Marchenko Methods Research Articles

Full wavefield inversion with multiples: Nonlinear Bayesian inverse multiple scattering theory beyond the Born approximation

Imaging earth’s structures is fundamental to the earth’s interior, yet how to reconstruct earth’s heterogeneities using full-waveform inversion of full wavefield data that includes primary reflections and multiples remains enigmatic. I develop a nonlinear Bayesian approach for full-waveform inversion method with multiple scattering. Instead of using single scattering Born approximation to formulate the sensitivity kernels, I develop multiple scattering sensitivity kernels using multiple scattering-based Green’s functions. It is based on the Lippmann-Schwinger integral and Marchenko methods, for which the Green’s functions are retrieved from reflection data by solving a Marchenko equation. To estimate the uncertainty of velocities, I apply a Bayesian framework to the inverse problem. Our results indicate that the method does not depend on the earth’s scattering potential and the full-waveform inversion method with multiple scattering is a good alternative approach to image the earth’s structures.

An analysis of acquisition-related subsampling effects on Marchenko focusing, redatuming, and primary estimation

Marchenko methods can retrieve Green’s functions and focusing functions from single-sided reflection data and a smooth velocity model, as essential components of a redatuming process. Recent studies also indicate that a modified Marchenko scheme can reconstruct primary-only reflection responses directly from reflection data without requiring a priori model information. To provide insight into the artifacts that arise when input data are not ideally sampled, we study the effects of subsampling in both types of Marchenko methods in 2D earth and data — by analyzing the behavior of Marchenko-based results on synthetic data subsampled in sources or receivers. With a layered model, we find that for Marchenko redatuming, subsampling effects jointly depend on the choice of integration variable and the subsampling dimension, originated from the integrand gather in the multidimensional convolution process. When reflection data are subsampled in a single dimension, integrating on the other yields spatial gaps together with artifacts, whereas integrating on the subsampled dimension produces aliasing artifacts but without spatial gaps. Our complex subsalt model indicates that the subsampling may lead to very strong artifacts, which can be further complicated by having limited apertures. For Marchenko-based primary estimation (MPE), subsampling below a certain fraction of the fully sampled data can cause MPE iterations to diverge, which can be mitigated to some extent by using more robust iterative solvers, such as least-squares QR. Our results, covering redatuming and primary estimation in a range of subsampling scenarios, provide insights that can inform acquisition sampling choices as well as processing parameterization and quality control, e.g., to set up appropriate data filters and scaling to accommodate the effects of dipole fields, or to help ensuring that the data interpolation achieves the desired levels of reconstruction quality that minimize subsampling artifacts in Marchenko-derived fields and images.

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.

Green’s function representations for Marchenko imaging without up/down decomposition

SUMMARY Marchenko methods are based on integral representations which express Green’s functions for virtual sources and/or receivers in the subsurface in terms of the reflection response at the surface. An underlying assumption is that inside the medium the wave field can be decomposed into downgoing and upgoing waves and that evanescent waves can be neglected. We present a new derivation of Green’s function representations which circumvents these assumptions, both for the acoustic and the elastodynamic situation. These representations form the basis for research into new Marchenko methods which have the potential to handle refracted and evanescent waves and to more accurately image steeply dipping reflectors.

Handling gaps in acquisition geometries — Improving Marchenko-based imaging using sparsity-promoting inversion and joint inversion of time-lapse data

Marchenko focusing and imaging are novel methods for correctly handling multiple scattered energy while processing seismic data. However, strict requirements in the acquisition geometry, specifically the colocation of sources and receivers as well as dense and regular sampling, currently constrain their practical applicability. We have reformulated the Marchenko equations to handle the case in which there are gaps in the source geometry while the receiver sampling remains regular (or the opposite, by means of reciprocity). Using synthetic data based on a velocity model that produces strong interbed multiples, we test different solvers for the newly formulated inversion problem, and we compare these results to results obtained by applying standard Marchenko inversion to a previously reconstructed data set. When using the unreconstructed data set, the ability of the Marchenko equations to retrace multiple reflected energy deteriorates. Sparsity-promoting Marchenko inversion, while improving the appearance of focusing functions, barely decreases multiple leakage in gathers and does not visibly improve the final image when compared to standard least-squares inversion. On the other hand, reconstructing the wavefield in advance restores the proper functioning of the Marchenko methods. Further, we test a joint inversion technique designed for time-lapse data with nonrepeated geometries and originally intended to be solved using sparsity-promoting inversion. Motivated by our previous results, we compare images produced by this method to images produced by solving the same joint inversion problem without sparsity constraint. We find that the joint inversion alone hardly improves the resulting images but, when combined with the sparsity constraint, it leads to better noise and multiple suppression and produces a clean time-lapse image. Overall, none of the results from sparsity-promoting inversion techniques match the results obtained when reconstructing the wavefield in advance. We show that this can be explained by the comparatively slow convergence rate of the sparsity-promoting Marchenko inversion.

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.

Three-dimensional Marchenko internal multiple attenuation on narrow azimuth streamer data of the Santos Basin, Brazil.

