Improving plane‐wave decomposition and migration

Geophysics ◽  
1997 ◽  
Vol 62 (1) ◽  
pp. 195-205 ◽  
Author(s):  
Hans J. Tieman
Keyword(s):  
Plane Wave ◽  
Computer Time ◽  
Lateral Velocity ◽  
High Quality ◽  
Depth Migration ◽  
Wave Data ◽  
Velocity Variations ◽  
And Migration ◽  
The One ◽  
Slant Stacking

Plane‐wave data can be produced by slant stacking common geophone gathers over source locations. Practical difficulties arise with slant stacks over common receiver gathers that do not arise with slant stacks over common‐midpoint gathers. New techniques such as hyperbolic velocity filtering allow the production of high‐quality slant stacks of common‐midpoint data that are relatively free of artifacts. These techniques can not be used on common geophone data because of the less predictive nature of data in this domain. However, unlike plane‐wave data, slant stacks over midpoint gathers cannot be migrated accurately using depth migration. A new transformation that links common‐midpoint slant stacks to common geophone slant stacks allows the use together of optimized methods of slant stacking and accurate depth migration in data processing. Accurate depth migration algorithms are needed to migrate plane‐wave data because of the potentially high angles of propagation exhibited by the data and because of any lateral velocity variations in the subsurface. Splitting the one‐way wave continuation operator into two components (one that is a function of a laterally independent velocity, and a residual term that handles lateral variations in subsurface velocities) results in a good approximation. The first component is applied in the wavenumber domain, the other is applied in the space domain. The approximation is accurate for any angle of propagation in the absence of lateral velocity variations, although with severe lateral velocity variations the accuracy is reduced to 50°. High‐quality plane‐wave data migrated using accurate wave continuation operators results in a high‐quality image of the subsurface. Because of the signal‐to‐noise content of this data the number of sections that need to be migrated can be reduced considerably. This not only saves computer time, more importantly it makes computer‐intensive tasks such as migration velocity analysis based on maximizing stack power more feasible.

3D plane-wave migration in tilted coordinates: A field data example

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA199-WCA209 ◽  
Author(s):  
Guojian Shan ◽  
Robert Clapp ◽  
Biondo Biondi
Keyword(s):  
Plane Wave ◽  
Field Data ◽  
Synthetic Data ◽  
Data Set ◽  
Wave Source ◽  
Surface Data ◽  
The One ◽  
Wave Migration ◽  
Wavefield Extrapolation ◽  
Slant Stacking

We have extended isotropic plane-wave migration in tilted coordinates to 3D anisotropic media and applied it on a Gulf of Mexico data set. Recorded surface data are transformed to plane-wave data by slant-stack processing in inline and crossline directions. The source plane wave and its corresponding slant-stacked data are extrapolated into the subsurface within a tilted coordinate system whose direction depends on the propagation direction of the plane wave. Images are generated by crosscorrelating these two wavefields. The shot sampling is sparse in the crossline direction, and the source generated by slant stacking is not really a plane-wave source but a phase-encoded source. We have discovered that phase-encoded source migration in tilted coordinates can image steep reflectors, using 2D synthetic data set examples. The field data example shows that 3D plane-wave migration in tilted coordinates can image steeply dipping salt flanks and faults, even though the one-way wave-equation operator is used for wavefield extrapolation.

Seismic depth migration with the Dirac equation

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 925-933 ◽  
Author(s):  
Ketil Hokstad ◽  
Rune Mittet
Keyword(s):  
Dirac Equation ◽  
Taylor Series ◽  
Rapid Expansion ◽  
Difference Operators ◽  
Square Root ◽  
Exact Linearization ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Spatial Derivatives ◽  
Velocity Variations

