Reply by the authors to Arthur E. Barnes

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1944-1946
Author(s):  
M. Tygel ◽  
J. Schleicher ◽  
P. Hubral
Keyword(s):  
Destructive Interference ◽  
Depth Migration ◽  
Time Migration ◽  
Pulse Distortion ◽  
Nmo Correction ◽  
Important Processing ◽  
Offset Time ◽  
Zero Offset ◽  
Seismic Reflections ◽  
Processing Steps

We highly appreciate the useful remarks of Dr. Barnes relating our work to well‐known practical seismic processing effects. This is of particular interest as normal‐moveout (NMO) correction and post‐stack time migration are still two very important processing steps. Most exploration geophysicists know about the significance of pulse distortions known as “NM0 stretch” and “frequency shifting due to zero‐offset time migration.” As a result of the discussion of Dr. Barnes, it should now be possible to better appreciate the importance of our very general formulas (27) describing the pulse distortion of seismic reflections from an arbitrarily curved subsurface reflector when subjected to a prestack depth migration in 3‐D laterally inhomogeneous media. This discussion thus relates in particular to such important questions as how to correctly sample signals in the time or depth domain in order to avoid spatial aliasing, or how to stack seismic data without loss of information due to destructive interference of wavelets of different lengths.

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.

Analyzing operators of 3-D imaging software with impulse responses

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1079-1092 ◽  
Author(s):  
William A. Schneider
Keyword(s):  
Impulse Response ◽  
Imaging Techniques ◽  
Velocity Model ◽  
Impulse Responses ◽  
Step Size ◽  
Data Set ◽  
Time Migration ◽  
Offset Time ◽  
Zero Offset ◽  
Algorithm Implementation

No processing step changes seismic data more than 3-D imaging. Imaging techniques such as 3-D migration and dip moveout (DMO) generally change the position, amplitude, and phase of reflections as they are converted into reflector images. Migration and DMO may be formulated in many different ways, and various algorithms are available for implementing each formulation. These algorithms all make physical approximations, causing imaging software to vary with algorithm choice. Imaging software also varies because of additional implementation approximations, such as those that trade accuracy for efficiency. Imaging fidelity, then, generally depends upon algorithm, implementation, specific software parameters (such as aperture, antialias filter settings, and downward‐continuation step size), specific acquisition parameters (such as nominal x- and y-direction trace spacings and wavelet frequency range), and, of course, the velocity model. Successfully imaging the target usually requires using appropriate imaging software, parameters, and velocities. Impulse responses provide an easy way to quantitatively understand the operators of imaging software and then predict how specific imaging software will perform with the chosen parameters. (An impulse response is the image computed from a data set containing only one nonzero trace and one arrival on that trace.) I have developed equations for true‐amplitude impulse responses of 3-D prestack time migration, 3-D zero‐offset time migration, 3-D exploding‐reflector time migration, and DMO. I use these theoretical impulse responses to analyze the operators of actual imaging software for a given choice of software parameters, acquisition parameters, and velocity model. The procedure is simple: compute impulse responses of some software; estimate position, amplitude, and phase of the impulse‐response events; and plot these against the theoretical values. The method is easy to use and has proven beneficial for analyzing general imaging software and for parameter evaluation with specific imaging software.

Plumes: Response of time migration to lateral velocity variation

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1118-1127 ◽  
Author(s):  
Dimitri Bevc ◽  
James L. Black ◽  
Gopal Palacharla
Keyword(s):  
Impulse Response ◽  
Synthetic Data ◽  
Migration Velocity ◽  
Velocity Variation ◽  
Velocity Dependence ◽  
Geometrical Construction ◽  
Lateral Velocity ◽  
Time Migration ◽  
Offset Time ◽  
Zero Offset

We analyze how time migration mispositions events in the presence of lateral velocity variation by examining the impulse response of depth modeling followed by time migration. By examining this impulse response, we lay the groundwork for the development of a remedial migration operator that links time and depth migration. A simple theory by Black and Brzostowski predicted that the response of zero‐offset time migration to a point diffractor in a v(x, z) medium would be a distinctive, cusp‐shaped curve called a plume. We have constructed these plumes by migrating synthetic data using several time‐migration methods. We have also computed the shape of the plumes by two geometrical construction methods. These two geometrical methods compare well and explain the observed migration results. The plume response is strongly influenced by migration velocity. We have studied this dependency by migrating synthetic data with different velocities. The observed velocity dependence is confirmed by geometrical construction. A simple first‐order theory qualitatively explains the behavior of zero‐offset time migration, but a more complete understanding of migration velocity dependence in a v(x, z) medium requires a higher order finite‐offset theory.

