Seismic depth imaging with the Gabor transform

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.

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The Hartley transform in seismic imaging

Geophysics ◽

10.1190/1.1487072 ◽

2001 ◽

Vol 66 (4) ◽

pp. 1251-1257 ◽

Cited By ~ 9

Author(s):

Henning Kühl ◽

Maurico D. Sacchi ◽

Jürgen Fertig

Keyword(s):

Fourier Transform ◽

Phase Shift ◽

Shift Operator ◽

Three Dimensions ◽

Lateral Velocity ◽

Data Set ◽

Prestack Migration ◽

Hartley Transform ◽

Alternative Means ◽

The Fourier Transform

Phase‐shift migration techniques that attempt to account for lateral velocity variations make substantial use of the fast Fourier transform (FFT). Generally, the Hermitian symmetry of the complex‐valued Fourier transform causes computational redundancies in terms of the number of operations and memory requirements. In practice a combination of the FFT with the well‐known real‐to‐complex Fourier transform is often used to avoid such complications. As an alternative means to the Fourier transform, we introduce the inherently real‐valued, non‐symmetric Hartley transform into phase‐shift migration techniques. By this we automatically avoid the Hermitian symmetry resulting in an optimized algorithm that is comparable in efficiency to algorithms based on the real‐to‐complex FFT. We derive the phase‐shift operator in the Hartley domain for migration in two and three dimensions and formulate phase shift plus interpolation, split‐step migration, and split‐step double‐square‐root prestack migration in terms of the Hartley transform as examples. We test the Hartley phase‐shift operator for poststack and prestack migration using the SEG/EAGE salt model and the Marmousi data set, respectively.

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Prestack migration by split‐step DSR

Geophysics ◽

10.1190/1.1444065 ◽

1996 ◽

Vol 61 (5) ◽

pp. 1412-1416 ◽

Cited By ~ 60

Author(s):

Alexander Mihai Popovici

Keyword(s):

Phase Shift ◽

Square Root ◽

Variable Velocity ◽

Step Method ◽

Lateral Velocity ◽

Data Set ◽

Prestack Migration ◽

Imaging Results ◽

Velocity Variations ◽

Migration Equation

The double‐square‐root (DSR) prestack migration equation, though defined for depth variable velocity, can be used to image media with strong velocity variations using a phase‐shift plus interpolation (PSPI) or split‐step correction. The split‐step method is based on applying a phase‐shift correction to the extrapolated wavefield—a correction that attempts to compensate for the lateral velocity variations. I show how to extend DSR prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the Marmousi data set. The split‐step DSR migration is very fast and offers excellent imaging results.

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True-amplitude one-way wave equation migration in the mixed domain

Geophysics ◽

10.1190/1.3478574 ◽

2010 ◽

Vol 75 (5) ◽

pp. S199-S209 ◽

Cited By ~ 6

Author(s):

Flor A. Vivas ◽

Reynam C. Pestana

Keyword(s):

Wave Equation ◽

Phase Shift ◽

Numerical Experiments ◽

Wave Equations ◽

Laplacian Operator ◽

Single Shot ◽

Lateral Velocity ◽

Data Set ◽

Classical Formulation ◽

Imaging Tool

One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium,the extrapolation of the wavefields with the true-amplitude one-way wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain true-amplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.

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Offset‐domain pseudoscreen prestack depth migration

Geophysics ◽

10.1190/1.1527089 ◽

2002 ◽

Vol 67 (6) ◽

pp. 1895-1902 ◽

Cited By ~ 26

Author(s):

Shengwen Jin ◽

Charles C. Mosher ◽

Ru‐Shan Wu

Keyword(s):

