Prestack phase-shift migration of separate offsets in laterally inhomogeneous media

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. S95-S101 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Synthetic Data ◽  
Real Data ◽  
Inhomogeneous Media ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Fast Wave ◽  
Lateral Inhomogeneity ◽  
Zero Offset

Using the stationary-phase method, prestack phase-shift migration is implemented one offset at a time. This separate-offset implementation allows for a Fourier (reasonably fast) wave-equation-type migration on data with irregular offset sampling. However, the separate-offset phase-shift migration, like its zero-offset counterpart, handles only vertically inhomogeneous media. Using the combination of the split-step and phase-shift-plus-interpolation (PSPI) approaches, the separate-offset phase-shift migration is extended to handle laterally inhomogeneous media. The cost of the separate-offset implementation is practically equivalent to that of the conventional zero-offset version. However, due to the lack of exact source and receiver ray-trajectory information in the separate-offset implementation, the combined split-step and PSPI handles only smooth lateral inhomogeneity. Specifically, it produces images equivalent to those resulting from smoothing the velocity model laterally over a window equal to the half offset. Thus, for zero-offset or laterally homogeneous media, the separate-offset migration is equivalent to any wave-equation-based migration. Errors might occur for finite-offset data in laterally inhomogeneous media. Such errors depend primarily on the strength of lateral inhomogeneity. Using this separate-offset phase-shift migration, accurate images of synthetic data of a model with large reflector dips and good images from real data from offshore Trinidad are obtained.

Large‐scale three‐dimensional seismic models and their interpretive significance

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1166-1182 ◽  
Author(s):  
Irshad R. Mufti
Keyword(s):  
Finite Difference ◽  
Wave Field ◽  
Large Scale ◽  
Three Dimensional ◽  
Synthetic Data ◽  
Real Data ◽  
The Real ◽  
Need To Evaluate ◽  
Zero Offset ◽  
Seismic Models

Finite‐difference seismic models are commonly set up in 2-D space. Such models must be excited by a line source which leads to different amplitudes than those in the real data commonly generated from a point source. Moreover, there is no provision for any out‐of‐plane events. These problems can be eliminated by using 3-D finite‐difference models. The fundamental strategy in designing efficient 3-D models is to minimize computational work without sacrificing accuracy. This was accomplished by using a (4,2) differencing operator which ensures the accuracy of much larger operators but requires many fewer numerical operations as well as significantly reduced manipulation of data in the computer memory. Such a choice also simplifies the problem of evaluating the wave field near the subsurface boundaries of the model where large operators cannot be used. We also exploited the fact that, unlike the real data, the synthetic data are free from ambient noise; consequently, one can retain sufficient resolution in the results by optimizing the frequency content of the source signal. Further computational efficiency was achieved by using the concept of the exploding reflector which yields zero‐offset seismic sections without the need to evaluate the wave field for individual shot locations. These considerations opened up the possibility of carrying out a complete synthetic 3-D survey on a supercomputer to investigate the seismic response of a large‐scale structure located in Oklahoma. The analysis of results done on a geophysical workstation provides new insight regarding the role of interference and diffraction in the interpretation of seismic data.

Vertical seismic profile migration using the Kirchhoff integral

Geophysics ◽  
1988 ◽  
Vol 53 (6) ◽  
pp. 786-799 ◽  
Author(s):  
P. B. Dillon
Keyword(s):  
Wave Equation ◽  
Wave Field ◽  
Near Field ◽  
Synthetic Data ◽  
Real Data ◽  
Seismic Profile ◽  
Processing Strategies ◽  
Receiver Array ◽  
Vsp Data ◽  
Kirchhoff Integral

Wave‐equation migration can form an accurate image of the subsurface from suitable VSP data. The image’s extent and resolution are determined by the receiver array dimensions and the source location(s). Experiments with synthetic and real data show that the region of reliable image extent is defined by the specular “zone of illumination.” Migration is achieved through wave‐field extrapolation, subject to an imaging procedure. Wave‐field extrapolation is based upon the scalar wave equation and, for VSP data, is conveniently handled by the Kirchhoff integral. The migration of VSP data calls for imaging very close to the borehole, as well as imaging in the far field. This dual requirement is met by retaining the near‐field term of the integral. The complete integral solution is readily controlled by various weighting devices and processing strategies, whose worth is demonstrated on real and synthetic data.

