First‐arrival traveltime calculation for anisotropic media

Geophysics ◽  
1993 ◽  
Vol 58 (9) ◽  
pp. 1349-1358 ◽  
Author(s):  
Fuhao Qin ◽  
Gerard T. Schuster
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Anisotropic Media ◽  
Data Inversion ◽  
Test Models ◽  
Transverse Isotropic ◽  
First Arrival ◽  
Finite Difference Solution ◽  
Good Agreement

Huygens’s principle is used to derive a method for calculating first arrival traveltimes in anisotropic media. Numerical results for several transverse isotropic (TI) models show that calculated Huygens traveltimes are in good agreement with traveltimes computed by a finite‐difference solution to the anisotropic wave equation. This Huygens method is stable and accurate for the test models and so it may be useful in anisotropic data inversion and wave propagation visualization.

On the construction and efficiency of staggered numerical differentiators for the wave equation

Geophysics ◽  
1990 ◽  
Vol 55 (1) ◽  
pp. 107-110 ◽  
Author(s):  
M. Kindelan ◽  
A. Kamel ◽  
P. Sguazzero
Keyword(s):  
Numerical Modeling ◽  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Computational Efficiency ◽  
Differential Operators ◽  
High Order

Finite‐difference (FD) techniques have established themselves as viable tools for the numerical modeling of wave propagation. The accuracy and the computational efficiency of numerical modeling can be enhanced by using high‐order spatial differential operators (Dablain,1986).

Temporal high-order staggered-grid finite-difference schemes for elastic wave propagation

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T207-T224 ◽  
Author(s):  
Zhiming Ren ◽  
Zhen Chun Li
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
High Order ◽  
Elastic Wave Propagation ◽  
Staggered Grid ◽  
S Wave ◽  
Order Accuracy ◽  
Spatial Derivatives

The traditional high-order finite-difference (FD) methods approximate the spatial derivatives to arbitrary even-order accuracy, whereas the time discretization is still of second-order accuracy. Temporal high-order FD methods can improve the accuracy in time greatly. However, the present methods are designed mainly based on the acoustic wave equation instead of elastic approximation. We have developed two temporal high-order staggered-grid FD (SFD) schemes for modeling elastic wave propagation. A new stencil containing the points on the axis and a few off-axial points is introduced to approximate the spatial derivatives. We derive the dispersion relations of the elastic wave equation based on the new stencil, and we estimate FD coefficients by the Taylor series expansion (TE). The TE-based scheme can achieve ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracy ([Formula: see text]). We further optimize the coefficients of FD operators using a combination of TE and least squares (LS). The FD coefficients at the off-axial and axial points are computed by TE and LS, respectively. To obtain accurate P-, S-, and converted waves, we extend the wavefield decomposition into the temporal high-order SFD schemes. In our modeling, P- and S-wave separation is implemented and P- and S-wavefields are propagated by P- and S-wave dispersion-relation-based FD operators, respectively. We compare our schemes with the conventional SFD method. Numerical examples demonstrate that our TE-based and TE + LS-based schemes have greater accuracy in time and better stability than the conventional method. Moreover, the TE + LS-based scheme is superior to the TE-based scheme in suppressing the spatial dispersion. Owing to the high accuracy in the time and space domains, our new SFD schemes allow for larger time steps and shorter operator lengths, which can improve the computational efficiency.

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.

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