Heterogeneous Media ◽

Fourier Transforms ◽

Data Migration ◽

Computationally Efficient ◽

Lateral Velocity ◽

Prestack Depth Migration ◽

Depth Migration ◽

Narrow Angle ◽

Domain Phase ◽

Wavefield Extrapolation

The double square root equation for laterally varying media in midpoint‐offset coordinates provides a convenient framework for developing efficient 3‐D prestack wave‐equation depth migrations with screen propagators. Offset‐domain pseudoscreen prestack depth migration downward continues the source and receiver wavefields simultaneously in midpoint‐offset coordinates. Wavefield extrapolation is performed with a wavenumber‐domain phase shift in a constant background medium followed by a phase correction in the space domain that accommodates smooth lateral velocity variations. An extra wide‐angle compensation term is also applied to enhance steep dips in the presence of strong velocity contrasts. The algorithm is implemented using fast Fourier transforms and tri‐diagonal matrix solvers, resulting in a computationally efficient implementation. Combined with the common‐azimuth approximation, 3‐D pseudoscreen migration provides a fast wavefield extrapolation for 3‐D marine streamer data. Migration of the 2‐D Marmousi model shows that offset domain pseudoscreen migration provides a significant improvement over first‐arrival Kirchhoff migration for steeply dipping events in strong contrast heterogeneous media. For the 3‐D SEG‐EAGE C3 Narrow Angle synthetic dataset, image quality from offset‐domain pseudoscreen migration is comparable to shot‐record finite‐difference migration results, but with computation times more than 100 times faster for full aperture imaging of the same data volume.

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Seismic depth migration with the Dirac equation

Geophysics ◽

10.1190/1.1444600 ◽

1999 ◽

Vol 64 (3) ◽

pp. 925-933 ◽

Cited By ~ 1

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.

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Analytic synthetic seismograms for depth migration testing

Geophysics ◽

10.1190/1.1443087 ◽

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.

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Overthrust imaging with tomo‐datuming: A case study

Geophysics ◽

10.1190/1.1444319 ◽

1998 ◽

Vol 63 (1) ◽

pp. 25-38 ◽

Cited By ~ 27

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.

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Improving plane‐wave decomposition and migration

Geophysics ◽

10.1190/1.1444119 ◽

1997 ◽

Vol 62 (1) ◽

pp. 195-205 ◽

Cited By ~ 3

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.

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

Geophysics ◽

10.1190/1.1487022 ◽

1995 ◽

Vol 60 (6) ◽

pp. 1947-1947 ◽

Cited By ~ 1

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.

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Stable wide‐angle Fourier finite‐difference downward extrapolation of 3‐D wavefields

Geophysics ◽

10.1190/1.1484530 ◽

2002 ◽

Vol 67 (3) ◽

pp. 872-882 ◽

Cited By ~ 53

Author(s):

Biondo Biondi

Keyword(s):

Finite Difference ◽

Continuation Method ◽

Phase Error ◽

Azimuthal Anisotropy ◽

Laplacian Operator ◽

Lateral Velocity ◽

Data Set ◽

Frequency Dependent ◽

Reference Velocity ◽

Velocity Variations

I present an unconditionally stable, implicit finite‐difference operator that corrects the constant‐velocity phase‐shift operator for lateral velocity variations. The method is based on the Fourier finite‐difference (FFD) method. Contrary to previous results, my correction operator is stable even when the medium velocity has sharp discontinuities, and the reference velocity is higher than the medium velocity. The stability of the new correction enables the definition of a new downward‐continuation method based on the interpolation of two wavefields: the first wavefield is obtained by applying the FFD correction starting from a reference velocity lower than the medium velocity; the second wavefield is obtained by applying the FFD correction starting from a reference velocity higher than the medium velocity. The proposed Fourier finite‐difference plus interpolation (FFDPI) method combines the advantages of the FFD technique with the advantages of interpolation. A simple and economical procedure for defining frequency‐dependent interpolation weight is presented. When the interpolation step is performed using these frequency‐dependent interpolation weights, it significantly reduces the residual phase error after interpolation, the frequency dispersion caused by the discretization of the Laplacian operator, and the azimuthal anisotropy caused by splitting. Tests on zero‐offset data from the SEG‐EAGE salt data set show that the FFDPI method improves the imaging of a fault reflection with respect to a similar interpolation scheme that uses a split‐step correction for adapting to lateral velocity variations.

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