High-order finite-difference approximations to solve pseudoacoustic equations in 3D VTI media

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T225-T235 ◽  
Author(s):  
Leandro Di Bartolo ◽  
Leandro Lopes ◽  
Luis Juracy Rangel Lemos
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Computational Cost ◽  
Transversely Isotropic ◽  
High Order ◽  
Staggered Grid ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Grid Scheme

Pseudoacoustic algorithms are very fast in comparison with full elastic ones for vertical transversely isotropic (VTI) modeling, so they are suitable for many applications, especially reverse time migration. Finite differences using simple grids are commonly used to solve pseudoacoustic equations. We have developed and implemented general high-order 3D pseudoacoustic transversely isotropic formulations. The focus is the development of staggered-grid finite-difference algorithms, known for their superior numerical properties. The staggered-grid schemes based on first-order velocity-stress wave equations are developed in detail as well as schemes based on direct application to second-order stress equations. This last case uses the recently presented equivalent staggered-grid theory, resulting in a staggered-grid scheme that overcomes the problem of large memory requirement. Two examples are presented: a 3D simulation and a prestack reverse time migration application, and we perform a numerical analysis regarding computational cost and precision. The errors of the new schemes are smaller than the existing nonstaggered-grid schemes. In comparison with existing staggered-grid schemes, they require 25% less memory and only have slightly greater computational cost.

Reverse-time migration in acoustic VTI media using a high-order stereo operator

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. WA3-WA11 ◽  
Author(s):  
Wei Xie ◽  
Dinghui Yang ◽  
Faqi Liu ◽  
Jingshuang Li
Keyword(s):  
Computational Cost ◽  
High Order ◽  
New Method ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Imaging Method ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Velocity Models ◽  
Time Migration

With the capability of handling complicated velocity models, reverse-time migration (RTM) has become a powerful imaging method. Improving imaging accuracy and computational efficiency are two significant but challenging tasks in the applications of RTM. Despite being the most popular numerical technique applied in RTM, finite-difference (FD) methods often suffer from undesirable numerical dispersion, leading to a noticeable loss of imaging resolution. A new and effective FD operator, called the high-order stereo operator, has been developed to approximate the partial differential operators in the wave equation, from which a numerical scheme called the three-step stereo method (TSM) has been developed and has shown effectiveness in suppressing numerical dispersion. Numerical results show that compared with the conventional numerical methods such as the Lax-Wendroff correction (LWC) scheme and the staggered-grid (SG) FD method, this new method significantly reduces numerical dispersion and computational cost. Tests on the impulse response and the 2D prestack Hess acoustic VTI model demonstrated that the TSM achieves higher image quality than the LWC and SG methods do, especially when coarse computation grids were used, which indicated that the new method can be a promising algorithm for large-scale anisotropic RTM.

First-order particle velocity equations of decoupled P- and S-wavefields and their application in elastic reverse time migration

Geophysics ◽  
2021 ◽  
pp. 1-78
Author(s):  
Zhiyuan Li ◽  
Youshan Liu ◽  
Guanghe Liang ◽  
Guoqiang Xue ◽  
Runjie Wang
Keyword(s):  
Particle Velocity ◽  
Computational Cost ◽  
Staggered Grid ◽  
Additional Stress ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Computational Accuracy ◽  
First Order ◽  
Time Migration ◽  
Stress Variable

The separation of P- and S-wavefields is considered to be an effective approach for eliminating wave-mode cross-talk in elastic reverse-time migration. At present, the Helmholtz decomposition method is widely used for isotropic media. However, it tends to change the amplitudes and phases of the separated wavefields compared with the original wavefields. Other methods used to obtain pure P- and S-wavefields include the application of the elastic wave equations of the decoupled wavefields. To achieve a high computational accuracy, staggered-grid finite-difference (FD) schemes are usually used to numerically solve the equations by introducing an additional stress variable. However, the computational cost of this method is high because a conventional hybrid wavefield (P- and S-wavefields are mixed together) simulation must be created before the P- and S-wavefields can be calculated. We developed the first-order particle velocity equations to reduce the computational cost. The equations can describe four types of particle velocity wavefields: the vector P-wavefield, the scalar P-wavefield, the vector S-wavefield, and the vector S-wavefield rotated in the direction of the curl factor. Without introducing the stress variable, only the four types of particle velocity variables are used to construct the staggered-grid FD schemes, so the computational cost is reduced. We also present an algorithm to calculate the P and S propagation vectors using the four particle velocities, which is simpler than the Poynting vector. Finally, we applied the velocity equations and propagation vectors to elastic reverse-time migration and angle-domain common-image gather computations. These numerical examples illustrate the efficiency of the proposed methods.

