Full-wave-equation depth extrapolation for migration using matrix multiplication

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. S395-S403
Author(s):  
Jiachun You ◽  
Junxing Cao
Keyword(s):  
Wave Equation ◽  
Data Storage ◽  
Matrix Multiplication ◽  
Evanescent Waves ◽  
Lateral Velocity ◽  
Full Wave ◽  
Time Migration ◽  
Imaging Result ◽  
Extrapolation Scheme ◽  
Migration Method

To investigate wavefield depth extrapolation using the full-wave equation, we have derived a new depth extrapolation scheme for migration using functions of the vertical wavenumber. We develop a complete matrix multiplication formulation and approach to calculate the related mathematical functions of the vertical wavenumber and perform depth extrapolation using matrix multiplication only. Because our depth extrapolation algorithm involves only matrix multiplication, it is naturally applicable to parallel computations. Impulse response experiments demonstrate that our proposed migration method can achieve the same accuracy as full-wave-equation migration using the finite-difference method, in terms of phase information, even for media with strong lateral velocity changes. In numerical experiments using a smoothed version of the 2D SEG/EAGE salt model, our migration method provides an equivalent imaging result compared with reverse time migration (RTM) and a more accurate imaging result than migration using one-way propagators. Our method has certain potential advantages over RTM using the same full-wave equation with fewer internal multiple scatterings and fewer data storage requirements. Our adopted method is a stable depth extrapolation scheme because the evanescent waves are well suppressed. The numerical experimental results on the synthetic model demonstrate the importance of suppressing evanescent waves in a full-wave-equation-based depth extrapolation scheme and migration for imaging quality and computation cost.

scholarly journals Depth-Extrapolation-Based True-Amplitude Full-Wave-Equation Migration from Topography

2021 ◽  
Vol 11 (7) ◽  
pp. 3010
Author(s):  
Hao Liu ◽  
Xuewei Liu
Keyword(s):  
Wave Equation ◽  
Partial Derivative ◽  
Model Experiment ◽  
Surface Velocity ◽  
Seismic Exploration ◽  
Initial Condition ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Full Wave ◽  
Time Migration

The lack of an initial condition is one of the major challenges in full-wave-equation depth extrapolation. This initial condition is the vertical partial derivative of the surface wavefield and cannot be provided by the conventional seismic acquisition system. The traditional solution is to use the wavefield value of the surface to calculate the vertical partial derivative by assuming that the surface velocity is constant. However, for seismic exploration on land, the surface velocity is often not uniform. To solve this problem, we propose a new method for calculating the vertical partial derivative from the surface wavefield without making any assumptions about the surface conditions. Based on the calculated derivative, we implemented a depth-extrapolation-based full-wave-equation migration from topography using the direct downward continuation. We tested the imaging performance of our proposed method with several experiments. The results of the Marmousi model experiment show that our proposed method is superior to the conventional reverse time migration (RTM) algorithm in terms of imaging accuracy and amplitude-preserving performance at medium and deep depths. In the Canadian Foothills model experiment, we proved that our method can still accurately image complex structures and maintain amplitude under topographic scenario.

scholarly journals Wave-equation time migration

Geophysics ◽  
2020 ◽  
pp. 1-58
Author(s):  
Sergey Fomel ◽  
Harpreet Kaur
Keyword(s):  
Wave Equation ◽  
Time Domain ◽  
Field Data ◽  
Model Building ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Time Migration ◽  
Expensive Model ◽  
Approximate Equations ◽  
Ray Coordinates

Time migration, as opposed to depth migration, suffers from two well-known shortcomings: (1)approximate equations are used for computing Green’s functions inside the imaging operator; (2) in case of lateral velocity variations, the transformation between the image ray coordinates andthe Cartesian coordinates is undefined in places where the image rays cross. We show that thefirst limitation can be removed entirely by formulating time migration through wave propagationin image-ray coordinates. The proposed approach constructs a time-migrated image without relyingon any kind of traveltime approximation by formulating an appropriate geometrically accurateacoustic wave equation in the time-migration domain. The advantage of this approach is that thepropagation velocity in image-ray coordinates does not require expensive model building and canbe approximated by quantities that are estimated in conventional time-domain processing. Synthetic and field data examples demonstrate the effectiveness of the proposed approach and show that theproposed imaging workflow leads to a significant uplift in terms of image quality and can bridge thegap between time and depth migrations. The image obtained by the proposed algorithm is correctlyfocused and mapped to depth coordinates it is comparable to the image obtained by depth migration.

