Marchenko Method Research Articles

Marchenko imaging assisted by vertical seismic profiling data for land seismic data in the Middle East

Attenuating interference from internal multiples has challenged seismic data imaging in the Middle East basins. The challenge results from the strong short-period internal multiples that exhibit nearly indistinguishable characteristics from the primaries reflected from the underlying reservoirs due to predominantly horizontal strata and occasional low-relief structures, as indicated in the Jurassic formations in Kuwait. To address the internal-multiple issues, multiple prediction followed by adaptive subtraction is the most common approach in the industry. However, due to the similarities between primaries and multiples, applying adaptive subtraction poses a high risk of primary-amplitude damage, preventing quantitative seismic data interpretation. Therefore, we examine the Marchenko method that retrieves Green’s functions from surface seismic data for target-oriented imaging without applying adaptive subtraction. Marchenko imaging has produced promising results on several offshore seismic data sets, but an onshore application is still needed. To better understand the effects of internal multiples and implement Marchenko imaging, we perform integrated analysis through well-log, vertical seismic profiling (VSP), and seismic data from a hydrocarbon field in Kuwait. In addition, we use VSP data to cross-check the retrieved Green’s functions and estimate the scaling factor of the Marchenko method. The results indicate that (1) the poor imaging at the center of the field is due to the destructive interference of internal multiples, (2) the reverberation of internal multiples between the evaporite formations of the overburden is the most likely candidate that affects the seismic images of the Jurassic reservoirs, (3) the retrieved Green’s functions conform to the recorded Green’s functions from VSP data, and (4) Marchenko imaging provides a means to improve the seismic images of the Jurassic formations in Kuwait.

Time‐lapse applications of the Marchenko method on the Troll field

AbstractThe 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.

The Marchenko method to solve the general system of derivative nonlinear Schrödinger equations

The Marchenko method to solve the general system of derivative nonlinear Schrödinger equations

Focusing functions correction in Marchenko imaging with deep learning and transfer learning

The Marchenko method is a subsurface approach to calculate Green's functions from the subsurface to the surface. It requires only reflection response from the surface and direct arrivals from the selected subsurface focal points to the surface. Marchenko imaging can use these obtained Green's functions to create images for all the focal points (image points). Focusing functions are defined to relate Green's functions with the surface reflection response. Estimating focusing functions is the first step in the calculation of Marchenko imaging. However, the estimation of incorrect direct arrivals with the initial model can lead to errors in the focusing functions, thus affecting the effect of the image. In addition, Marchenko imaging is not an efficient imaging method, especially when the imaging area is particularly large, since the number of image points corresponds to the times of solving Marchenko equations. Therefore, we introduce a deep learning and transfer learning approach for the focusing functions correction. We construct a neural network and train the model in a small imaging region by deep learning. Then update this model for extended area by transfer learning which enables the network to work on new data dissimilar to the training data. Hence, our work can quickly obtain accurate focusing functions in a larger region instead of training an extra model for correcting new focusing functions. Numerical examples shows that our network not only helps to reduce errors in the focusing functions, but also can be applied to cope with other limitations of the Marchenko method like travel time error and imperfectly sampled reflection response.

Crosscorrelation-based Marchenko imaging with an optimal aperture

The Marchenko method can retrieve Green’s functions among virtual sources in the subsurface and receivers at the surface from single-sided reflection data. This process, called Marchenko redatuming, allows the estimation of the full-wavefield information in the inhomogeneous subsurface. The retrieved Green’s functions form the input for creating the subsurface image free of artifacts caused by internal multiples. However, when using the crosscorrelation imaging condition in Marchenko imaging, the shallower regions of the obtained image suffer from reduced resolution due to unwanted interference, which occurs in the crosscorrelation gathers at far offsets. With a simple model, we have found that the image resolution obtained using the crosscorrelation Marchenko imaging (CCMI) method is influenced by image position and the integral range of the crosscorrelation function. To overcome this problem, we develop an optimal aperture-bounded crosscorrelation imaging condition using only the constructive wavefields during integration. The optimal aperture is defined as an intercept at which the time difference between the first arrivals of up- and downgoing Green’s functions equals one wavelet period. The integration of the crosscorrelation function is then performed over the determined optimal range that varies with the image position rather than a fixed aperture. As a result, we exclude the unwanted destructive interference from the crosscorrelation gather, resulting in a high-resolution image. The effectiveness of our method has been validated with its applications to synthetic and field data tests.