Wave equation migration with the phase‐shift method

Geophysics ◽  
1978 ◽  
Vol 43 (7) ◽  
pp. 1342-1351 ◽  
Author(s):  
Jenö Gazdag
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Finite Difference Methods ◽  
Migration Process ◽  
Asymptotic Equation ◽  
Shift Method ◽  
Seismic Records ◽  
Zero Offset ◽  
Degree Equation ◽  
Phase Shift Method

Accurate methods for the solution of the migration of zero‐offset seismic records have been developed. The numerical operations are defined in the frequency domain. The source and recorder positions are lowered by means of a phase shift, or a rotation of the phase angle of the Fourier coefficients. For applications with laterally invariant velocities, the equations governing the migration process are solved very accurately by the phase‐shift method. The partial differential equations considered include the 15 degree equation, as well as higher order approximations to the exact migration process. The most accurate migration is accomplished by using the asymptotic equation, whose dispersion relation is the same as that of the full wave equation for downward propagating waves. These equations, however, do not account for the reflection and transmission effects, multiples, or evanescent waves. For comparable accuracy, the present approach to migration is expected to be computationally more efficient than finite‐difference methods in general.

Efficient simultaneous wave-equation statics and trace regularization by series approximation

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. U19-U28 ◽  
Author(s):  
Robert J. Ferguson
Keyword(s):  
Wave Equation ◽  
Gradient Methods ◽  
Synthetic Data ◽  
Real Data ◽  
Three Dimensions ◽  
Velocity Variation ◽  
Acceptable Solution ◽  
Direct Inversion ◽  
Data Inversion ◽  
Series Approximation

To address the problems of irregular trace spacing and statics correction, simultaneous regularization and wave-equation statics (WE statics) are implemented by least-squares inversion. In general, inversion is found to be intractable in three dimensions, so series approximation is made to reduce significantly the number of required integrals. The resulting operator is suitable for direct inversion or for use with gradient methods. Real and synthetic data are used to determine the viability of the inversion. For synthetic data, even for severe velocity variation and topography, inversion converges to an acceptable solution, and aliasing is reduced significantly. Similarly, for real data, inversion is found to return an antialiased, regularized result with WE statics applied.

Near-surface imaging using ambient-noise body waves

2016 ◽  
Vol 4 (3) ◽  
pp. SJ55-SJ65 ◽  
Author(s):  
Pascal Edme ◽  
David F. Halliday
Keyword(s):  
Ambient Noise ◽  
Body Wave ◽  
Synthetic Data ◽  
Real Data ◽  
Body Waves ◽  
Seismic Survey ◽  
Subsurface Imaging ◽  
Near Surface ◽  
Deconvolution Technique ◽  
Zero Offset

We have introduced a workflow that allows subsurface imaging using upcoming body-wave arrivals extracted from ambient-noise land seismic data. Rather than using the conventional seismic interferometry approach based on correlation, we have developed a deconvolution technique to extract the earth response from the observed periodicity in the seismic traces. The technique consists of iteratively applying a gapped spiking deconvolution, providing multiple-free images with higher resolution than conventional correlation. We have validated the workflow for zero-offset traces with simple synthetic data and real data recorded during a small point-receiver land seismic survey.

scholarly journals Automatic wave-equation migration velocity analysis by focusing subsurface virtual sources

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. U1-U8 ◽  
Author(s):  
Bingbing Sun ◽  
Tariq Alkhalifah
Keyword(s):  
Wave Equation ◽  
Model Building ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Real Data ◽  
Migration Velocity ◽  
Velocity Analysis ◽  
Data Set ◽  
Migration Velocity Analysis ◽  
Band Limited

