-reciprocal medium and to form an image, free from artefacts related to multiple scattering. The proposed method is illustrated at the hand of the numerically modeled reflection response of a horizontally layered medium.Accepted Author ManuscriptImPhys/Acoustical Wavefield ImagingApplied Geophysics and Petrophysic

Elastodynamic single-sided homogeneous Green’s function representation: Theory and numerical examples

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.Accepted Author ManuscriptApplied Geophysics and PetrophysicsImPhys/Acoustical Wavefield Imagin

Internal multiple elimination: Can we trust an acoustic approximation?

Correct handling of strong elastic, internal, multiples remains a challenge for seismic imaging. Methods aimed at eliminating them are currently limited by monotonicity violations, a lack of a-priori knowledge about mode conversions, or unavailability of multi-component sources and receivers for not only particle velocities but also the traction vector. Most of these challenges vanish in acoustic media such that Marchenko-equation-based methods are able in theory to remove multiples exactly (within a certain wavenumber-frequency band). In practice, however, when applied to (elastic) field data, mode conversions are unaccounted for. Aiming to support a recently published marine field data study, we build a representative synthetic model. For this setting, we demonstrate that mode conversions can have a substantial impact on the recovered multiple-free reflection response. Nevertheless, the images are significantly improved by acoustic multiple elimination. Moreover, after migration the imprint of elastic effects is considerably weaker and unlikely to alter the seismic interpretation.Accepted Author ManuscriptApplied Geophysics and PetrophysicsImPhys/Medical Imagin

Tackling Different Velocity Borne Challenges of the Elastodynamic Marchenko Method

The elastodynamic Marchenko method removes overburden interactions obscuring the target information. This method either relies on separability of the so-called focusing and Green's functions or requires an accurate initial estimate of the focusing and Green's function overlap. Hitherto, F1- and G-+ have been assumed separable, whereas F1+ and (G-)* share an unavoidable overlap, which has been considered understood but hard to predict without knowing the model. However, velocity differences between P- and S-waves cause so far unexplored fundamental challenges for elastodynamic Marchenko autofocusing. These challenges are analysed for horizontally-layered media. First, the F1-/G-+ separability assumption can be violated depending on the medium, the redatuming depth and the angle of incidence. Second, the initial estimate of the said unavoidable overlap can be even more complicated than originally thought, including some of the internal multiples. We propose a strategy where we trade-off this sophisticated initial estimate with a trivial one at the cost of a more restrictive F1-/G-+ separability assumption, or at the cost of introducing an overlap between F1- and G-+ instead. The proposed method finds the desired solutions convolved by an unknown matrix which we can hope to remove by exploiting energy conservation and minimum-phase properties of the focusing functions.Accepted author manuscriptApplied Geophysics and PetrophysicsQN/Theoretical PhysicsImPhys/Acoustical Wavefield Imagin

Comparison of monotonicity challenges encountered by the inverse scattering series and the Marchenko de-multiple method for elastic waves

The reflection response of strongly scattering media often contains complicated interferences between primaries and (internal) multiples, which can lead to imaging artifacts unless handled correctly. Internal multiples can be kinematically predicted, for example by the Jakubowicz method or by the inverse scattering series (ISS), as long as monotonicity, that is, "correct"temporal event ordering, is obeyed. Alternatively, the (conventional) Marchenko method removes all overburden-related wavefield interactions by formulating an inverse problem that can be solved if the Green's and the so-called focusing functions are separable in the time domain, except for an overlap that must be predicted. For acoustic waves, the assumptions of the aforementioned methods are often satisfied within the recording regimes used for seismic imaging. However, elastic media support wave propagation via coupled modes that travel with distinct velocities. Compared to the acoustic case, not only does the multiple issue become significantly more severe, but also violation of monotonicity becomes much more likely. By quantifying the assumptions of the conventional Marchenko method and the ISS, unexpected similarities as well as differences between the requirements of the two methods come to light. Our analysis demonstrates that the conventional Marchenko method relies on a weaker form of monotonicity. However, this advantage must be compensated by providing more prior information, which in the elastic case is an outstanding challenge. Rewriting, or remixing, the conventional Marchenko scheme removes the need for prior information but leads to a stricter monotonicity condition, which is now almost as strict as for the ISS. Finally, we introduce two strategies on how the remixed Marchenko solutions can be used for imperfect, but achievable, demultiple purposes.Accepted Author ManuscriptApplied Geophysics and PetrophysicsImPhys/Medical Imagin

Elastodynamic Plane Wave Marchenko Redatuming: Theory and Examples

The Marchenko method is capable to create virtual sources inside a medium that is only accessible from an openboundary. The resulting virtual data can be used to retrieve images free of artefacts caused by internal multiples. Conventionally, the Marchenko method retrieves a so-called focusing wavefield that focuses the data from the recording surface to a point inside the medium. Recently, it was suggested to modify the focusing condition such that the new focusing wavefield creates a virtual plane wave source inside the medium, instead of a virtual point source. The virtual plane wave data can be used to image an entire surface inside the medium in a single step rather than imaging individual points on the surface. Consequently, the imaging process is accelerated significantly. We provide an extension of plane wave Marchenko redatuming for elastodynamic waves and demonstrate itsperformance numerically.Accepted author manuscriptApplied Geophysics and PetrophysicsImPhys/Acoustical Wavefield Imagin