We demonstrate the applicability of the Dirac equation in seismic wavefield extrapolation by presenting a new explicit one‐way prestack depth migration scheme. The method is in principle accurate up to 90° from the vertical, and it tolerates lateral velocity variations. This is achieved by performing the extrapolation step of migration with the Dirac equation, implemented in the space‐frequency domain. The Dirac equation is an exact linearization of the square‐root wave equation and is equivalent to keeping infinitely many terms in a Taylor series or continued‐fraction expansion of the square‐root operator. An important property of the new method is that the local velocity and the spatial derivatives decouple in separate terms within the extrapolation operator. Therefore, we do not need to precompute and store large tables of convolutional extrapolator coefficients depending on velocity. The main drawback of the explicit scheme is that evanescent energy must be removed at each depth step to obtain numerical stability. We have tested two numerical implementations of the migration scheme. In the first implementation, we perform depth stepping using the Taylor series approximation and compute spatial derivatives with high‐order finite difference operators. In the second implementation, we perform depth stepping with the Rapid expansion method and numerical differentiation with the pseudospectral method. The imaging condition is a generalization of Claerbout’s U / D principle. For both implementations, the impulse response is accurate up to 80° from the vertical. Using synthetic data from a simple fault model, we test the depth migration scheme in the presence of lateral velocity variations. The results show that the proposed migration scheme images dipping reflectors and the fault plane in the correct positions.

Analytic synthetic seismograms for depth migration testing

Geophysics ◽  
1991 ◽  
Vol 56 (5) ◽  
pp. 697-700
Author(s):  
Samuel H. Gray ◽  
Chester A. Jacewitz ◽  
Michael E. Epton
Keyword(s):  
Velocity Field ◽  
Acoustic Velocity ◽  
Synthetic Seismograms ◽  
Lateral Velocity ◽  
Seismic Migration ◽  
Depth Migration ◽  
Circular Arcs ◽  
Velocity Variations ◽  
Migration Testing ◽  
Speed And Accuracy

By using the fact that raypaths in a linear acoustic velocity field are circular arcs, we analytically generate a number of distinct nontrivial synthetic seismograms. The seismograms yield accurate traveltimes from reflection events, but they do not give reflection amplitudes. The seismograms are useful for testing seismic migration programs for both speed and accuracy, in settings where lateral velocity variations can be arbitrarily high and dipping reflectors arbitrarily steep. Two specific examples are presented as illustrations.

Overthrust imaging with tomo‐datuming: A case study

Geophysics ◽  
1998 ◽  
Vol 63 (1) ◽  
pp. 25-38 ◽  
Author(s):  
Xianhuai Zhu ◽  
Burke G. Angstman ◽  
David P. Sixta
Keyword(s):  
Velocity Model ◽  
Surface Velocity ◽  
Time Shift ◽  
Lateral Velocity ◽  
Prestack Depth Migration ◽  
Near Surface ◽  
Depth Migration ◽  
Static Corrections ◽  
Velocity Variations ◽  
Kirchhoff Method

Through the use of iterative turning‐ray tomography followed by wave‐equation datuming (or tomo‐datuming) and prestack depth migration, we generate accurate prestack images of seismic data in overthrust areas containing both highly variable near‐surface velocities and rough topography. In tomo‐datuming, we downward continue shot records from the topography to a horizontal datum using velocities estimated from tomography. Turning‐ray tomography often provides a more accurate near‐surface velocity model than that from refraction statics. The main advantage of tomo‐datuming over tomo‐statics (tomography plus static corrections) or refraction statics is that instead of applying a vertical time‐shift to the data, tomo‐datuming propagates the recorded wavefield to the new datum. We find that tomo‐datuming better reconstructs diffractions and reflections, subsequently providing better images after migration. In the datuming process, we use a recursive finite‐difference (FD) scheme to extrapolate wavefield without applying the imaging condition, such that lateral velocity variations can be handled properly and approximations in traveltime calculations associated with the raypath distortions near the surface for migration are avoided. We follow the downward continuation step with a conventional Kirchhoff prestack depth migration. This results in better images than those migrated from the topography using the conventional Kirchhoff method with traveltime calculation in the complicated near surface. Since FD datuming is only applied to the shallow part of the section, its cost is much less than the whole volume FD migration. This is attractive because (1) prestack depth migration usually is used iteratively to build a velocity model, so both efficiency and accuracy are important factors to be considered; and (2) tomo‐datuming can improve the signal‐to‐noise (S/N) ratio of prestack gathers, leading to more accurate migration velocity analysis and better images after depth migration. Case studies with synthetic and field data examples show that tomo‐datuming is especially helpful when strong lateral velocity variations are present below the topography.