Reflection angle/azimuth-dependent least-squares reverse-time migration

Geophysics ◽  
2021 ◽  
pp. 1-68
Author(s):  
Eric Duveneck ◽  
Michael Kiehn ◽  
Anu Chandran ◽  
Thomas Kühnel
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Least Squares ◽  
Reflection Coefficients ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Linear Inverse Problem ◽  
Full Wave ◽  
Time Migration ◽  
Reflection Angle

Seismic images under complex overburdens like salt are strongly affected by illumination variations due to overburden velocity variations and imperfect acquisition geometries, making it difficult to obtain reliable image amplitudes. Least-squares reverse-time migration (LSRTM) addresses these issues by formulating full wave-equation imaging as a linear inverse problem and solving for a reflectivity model that explains the recorded seismic data. Because subsurface reflection coefficients depend on the incident angle, and possibly on azimuth, quantitative interpretation under complex overburdens requires LSRTM with output in terms of image gathers, e.g., as a function of reflection angle or angle and azimuth. We present a reflection angle- or angle/azimuth-dependent LSRTM method aimed at obtaining physically meaningful image amplitudes interpretable in terms of angle- or angle/azimuth-dependent reflection coefficients. The method is formulated as a linear inverse problem solved iteratively with the conjugate gradient method. It requires an adjoint pair of linear operators for reflection angle/azimuth-dependent migration and demigration based on full wave-equation propagation. We implement these operators in an efficient way by using a mapping approach between migrated shot gathers and subsurface reflection angle/azimuth gathers. To accelerate convergence of the iterative inversion, we apply image-domain preconditioning operators computed from a single de-remigration step. An angle continuity constraint and a structural dip constraint, implemented via shaping regularization, are used to stabilize the solution in the presence of limited illumination and to control the effects of coherent noise. We demonstrate the method on a synthetic data example and on a wide-azimuth streamer dataset from the Gulf of Mexico, where we show that angle/azimuth-dependent LSRTM can achieve significant uplift in subsalt image quality, with overburden- and acquisition-related illumination variation effects on angle/azimuth-dependent image amplitudes largely removed.

One-step wave extrapolation matrix method for reverse time migration

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. S359-S366 ◽  
Author(s):  
Daniel E. Revelo ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Chebyshev Polynomial ◽  
Matrix Method ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
First Order ◽  
Full Wave ◽  
Time Migration ◽  
The Matrix ◽  
One Step

We have developed a new method for solving the acoustic full-wave equation, which we call the one-step wave extrapolation (OSE) matrix method. In our method, the wave equation is redefined by introducing a complex (analytic) wavefield and reformulating the traditional acoustic full-wave equation as a first-order partial differential equation in time. Afterward, the analytical wavefield is separated to its real and imaginary parts, and the resulting first-order coupled set of equations is solved by the Tal-Ezer’s technique, which consists of using the Chebyshev polynomial expansion to approximate the matrix exponential operator. The matrix is antisymmetrical with a square-root pseudodifferential operator, which is computed using the Fourier method. In this way, the implementation of the proposed method is straightforward and if the appropriate number of Chebyshev polynomial expansion terms is chosen, the proposed numerical algorithm is unconditionally stable and propagates seismic waves free of numerical dispersion for any seismic velocity variation in a recursive manner. Moreover, in our method, the number of Fourier transforms is explicitly determined and it is a function of the maximum eigenvalue of the matrix operator and time-step size. A numerical modeling example is shown to demonstrate that the proposed method has the capability to extrapolate waves using a time stepping up to Nyquist limit. We have also developed a reverse time migration example with illumination compensation. The migration results based on the OSE method demonstrate the capability of this new method to image complex structures in the presence of strong velocity contrasts.

Least-squares migration of multisource data with a deblurring filter

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. R135-R146 ◽  
Author(s):  
Wei Dai ◽  
Xin Wang ◽  
Gerard T. Schuster
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Computational Efficiency ◽  
Computational Cost ◽  
Processing Technique ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Migration Method ◽  
Least Squares Migration

Least-squares migration (LSM) has been shown to be able to produce high-quality migration images, but its computational cost is considered to be too high for practical imaging. We have developed a multisource least-squares migration algorithm (MLSM) to increase the computational efficiency by using the blended sources processing technique. To expedite convergence, a multisource deblurring filter is used as a preconditioner to reduce the data residual. This MLSM algorithm is applicable with Kirchhoff migration, wave-equation migration, or reverse time migration, and the gain in computational efficiency depends on the choice of migration method. Numerical results with Kirchhoff LSM on the 2D SEG/EAGE salt model show that an accurate image is obtained by migrating a supergather of 320 phase-encoded shots. When the encoding functions are the same for every iteration, the input/output cost of MLSM is reduced by 320 times. Empirical results show that the crosstalk noise introduced by blended sources is more effectively reduced when the encoding functions are changed at every iteration. The analysis of signal-to-noise ratio (S/N) suggests that not too many iterations are needed to enhance the S/N to an acceptable level. Therefore, when implemented with wave-equation migration or reverse time migration methods, the MLSM algorithm can be more efficient than the conventional migration method.