Elastic reverse time migration using acoustic propagators

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. S399-S408 ◽  
Author(s):  
Yunyue Elita Li ◽  
Yue Du ◽  
Jizhong Yang ◽  
Arthur Cheng ◽  
Xinding Fang
Keyword(s):  
Wave Equations ◽  
Mode Conversion ◽  
Computational Cost ◽  
Body Waves ◽  
Constant Density ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Isotropic Constant ◽  
Time Migration

Elastic wave imaging has been a significant challenge in the exploration industry due to the complexities in wave physics and numerical implementation. We have separated the governing equations for P- and S-wave propagation without the assumptions of homogeneous Lamé parameters to capture the mode conversion between the two body waves in an isotropic, constant-density medium. The resulting set of two coupled second-order equations for P- and S-potentials clearly demonstrates that mode conversion only occurs at the discontinuities of the shear modulus. Applying the Born approximation to the new equations, we derive the PP, PS, SP, and SS imaging conditions from the first gradients of waveform matching objective functions. The resulting images are consistent with the physical perturbations of the elastic parameters, and, hence, they are automatically free of the polarity reversal artifacts in the converted images. When implementing elastic reverse time migration (RTM), we find that scalar wave equations can be used to back propagate the recorded P-potential, as well as individual components in the vector field of the S-potential. Compared with conventional elastic RTM, the proposed elastic RTM implementation using acoustic propagators not only simplifies the imaging condition, it but also reduces the computational cost and the artifacts in the images. We have determined the accuracy of our method using 2D and 3D numerical examples.

An optimized wave equation for seismic modeling and reverse time migration

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA153-WCA158 ◽  
Author(s):  
Faqi Liu ◽  
Guanquan Zhang ◽  
Scott A. Morton ◽  
Jacques P. Leveille
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Computational Cost ◽  
Numerical Dispersion ◽  
New Wave ◽  
Acoustic Wave Equation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration

The acoustic wave equation has been widely used for the modeling and reverse time migration of seismic data. Numerical implementation of this equation via finite-difference techniques has established itself as a valuable approach and has long been a favored choice in the industry. To ensure quality results, accurate approximations are required for spatial and time derivatives. Traditionally, they are achieved numerically by using either relatively very fine computation grids or very long finite-difference operators. Otherwise, the numerical error, known as numerical dispersion, is present in the data and contaminates the signals. However, either approach will result in a considerable increase in the computational cost. A simple and computationally low-cost modification to the standard acoustic wave equation is presented to suppress numerical dispersion. This dispersion attenuator is one analogy of the antialiasing operator widely applied in Kirchhoff migration. When the new wave equation is solved numerically using finite-difference schemes, numerical dispersion in the original wave equation is attenuated significantly, leading to a much more accurate finite-difference scheme with little additional computational cost. Numerical tests on both synthetic and field data sets in both two and three dimensions demonstrate that the optimized wave equation dramatically improves the image quality by successfully attenuating dispersive noise. The adaptive application of this new wave equation only increases the computational cost slightly.

High Order Finite Difference Modeling and Reverse Time Migration

1991 ◽  
Vol 22 (3) ◽  
pp. 533-545 ◽  
Author(s):  
D. Loewenthal ◽  
C. J. Wang ◽  
O. G. Johnson ◽  
C. Juhlin
Keyword(s):  
Finite Difference ◽  
High Order ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
High Order Finite Difference
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