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.

3-D prestack Kirchhoff beam migration for depth imaging

Geophysics ◽  
2000 ◽  
Vol 65 (5) ◽  
pp. 1592-1603 ◽  
Author(s):  
Yonghe Sun ◽  
Fuhao Qin ◽  
Steve Checkles ◽  
Jacques P. Leveille
Keyword(s):  
Distributed Environment ◽  
Small Surface ◽  
Depth Imaging ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Surface Patches ◽  
Image Volume ◽  
Kirchhoff Depth Migration ◽  
Full Volume ◽  
Single Set

A beam implementation is presented for efficient full‐volume 3-D prestack Kirchhoff depth migration of seismic data. Unlike conventional Kirchhoff migration in which the input seismic traces in time are migrated one trace at a time into the 3-D image volume for the earth’s subsurface, the beam migration processes a group of input traces (a supergather) together. The requirement for a supergather is that the source and receiver coordinates of the traces fall into two small surface patches. The patches are small enough that a single set of time maps pertaining to the centers of the patches can be used to migrate all the traces within the supergather by Taylor expansion or interpolation. The migration of a supergather consists of two major steps: stacking the traces into a τ-P beam volume, and mapping the beams into the image volume. Since the beam volume is much smaller than the image volume, the beam migration cost is roughly proportional to the number of input supergathers. The computational speedup of beam migration over conventional Kirchhoff migration is roughly proportional to [Formula: see text], the average number of traces per supergather, resulting a theoretical speedup up to two orders of magnitudes. The beam migration was successfully implemented and has been in production use for several years. A factor of 5–25 speedup has been achieved in our in‐house depth migrations. The implementation made 3-D prestack full‐volume depth imaging feasible in a parallel distributed environment.

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.

Parsimonious 2D prestack Kirchhoff depth migration

Geophysics ◽  
2003 ◽  
Vol 68 (3) ◽  
pp. 1043-1051 ◽  
Author(s):  
Biaolong Hua ◽  
George A. McMechan
Keyword(s):  
Fresnel Zone ◽  
Computation Time ◽  
Structural Features ◽  
Common Source ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Migration Velocity Analysis ◽  
Reflector Surface ◽  
Kirchhoff Depth Migration ◽  
Ray Parameter

The efficiency of prestack Kirchhoff depth migration is much improved by using ray parameter information measured from prestack common‐source and common‐receiver gathers. Ray tracing is performed only back along the emitted and emergent wave directions, and so is much reduced. The position of the intersection of the source and receiver rays is adjusted to satisfy the image time condition. The imaged amplitudes are spread along the local reflector surface only within the first Fresnel zone. There is no need to build traveltime tables before migration because the traveltime calculation is embedded into the migration. To further reduce the computation time, the input data are decimated by applying an amplitude threshold before the estimation of ray parameters, and only peak and trough points on each trace are searched for ray parameters. Numerical results show that the proposed implementation is typically 50–80 times faster than traditional Kirchhoff migration for synthetic 2D prestack data. The migration speed improvement is obtained at the expense of some reduction in migration quality; the optimal compromise is implemented by the choice of migration parameters. The main uses of the algorithm will be to get a fast first look at the main structural features and for iterative migration velocity analysis.

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.

Imaging complex structures with semirecursive Kirchhoff migration

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 577-588 ◽  
Author(s):  
Dimitri Bevc
Keyword(s):  
Wave Equation ◽  
Adverse Effects ◽  
Velocity Structure ◽  
Computational Cost ◽  
Downward Continuation ◽  
Complex Structures ◽  
Complex Velocity ◽  
Data Set ◽  
Kirchhoff Migration ◽  
Small Depth

I present a semirecursive Kirchhoff migration algorithm that is capable of obtaining accurate images of complex structures by combining wave‐equation datuming and Kirchhoff migration. The method is successful because breaking up the complex velocity structure into small depth regions allows traveltimes to be calculated in regions where the computation is well‐behaved and where the computation corresponds to energetic arrivals. The traveltimes computed in such a region are used first for imaging and second for downward continuation of the entire survey (shots and receivers) to the boundary of the next region. This process results in images comparable to those obtained by shot‐profile migration, but at reduced computational cost. Because traveltimes are computed for small depth domains, the adverse effects of caustics, headwaves, and multiple arrivals do not develop. In principle, this method requires only the same number of traveltime calculations as a standard migration. Tests on the Marmousi data set produce excellent results.

Parsimonious 2‐D poststack Kirchhoff depth migration

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1497-1503 ◽  
Author(s):  
Biao‐Long Hua ◽  
George A. McMechan
Keyword(s):  
Incident Wave ◽  
Input Data ◽  
Fresnel Zone ◽  
Computation Time ◽  
Structural Features ◽  
Seismic Section ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Ray Path ◽  
Kirchhoff Depth Migration

Efficiency of Kirchhoff migration can be much improved by using slope information from the seismic section to estimate the incident wave directions. Ray tracing is performed only back along the incident wave directions and so is much reduced. Unlike in conventional Kirchhoff implementations, there is no need to build traveltime tables, so relatively little memory and input/output use are required. Compression of the input data and restricting the contribution of each time sample to the image to lie within a Fresnel zone of its ray path further reduces the computation time. Synthetic and field data tests show that the new algorithm is about 30 times faster than traditional Kirchhoff migration for 2‐D poststack data. The main structural features may be imaged very quickly at the expense of some details. There is a tradeoff between speed and image quality; the optimal compromise is implemented by the choice of migration parameters.