Macro-velocity model building is important for subsequent prestack depth migration and full-waveform inversion. Wave-equation migration velocity analysis uses the band-limited waveform to invert for velocity. Normally, inversion would be implemented by focusing the subsurface offset common-image gathers. We reexamine this concept with a different perspective: In the subsurface offset domain, using extended Born modeling, the recorded data can be considered as invariant with respect to the perturbation of the position of the virtual sources and velocity at the same time. A linear system connecting the perturbation of the position of those virtual sources and velocity is derived and solved subsequently by the conjugate gradient method. In theory, the perturbation of the position of the virtual sources is given by the Rytov approximation. Thus, compared with the Born approximation, it relaxes the dependency on amplitude and makes the proposed method more applicable for real data. We determined the effectiveness of the approach by applying the proposed method on isotropic and anisotropic vertical transverse isotropic synthetic data. A real data set example verifies the robustness of the proposed method.

Born theory of wave‐equation dip moveout

Geophysics ◽  
1991 ◽  
Vol 56 (2) ◽  
pp. 182-189 ◽  
Author(s):  
Christopher L. Liner
Keyword(s):  
Wave Equation ◽  
Synthetic Data ◽  
Operator Method ◽  
Scattering Data ◽  
Prestack Migration ◽  
Theoretical Approaches ◽  
Geometric Spreading ◽  
Total Process ◽  
Zero Offset ◽  
Amplitude Preservation

Wave‐equation dip moveout (DMO) addresses the DMO amplitude problem of finding an algorithm which faithfully preserves angular reflectivity while processing data to zero offset. Only three fundamentally different theoretical approaches to the DMO amplitude problem have been proposed: (1) mathematical decomposition of a prestack migration operator; (2) intuitively accounting for specific amplitude factors; and (3) cascading operators for prestack migration (or inversion) and zero‐offset forward modeling. Pursuing the cascaded operator method, wave‐equation DMO for shot profiles has been developed. In this approach, a prestack common‐shot inversion operator is combined with a zero‐offset modeling operator. Both integral operators are theoretically based on the Born asymptotic solution to the point‐source, scalar wave equation. This total process, termed Born DMO, simultaneously accomplishes geometric spreading corrections, NMO, and DMO in an amplitude‐preserving manner. The theory is for constant velocity and density, but variable velocity can be approximately incorporated. Common‐shot Born DMO can be analytically verified by using Kirchhoff scattering data for a horizontal plane. In this analytic test, Born DMO yields the correct zero‐offset reflector with amplitude proportional to the angular reflection coefficient. Numerical tests of common‐shot Born DMO on synthetic data suggest that angular reflectivity is successfully preserved. In those situations where amplitude preservation is important, Born DMO is an alternative to conventional NMO + DMO processing.

Datum correction by wave‐equation extrapolation

Geophysics ◽  
1988 ◽  
Vol 53 (10) ◽  
pp. 1311-1322 ◽  
Author(s):  
V. Shtivelman ◽  
A. Canning
Keyword(s):  
Wave Equation ◽  
Synthetic Data ◽  
Real Data ◽  
Static Solution ◽  
Velocity Model ◽  
Lateral Velocity ◽  
Static Correction ◽  
Slow Velocity ◽  
Velocity Variations ◽  
Seismic Sections

Seismic sections are usually datum corrected by static shifting. For small differences in elevation and slow velocity variations between the input datum and the output datum, static shifting is a sufficiently accurate datum correction procedure. However, for significant differences in elevations and a more complicated velocity model, the accuracy of the static solution may prove to be insufficient; and a more exact method should be used. In this paper, we study the limitations of the static method of datum correction and develop simple and effective extrapolation schemes based on the wave equation, schemes which lead to more accurate datum correction. The distortions of seismic events caused by static correction are illustrated by a number of simple examples. To reduce the distortions, we propose a number of extrapolation schemes based on the asymptories of the Kirchhoff integral solution of the 2-D scalar wave equation. Application of the extrapolation algorithms to synthetic data shows that they provide accurate datum corrections even for a nonplanar input datum and vertical and lateral velocity variations. The algorithms have been successfully applied to real data.