Accurate depth migration by a generalized phase‐shift method

Geophysics ◽  
1987 ◽  
Vol 52 (8) ◽  
pp. 1074-1084 ◽  
Author(s):  
Dan Kosloff ◽  
David Kessler
Keyword(s):  
Phase Shift ◽  
Velocity Structure ◽  
Shift Method ◽  
Depth Migration ◽  
Integration Techniques ◽  
Matrix Diagonalization ◽  
Offset Time ◽  
Zero Offset ◽  
Spatially Variant ◽  
Phase Shift Method

A new depth migration method derived in the space‐frequency domain is based on a generalized phase‐shift method for the downward continuation of surface data. For a laterally variable velocity structure, the Fourier spatial components are no longer eigenvectors of the wave equation, and therefore a rigorous application of the phase‐shift method would seem to require finding the eigenvectors by a matrix diagonalization at every depth step. However, a recently derived expansion technique enables phase‐shift accuracy to be obtained without resorting to a costly matrix diagonalization. The new technique is applied to the migration of zero‐offset time sections. As with the laterally uniform velocity case, the evanescent components of the solution need to be isolated and eliminated, in this case by the application of a spatially variant high‐cut filter. Tests performed on the new method show that it is more accurate and efficient than standard integration techniques such as the Runge‐Kutta method or the Taylor method.

Generalized nonhyperbolic moveout approximation

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. U9-U18 ◽  
Author(s):  
Sergey Fomel ◽  
Alexey Stovas
Keyword(s):  
Functional Form ◽  
Velocity Analysis ◽  
Numerical Examples ◽  
Time Migration ◽  
Analytical Approximations ◽  
Offset Time ◽  
Zero Offset ◽  
Two Parameters

Reflection moveout approximations are commonly used for velocity analysis, stacking, and time migration. A novel functional form for approximating the moveout of reflection traveltimes at large offsets is introduced. In comparison with the classic hyperbolic approximation, which uses only two parameters (zero-offset time and moveout velocity), this form involves five parameters that can be determined, in a known medium, from zero-offset computations and from tracing one nonzero-offset ray. It is called a generalized approximation because it reduces to some known three-parameter forms with a particular choice of coefficients. By testing the accuracy of the proposed approximation with analytical and numerical examples, the new approximation is shown to bring an improvement in accuracy of several orders of magnitude compared to known analytical approximations, which makes it as good as exact for many practical purposes.

On: “Pulse distortion in depth migration,” M. Tygel, J. Schleicher, and P. Hubral (October 1994 GEOPHYSICS, 59, p. 1561–1569).

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1942-1944 ◽  
Author(s):  
Arthur E. Barnes
Keyword(s):  
Seismic Reflection ◽  
Prestack Depth Migration ◽  
Seismic Reflection Data ◽  
Depth Migration ◽  
Reflection Data ◽  
Time Migration ◽  
Pulse Distortion ◽  
Nmo Stretch ◽  
Frequency Shifting

Tygel et al. have written an excellent and rigorous discussion of pulse distortion in seismic reflection data caused by prestack depth migration. Such distortion is easily understood by recognizing that it is more or less the same effect as normal moveout (NMO) stretch combined with frequency shifting due to poststack time migration.

Finite‐element prestack reverse‐time migration for elastic waves

Geophysics ◽  
1989 ◽  
Vol 54 (9) ◽  
pp. 1204-1208 ◽  
Author(s):  
Yu‐chiung Teng ◽  
Ting‐fang Dai
Keyword(s):  
Acoustic Waves ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Initial Value ◽  
Depth Migration ◽  
Time Migration ◽  
Surface Data ◽  
Zero Offset ◽  
Dip Angle ◽  
Well Posed

Reverse‐time migration of zero‐offset data for acoustic waves has been successfully implemented by Whitmore (1983), Baysal et al. (1983), McMechan (1983), and Loewenthal and Mufti (1983). In reverse‐time migration, data recorded on the surface are used as the boundary condition and are extrapolated backward in time (Whitmore, 1983; Levin, 1984). Reverse‐time migration is mathematically a well‐posed problem. This is in contrast to conventional depth‐extrapolation‐migration schemes, in which the surface data are initial‐value conditions for solving the wave equation. Reverse‐time migration may offer improvements over conventional depth migration due to its freedom from dip‐angle limitations.

IMAGING BENEATH COMPLEX STRUCTURE: A CASE HISTORY

1981 ◽  
Vol 21 (1) ◽  
pp. 112
Author(s):  
K. Lamer ◽  
B. Gibson ◽  
R. Chambers
Keyword(s):  
Velocity Structure ◽  
Deep Structure ◽  
Complex Structure ◽  
Velocity Model ◽  
Important Conclusion ◽  
Depth Migration ◽  
Interval Velocity ◽  
Velocity Models ◽  
Time Migration ◽  
Zero Offset