Response by Arthur E. Barnes to the reply by the authors

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1947-1947 ◽  
Author(s):  
Arthur E. Barnes
Keyword(s):  
Lateral Velocity ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Time Migration ◽  
Pulse Distortion ◽  
Velocity Variations ◽  
Zero Offset ◽  
Complex Geology

I appreciate the thoughtful and thorough response given by Tygel et al. They point out that even for a single dipping reflector imaged by a single non‐zero offset raypath, pulse distortion caused by “standard processing” (NM0 correction‐CMP sort‐stack‐time migration) and pulse distortion caused by prestack depth migration are not really the same, because the reflecting point is mispositioned in standard processing. Within a CMP gather, this mispositioning increases with offset, giving rise to “CMP smear.” CMP smear degrades the stack, introducing additional pulse distortion. Where i‐t is significant, and where lateral velocity variations or reflection curvature are large, such as for complex geology, the pulse distortion of standard processing can differ greatly from that of prestack depth migration.

Including lateral velocity variations into true-amplitude common-shot wave-equation migration

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S175-S186 ◽  
Author(s):  
Daniela Amazonas ◽  
Rafael Aleixo ◽  
Gabriela Melo ◽  
Jörg Schleicher ◽  
Amélia Novais ◽  
...  
Keyword(s):  
Wave Equations ◽  
Heterogeneous Media ◽  
Synthetic Data ◽  
Approximate Solutions ◽  
Vertical Gradient ◽  
Correct Description ◽  
Lateral Velocity ◽  
One Dimensional ◽  
Velocity Variations ◽  
The One

In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-difference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data examples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves amplitude recovery in the migrated images in more realistic situations.

Seismic depth imaging with the Gabor transform

Geophysics ◽  
2008 ◽  
Vol 73 (3) ◽  
pp. S91-S97 ◽  
Author(s):  
Yongwang Ma ◽  
Gary F. Margrave
Keyword(s):  
Fourier Transform ◽  
Phase Shift ◽  
Gabor Transform ◽  
Positioning Error ◽  
Lateral Velocity ◽  
Data Set ◽  
Depth Migration ◽  
Velocity Variations ◽  
Spatial Positioning ◽  
Wavefield Extrapolation

Wavefield extrapolation in depth, a vital component of wave-equation depth migration, is accomplished by repeatedly applying a mathematical operator that propagates the wavefield across a single depth step, thus creating a depth marching scheme. The phase-shift method of wavefield extrapolation is fast and stable; however, it can be cumbersome to adapt to lateral velocity variations. We address the extension of phase-shift extrapolation to lateral velocity variations by using a spatial Gabor transform instead of the normal Fourier transform. The Gabor transform, also known as the windowed Fourier transform, is applied to the lateral spatial coordinates as a windowed discrete Fourier transform where the entire set of windows is required to sum to unity. Within each window, a split-step Fourier phase shift is applied. The most novel element of our algorithm is an adaptive partitioning scheme that relates window width to lateral velocity gradient such that the estimated spatial positioning error is bounded below a threshold. The spatial positioning error is estimated by comparing the Gabor method to its mathematical limit, called the locally homogeneous approximation — a frequency-wavenumber-dependent phase shift that changes according to the local velocity at each position. The assumption of local homogeneity means this position-error estimate may not hold strictly for large scattering angles in strongly heterogeneous media. The performance of our algorithm is illustrated with imaging results from prestack depth migration of the Marmousi data set. With respect to a comparable space-frequency domain imaging method, the proposed method improves images while requiring roughly 50% more computing time.

Plane-wave depth migration

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. S261-S272 ◽  
Author(s):  
Paul L. Stoffa ◽  
Mrinal K. Sen ◽  
Roustam K. Seifoullaev ◽  
Reynam C. Pestana ◽  
Jacob T. Fokkema
Keyword(s):  
Phase Space ◽  
Plane Wave ◽  
Plane Waves ◽  
Computation Time ◽  
Depth Migration ◽  
Velocity Modeling ◽  
Natural Domain ◽  
Data Volume ◽  
Wave Migration ◽  
Slant Stacking

