Post-stack depth migration in the frequency–space domain

1986 ◽  
Vol 23 (6) ◽  
pp. 839-848 ◽  
Author(s):  
Panos G. Kelamis ◽  
Einar Kjartansson ◽  
E George Marlin
Keyword(s):  
Synthetic Data ◽  
Real Data ◽  
Wave Number ◽  
Lateral Velocity ◽  
Space Domain ◽  
Frequency Wave ◽  
Depth Migration ◽  
Frequency Space ◽  
Time Space ◽  
Z Domain

The 45 °monochromatic one-way wave equation, along with the thin-lens term, is used, and a depth-migration algorithm is developed in the frequency–space (ω, x) domain. Using this approach, an unmigrated stack section is directly transformed into a depth-migrated section taking into account both vertical and lateral velocity variations. In practice, the algorithm can accommodate steep events with dips of the order of 60–65°. The use of the frequency–space domain offers several advantages over the conventional time–space and frequency–wave-number domains. Time derivatives are evaluated exactly by a simple multiplication, while the use of the space (x, z) domain facilitates the handling of lateral velocity inhomogeneities. The performance of the depth-migration algorithm is tested with synthetic data from complicated models and real data from the Foothills area of western Canada.

A method for computation of velocity profiles by inversion of large‐offset records

Geophysics ◽  
1984 ◽  
Vol 49 (8) ◽  
pp. 1249-1258 ◽  
Author(s):  
Philip M. Carrion ◽  
John T. Kuo
Keyword(s):  
Wave Field ◽  
Critical Path ◽  
Real Data ◽  
Velocity Profiles ◽  
Spatial Transformation ◽  
Lateral Velocity ◽  
Minimization Procedure ◽  
Z Domain ◽  
Minimization Technique ◽  
Lateral Extent

This paper describes a new method for recovering velocity profiles utilizing both phase and amplitude information including wide‐angle arrivals, post‐ and precritical reflections. This method is based on a double spatial transformation with a minimization procedure. The first transformation is slant stacking of the observed wave field (seismogram). The second is projecting the slant stacked wave field into the domain of horizontal slowness p and depth z. In this domain the inverse problem is reduced to finding the critical path [Formula: see text] where V(z) is the true velocity of the compressional waves. A numerical algorithm based on a minimization technique is used to find the critical path, which is equivalent to the set of turning points of the critically reflected rays. When this path is found, then the following criteria are satisfied: (1) most of the energy is concentrated away from the precritical region; (2) the computed reflection coefficients reach their maximum on this path; and (3) for horizontally stratified media or CMP data, the reflectors are aligned in the p-z domain. In tests, this method has been shown to recover the velocity profile from both synthetic and real data. It is shown that the method is able to recover accurately velocity profiles even if only part of the data are given. For example, only part of the data are available when low‐ and high‐frequency components are missing or when the data are truncated in lateral extent due to the finite length of the recording system. Moreover, the method is able to handle virtually any vertical velocity gradients in a medium; therefore, it can be applied to complicated geologic structures. The method does not require elimination of multiples, but it is not applicable to the case of a medium with a large lateral velocity gradient. It can be used even for an elastic medium when the mode‐converted energy is not small.

Parallel Butterworth and Chebyshev dip filters with applications to 3-D seismic migration

Geophysics ◽  
1999 ◽  
Vol 64 (5) ◽  
pp. 1573-1578 ◽  
Author(s):  
Hongbo Zhou ◽  
George A. McMechan
Keyword(s):  
Transfer Function ◽  
Phase Compensation ◽  
Space Domain ◽  
Seismic Migration ◽  
Butterworth Filter ◽  
Depth Migration ◽  
Frequency Space ◽  
Chebyshev Filters ◽  
The Cost ◽  
Chebyshev Filter

Dip filtering is a necessary part of accurate frequency‐space domain migration, so design and application of reliable and efficient filters are of practical as well as theoretical importance. Frequency‐space domain dip filters are implemented using Butterworth and Chebyshev algorithms. By transforming the product terms of the filter transfer function into summations, a cascaded (serial) Butterworth or Chebyshev dip filter can be made parallel, which improves computational efficiency. For a given order of filter, the cost of the Butterworth and Chebyshev filters is the same. However, the Chebyshev filter has a sharper transition zone than that of a Butterworth filter with the same order, which makes it more effective for phase compensation than a Butterworth filter, but at the expense of some wavenumber‐dependent amplitude ripples. Both implementations have been incorporated into 3-D one‐way frequency‐space depth migration for evanescent energy removal and for phase compensation of splitting errors; a single filter achieves both goals.

