Analytic synthetic seismograms for depth migration testing

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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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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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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Seismic depth imaging with the Gabor transform

Geophysics ◽

10.1190/1.2903821 ◽

2008 ◽

Vol 73 (3) ◽

pp. S91-S97 ◽

Cited By ~ 6

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.

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Tomographic velocity analysis and wave-equation depth migration in an overthrust terrain: A case study from the Tuha Basin, China

Interpretation ◽

10.1190/int-2017-0053.1 ◽

2018 ◽

Vol 6 (1) ◽

pp. T1-T13

Author(s):

Bin Lyu ◽

Qin Su ◽

Kurt J. Marfurt

Keyword(s):

Wave Equation ◽

Data Quality ◽

Model Building ◽

Velocity Model ◽

Lateral Velocity ◽

Near Surface ◽

Depth Migration ◽

Time Migration ◽

Head Waves ◽

Velocity Variations

Although the structures associated with overthrust terrains form important targets in many basins, accurately imaging remains challenging. Steep dips and strong lateral velocity variations associated with these complex structures require prestack depth migration instead of simpler time migration. The associated rough topography, coupled with older, more indurated, and thus high-velocity rocks near or outcropping at the surface often lead to seismic data that suffer from severe statics problems, strong head waves, and backscattered energy from the shallow section, giving rise to a low signal-to-noise ratio that increases the difficulties in building an accurate velocity model for subsequent depth migration. We applied a multidomain cascaded noise attenuation workflow to suppress much of the linear noise. Strong lateral velocity variations occur not only at depth but near the surface as well, distorting the reflections and degrading all deeper images. Conventional elevation corrections followed by refraction statics methods fail in these areas due to poor data quality and the absence of a continuous refracting surface. Although a seismically derived tomographic solution provides an improved image, constraining the solution to the near-surface depth-domain interval velocities measured along the surface outcrop data provides further improvement. Although a one-way wave-equation migration algorithm accounts for the strong lateral velocity variations and complicated structures at depth, modifying the algorithm to account for lateral variation in illumination caused by the irregular topography significantly improves the image, preserving the subsurface amplitude variations. We believe that our step-by-step workflow of addressing the data quality, velocity model building, and seismic imaging developed for the Tuha Basin of China can be applied to other overthrust plays in other parts of the world.

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A strategy for computing seismic attributes on depth-migrated data using time-shift gathers

Geophysics ◽

10.1190/geo2016-0335.1 ◽

2017 ◽

Vol 82 (3) ◽

pp. O37-O46

Author(s):

Mark Roberts

Keyword(s):

Interpolation Method ◽

Seismic Attributes ◽

Depth Image ◽

Time Shift ◽

Lateral Velocity ◽

Prestack Depth Migration ◽

Depth Migration ◽

Velocity Variations ◽

Made In

Assumptions made during postprocessing can be as important as those made during migration. Prestack depth migration (PSDM) is often used due to its ability to handle lateral velocity variations and dipping events. However, most postprocessing flows still use a simple 1D “depth-to-time” vertical stretch, violating the very assumptions that led us to use PSDM. Postprocessing workflows based on time-shift depth image gathers allow for postprocessing flows that make no further approximations other than those made in migration. For cases in which time-shift gathers are not computed during migration, they can be approximated by use of the “exploding-reflector” model or through an orthogonal shift and interpolation method.

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Depth Imaging Sub-salt Structures: A Case Study in the Midyan Peninsula (Red Sea)

GeoArabia ◽

10.2113/geoarabia0404445 ◽

1999 ◽

Vol 4 (4) ◽

pp. 445-464 ◽

Cited By ~ 2

Author(s):

Denis Mougenot ◽

Amir A. Al-Shakhis

Keyword(s):

Red Sea ◽

Structural Model ◽

Oil And Gas ◽

Imaging Techniques ◽

Depth Image ◽

Lateral Velocity ◽

Seismic Line ◽

Depth Imaging ◽

Depth Migration ◽

Velocity Variations

ABSTRACT In the Midyan Peninsula (onshore northern Red Sea, Saudi Arabia), the current prospective oil and gas exploration targets are sub-salt structures. In this region, conventional time-migrated seismic sections are distorted due to the presence of salt diapirs, faults, and related lateral velocity variations. As demonstrated in other sub-salt prospects (North Sea, Gulf of Suez, and Gulf of Mexico), pre-stack depth migration can remove these distortions and accurately focus the structural image. Depth migration, however, requires a model which includes both lateral and vertical velocity variations to compensate for ray bending. Building such a velocity model is an iterative process which involves integration of various time/depth processing and interpretation skills. A 2-D seismic line, crossing various extensional structures in the dip direction, is used to illustrate these depth-imaging techniques. At the location of the sub-salt prospect, the depth image is improved and the lateral position of the main fault is shifted by 345 meters. The resulting structural model has refined the target definition and well position. This imaging approach is compared with the different steps of the seismic processing/interpretation flow.

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