Vertical Seismic-Profile Data Local Full-Waveform Inversion Based on Marchenko Redatuming

Local full-waveform inversion (FWI) methods use redatumed seismic responses of virtual receivers within the subsurface to build the local objective function based on the convolution-type representation theorem. The Marchenko method is widely used to obtain the redatumed data. The method only requires a smoothed velocity model with correct kinematic characteristics of seismic responses for redatuming of the single-sided reflection data. However, the standard Marchenko method is insensitive to lateral propagation of the wavefield. By injecting the standard Marchenko redatumed wavefield along the boundary of the target, the local wavefield propagation modeling produces errors, which affects the accuracy of the local FWI. In this paper, a method to obtain more accurate Green’s functions is proposed by incorporating vertical seismic profile data (VSP) into the calculation process of the Marchenko source-receiver redatuming. This method allows one to obtain the accurate laterally propagating waveform, resulting in a significant improvement of lateral resolution. The proposed method is applied to a benchmark model dataset and compared with the local FWI based on standard Marchenko redatuming.

A note on Marchenko-linearised full waveform inversion for imaging

SUMMARY Full waveform inversion and least-squares reverse time migration are the leading technologies for imaging with seismic waves. Both of them usually rely (in one way or another) on a single-scattering approximation, i.e. the Born approximation, to compute gradients and obtain an updated model. This approximation linearises the relation between modelled data and model by ignoring multiple scattering. We propose to use the Marchenko integral, an equation originating from inverse scattering theory, to obtain an alternative linear equation. Using the Marchenko method we can retrieve Green’s functions, including all orders of scattering, for virtual sources anywhere within the volume of interest – without prior knowledge of the high-wavelength model variations that induce scattering. Plugging these estimated Green’s functions into the Lippmann–Schwinger integral delivers a Marchenko-linearised relation between the full waveform data and the model. We present this new linearisation strategy and illustrate its advantages and disadvantages by comparing numerical results for different inversion kernels. Our new linearisation is exact, i.e. it does not exclude any orders of scattering, however, it relies on the quality of the Marchenko-derived Green’s functions. These Marchenko-based Green’s functions require an estimate of the first arrivals of the Green’s functions – commonly obtained by modelling in a background medium. Although these first arrival estimates strongly bias our results for inaccurate background models, we find the Marchenko-linearisation to deliver overall slightly better inverted models than the single-scattering approximation.

Extracting small time-lapse traveltime changes in a reservoir using primaries and internal multiples after Marchenko-based target zone isolation

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.

The Generalized Marchenko Method in the Inverse Scattering Problem for a First-Order Linear System with Energy-Dependent Potentials

The Generalized Marchenko Method in the Inverse Scattering Problem for a First-Order Linear System with Energy-Dependent Potentials

Sparse Inversion for the Iterative Marchenko Scheme of Irregularly Sampled Data

The Marchenko method is a data-driven way that makes it possible to calculate Green’s functions from virtual points in the subsurface using the reflection data at the surface and requiring only a macro velocity model. This method requires collocated sources and receivers. However, in practice, irregular sampling of sources or receivers will cause gaps and distortions in the obtained focusing functions and Green’s functions. To solve this problem, this paper proposes to integrate a sparse inversion into the iterative Marchenko scheme. Specifically, we add sparsity constraints to the Marchenko equations and apply the sparse inversion during the iterative process. To reduce the strict requirements on acquisition geometries, our work deals with the situation in which the sources are subsampled where the integrations are carried out over the receivers, while the existing point spread function method solves the situation where the receivers are subsampled. We make a step to handle both situations at the same time by integrating this method with our work because of the same iterative framework. Our new method is applied to a two-dimensional numerical example with irregularly sampled data. The result shows that it can effectively fill gaps in the obtained focusing functions and Green’s functions in the Marchenko method.

