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.

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.

Q least-squares reverse time migration based on the first-order viscoacoustic quasidifferential equations

Geophysics ◽  
2021 ◽  
pp. 1-65
Author(s):  
Yingming Qu ◽  
Yixin Wang ◽  
Zhenchun Li ◽  
Chang Liu
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Surface Topography ◽  
Density Perturbation ◽  
Second Order ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
First Order ◽  
Time Migration ◽  
Grid Scheme

Seismic wave attenuation caused by subsurface viscoelasticity reduces the quality of migration and the reliability of interpretation. A variety of Q-compensated migration methods have been developed based on the second-order viscoacoustic quasidifferential equations. However, these second-order wave-equation-based methods are difficult to handle with density perturbation and surface topography. In addition, the staggered grid scheme, which has an advantage over the collocated grid scheme because of its reduced numerical dispersion and enhanced stability, works in first-order wave-equation-based methods. We have developed a Q least-squares reverse time migration method based on the first-order viscoacoustic quasidifferential equations by deriving Q-compensated forward-propagated operators, Q-compensated adjoint operators, and Q-attenuated Born modeling operators. Besides, our method using curvilinear grids is available even when the attenuating medium has surface topography and can conduct Q-compensated migration with density perturbation. The results of numerical tests on two synthetic and a field data sets indicate that our method improves the imaging quality with iterations and produces better imaging results with clearer structures, higher signal-to-noise ratio, higher resolution, and more balanced amplitude by correcting the energy loss and phase distortion caused by Q attenuation. It also suppresses the scattering and diffracted noise caused by the surface topography.

Least-squares reverse time migration in TTI media using a pure qP-wave equation

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. S199-S216
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Jidong Yang ◽  
Xu Guo ◽  
Yundong Guo
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Seismic Imaging ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
The Matrix ◽  
The Stability

Anisotropy is a common phenomenon in subsurface strata and should be considered in seismic imaging and inversion. Seismic imaging in a vertical transversely isotropic (VTI) medium does not take into account the effects of the tilt angles, which can lead to degraded migrated images in areas with strong anisotropy. To correct such waveform distortion, reduce related image artifacts, and improve migration resolution, a tilted transversely isotropic (TTI) least-squares reverse time migration (LSRTM) method is presented. In the LSRTM, a pure qP-wave equation is used and solved with the finite-difference method. We have analyzed the stability condition for the pure qP-wave equation using the matrix method, which is used to ensure the stability of wave propagation in the TTI medium. Based on this wave equation, we derive a corresponding demigration (Born modeling) and adjoint migration operators to implement TTI LSRTM. Numerical tests on the synthetic data show the advantages of TTI LSRTM over VTI RTM and VTI LSRTM when the recorded data contain strong effects caused by large tilt angles. Our numerical experiments illustrate that the sensitivity of the adopted TTI LSRTM to the migration velocity errors is much higher than that to the anisotropic parameters (including epsilon, delta, and tilted angle parameters), and its sensitivity to the epsilon model and tilt angle is higher than that to the delta model.

scholarly journals Reverse Time Migration Based on the Pseudo-Space-Domain First-Order Velocity-Stress Acoustic Wave Equation

2021 ◽  
Vol 9 ◽  
Author(s):  
Xiaobo Zhang ◽  
Xiutian Wang ◽  
Baohua Liu ◽  
Peng Song ◽  
Jun Tan ◽  
...  
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Imaging Method ◽  
Acoustic Wave Equation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Space Domain ◽  
First Order ◽  
Time Migration ◽  
Rectangular Mesh

Reverse time migration (RTM) is an ideal seismic imaging method for complex structures. However, in conventional RTM based on rectangular mesh discretization, the medium interfaces are usually distorted. Besides, reflected waves generated by the two-way wave equation can cause artifacts during imaging. To overcome these problems, a high-order finite-difference (FD) scheme and stability condition for the pseudo-space-domain first-order velocity-stress acoustic wave equation were derived, and based on the staggered-grid FD scheme, the RTM of the pseudo-space-domain acoustic wave equation was implemented. Model experiments showed that the proposed RTM of the pseudo-space-domain acoustic wave equation could systematically avoid the interface distortion problem when the velocity interfaces were considered to compute the pseudo-space-domain intervals. Moreover, this method could effectively suppress the false scattering of dipping interfaces and reflections during wavefield extrapolation, thereby reducing migration artifacts on the profile and significantly improving the quality of migration imaging.

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.

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