Finite‐difference modeling in transversely isotropic media

Geophysics ◽  
1994 ◽  
Vol 59 (2) ◽  
pp. 282-289 ◽  
Author(s):  
Eduardo L. Faria ◽  
Paul L. Stoffa
Keyword(s):  
Finite Difference ◽  
Time Derivative ◽  
Computational Cost ◽  
Vertical Component ◽  
Transversely Isotropic ◽  
Staggered Grid ◽  
Difference Operators ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Modeling Algorithm

We developed a modeling algorithm for transversely isotropic media that uses finite‐difference operators in a staggered grid. Staggered grid schemes are more stable than the conventional finite‐difference methods because the differences are actually based on half the grid spacing. This modeling algorithm uses the full elastic wave equation that makes possible the modeling of all kinds of waves propagating in transversely isotropic media. The spatial derivatives are represented by fourth‐order, finite‐difference operators while the time derivative is represented by a secondorder, finite‐difference operator. The algorithm has no limitation on the acquisition geometry or on the heterogeneity of the media. The program is currently formulated to work in a 2-D transversely isotropic medium but can readily be extended to 3-D. Snapshots can be obtained at any time with no additional computational cost. A four‐layer model is used to show the usefulness of the method. Horizontal and vertical component seismograms are modeled in transversely isotropic media and compared with seismograms modeled in the corresponding isotropic media.

Finite Difference Modeling of Elastic Wave Propagation

2012 ◽  
Vol 433-440 ◽  
pp. 4656-4661
Author(s):  
Qiang Zhang ◽  
Qi Zhen Du ◽  
Xu Fei Gong
Keyword(s):  
Finite Difference ◽  
Elastic Wave ◽  
Difference Scheme ◽  
Transversely Isotropic ◽  
Second Order ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Order Difference ◽  
Transversely Isotropic Media ◽  
Isotropic Media

We present a staggered-grid finite difference scheme for velocity-stress equations to simulate the elastic wave propagating in transversely isotropic media. Instead of the widely used temporally second-order difference scheme, a temporally fourth-order scheme is obtained in this paper. We approximate the third-order spatial derivatives with 2N-order difference rather than second-order or other fixed order difference as before. Thus, it could be possible to make a balanced accuracy of O (Δt4+Δx2N) with arbitrary N. Related issues such as stability criterion, numerical dispersion, source loading and boundary condition are also discussed in this paper. The numerical modeling result indicates that the scheme is reliable.

A stable one-way wave propagator for VTI media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB3-WB17 ◽  
Author(s):  
Peter M. Bakker
Keyword(s):  
Phase Shift ◽  
Finite Difference ◽  
Vertical Direction ◽  
Transversely Isotropic ◽  
Difference Operators ◽  
Wide Angle ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
Reference Medium

For the purpose of one-way wave-equation imaging, a pseudoscreen propagator is developed for transversely isotropic media with vertical axes of symmetry. The phase shift for propagation through a depth slice is decomposed into three terms: a Gazdag phase shift for propagation in a laterally homogeneous reference medium, a correction for lateral variability of vertical propagation, and a remaining wide-angle term for oblique directions of propagation. Based on rational function approximation for this remaining wide-angle term, a Fourier finite-difference (FFD) approach with four-way splitting is applied. Fourth-order Padé approximation is unsatisfactory in anelliptic media for large propagation angles with respect to the vertical direction. Therefore, a method of coefficient optimization is developed in conjunction with a method of choosing an adequate homogeneous reference medium in a depth slice. By symmetrizing the finite-difference operators, and because of the choice of the optimized coefficients, the propagator is stable in the sense that the least-squares norm of the wavefield, measured for a frequency-depth slice, does not grow with increasing depth of propagation. A small amount of artificial damping is applied to suppress artifacts that appear at the critical angle defined by the velocities in the reference medium and the actual medium. Synthetic examples confirm that good kinematic accuracy is achieved for a wide range of propagation angles (typically up to 60°).

Anisotropic complex Padé hybrid finite-difference depth migration

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. S51-S59 ◽  
Author(s):  
Daniela Amazonas ◽  
Rafael Aleixo ◽  
Jörg Schleicher ◽  
Jessé C. Costa
Keyword(s):  
Finite Difference ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Evanescent Waves ◽  
Acoustic Wave Equation ◽  
Depth Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Numerical Instabilities ◽  
Branch Cut

Standard real-valued finite-difference (FD) and Fourier finite-difference (FFD) migrations cannot handle evanescent waves correctly, which can lead to numerical instabilities in the presence of strong velocity variations. A possible solution to these problems is the complex Padé approximation, which avoids problems with evanescent waves by rotating the branch cut of the complex square root. We have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable FD and hybrid FD/FFD migrations for such media. Our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips. Our studies lead to the conclusion that the rotation angle of the branch cut that should yield the most stable image is 60° for FD migration, as confirmed by numerical impulse responses and work with synthetic data.

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.

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