Algorithm for accurate velocity analysis of 1D acoustic reflection and transmission responses by projection constrained multi-physics inversion

Velocity analysis is critical in seismic data interpretation. In sedimentary environment, 1D geometry is a reasonable assumption. Velocity analysis under such assumption has been done through the interpretation of either reflection or transmission responses. Recently, Marchenko method has also been applied to provide 1D velocity model from reflection data. However, none of the previous researchers attempted to invert the velocity model from combined physics from the reflection and transmission data with a projection constraint. The projection constraint is computed from the determinant of matrices of dot products of the velocity models. Although, the MATLAB code for the forward responses is given, the algorithm for performing the multi-physics inversion of 1D reflection and transmission data, with projection constraint, is presented. The concept is tested on two different velocity model examples with varying overburden thicknesses and layer thicknesses. Results are consistent even with up to 5% random noise. It is also worthy of note that while the near-surface velocity section remains less affected in the reflection response data, the deep-subsurface velocity values are equally resilient from the transmission response. This presents a much-desired complementary benefit for velocity analysis from a multi-physics interpretation.

Marchenko redatuming, imaging, and multiple elimination and their mutual relations

With the Marchenko method, it is possible to retrieve Green’s functions between virtual sources in the subsurface and receivers at the surface from reflection data at the surface and focusing functions. A macro model of the subsurface is needed to estimate the first arrival; the internal multiples are retrieved entirely from the reflection data. The retrieved Green’s functions form the input for redatuming by multidimensional deconvolution (MDD). The redatumed reflection response is free of internal multiples related to the overburden. Alternatively, the redatumed response can be obtained by applying a second focusing function to the retrieved Green’s functions. This process is called Marchenko redatuming by double focusing. It is more stable and better suited for an adaptive implementation than Marchenko redatuming by MDD, but it does not eliminate the multiples between the target and the overburden. An attractive efficient alternative is plane-wave Marchenko redatuming, which retrieves the responses to a limited number of plane-wave sources at the redatuming level. In all cases, an image of the subsurface can be obtained from the redatumed data, free of the artifacts caused by internal multiples. Another class of Marchenko methods aims at eliminating the internal multiples from the reflection data, while keeping the sources and receivers at the surface. A specific characteristic of this form of multiple elimination is that it predicts and subtracts all orders of internal multiples with the correct amplitude, without needing a macro subsurface model. Like Marchenko redatuming, Marchenko multiple elimination can be implemented as an MDD process, a double dereverberation process, or an efficient plane-wave oriented process. We have systematically examined the different approaches to Marchenko redatuming, imaging, and multiple elimination using a common mathematical framework.

Direct and inverse scattering problems for the first‐order discrete system associated with the derivative NLS system

AbstractThe direct and inverse scattering problems are analyzed for a first‐order discrete system associated with the semi‐discrete version of the derivative nonlinear Schrödinger (NLS) system. The Jost solutions, the scattering coefficients, the bound‐state dependency and norming constants are investigated and related to the corresponding quantities for two particular discrete linear systems associated with the semi‐discrete version of the NLS system. The bound‐state data set with any multiplicities is described in an elegant manner in terms of a pair of constant matrix triplets. Several methods are presented to solve the inverse problem to recover the potential values in the first‐order discrete system. One of these methods uses a newly derived, standard discrete Marchenko system using as input the scattering data directly coming from the first‐order discrete system. This new Marchenko method is presented in a way that it is generalizable to other first‐order systems both in the discrete and continuous cases for which a Marchenko system and a Marchenko theory are not yet available. Finally, using the time‐evolved scattering data set, the inverse scattering transform is applied on the corresponding semi‐discrete derivative NLS system, and in the reflectionless case certain explicit solution formulas are presented in closed form expressed in terms of the two matrix triplets.

Subbasalt Marchenko imaging with offshore Brazil field data

Subbasalt imaging for hydrocarbon exploration faces challenges with the presence of multiple scattering, attenuation, and mode conversion as seismic waves encounter highly heterogeneous and rugose basalt layers. A combination of modern seismic acquisition that can record densely sampled data and advanced imaging techniques make imaging through basalt feasible. Yet, the internal multiples, if not properly handled during seismic processing, can be mapped to reservoir layers by conventional imaging methods, misguiding geologic interpretation. Traditional internal multiple elimination methods suffer from the requirement of picking horizons of multiple generators and/or a top-down adaptive subtraction process. Marchenko imaging provides an alternative solution to directly remove the artifacts due to internal multiples, without the need of horizon picking or subtraction. In this paper, we evaluate a successful application of direct Marchenko imaging for subbasalt demultiple and imaging with an offshore Brazil field data set. The internal multiples in this example are generated from the seabed and basalt layers, causing severe artifacts in conventional seismic images. We have determined that these artifacts are largely suppressed with Marchenko imaging, and we have developed a general workflow for data preprocessing and regularization of marine streamer data sets. We find that horizontally propagating waves also can be reconstructed by the Marchenko method at far offsets.

