Finite-difference solution of linearized eikonal equation for transversely isotropic media

2019 ◽  
Author(s):  
Yogesh Arora ◽  
Ilya Tsvankin
Keyword(s):  
Finite Difference ◽  
Eikonal Equation ◽  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Finite Difference Solution ◽  
Difference Solution

scholarly journals A complex point-source solution of the acoustic eikonal equation for Gaussian beams in transversely isotropic media

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. T191-T207
Author(s):  
Xingguo Huang ◽  
Hui Sun ◽  
Zhangqing Sun ◽  
Nuno Vieira da Silva
Keyword(s):  
Point Source ◽  
Eikonal Equation ◽  
Taylor Series Expansion ◽  
Transversely Isotropic ◽  
Complex Point ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Complex Point Source ◽  
Complex Eikonal

The complex traveltime solutions of the complex eikonal equation are the basis of inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography. We have developed analytic approximations for the complex traveltime in transversely isotropic media with a titled symmetry axis, which is defined by a Taylor series expansion over the anisotropy parameters. The formulation for the complex traveltime is developed using perturbation theory and the complex point-source method. The real part of the complex traveltime describes the wavefront, and the imaginary part of the complex traveltime describes the decay of the amplitude of waves away from the central ray. We derive the linearized ordinary differential equations for the coefficients of the Taylor-series expansion using perturbation theory. The analytical solutions for the complex traveltimes are determined by applying the complex point-source method to the background traveltime formula and subsequently obtaining the coefficients from the linearized ordinary differential equations. We investigate the influence of the anisotropy parameters and of the initial width of the ray tube on the accuracy of the computed traveltimes. The analytical formulas, as outlined, are efficient methods for the computation of complex traveltimes from the complex eikonal equation. In addition, those formulas are also effective methods for benchmarking approximated solutions.

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.

On: “3-D traveltime compuation using the fast marching method” (J. A. Sethian and A. M. Popovici, GEOPHYSICS, 64, 516–523).

Geophysics ◽  
2000 ◽  
Vol 65 (2) ◽  
pp. 682-682
Author(s):  
Fuhao Qin
Keyword(s):  
Finite Difference ◽  
Eikonal Equation ◽  
Fast Marching Method ◽  
Fast Marching ◽  
First Arrival ◽  
Finite Difference Solution ◽  
Marching Method ◽  
Difference Solution

The Sethian and Popovici paper “3-D traveltime computation using the fast marching method” that appeared in Geophysics, Vol. 64, 516–523, discussed a method to solve the eikonal equation for first arrival traveltimes which was called the “fast marching” method. The method, as the authors demonstrated, is very fast and stable. However, their method is very similar to the method discussed by F. Qin et al. (1992), entitled “Finite difference solution of the eikonal equation along expanding wavefronts,” Geophysics, Vol. 57, 478–487. F. Qin et al. first proposed the “expanding wavefront” method for solving eikonal equation in the 60th Ann. Internat. Mtg. of the SEG in 1990.

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.

scholarly journals An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. C9-C20 ◽  
Author(s):  
Qi Hao ◽  
Tariq Alkhalifah
Keyword(s):  
Eikonal Equation ◽  
Wave Attenuation ◽  
Transversely Isotropic ◽  
S Wave ◽  
P Waves ◽  
The Earth ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Vti Media ◽  
Complex Valued

Seismic-wave attenuation is an important component of describing wave propagation. Certain regions, such as gas clouds inside the earth, exert highly localized attenuation. In fact, the anisotropic nature of the earth induces anisotropic attenuation because the quasi P-wave dispersion effect should be profound along the symmetry direction. We have developed a 2D acoustic eikonal equation governing the complex-valued traveltime of quasi P-waves in attenuating, transversely isotropic media with a vertical-symmetry axis (VTI). This equation is derived under the assumption that the complex-valued traveltime of quasi P-waves in attenuating VTI media are independent of the S-wave velocity parameter [Formula: see text] in Thomsen’s notation and the S-wave attenuation coefficient [Formula: see text] in Zhu and Tsvankin’s notation. We combine perturbation theory and Shanks transform to develop practical approximations to the acoustic attenuating eikonal equation, capable of admitting an analytical description of the attenuation in homogeneous media. For a horizontal-attenuating VTI layer, we also derive the nonhyperbolic approximations for the real and imaginary parts of the complex-valued reflection traveltime. These equations reveal that (1) the quasi SV-wave velocity and the corresponding quasi SV-wave attenuation coefficient given as part of Thomsen-type notation barely affect the ray velocity and ray attenuation of quasi P-waves in attenuating VTI media; (2) combining the perturbation method and Shanks transform provides an accurate analytic eikonal solution for homogeneous attenuating VTI media; (3) for a horizontal attenuating VTI layer with weak attenuation, the real part of the complex-valued reflection traveltime may still be described by the existing nonhyperbolic approximations developed for nonattenuating VTI media, and the imaginary part of the complex-valued reflection traveltime still has the shape of nonhyperbolic curves. In addition, we have evaluated the possible extension of the proposed eikonal equation to realistic attenuating media, an alternative perturbation solution to the proposed eikonal equation, and the feasibility of applying the proposed nonhyperbolic equation for the imaginary part of the complex-valued traveltime to invert for interval attenuation parameters.

Finite‐difference solution of the eikonal equation using an efficient, first‐arrival, wavefront tracking scheme

Geophysics ◽  
1994 ◽  
Vol 59 (4) ◽  
pp. 632-643 ◽  
Author(s):  
Shunhua Cao ◽  
Stewart Greenhalgh
Keyword(s):  
Finite Difference ◽  
Eikonal Equation ◽  
Discretization Scheme ◽  
Efficiency Improvement ◽  
Average Value ◽  
Velocity Models ◽  
First Arrival ◽  
Finite Difference Solution ◽  
Corner Node ◽  
Difference Solution

First‐break traveltimes can be accurately computed by the finite‐difference solution of the eikonal equation using a new corner‐node discretization scheme. It offers accuracy advantages over the traditional cell‐centered node scheme. A substantial efficiency improvement is achieved by the incorporation of a wavefront tracking algorithm based on the construction of a minimum traveltime tree. For the traditional discretization scheme, an accurate average value for the local squared slowness is found to be crucial in stabilizing the numerical scheme for models with large slowness contrasts. An improved method based on the traditional discretization scheme can be used to calculate traveltimes in arbitrarily varying velocity models, but the method based on the corner‐node discretization scheme provides a much better solution.

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