Modeling of frequency-domain elastic-wave equation with an average-derivative optimal method

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. T339-T356 ◽  
Author(s):  
Jing-Bo Chen ◽  
Jian Cao
Keyword(s):  
Wave Equation ◽  
Phase Velocity ◽  
Elastic Wave ◽  
Frequency Domain ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Classical Scheme ◽  
Wave Phase Velocity ◽  
Grid Points ◽  
Average Derivative

Based on an average-derivative method, we developed a new nine-point numerical scheme for a frequency-domain elastic-wave equation. Compared with the classic nine-point scheme, this scheme reduces the required number of grid points per wavelength for equal and unequal directional spacings. The reduction in the number of grid points increases as the Poisson’s ratio becomes larger. In particular, as the Poisson’s ratio reaches 0.5, the numerical S-wave phase velocity of this new scheme becomes zero, whereas the classical scheme produces spurious numerical S-wave phase velocity. Numerical examples demonstrate that this new scheme produces more accurate results than the classical scheme at approximately the same computational cost.

An average-derivative optimal scheme for modeling of the frequency-domain 3D elastic wave equation

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T209-T234 ◽  
Author(s):  
Jing-Bo Chen ◽  
Jian Cao
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Frequency Domain ◽  
Poisson’S Ratio ◽  
Poisson's Ratio ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Wave Phase Velocity ◽  
Point Scheme ◽  
Grid Points

Because of its high computational cost, we needed to develop an efficient numerical scheme for the frequency-domain 3D elastic wave equation. In addition, the numerical scheme should be applicable to media with a liquid-solid interface. To address these two issues, we have developed a new average-derivative optimal 27-point scheme with arbitrary directional grid intervals and a corresponding numerical dispersion analysis for the frequency-domain 3D elastic wave equation. The novelty of this scheme is that its optimal coefficients depend on the ratio of the directional grid intervals and Poisson’s ratio. In this way, this scheme can be applied to media with a liquid-solid interface and a computational grid with arbitrary directional grid intervals. For media with a variable Poisson’s ratio, we have developed an effective and stable interpolation method for optimization coefficients. Compared with the classic 19-point scheme, this new scheme reduces the required number of grid points per wavelength for equal and unequal directional grid intervals. The reduction of the number of grid points increases as the Poisson’s ratio becomes larger. In particular, the numerical S-wave phase velocity of this new scheme becomes zero, whereas the classic 19-point scheme produces a spurious numerical S-wave phase velocity, as Poisson’s ratio reaches 0.5. We have performed numerical examples to develop the theoretical analysis.

True-amplitude linearized waveform inversion with the quasi-elastic wave equation

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R827-R844 ◽  
Author(s):  
Zongcai Feng ◽  
Gerard Schuster
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Waveform Inversion ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Elastic Wave Equation ◽  
P Waves ◽  
Time Migration

We present a quasi-elastic wave equation as a function of the pressure variable, which can accurately model PP reflections with elastic amplitude variation with offset effects under the first-order Born approximation. The kinematic part of the quasi-elastic wave equation accurately models the propagation of P waves, whereas the virtual-source part, which models the amplitudes of reflections, is a function of the perturbations of density and Lamé parameters [Formula: see text] and [Formula: see text]. The quasi-elastic wave equation generates a scattering radiation pattern that is exactly the same as that for the elastic wave equation, and only requires the solution of two acoustic wave equations for each shot gather. This means that the quasi-elastic wave equation can be used for true-amplitude linearized waveform inversion (also known as least-squares reverse time migration) of elastic PP reflections, in which the corresponding misfit gradients are with respect to the perturbations of density and the P- and S-wave impedances. The perturbations of elastic parameters are iteratively updated by minimizing the [Formula: see text]-norm of the difference between the recorded PP reflections and the predicted pressure data modeled from the quasi-elastic wave equation. Numerical tests on synthetic and field data indicate that true-amplitude linearized waveform inversion using the quasi-elastic wave equation can account for the elastic PP amplitudes and provide a robust estimate of the perturbations of P- and S-wave impedances and, in some cases, the density. In addition, true-amplitude linearized waveform inversion provides images with a wider bandwidth and fewer artifacts because the PP amplitudes are accurately explained. We also determine the 2D scalar quasi-elastic wave equation for P-SV reflections and the 3D vector equation for PS reflections.