ABSTRACTIn recent years, a variety of Marchenko methods for the attenuation of internal multiples has been developed. These methods have been extensively tested on two‐dimensional synthetic data and applied to two‐dimensional field data, but only little is known about their behaviour on three‐dimensional synthetic data and three‐dimensional field data. Particularly, it is not known whether Marchenko methods are sufficiently robust for sparse acquisition geometries that are found in practice. Therefore, we start by performing a series of synthetic tests to identify the key acquisition parameters and limitations that affect the result of three‐dimensional Marchenko internal multiple prediction and subtraction using an adaptive double‐focusing method. Based on these tests, we define an interpolation strategy and use it for the field data application. Starting from a wide azimuth dense grid of sources and receivers, a series of decimation tests are performed until a narrow azimuth streamer geometry remains. We evaluate the effect of the removal of sail lines, near offsets, far offsets and outer cables on the result of the adaptive double‐focusing method. These tests show that our method is most sensitive to the limited aperture in the crossline direction and the sail line spacing when applying it to synthetic narrow azimuth streamer data. The sail line spacing can be interpolated, but the aperture in the crossline direction is a limitation of the acquisition. Next, we apply the adaptive Marchenko double‐focusing method to the narrow azimuth streamer field data from the Santos Basin, Brazil. Internal multiples are predicted and adaptively subtracted, thereby improving the geological interpretation of the target area. These results imply that our adaptive double‐focusing method is sufficiently robust for the application to three‐dimensional field data, although the key acquisition parameters and limitations will naturally differ in other geological settings and for other types of acquisition.

Imaging vertical structures using Marchenko methods with vertical seismic-profile data

Marchenko methods use seismic data acquired at or near the surface of the earth to estimate seismic signals as if the receiver (now a virtual receiver) was at an arbitrary point inside the subsurface of the earth. This process is called redatuming, and it is central to subsurface imaging. Marchenko methods estimate the multiply scattered components of these redatumed signals, which is not the case for most other redatuming techniques that are based on single-scattering assumptions. As a result, images created using Marchenko redatumed signals contain a reduction in the artifacts that usually contaminate migrated seismic images due to improper handling of internal multiples. We exploit recent theoretical advances that enable virtual sources and virtual receivers to be placed at arbitrary points inside the subsurface as a means to incorporate vertical seismic profile (VSP) data into Marchenko methods. The advantage of including this type of data is that the additional acquisition boundary increases subsurface illumination, which in turn enables vertical interfaces and steeply dipping structures to be imaged. We develop this methodology using two synthetic data sets. The first is created using a simple variable density but constant velocity subsurface model. In this example, we find that our newly devised VSP Marchenko imaging methodology enables imaging of horizontal and vertical structures and that optimal results are achieved by combining these images with those created using standard Marchenko imaging. A second example demonstrates that the method can be applied to more realistic subsurface structures, in this case a modified version of the Marmousi 2 model. We determine the applicability of the methods to image fault structures with the final imaging result containing reduced contamination due to internal multiples and an improvement in the imaging of fault structures when compared to other standard imaging methods alone.

Marchenko methods in a 3-D world

SUMMARYMarchenko methods are a suite of geophysical techniques that convert seismograms of energy created by surface sources and measured by surface receivers into seismograms that would have been recorded by a virtual receiver at an arbitrary point inside the subsurface—an operation called redatuming. In principle these redatumed seismograms contain all contributions from direct, primary (singly-reflected) and multiply-reflected waves that would have been recorded by a real subsurface receiver, without requiring prior information about interfaces that generated the reflections. The potential of these methods for seismic imaging and redatuming has been demonstrated extensively in previous literature, but only using 1-D and 2-D Marchenko methods. There remain aspects of the methods that are poorly understood when applied in a 3-D world, so we investigate the application of Marchenko methods to 3-D data, subsurface structures and wavefields. We first show that for waves propagating in three dimensions, Marchenko methods can be applied to seismic data collected using both linear (so-called 2-D seismic) and areal (3-D seismic) acquisition arrays. However, for 2-D acquisition arrays the Marchenko workflow requires additional dimensionality correction factors to obtain accurate solutions, even in a subsurface that only varies with depth. Without these correction factors phase errors occur in redatumed Marchenko estimates; these errors propagate through the Marchenko algorithm and create depth errors in the Marchenko images. Furthermore, applying Marchenko methods to fully 3-D seismic wavefields recorded by linear (2-D seismic) arrays that contain out-of-plane reflections deteriorates surface-to-subsurface Green’s function estimates with spurious energy and resulting images are less accurate than those created using ‘conventional’ imaging methods. The application of fully 3-D Marchenko methods using data recorded on areal arrays solves both of the above problems, creating accurately redatumed wavefields and images with reduced artefact contamination. However, it appears that source–receiver spacing at most of $\lambda _A /4$ is required for accurate results using existing Marchenko methods, where λA is the dominant wavelength and in many real 3-D seismic acquisition scenarios this is impractical.