Migration is recognised as the essential step in converting seismic, data into a representation of the earth's subsurface structure. Ironically, conventional migration often fails where migration is needed most—when the data are recorded over complex structures. Processing field data shot in Central America, and synthetic data derived for that section, demonstrates that time migration actually degrades the image of the deep structure that lies below a complicated overburden.In the Central American example, velocities increase nearly two-fold across an arched and thrust-faulted interface. Wavefront distortion introduced by this feature gives rise to distorted reflections from depth. Even with interval velocity known perfectly, no velocity is proper for time migrating the data here; time migration is the wrong process because it does not honour Snell's Law. Depth migration of the stacked data, on the other hand, produces a reasonable image of the deeper section. The depth migration, however, leaves artifacts that could be attributed to problems that are common in structurally complicated areas: (1) departures of the stacked section from the ideal, a zero-offset section; (2) incorrect specification of velocities; and (3) loss of energy transmitted through the complex zoneFor such an inhomogeneous velocity structure, shortcomings in CDP stacking are directly related to highly non- hyperbolic moveout. As with migration velocity, no proper stacking velocity can be developed for these data, even from the known interval-velocity model. Proper treatment of nonzero-offset reflection data could be accomplished by depth migration before stacking. Simple ray-theoretical correction of the complex moveouts, however, can produce a stack that is similar to the desired zero-offset section.Overall, the choice of velocity model most strongly influences the results of depth migration. Processing the data with a range of plausible velocity models, however, leads to an important conclusion: although the velocities can never be known exactly, depth migration is essential for clarifying structure beneath complex overburden.

An inverse ray method for computing geologic structures from seismic reflections—Zero‐offset case

Geophysics ◽  
1981 ◽  
Vol 46 (3) ◽  
pp. 268-287 ◽  
Author(s):  
B. T. May ◽  
J. D. Covey
Keyword(s):  
Inverse Modeling ◽  
Fault Model ◽  
Ray Method ◽  
Practical Procedure ◽  
Time Migration ◽  
Rapid Construction ◽  
Interactive Algorithm ◽  
Zero Offset ◽  
The Time Domain ◽  
Seismic Reflections

Seismic interpretation of structures usually involves identifying and mapping marker reflections in the time domain; however, forward modeling has shown that it can be difficult to map the complex reflection images arising from geologic structures. Inverse modeling by ray techniques offers the potential of computing a structure in the depth domain where it is comparatively easy to evaluate a structural target. An interactive algorithm is presented which has its basis in the eikonal equation and results in a practical procedure to compute models with complex geometries and inhomogeneous layers. Input consists of interpreted reflection times from a CDP‐stacked section and spatial velocity functions determined externally to the algorithm. Output is a two‐dimensional (2-D) model having curvilinear reflectors that can terminate within the model, at faults, and at unconformities. Benefits of structural inverse modeling are realized in the rapid construction of models that have velocity fields defined in the depth domain, explicitly accounting for ray curvature and ray kinking. To illustrate the inverse technique, examples of a complex synthetic thrust fault model and a field‐recorded growth fault model are included. The capability to inverse model steeply dipping structures is of particular interest because it completes a full modeling cycle of (1) theoretical prediction that steep‐dip reflections should be observable, (2) processing of field‐recorded CDP trace data to produce interpretable steep‐dip reflections, and finally (3) computation of steep‐dip reflector positions in the depth domain. An interesting benefit is the application of this algorithm to computing image rays on complex structures and the subsequent implications about time migration of CDP‐stacked sections.

A two‐pass approximation to 3-D prestack migration

Geophysics ◽  
1996 ◽  
Vol 61 (2) ◽  
pp. 409-421 ◽  
Author(s):  
Anat Canning ◽  
Gerald H. F. Gardner
Keyword(s):  
Computation Time ◽  
Three Dimensions ◽  
Velocity Analysis ◽  
Depth Migration ◽  
Prestack Migration ◽  
Time Migration ◽  
Data Volume ◽  
Land Survey ◽  
Zero Offset ◽  
The One

A two‐pass approximation to 3-D Kirchhoff migration simplifies the migration procedure by reducing it to a succession of 2-D operations. This approach has proven very successful in the zero‐offset case. A two‐pass approximation to 3-D migration is described here for the prestack case. Compared to the one‐pass approach, the scheme presented here provides significant reduction in computation time and a relatively simple data manipulation scheme. The two‐pass method was designed using velocity independent prestack time migration (DMO‐PSI) applied in the crossline direction, followed by conventional prestack depth migration in the inline direction. Velocity analysis, an important part of prestack migration, is also included in the two‐pass scheme. It is carried out as a 2-D procedure after 3-D effects are removed from the data volume. The procedure presented here is a practical full volume 3-D prestack migration. One of its main benefits is a realistic and efficient iterative velocity analysis procedure in three dimensions. The algorithm was designed in the frequency domain and the computational scheme was optimized by processing individual frequency slices independently. Irregular trace distribution, a feature that characterizes most 3-D seismic surveys, is implicitly accounted for within the two‐pass algorithm. A numerical example tests the performance of the two‐pass 3-D prestack migration program in the presence of a vertical velocity gradient. A 3-D land survey from a fold and thrust belt region was used to demonstrate the algorithm in a complex geological setting. The results were compared with images from other 2-D and 3-D migration schemes and show improved resolution and higher signal content.

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