scholarly journals SEISMIC IMAGING OF HIGH SLOPE STRUCTURES USING ONE-WAY WAVE EQUATION MIGRATION TECHNIQUES

2015 ◽  
Vol 33 (1) ◽  
pp. 71 ◽  
Author(s):  
Adeilton Rigaud Lucas Santos ◽  
Reynam Da Cruz Pestana ◽  
Gary Corey Aldunate
Keyword(s):  
Wave Equation ◽  
Fourier Domain ◽  
Complex Structures ◽  
Lateral Velocity ◽  
Seismic Migration ◽  
Salt Domes ◽  
Vertical Fault ◽  
Migration Method ◽  
The One ◽  
Imaging Conditions

ABSTRACT. The two-way wave equation deal with the wavefield in all of its forms, including multiples, refractions and internal reflections in more than one layer. Disregarding the downgoing wavefield, that is, considering only the propagation of the upgoing wavefield, starts using the one-way wave equation and some of these events are not considered. Thus, in areas of simple geology, the solution of the one-way wave equation is a good approximation. However, if the geology is complex, with abrupt lateral velocity variations and sub-vertical interfaces, such as in the presence of salt domes, methods of migration employing the one-way wave equation fail in imaging this kind of structures. In this paper, we present a way to overcome this limitation, by propagating the downgoing wavefield in following step upgoing wavefield. From these two extrapolated wavefields we apply four different imaging conditions, generating four intermediate sections, and the migrated section is formed from the weighted sum between the previous intermediate sections. This migration method was tested in two geological models (vertical fault and in a representative section of the Santos basin) and was able to reconstruct the complex structures existing in the models.Keywords: seismic migration, Fourier domain, sub-vertical reflectors.RESUMO. A equação completa da onda trata da propagação do campo de ondas em todas as suas formas, incluindo múltiplas, refrações e reflexões internas em mais de uma camada. Ao desprezar o campo de ondas descendente, ou seja, ao considerar a propagação apenas do campo ascendente, passa-se a empregar a equação unidirecional da onda e parte destes eventos não são considerados. Desta forma, em áreas de geologia simples, a solução da equação unidirecional é uma boa aproximação, entretanto, caso a geologia seja complexa, apresentando variações laterais bruscas de velocidade e interfaces subverticais, tal como na presença de domos salinos, os métodos de migração que empregam a equação unidirecional falham no imageamento das estruturas. Neste trabalho, apresenta-se uma forma de transpor esta limitação, através da propagação do campo de ondas descendente, numa etapa seguinte à do campo ascendente. A partir desses dois campos extrapolados, aplicam-se quatro condições de imagem distintas, gerando quatro seções intermediárias, e a seção migrada será formada a partir da soma ponderada entre as seções anteriores. Este método de migração foi testado em dois modelos geológicos (falha vertical e seção tipo da bacia de Santos) e mostrou-se capaz de reconstruir as estruturas mais complexas existentes nos modelos.Palavras-chave: migração sísmica, domínio de Fourier, refletores subverticais.

Full wave-equation based optimal shot selection for least-squares reverse time migration

2019 ◽  
Author(s):  
Anu Chandran ◽  
Thomas Kühnel ◽  
Farhad Bazargani ◽  
Michael Kiehn ◽  
Dung Nguyen ◽  
...  
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Full Wave ◽  
Time Migration ◽  
Selection For

Tomographic velocity analysis and wave-equation depth migration in an overthrust terrain: A case study from the Tuha Basin, China

2018 ◽  
Vol 6 (1) ◽  
pp. T1-T13
Author(s):  
Bin Lyu ◽  
Qin Su ◽  
Kurt J. Marfurt
Keyword(s):  
Wave Equation ◽  
Data Quality ◽  
Model Building ◽  
Velocity Model ◽  
Lateral Velocity ◽  
Near Surface ◽  
Depth Migration ◽  
Time Migration ◽  
Head Waves ◽  
Velocity Variations

Although the structures associated with overthrust terrains form important targets in many basins, accurately imaging remains challenging. Steep dips and strong lateral velocity variations associated with these complex structures require prestack depth migration instead of simpler time migration. The associated rough topography, coupled with older, more indurated, and thus high-velocity rocks near or outcropping at the surface often lead to seismic data that suffer from severe statics problems, strong head waves, and backscattered energy from the shallow section, giving rise to a low signal-to-noise ratio that increases the difficulties in building an accurate velocity model for subsequent depth migration. We applied a multidomain cascaded noise attenuation workflow to suppress much of the linear noise. Strong lateral velocity variations occur not only at depth but near the surface as well, distorting the reflections and degrading all deeper images. Conventional elevation corrections followed by refraction statics methods fail in these areas due to poor data quality and the absence of a continuous refracting surface. Although a seismically derived tomographic solution provides an improved image, constraining the solution to the near-surface depth-domain interval velocities measured along the surface outcrop data provides further improvement. Although a one-way wave-equation migration algorithm accounts for the strong lateral velocity variations and complicated structures at depth, modifying the algorithm to account for lateral variation in illumination caused by the irregular topography significantly improves the image, preserving the subsurface amplitude variations. We believe that our step-by-step workflow of addressing the data quality, velocity model building, and seismic imaging developed for the Tuha Basin of China can be applied to other overthrust plays in other parts of the world.

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