Accelerating Seismic Redatuming Using Tile Low-Rank Approximations on NEC SX-Aurora TSUBASA

With the aim of imaging subsurface discontinuities, seismic data recorded at the surface of the Earth must be numerically re-positioned inside the subsurface where reflections have originated, a process referred to as redatuming. The recently developed Marchenko method is able to handle full-wavefield data including multiple arrivals. A downside of this approach is that a multi-dimensional convolution operator must be repeatedly evaluated to solve an expensive inverse problem. As such an operator applies multiple dense matrix-vector multiplications (MVM), we identify and leverage the data sparsity structure for each frequency matrix and propose to accelerate the MVM step using tile low-rank (TLR) matrix approximations. We study the TLR impact on time-to-solution for the MVM using different accuracy thresholds whilst at the same time assessing the quality of the resulting subsurface seismic wavefields and show that TLR leads to a minimal degradation in terms of signal-to-noise ratio on a 3D synthetic dataset. We mitigate the load imbalance overhead and provide performance evaluation on two distributed-memory systems. Our MPI+OpenMP TLR-MVM implementation reaches up to 3X performance speedup against the dense MVM counterpart from NEC scientific library on 128 NEC SX-Aurora TSUBASA cards. Thanks to the second generation of high bandwidth memory technology, it further attains up to 67X performance speedup compared to the dense MVM from Intel MKL when running on 128 dual-socket 20-core Intel Cascade Lake nodes with DDR4 memory. This corresponds to 110 TB/s of aggregated sustained bandwidth for our TLR-MVM implementation, without suffering deterioration in the quality of the reconstructed seismic wavefields.

3-D Virtual Seismology

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.

Adaptation of the iterative Marchenko scheme for imperfectly sampled data

SUMMARY 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 colocated 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 2-D numerical data example. The results show clear improvement of the proposed scheme over the standard iterative scheme. By removing the requirement for perfect geometries, the Marchenko method can be more widely applied to field data.

The Marchenko method for evanescent waves

SUMMARY 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.

A modified Marchenko method to retrieve the wave field inside layered metamaterial from reflection measurements at the surface.

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. This study extends the Marchenko method for applications in layered metamaterials with, in the low-frequency limit, effective negative constitutive parameters. It illustrates the method with a numerical example, which confirms that the method properly accounts for multiple scattering. The proposed method has potential applications, for example, in non-destructive testing of layered materials.

Target-oriented full-waveform inversion using Marchenko redatumed wavefields

SUMMARY Full-waveform inversion (FWI) has demonstrated increasing success in estimating medium properties, but its computational cost still poses challenges in moving towards high-resolution imaging of targets at depth. Here, we propose a target-oriented FWI method that inverts for the medium parameters confined within an arbitrary region of interest. Our method is novel in terms of both local wavefield modelling and data redatuming, in order to build a target-oriented objective function which is sensitive to the target medium only without further assumptions about the medium outside. Based on the convolution-type representation theorem, our local forward modelling operator propagates wavefields within the target medium only while providing full acoustic coupling between the target medium and the surrounding geology. A key requirement of our local FWI method is that the subsurface wavefields surrounding and inside the target be as accurate as possible. As such, the subsurface wavefields are retrieved by the Marchenko method, which can redatum the single-sided surface reflection data to the target zone while preserving both primary and multiple reflections, with minimal a priori knowledge of the full-domain medium. Given a sufficiently accurate initial velocity macromodel, our numerical examples show that our local FWI method resolves the reservoir zone of a 2-D Barrett Unconventional P-wave velocity model much more efficiently than the conventional full-domain FWI without significantly sacrificing accuracy. Our method may further enable FWI approaches to high-resolution imaging of subsurface targets.