To: “An analytical expression for the gravity field of a polyhedral body with linearly varying density” R. O. Hansen (GEOPHYSICS, 64, 75–77)

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 992-992
Keyword(s):  
Finite Difference ◽  
Gravity Field ◽  
Eikonal Equation ◽  
Point Of View ◽  
Geophysical Prospecting ◽  
Numerical Techniques ◽  
Field Equations ◽  
Domain Of Applicability ◽  
First Arrival ◽  
Finite Difference Solution

The domain of applicability of the expressions derived in this paper is stated incorrectly. The second‐last sentence of the section “Derivation of the Field Equations” should read “Using the numerical techniques discussed by Pohanka for circumventing this problem, an expression applicable everywhere in space can be obtained.” The following sentence is incorrect and should be deleted. I thank Dr. Marion Ivan for point this error out to me. After acceptance of the paper, an article covering substantially the same material, though with a somewhat different point of view, appeared. The article is: Pohanka, V., 1998, Optimum expression for computation of the gravity field of a polyhedral body with linearly increasing density: Geophysical Prospecting, 46, 391–404. Were I aware of Dr. Pohanka’s paper, it would of course have been referenced in mine. To: “Finite‐difference solution of the eikonal equation using an efficient, first‐arrival, wavefront tracking scheme” S. Cao and S. Greenhalgh (Geophysics, 59, 635) Equation 8a is in error. The parenthetical expressions on the left should be squared to make it dimensionally correct.

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 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.

3-D traveltime computation using Huygens wavefront tracing

Geophysics ◽  
2001 ◽  
Vol 66 (3) ◽  
pp. 883-889 ◽  
Author(s):  
Paul Sava ◽  
Sergey Fomel
Keyword(s):  
Finite Difference ◽  
Ray Tracing ◽  
Eikonal Equation ◽  
Numerical Solutions ◽  
Time Step ◽  
Computational Speed ◽  
First Arrival ◽  
Finite Difference Solution ◽  
Explicit Finite Difference ◽  
Ray Coordinates

Traveltime computation is widely used in seismic modeling, imaging, and velocity analysis. The two most commonly used methods are ray tracing and numerical solutions to the eikonal equation. Eikonal solvers are fast and robust but are limited to computing only the first‐arrival traveltimes. Ray tracing can compute multiple arrivals but lacks the robustness of eikonal solvers. We propose a robust and complete method of traveltime computation. It is based on a system of partial differential equations, which is equivalent to the eikonal equation but formulated in the ray‐coordinates system. We use a first‐order discretization scheme that is interpreted very simply in terms of the Huygens’s principle. Our explicit finite‐difference solution to the eikonal equation solved in the ray‐coordinates system delivers both computational speed and stability since we use more than one point on the current wavefront at every time step. The finite‐difference method has proven to be a robust alternative to conventional ray tracing, while being faster and having a better ability to handle rough velocity media and penetrate shadow zones.

Maximum energy traveltimes calculated in the seismic frequency band

Geophysics ◽  
1996 ◽  
Vol 61 (1) ◽  
pp. 253-263 ◽  
Author(s):  
Dave E. Nichols
Keyword(s):  
Finite Difference ◽  
Frequency Band ◽  
High Frequency ◽  
Eikonal Equation ◽  
Complex Structure ◽  
Maximum Energy ◽  
Imaging Algorithm ◽  
Kirchhoff Migration ◽  
First Arrival ◽  
Frequency Approximation

Prestack Kirchhoff migration using first‐arrival traveltimes has been shown to produce poor images in areas of complex structure. To avoid this problem, I propose a new method for calculating traveltimes that estimates the traveltime of the maximum energy arrival, rather than the first arrival. This method estimates a traveltime that is valid in the seismic frequency band, not the usual high‐frequency approximation. Instead of solving the eikonal equation for the traveltime, I solve the Helmholtz equation to estimate the wavefield for a few frequencies. I then perform a parametric fit to the wavefield to estimate a traveltime, amplitude, and phase. The images created by using these parameters in a Kirchhoff imaging algorithm are comparable in quality to those created using full‐wavefield, finite‐difference, shot‐profile migration.

scholarly journals Finite‐difference quasi‐P traveltimes for anisotropic media

Geophysics ◽  
2002 ◽  
Vol 67 (1) ◽  
pp. 147-155 ◽  
Author(s):  
Jianliang Qian ◽  
William W. Symes
Keyword(s):  
Finite Difference ◽  
Group Velocity ◽  
Velocity Vector ◽  
Eikonal Equation ◽  
Nonlinear Partial Differential Equation ◽  
Elastic Solid ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
P Wave ◽  
First Arrival

The first‐arrival quasi‐P wave traveltime field in an anisotropic elastic solid solves a first‐order nonlinear partial differential equation, the q P eikonal equation. The difficulty in solving this eikonal equation by a finite‐difference method is that for anisotropic media the ray (group) velocity direction is not the same as the direction of the traveltime gradient, so that the traveltime gradient can no longer serve as an indicator of the group velocity direction in extrapolating the traveltime field. However, establishing an explicit relation between the ray velocity vector and the phase velocity vector overcomes this difficulty. Furthermore, the solution of the paraxial q P eikonal equation, an evolution equation in depth, gives the first‐arrival traveltime along downward propagating rays. A second‐order upwind finite‐difference scheme solves this paraxial eikonal equation in O(N) floating point operations, where N is the number of grid points. Numerical experiments using 2‐D and 3‐D transversely isotropic models demonstrate the accuracy of the scheme.

Finite‐difference solution of the eikonal equation along expanding wavefronts

Geophysics ◽  
1992 ◽  
Vol 57 (3) ◽  
pp. 478-487 ◽  
Author(s):  
Fuhao Qin ◽  
Yi Luo ◽  
Kim B. Olsen ◽  
Wenying Cai ◽  
Gerard T. Schuster
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Scheme ◽  
Eikonal Equation ◽  
Computational Cost ◽  
Velocity Models ◽  
Large Velocity ◽  
Programming Effort ◽  
Finite Difference Solution ◽  
Solution Region

We show that a scheme to solve the 2-D eikonal equation by a finite‐difference method can violate causality for moderate to large velocity contrasts [Formula: see text]. As an alternative, we present a finite‐difference scheme in which the solution region progresses outward from an “expanding wavefront” rather than an “expanding square,” and therefore honors causality. Our method appears to be stable and reasonably accurate for a variety of velocity models with moderate to large velocity contrasts. The penalty is a large increase in computational cost and programming effort.

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.

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