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.

Finite‐difference calculation of direct‐arrival traveltimes using the eikonal equation

Geophysics ◽  
2002 ◽  
Vol 67 (4) ◽  
pp. 1270-1274 ◽  
Author(s):  
Le‐Wei Mo ◽  
Jerry M. Harris
Keyword(s):  
Finite Difference ◽  
Ray Tracing ◽  
Finite Differences ◽  
Eikonal Equation ◽  
Velocity Model ◽  
Uniform Grid ◽  
Square Grid ◽  
Kirchhoff Migration ◽  
Difference Calculation

Traveltimes of direct arrivals are obtained by solving the eikonal equation using finite differences. A uniform square grid represents both the velocity model and the traveltime table. Wavefront discontinuities across a velocity interface at postcritical incidence and some insights in direct‐arrival ray tracing are incorporated into the traveltime computation so that the procedure is stable at precritical, critical, and postcritical incidence angles. The traveltimes can be used in Kirchhoff migration, tomography, and NMO corrections that require traveltimes of direct arrivals on a uniform grid.

Kirchhoff migration using eikonal-based computation of traveltime source derivatives

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. S211-S219 ◽  
Author(s):  
Siwei Li ◽  
Sergey Fomel
Keyword(s):  
Differential Equation ◽  
Eikonal Equation ◽  
Order Partial Differential Equation ◽  
Fast Marching ◽  
First Order ◽  
Velocity Models ◽  
Kirchhoff Migration ◽  
Input Source ◽  
First Arrival ◽  
Synthetic Models

The computational efficiency of Kirchhoff-type migration can be enhanced by using accurate traveltime interpolation algorithms. We addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location. We adopted a first-order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently. Unlike methods that rely on finite-difference estimations, the accuracy of the eikonal-based derivative did not depend on input source sampling. For smooth velocity models, the first-order traveltime source derivatives enabled a cubic Hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into Kirchhoff antialiasing schemes. We provided an implementation of the proposed method to first-arrival traveltimes by modifying the fast-marching eikonal solver. Several simple synthetic models and a semirecursive Kirchhoff migration of the Marmousi model demonstrated the applicability of the proposed method.

Can we image complex structures with first‐arrival traveltime?

Geophysics ◽  
1993 ◽  
Vol 58 (4) ◽  
pp. 564-575 ◽  
Author(s):  
Sébastien Geoltrain ◽  
Jean Brac
Keyword(s):  
Eikonal Equation ◽  
Geological Structure ◽  
Complex Structures ◽  
Complex Velocity ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Finite Differencing ◽  
Explicit Reference ◽  
First Arrival ◽  
Kirchhoff Depth Migration

We experienced difficulties when attempting to perform seismic imaging in complex velocity fields using prestack Kirchhoff depth migration in conjunction with traveltimes computed by finite‐differencing the eikonal equation. The problem arose not because of intrinsic limitations of Kirchhoff migration, but rather from the failure of finite‐differencing to compute traveltimes representative of the energetic part of the wavefield. Further analysis showed that the first arrival is most often associated with a marginally energetic event wherever subsequent arrivals exist. The consequence is that energetic seismic events are imaged with a kinematically incorrect operator and turn out mispositioned at depth. We therefore recommend that first‐arrival traveltime fields, such as those computed by finite‐differencing the eikonal equation, be used in Kirchhoff migration only with great care when the velocity field hosts multiple transmitted arrivals; such a situation is typically met where geological structure creates strong and localized velocity heterogeneities, which partition the incident and reflected wavefields into multiple arrivals; in such an instance, imaging cannot be strictly considered a kinematic process, as it must be performed with explicit reference to the relative amplitudes of multiple arrivals.

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.

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.

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.

scholarly journals Imaging complex geologic structure with single‐arrival Kirchhoff prestack depth migration

Geophysics ◽  
1997 ◽  
Vol 62 (5) ◽  
pp. 1533-1543 ◽  
Author(s):  
François Audebert ◽  
Dave Nichols ◽  
Thorbjørn Rekdal ◽  
Biondo Biondi ◽  
David E. Lumley ◽  
...  
Keyword(s):  
Finite Difference ◽  
Green’S Function ◽  
Green's Function ◽  
Continuous Phase ◽  
Maximum Energy ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Full Waveform ◽  
First Arrival ◽  
Paraxial Ray

We compare various forms of single‐arrival Kirchhoff prestack depth migration to a full‐waveform, finite‐difference migration image, using synthetic seismic data generated from the structurally complex 2-D Marmousi velocity model. First‐arrival‐traveltime Kirchhoff migration produces severe artifacts and image contamination in regions of the depth model where significant reflection energy propagates as late or multiple arrivals in the total reflection wavefield. Kirchhoff migrations using maximum‐energy‐arrival traveltime trajectories significantly improve the image in the complex zone of the Marmousi model, but are not as coherent as the finite‐difference migration image. By carefully incorporating continuous phase estimates with the associated maximum‐energy arrival traveltimes, we obtain single‐arrival Kirchhoff images that are similar in quality to the finite‐difference migration image. Furthermore, maximum‐energy Green's function traveltime and phase values calculated within the seismic frequency band give a Kirchhoff image that is (1) far superior to a first‐arrival—based image, (2) much better than the analogous high‐frequency paraxial‐ray Green's function image, and (3) closely matched in quality to the full‐waveform finite‐difference migration image.

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