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.

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.

Anisotropic finite-difference algorithm for modeling elastic wave propagation in fractured coalbeds

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. C13-C26 ◽  
Author(s):  
Zhenglin Pei ◽  
Li-Yun Fu ◽  
Weijia Sun ◽  
Tao Jiang ◽  
Binzhong Zhou
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Taylor Series Expansion ◽  
Real Data ◽  
Elastic Wave Propagation ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
S Wave ◽  
Data Set

The simulation of wave propagations in coalbeds is challenged by two major issues: (1) strong anisotropy resulting from high-density cracks/fractures in coalbeds and (2) numerical dispersion resulting from high-frequency content (the dominant frequency can be higher than 100 Hz). We present a staggered-grid high-order finite-difference (FD) method with arbitrary even-order ([Formula: see text]) accuracy to overcome the two difficulties stated above. First, we derive the formulae based on the standard Taylor series expansion but given in a neat and explicit form. We also provide an alternative way to calculate the FD coefficients. The detailed implementations are shown and the stability condition for anisotropic FD modeling is examined by the eigenvalue analysis method. Then, we apply the staggered-grid FD method to 2D and 3D coalbed models with dry and water-saturated fractures to study the characteristics of the 2D/3C elastic wave propagation in anisotropic media. Several factors, like density and direction of vertical cracks, are investigated. Several phenomena, like S-wave splitting and waveguides, are observed and are consistent with those observed in a real data set. Numerical results show that our formulae can correlate the amplitude and traveltime anisotropies with the coal seam fractures.

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.

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