An analytical approach to migration velocity analysis

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1238-1249 ◽  
Author(s):  
Zhenyue Liu
Keyword(s):  
Synthetic Data ◽  
Migration Velocity ◽  
Velocity Estimation ◽  
Velocity Analysis ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Curvature Analysis ◽  
Derivative Function ◽  
Migration Velocity Analysis ◽  
Residual Curvature

Prestack depth migration provides a powerful tool for velocity analysis in complex media. Both prominent approaches to velocity analysis—depth‐focusing analysis and residual‐curvature analysis, rely on approximate formulas to update velocity. Generally, these formulas are derived under the assumptions of horizontal reflector, lateral velocity homogeneity, or small offset. Therefore, the conventional methods for updating velocity lack accuracy and computational efficiency when velocity has large, lateral variations. Here, based on ray theory, I find the analytic representation for the derivative of imaged depths with respect to migration velocity. This derivative function characterizes a general relationship between residual moveout and residual velocity. Using the derivative function and the perturbation method, I derive a new formula to update velocity from residual moveout. In the derivation, I impose no limitation on offset, dip, or velocity distribution. Consequently, I revise the residual‐curvature‐analysis method for velocity estimation in the postmigrated domain. Furthermore, my formula provides sensitivity and error estimation for migration‐based velocity analysis, which is helpful in quantifying the reliability of the estimated velocity. The theory and methodology in this paper have been tested on synthetic data (including the Marmousi data).

scholarly journals 3-D Butterworth Filtering for 3-D High-density Onshore Seismic Field Data

2018 ◽  
Vol 23 (2) ◽  
pp. 223-233
Author(s):  
Jianping Liao ◽  
Hexiu Liu ◽  
Weibo Li ◽  
Zhenwei Guo ◽  
Lixin Wang ◽  
...  
Keyword(s):  
Field Data ◽  
Synthetic Data ◽  
High Density ◽  
Western China ◽  
Seismic Survey ◽  
Apparent Velocity ◽  
Coherent Noise ◽  
Space Domain ◽  
Filtering Method ◽  
Time Space

Three-dimensional seismic survey is widely applied, but 3-D filtering technology has yet to be fully utilized for the analysis of seismic field data. The common approach is to first filter inline and then crossline. However, an effective 3-D filtering method is expected to eliminate coherent noise, such as the ground roll. We propose a 3-D Butterworth filtering method in the time-space domain. Firstly, a Butterworth-type filter in the frequency-wavenumber-domain is designed to suppress the linear noise with a specific apparent velocity. Secondly, transforming this filter to the time and space domain yields 3-D partial differential equations (PDEs), which are applied to suppress the linear noise. Factorizing the finite-difference equations in a different direction other than decreasing the 3-D PDEs to 2-D PDEs produces a highly accurate and efficient algorithm. Designing the 3-D Butterworth filter, selecting the filtering parameters, and showing its application to synthetic data and a 3-D high-density onshore seismic field data from a region in western China are discussed in detail. Numerical experiments with 3-D high-density onshore seismic field data demonstrate that it is more effective than the 3-D frequency-wavenumber-wavenumber (FKK) filtering method.

Adaptive prediction filtering in t-x-y domain for random noise attenuation using regularized nonstationary autoregression

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. V13-V21 ◽  
Author(s):  
Yang Liu ◽  
Ning Liu ◽  
Cai Liu
Keyword(s):  
Dimensional Space ◽  
Random Noise ◽  
Three Dimensional ◽  
Synthetic Data ◽  
Noise Attenuation ◽  
Random Noise Attenuation ◽  
Frequency Space ◽  
Adaptive Prediction ◽  
Time Space ◽  
Prediction Filter

Many natural phenomena, including geologic events and geophysical data, are fundamentally nonstationary. They may exhibit stationarity on a short timescale but eventually alter their behavior in time and space. We developed a 2D [Formula: see text] adaptive prediction filter (APF) and further extended this to a 3D [Formula: see text] version for random noise attenuation based on regularized nonstationary autoregression (RNA). Instead of patching, a popular method for handling nonstationarity, we obtained smoothly nonstationary APF coefficients by solving a global regularized least-squares problem. We used shaping regularization to control the smoothness of the coefficients of APF. Three-dimensional space-noncausal [Formula: see text] APF uses neighboring traces around the target traces in the 3D seismic cube to predict noise-free signal, so it provided more accurate prediction results than the 2D version. In comparison with other denoising methods, such as frequency-space deconvolution, time-space prediction filter, and frequency-space RNA, we tested the feasibility of our method in reducing seismic random noise on three synthetic data sets. Results of applying the proposed method to seismic field data demonstrated that nonstationary [Formula: see text] APF was effective in practice.

Seismic imaging using lateral adaptive windows

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. S73-S79
Author(s):  
Ørjan Pedersen ◽  
Sverre Brandsberg-Dahl ◽  
Bjørn Ursin
Keyword(s):  
Seismic Data ◽  
Seismic Imaging ◽  
Synthetic Data ◽  
Real Data ◽  
Extrapolation Methods ◽  
Depth Migration ◽  
Large Velocity ◽  
Migration Method ◽  
Wavefield Extrapolation ◽  
The Impact

One-way wavefield extrapolation methods are used routinely in 3D depth migration algorithms for seismic data. Due to their efficient computer implementations, such one-way methods have become increasingly popular and a wide variety of methods have been introduced. In salt provinces, the migration algorithms must be able to handle large velocity contrasts because the velocities in salt are generally much higher than in the surrounding sediments. This can be a challenge for one-way wavefield extrapolation methods. We present a depth migration method using one-way propagators within lateral windows for handling the large velocity contrasts associated with salt-sediment interfaces. Using adaptive windowing, we can handle large perturbations locally in a similar manner as the beamlet propagator, thus limiting the impact of the errors on the global wavefield. We demonstrate the performance of our method by applying it to synthetic data from the 2D SEG/EAGE [Formula: see text] salt model and an offshore real data example.