Fast migration/inversion with multivalued ray fields: Part 2—Applications to the 3D SEG/EAGE salt model

Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1320-1328 ◽  
Author(s):  
Sheng Xu ◽  
Gilles Lambaré ◽  
Henri Calandra
Keyword(s):  
Three Dimensional ◽  
Velocity Model ◽  
Three Dimensions ◽  
Complex Media ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Convenient Approach ◽  
First Arrival ◽  
Subsalt Imaging ◽  
Lateral Variations

Three‐dimensional prestack depth migration is the convenient approach for seismic imaging in the case of strong lateral variations of the velocity. Because of computing limitations, it has been limited to single‐arrival kinematic Kirchhoff migration until recently. This approach fails in the case of complex media characterized by multiarrival traveltimes. We present numerical strategies for extending in three dimensions first‐arrival kinematic Kirchhoff migration to multiarrival quantitative ray‐based migration (preserved amplitude migration). We rely on wavefront construction in a smooth velocity model to compute the multivalued traveltime and amplitude maps, and the CPU efficiency of migration itself is ensured by efficient and robust interpolation or extrapolation strategies. We present an application to the synthetic 3D SEG/EAGE salt model. Taking into account multiarrivals clearly improves subsalt imaging at the price of quite limited computing costs (a 20% increase in our case, with respect to a preserved‐amplitude single‐arrival migration).

Efficient 2.5-D true‐amplitude migration

Geophysics ◽  
2000 ◽  
Vol 65 (3) ◽  
pp. 943-950 ◽  
Author(s):  
Joe A. Dellinger ◽  
Samuel H. Gray ◽  
Gary E. Murphy ◽  
John T. Etgen
Keyword(s):  
Seismic Data ◽  
Constant Velocity ◽  
Three Dimensions ◽  
Additional Source ◽  
Standard Methods ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Out Of Plane ◽  
Kirchhoff Method ◽  
Kirchhoff Depth Migration

Kirchhoff depth migration is a widely used algorithm for imaging seismic data in both two and three dimensions. To perform the summation at the heart of the algorithm, standard Kirchhoff migration requires a traveltime map for each source and receiver. True‐amplitude Kirchhoff migration in 2.5-D υ(x, z) media additionally requires maps of amplitudes, out‐of‐plane spreading factors, and takeoff angles; these quantities are necessary for calculating the true‐amplitude weight term in the summation. The increased input/output (I/O) and computational expense of including the true‐amplitude weight term is often not justified by significant improvement in the final muted and stacked image. For this reason, some authors advocate neglecting the weight term in the Kirchhoff summation entirely for most everyday imaging purposes. We demonstrate that for nearly the same expense as ignoring the weight term, a much better solution is possible. We first approximate the true‐amplitude weight term by the weight term for constant‐velocity media; this eliminates the need for additional source and receiver maps. With one small additional approximation, the weight term can then be moved entirely outside the innermost loop of the summation. The resulting Kirchhoff method produces images that are almost as good as for exact true‐amplitude Kirchhoff migration and at almost the same cost as standard methods that do not attempt to preserve amplitudes.

scholarly journals Traveltime and amplitude calculations using the damped wave solution

Geophysics ◽  
2002 ◽  
Vol 67 (5) ◽  
pp. 1637-1647 ◽  
Author(s):  
Changsoo Shin ◽  
Dong‐Joo Min ◽  
Kurt J. Marfurt ◽  
Harry Y. Lim ◽  
Dongwoo Yang ◽  
...  
Keyword(s):  
Finite Element ◽  
Wave Equation ◽  
Finite Element Modeling ◽  
Numerical Solutions ◽  
Geological Structure ◽  
Earth Model ◽  
Reference Frequency ◽  
Depth Migration ◽  
Element Modeling ◽  
First Arrival

Because of its computational efficiency, prestack Kirchhoff depth migration remains the method of choice for all but the most complicated geological depth structures. Further improvement in computational speed and amplitude estimation will allow us to use such technology more routinely and generate better images. To this end, we developed a new, accurate, and economical algorithm to calculate first‐arrival traveltimes and amplitudes for an arbitrarily complex earth model. Our method is based on numerical solutions of the wave equation obtained by using well‐established finite‐difference or finite‐element modeling algorithms in the Laplace domain, where a damping term is naturally incorporated in the wave equation. We show that solving the strongly damped wave equation is equivalent to solving the eikonal and transport equations simultaneously at a fixed reference frequency, which properly accounts for caustics and other problems encountered in ray theory. Using our algorithm, we can easily calculate first‐arrival traveltimes for given models. We present numerical examples for 2‐D acoustic models having irregular topography and complex geological structure using a finite‐element modeling code.

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.

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