Constraints on the anisotropic parameters for pseudoelastic vertical transverse isotropy wave equations and the applications on imaging carbonate reservoirs

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. C85-C94 ◽  
Author(s):  
Houzhu (James) Zhang ◽  
Hongwei Liu ◽  
Yang Zhao
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Velocity ◽  
Wave Equations ◽  
Transverse Isotropy ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Computation Efficiency ◽  
The Stability ◽  
Vertical Transverse Isotropy

Seismic anisotropy is an intrinsic elastic property. Appropriate accounting of anisotropy is critical for correct and accurate positioning seismic events in reverse time migration. Although the full elastic wave equation may serve as the ultimate solution for modeling and imaging, pseudoelastic and pseudoacoustic wave equations are more preferable due to their computation efficiency and simplicity in practice. The anisotropic parameters and their relations are not arbitrary because they are constrained by the energy principle. Based on the investigation of the stability condition of the pseudoelastic wave equations, we have developed a set of explicit formulations for determining the S-wave velocity from given Thomsen’s parameters [Formula: see text] and [Formula: see text] for vertical transverse isotropy and tilted transverse isotropy media. The estimated S-wave velocity ensures that the wave equations are stable and well-posed in the cases of [Formula: see text] and [Formula: see text]. In the case of [Formula: see text], a common situation in carbonate, a positive value of S-wave velocity is needed to avoid the wavefield instability. Comparing the stability constraints of the pseudoelastic- with the full-elastic wave equation, we conclude that the feasible range of [Formula: see text] and [Formula: see text] was slightly larger for the pseudoelastic assumption. The success of achieving high-accuracy images and high-quality angle gathers using the proposed constraints is demonstrated in a synthetic example and a field example from Saudi Arabia.

An affine generalized optimal scheme with improved free-surface expression using adaptive strategy for frequency-domain elastic wave equation

Geophysics ◽  
2022 ◽  
pp. 1-71
Author(s):  
Shu-Li Dong ◽  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Free Surface ◽  
Elastic Wave ◽  
Rotation Angle ◽  
Surface Expression ◽  
Coordinate Systems ◽  
Elastic Wave Equation ◽  
Point Scheme ◽  
Sampling Intervals ◽  
Grid Points

Effective frequency-domain numerical schemes were central for forward modeling and inversion of the elastic wave equation. The rotated optimal nine-point scheme was a highly used finite-difference numerical scheme. This scheme made a weighted average of the derivative terms of the elastic wave equations in the original and the rotated coordinate systems. In comparison with the classical nine-point scheme, it could simulate S-waves better and had higher accuracy at nearly the same computational cost. Nevertheless, this scheme limited the rotation angle to 45°; thus, the grid sampling intervals in the x- and z-directions needed to be equal. Otherwise, the grid points would not lie on the axes, which dramatically complicates the scheme. Affine coordinate systems did not constrain axes to be perpendicular to each other, providing enhanced flexibility. Based on the affine coordinate transformations, we developed a new affine generalized optimal nine-point scheme. At the free surface, we applied the improved free-surface expression with an adaptive parameter-modified strategy. The new optimal scheme had no restriction that the rotation angle must be 45°. Dispersion analysis found that our scheme could effectively reduce the required number of grid points per shear wavelength for equal and unequal sampling intervals compared to the classical nine-point scheme. Moreover, this reduction improved with the increase of Poisson’s ratio. Three numerical examples demonstrated that our scheme could provide more accurate results than the classical nine-point scheme in terms of the internal and the free-surface grid points.

Elastic wave-equation datuming

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. U51-U61
Author(s):  
Xufei Gong ◽  
Qizhen Du ◽  
Qiang Zhao ◽  
Pengyuan Sun ◽  
Jianlei Zhang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Time Shift ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Wave Data ◽  
Static Corrections ◽  
The One ◽  
Processing Techniques ◽  
Multicomponent Data

Wave-equation datuming (WED) techniques have demonstrated superiority when waves occur on the acquisition surface nonvertically, and traditional static corrections based on the time shift become inaccurate. Meanwhile, as for multicomponent data, those scalar techniques can hardly maintain the vector characteristics for the following multicomponent data processing flows. Considering this, we have developed an elastic-wave datuming approach to handle the static corrections for multicomponent data. Different from those existing scalar WED techniques, the multicomponent data are first decomposed into multicomponent P- and S-wave data. Then, the decomposed data are transformed into the [Formula: see text]-[Formula: see text] domain, and they are extrapolated from the acquisition surface to the datum using the one-way elastic-wave continuation. Finally, the datumed multicomponent data are reconstructed at the output datum by adding up the datumed P- and S-wave data. This elastic WED can guarantee that the same wave modes on different components are equally datumed, and the data remain multicomponent so that they are still applicable to multicomponent-joint processing techniques. Finally, several test examples involved in this paper have proved our method’s effectiveness in multicomponent data datuming application.