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.

scholarly journals Deconvolution-type imaging condition effects on shot-profile migration amplitudes

2012 ◽  
Vol 5 (1) ◽  
pp. 05-18
Author(s):  
Flor-A. Vivas-Mejía ◽  
Herling González-Alvarez ◽  
Ligia-E. Jaimes-Osorio ◽  
Nancy Espindola-López
Keyword(s):  
Spectral Density ◽  
Weighting Function ◽  
Synthetic Data ◽  
Velocity Model ◽  
Single Shot ◽  
Lateral Velocity ◽  
Imaging Condition ◽  
Depth Migration ◽  
Reflectivity Function ◽  
Imaging Conditions

Amplitude preservation in Pre-Stack Depth Migration (PSDM) processes that use wavefield extrapolation must be ensured – first, in the operators used to continue the wavefield in time or depth, and second, in the imaging condition used to estimate the reflectivity function. In the later point, the conventional correlation-type imaging condition must be replaced by a deconvolution-type imaging condition. Migration performed in common-shot profile domain obtains the final migrated image as the superposition of images resulting of migrate each shot separately. The amplitude obtained in a point of the migrated image corresponds to the sum of the reflectivities for each shot which has illuminated such point, along the angles determined by the velocity model and the positions of the source and the receiver. The deeper the reflector, the lower the amplitude of the illumination field will be. As result, the correlation-type imaging condition produces images with an unbalanced amplitude decrease with depth. A deconvolution-type imaging condition scales the amplitudes through a correlation, using the weighting function dependent on the spectral density or the illumination of the downgoing wave field. In this article, two possible scaling functions have been used in the case of a single shot. In the case of data with multiple shots, five scaling possibilities are presented with the spectral density or the illumination function. The results of applying these imaging conditions to synthetic data with multiple shots show that the values of the amplitude in the migrated images are influenced by the coverage of the common midpoint, compensating this effect only in one of the imaging conditions described. Numerical experiments with synthetic data generated using Seismic Unix and the Sigsbee2a data are presented, highlighting that in velocity fields with strong vertical and lateral velocity variations, the balance of the amplitudes of the deep reflectors relative to the shallow reflectors is strongly influenced by the imaging condition applied.

Lateral prediction for noise attenuation by t-x and f-x techniques

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1887-1896 ◽  
Author(s):  
Ray Abma ◽  
Jon Claerbout
Keyword(s):  
Random Noise ◽  
High Amplitude ◽  
Noise Attenuation ◽  
Space Domain ◽  
Frequency Space ◽  
Amplitude Noise ◽  
Time Space ◽  
Domain Prediction ◽  
Prediction Techniques ◽  
Prediction Filter

Attenuating random noise with a prediction filter in the time‐space domain generally produces results similar to those of predictions done in the frequency‐space domain. However, in the presence of moderate‐ to high‐amplitude noise, time‐space or t-x prediction passes less random noise than does frequency‐space, or f-x prediction. The f-x prediction may also produce false events in the presence of parallel events where t-x prediction does not. These advantages of t-x prediction are the result of its ability to control the length of the prediction filter in time. An f-x prediction produces an effective t-x domain filter that is as long in time as the input data. Gulunay’s f-x domain prediction tends to bias the predictions toward the traces nearest the output trace, allowing somewhat more noise to be passed, but this bias may be overcome by modifying the system of equations used to calculate the filter. The 3-D extension to the 2-D t-x and f-x prediction techniques allows improved noise attenuation because more samples are used in the predictions, and the requirement that events be strictly linear is relaxed.

Stability versus accuracy for an explicit wavefield extrapolation operator

Geophysics ◽  
1993 ◽  
Vol 58 (2) ◽  
pp. 277-283 ◽  
Author(s):  
Atul Nautiyal ◽  
Samuel H. Gray ◽  
N. D. Whitmore ◽  
John D. Garing
Keyword(s):  
Fourier Transform ◽  
Unconditional Stability ◽  
Space Domain ◽  
Stability Properties ◽  
Depth Migration ◽  
Frequency Space ◽  
Wavefield Extrapolation ◽  
Extrapolation Operator ◽  
Accuracy And Stability

Wavefield extrapolation by recursive (depth‐by‐ depth) application of a convolutional operator in the frequency‐space domain, commonly used for depth migration in a laterally‐varying earth, has interesting accuracy and stability properties. We analyze these properties by investigating the operator and its spatial Fourier transform. In particular, we show that the instability caused by spatially truncating the operator can be remedied unconditionally by applying an appropriately chosen spatial taper. However, unconditional stability is gained only at the expense of accuracy. We also identify frequencies and depth extrapolation step sizes for which the problems of accuracy or stability are the most pronounced.

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