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°).

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.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

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.

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.

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