An elongated finite-difference scheme for 2D acoustic and elastic wave simulations in the frequency domain

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. T29-T45
Author(s):  
Junichi Takekawa ◽  
Hitoshi Mikada
Keyword(s):  
Finite Difference ◽  
Phase Velocity ◽  
Elastic Wave ◽  
Frequency Domain ◽  
Impedance Matrix ◽  
Memory Usage ◽  
Numerical Accuracy ◽  
Wave Modeling ◽  
Computational Costs ◽  
Grid Points

We have developed a novel scheme to simulate acoustic and elastic wave propagation in the frequency-domain using a rectangular finite-difference (FD) stencil. One of the main problem of the frequency-domain modeling is its huge computational costs, i.e., the calculation time and memory usage. To overcome this problem, researchers have proposed many schemes to reduce the number of grid points in a wavelength. In general, high-accuracy schemes require large-sized stencils that cause increment in the bandwidth of the impedance matrix. It is, therefore, important to improve the accuracy of numerical schemes without increasing the bandwidth. We have applied an elongated stencil with different sampling ratio between horizontal and vertical directions to circumvent extra numerical bandwidth in the impedance matrix. Optimal FD coefficients and the aspect ratio of the grid cell are determined to minimize the error of the phase velocity. We investigate the dispersion property of the proposed scheme using plane-wave analysis. The dispersion analysis indicates that we could reduce the number of grid points in a wavelength by approximately 2.78 for acoustic wave modeling and by approximately 3.15 for elastic wave modeling so that the error of the phase velocity is less than 1%. We also conduct numerical simulations using homogeneous and inhomogeneous models to demonstrate the effectiveness of our scheme. The comparison of numerical accuracy and computational costs between our scheme and the conventional ones indicates that the computational costs (calculation time, memory usage) can be reduced with high numerical accuracy especially in elastic wave modeling. Because our technique is a simple and a powerful cost-efficient frequency-domain method, the elongated stencil can be an alternative scheme to the conventional ones for acoustic and elastic wave modeling.

Modeling of frequency-domain elastic-wave equation with a general optimal scheme

2018 ◽  
Vol 159 ◽  
pp. 1-15 ◽  
Author(s):  
Aman Li ◽  
Hong Liu ◽  
Yuxin Yuan ◽  
Ting Hu ◽  
Xuebao Guo
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Frequency Domain ◽  
Elastic Wave Equation ◽  
Optimal Scheme

scholarly journals Pseudoelastic pure P-mode wave equation

Geophysics ◽  
2021 ◽  
Vol 86 (6) ◽  
pp. T469-T485
Author(s):  
Bingbing Sun ◽  
Tariq Alkhalifah
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Waveform Inversion ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Amplitude Variation With Offset ◽  
S Waves ◽  
Mode Wave ◽  
Wave Velocities ◽  
Simulation Engine

We have developed a pseudoelastic wave equation describing pure pressure waves propagating in elastic media. The pure pressure-mode (P-mode) wave equation uses all of the elastic parameters (such as density and the P- and S-wave velocities). It produces the same amplitude variation with offset (AVO) effects as PP-reflections computed by the conventional elastic wave equation. Because the new wave equation is free of S-waves, it does not require finer grids for simulation. This leads to a significant computational speedup when the ratio of pressure to S-wave velocities is large. We test the performance of our method on a simple synthetic model with high-velocity contrasts. The amplitude admitted by the pseudoelastic pure P-mode wave equation is highly consistent with that associated with the conventional elastic wave equation over a large range of incidence angles. We further verify our method’s robustness and accuracy using a more complex and realistic 2D salt model from the SEG Advanced Modeling Program. The ideal AVO behavior and computational advantage make our wave equation a good candidate as a forward simulation engine for performing elastic full-waveform inversion, especially for marine streamer data sets.

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