An introduction to Marchenko methods for imaging

Geoscientists often have little information about earth’s subsurface heterogeneities prior to mapping them using seismic or other geophysical data. Marchenko methods are a set of novel, data-driven techniques that help us to project surface seismic data to points in the subsurface, to form seismograms as though they had been created at each point. In so doing, Marchenko methods account for many of the complex, multiply reflected seismic wave interactions that take place in the real earth’s subsurface. The resulting seismograms are the information required to create subsurface images that are more accurate than standard methods. Our aim is to introduce these concepts with the minimum amount of mathematics required to understand how the Marchenko method can iterate to a solution and to provide a well-commented, easily editable MATLAB code package for demonstration and training purposes. Green’s function estimation using the Marchenko method is first illustrated for a constant velocity, variable density, 1D medium, with results that indicate a near-perfect match when compared with true, synthetically modeled solutions. Similar quality results are shown for variable velocity, 2D Green’s function estimation. Finally, we determine how these estimates could be used to create images of the subsurface, which, when compared with standard methods, contain reduced contamination due to multiple-related artifacts. Our code package includes the 2D data set required to reconstruct the relevant figures that we present, and it allows for experimentation with the implementation of the Marchenko method and the application of Marchenko imaging.

A practical implementation of subsalt Marchenko imaging with a Gulf of Mexico data set

Marchenko redatuming allows one to use surface seismic reflection data to generate the seismic response from sources at the surface to any point in the subsurface. Without requiring much information about the earth’s properties, the seismic response generated by Marchenko redatuming contains accurate estimates of not only the primaries but also the internal multiples. A target-oriented imaging method, referred to as Marchenko imaging, was implemented for imaging complex structures of the earth using the seismic response obtained through Marchenko redatuming. Taking account of the contribution of primaries and internal multiples, Marchenko imaging produces images that contain fewer artifacts than the images obtained using conventional imaging methods (e.g., reverse time migration) with the same input data. In this study, we applied Marchenko imaging to a field data set acquired at the Gulf of Mexico to produce an image of a subsalt area. We investigated two important and practical aspects of the Marchenko framework: (1) the missing near offsets in marine shot records and (2) the calibration of the reflection data. Finally, we suggested a workflow for processing the marine towed-streamer field data set acquired at the Gulf of Mexico, and we have developed a complete theoretical and practical framework to produce a target-oriented subsalt image using the Marchenko methods. The images obtained from Marchenko imaging are consistent and comparable, for the most part, with conventional migration methods. However, Marchenko imaging achieves improvements in the continuity of the geologic structures and in suppressing the artifacts that are caused by internal multiples.

Elastic internal multiple analysis and attenuation using Marchenko and interferometric methods

Conventional seismic processing aims to create data that contain only primary reflections, whereas real seismic recordings also contain multiples. As such, it is desirable to predict, identify, and attenuate multiples in seismic data. This task is more difficult in elastic (solid) media because mode conversions create families of internal multiples not present in the acoustic case. We have developed a method to predict prestack internal multiples in general elastic media based on the Marchenko method and convolutional interferometry. It can be used to identify multiples directly in prestack data or migrated sections, as well as to attenuate internal multiples by adaptively subtracting them from the original data set. We developed the method on two synthetic data sets, the first composed of horizontal density layers and constant velocities, and the second containing horizontal and vertical density and velocity variations. The full-elastic method is computationally expensive and ideally uses data components that are not usually recorded. We therefore tested an acoustic approximation to the method on the synthetic elastic data from the second model and find that although the spatial resolution of the resulting image is reduced by this approximation, it provides images with relatively fewer artifacts. We conclude that in most cases where cost is a factor and we are willing to sacrifice some resolution, it may be sufficient to apply the acoustic version of this demultiple method.

Des équations de Dirac et de Schrödinger pour la transformation de Fourier

Dyson has associated with the Fredholm determinants of the even (resp. odd) Dirichlet kernels a Schrödinger equation on the half-axis and has used, in tandem, the Gel'fand–Levitan and Marchenko methods of inverse scattering theory to study the asymptotics of these determinants. We have proposed following our unearthing of the conductor operator to seek to realize the Fourier transform itself as a scattering, and we obtain here to this end two Dirac systems on the entire real axis which are intrinsically associated, respectively, to the cosine and to the sine transforms. To cite this article: J.-F. Burnol, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Review of Some Exact Methods for the Solution of the One-Dimensional Inverse Problems

The inverse problem in geophysics is the determination of the inhomogeneous structure of an elastic medium from experimental observations. In this presentation, we first briefly review the Gelfand-Levitan and the Marchenko methods for the solution of the related quantum inverse problem. In these approaches, as well as in the methods for the geophysical problem we will discuss, the desired information is recovered from the solution of certain integral equations. Next, we consider the construction of a computational scheme for the inversion of geophysical data based on one of the methods. We will further show that the resulting algorithm is stable under small perturbations to the data. We will finally demonstrate the inversion scheme by inverting noiseless and noise-corrupted synthetic seismograms.