We present fast and efficient plane-wave migration methods for densely sampled seismic data in both the source and receiver domains. The methods are based on slant stacking over both shot and receiver positions (or offsets) for all the recorded data. If the data-acquisition geometry permits, both inline and crossline source and receiver positions can be incorporated into a multidimensional phase-velocity space, which is regular even for randomly positioned input data. By noting the maximum time dips present in the shot and receiver gathers and constant-offset sections, the number of plane waves required can be estimated, and this generally results in a reduction of the data volume used for migration. The required traveltime computations for depth imaging are independent for each particular plane-wave component. It thus can be used for either the source or the receiver plane waves during extrapolation in phase space, reducing considerably the computational burden. Since only vertical delay times are required, many traveltime techniques can be employed, and the problems with multipathing and first arrivals are either reduced or eliminated. Further, the plane-wave integrals can be pruned to concentrate the image on selected targets. In this way, the computation time can be further reduced, and the technique lends itself naturally to a velocity-modeling scheme where, for example, horizontal and then steeply dipping events are gradually introduced into the velocity analysis. The migration method also lends itself to imaging in anisotropic media because phase space is the natural domain for such an analysis.

A comparison of common‐midpoint, single‐shot, and plane‐wave depth migration

Geophysics ◽  
1984 ◽  
Vol 49 (11) ◽  
pp. 1896-1907 ◽  
Author(s):  
P. Temme
Keyword(s):  
Reflection Coefficient ◽  
Finite Difference ◽  
Plane Wave ◽  
Wave Field ◽  
Single Shot ◽  
Imaging Condition ◽  
Depth Migration ◽  
Finite Aperture ◽  
Wave Migration ◽  
Slant Stacking

A comparison of common‐midpoint (CMP), single‐shot, and plane‐wave migration was made for simple two‐dimensional structures such as a syncline and a horizontal reflector with a laterally variable reflection coefficient by using synthetic seismograms. The seismograms were calculated employing the finite‐difference technique. CMP sections were simulated by 18-fold stacking and plane‐wave sections by slant stacking. By applying a finite‐difference scheme, the synthetic wave field was continued downward. The usual imaging condition of CMP migration was extended in order to carry out migration of single‐shot and plane‐wave sections. The reflection coefficient was reconstructed by comparing the migrated wave field with the incident wave field at the reflector. The results are: (1) all three migration techniques succeeded in reconstructing the reflector position; (2) as a consequence of the finite aperture of the geophone spread, only segments of the reflector could be reconstructed by single‐shot and plane‐wave migration; (3) for single‐shot and plane‐wave migration the reflection coefficient could be obtained; and (4) CMP migration may lead to incorrect conclusions regarding the reflection coefficient.

Velocity analysis by iterative profile migration

Geophysics ◽  
1989 ◽  
Vol 54 (6) ◽  
pp. 718-729 ◽  
Author(s):  
Kamal Al‐Yahya
Keyword(s):  
Initial Velocity ◽  
Velocity Model ◽  
Migration Velocity ◽  
Velocity Analysis ◽  
Lateral Velocity ◽  
Prestack Migration ◽  
Subsequent Step ◽  
One Step ◽  
Velocity Variations ◽  
And Migration

In conventional seismic processing, velocity analysis is performed by using the normal moveout (NMO) equation which is based on the assumption of flat, horizontal reflectors. Imaging by migration (either before or after stack) is done normally in a subsequent step using these velocities. In this paper, velocity analysis and imaging are combined in one step, and migration itself is used as a velocity indicator. Because, unlike NMO, migration can be formulated for any velocity function, migration‐based velocity analysis methods are capable of handling arbitrary structures, i.e., those with lateral velocity variations. In the proposed scheme, each shot gather (profile) is migrated with an initial depth‐velocity model. Profile migration is implemented in the (x, ω) domain, but the actual implementation of profile migration is not critical, as long as it is not done in a spatial‐wavenumber domain, which would preclude handling of lateral velocity variations. After migration with an initial velocity model, the velocity error is estimated, and the initial velocity model is updated; the process is repeated until convergence is achieved. The velocity analysis is based on the principle that after prestack migration with the correct velocity model, an image in a common‐receiver gather (CRG) is aligned horizontally regardless of structure. The deviation from horizontal alignment is therefore a measure of the error in velocity. If the migration velocity is lower than the velocity of the medium, events curve upward, whereas if the migration velocity is higher than the velocity of the medium, events curve downward.

Sign in / Sign up

Export Citation Format

Share Document