Prestack phase‐shift migration of separate offsets

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1179-1194 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Phase Shift ◽  
Analytical Solutions ◽  
Stationary Point ◽  
Anisotropic Media ◽  
Phase Method ◽  
Velocity Models ◽  
Analytical Approximations ◽  
Isotropic Media ◽  
Point Solution ◽  
Zero Offset

Prestack phase‐shift migration is implemented by evaluating the offset‐wavenumber ([Formula: see text]) integral using the stationary‐phase method. Thus, the stationary point along [Formula: see text] must be calculated prior to applying the phase shift. This type of implementation allows for migration of separate offsets, as opposed to migration of the whole prestack data when using the original formulas. For zero‐offset data, the stationary point ([Formula: see text]) is known in advance, and, therefore, the phase‐shift migration can be implemented directly. For nonzero‐offset data, we first evaluate the [Formula: see text] that corresponds to the stationary point solution either numerically or through analytical approximations. The insensitivity of the phase to [Formula: see text] around the stationary point solution (its maximum) implies that even an imperfect [Formula: see text] obtained analytically can go a long way to getting an accurate image. In transversely isotropic media, the analytical solutions of the stationary point ([Formula: see text]) include more approximations than those corresponding to isotropic media (i.e., approximations corresponding to weaker anisotropy). Nevertheless, the resultant equations, obtained using Shanks transform and perturbation theory, produce accurate migration signatures for strong anisotropy (η ≈ 0.3) and even large offset‐to‐depth ratios (>2). The analytical solutions are particularly accurate in predicting the nonhyperbolic moveout behavior associated with anisotropic media, a key ingredient to performing an accurate nonhyperbolic moveout inversion for strongly anisotropic media. Although the prestack correction achieved using the phase‐shift method can also be obtained using a cascade of NMO correction, dip‐moveout (DMO) correction, and zero‐offset time migration, the prestack approach can handle sharper velocity models more efficiently. In addition, the resulting operator is sharper than that obtained from the DMO method. Synthetic, including the Marmousi, data applications of the proposed prestack migration demonstrate its usefulness.

Nonhyperbolic stretch-free normal moveout correction

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. U87-U95 ◽  
Author(s):  
Mohammad Mahdi Abedi ◽  
Mohammad Ali Riahi
Keyword(s):  
Synthetic Data ◽  
Real Data ◽  
Oil Field ◽  
New Method ◽  
Mapping Data ◽  
Data Set ◽  
Nmo Correction ◽  
Natural Result ◽  
Zero Offset ◽  
Normal Moveout Correction

Normal moveout (NMO) correction is routinely applied to traces of each common-midpoint (CMP) gather before forming a stack section. Conventional NMO correction has the drawback of producing stretching as a natural result of convergence of the NMO trajectories. Although this problem exists on completely hyperbolic reflections, the reflections will be further deviated from the desirable zero-offset equivalent if they indicate nonhyperbolic behavior. We have addressed this issue and developed a new method of stretch-free NMO correction in two steps: first, a novel way of rectifying NMO correction trajectories in a shifted hyperbolic NMO base, and second, a prioritized successive process of mapping data samples into an NMO-corrected gather. We have determined the advantage of the proposed method over two preceding methods: isomoveout and local stretch zeroing. The effectiveness of the new method in producing a stretch-free NMO gather was tested on synthetic data generated by ray tracing and a real data set of 200 CMP gathers of an Iranian oil field. The proposed method can be used in the presence of hyperbolic and nonhyperbolic events, and it recovers the amplitudes of interfering reflections to extend the usable offsets.

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