Illumination‐based normalization for wave‐equation depth migration

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1371-1379 ◽  
Author(s):  
James E. Rickett
Keyword(s):  
Wave Equation ◽  
Reference Model ◽  
Synthetic Data ◽  
Data Migration ◽  
Synthetic Image ◽  
Lateral Velocity ◽  
Depth Migration ◽  
True Model ◽  
The Cost ◽  
Normalization Scheme

Illumination problems caused by finite‐recording aperture and lateral velocity lensing can bias amplitudes in migration results. In this paper, I develop a normalization scheme appropriate for wave‐equation migration algorithms that compensates for irregular illumination. I generate synthetic seismic data over a reference reflectivity model, using the adjoint of wave‐equation shot‐profile migration as the forward modeling operator. I then migrate the synthetic data with the same shot‐profile algorithm. The ratio between the synthetic migration result and the initial reference model is a measure of seismic illumination. Dividing the true data migration result by this illumination function mitigates the illumination problems. The methodology can take into account reflector dip as well as both shot and receiver geometries, and, because it is based on wave‐equation migration, it naturally models the finite‐frequency effects of wave propagation. The reference model should be as close to the true model as possible; good choices include the migrated image, or a synthetic image with a single known dip that corresponds to the expected dip of a reflector of interest. Computational shortcuts allow the illumination functions to be computed at about the cost of a single migration. Results indicate that normalization can significantly reduce amplitude distortions due to irregular subsurface illumination.

Offset‐domain pseudoscreen prestack depth migration

Geophysics ◽  
2002 ◽  
Vol 67 (6) ◽  
pp. 1895-1902 ◽  
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.

Application of an integrated wave-equation datuming scheme to overthrust data: A case history from the Chinese foothills

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. B153-B165 ◽  
Author(s):  
Kai Yang ◽  
Hong-Ming Zheng ◽  
Li Wang ◽  
Yu-Zhu Liu ◽  
Fan Jiang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Synthetic Data ◽  
Western China ◽  
Velocity Variation ◽  
Lateral Velocity ◽  
Near Surface ◽  
Depth Images ◽  
Static Corrections ◽  
Kirchhoff Integral

An integrated wave-equation datuming scheme improves the imaging quality of seismic data from overthrust areas. It can be regarded as integrated because upward-layer replacement is included. In this scheme, data are downward continued to a nonplanar datum (such as the base of the weathering layer), followed by upward continuation from the nonplanar datum to a final planar datum using a one-way extrapolator. When compared with a Kirchhoff integral, this method can deal better with the strong lateral velocity variation within the near surface. After a test on synthetic data, the scheme is applied successfully to real 2D overthrust data acquired in the Qi-Lian foothills, western China. Compared with results using static corrections, integrated wave-equation datuming results lead to better reconstruction of the diffractions and reflections, more reliable migration-velocity analyses, and stronger stack and final depth images.

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).

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.

The finite-frequency sensitivity kernel for migration residual moveout and its applications in migration velocity analysis

Geophysics ◽  
2008 ◽  
Vol 73 (6) ◽  
pp. S241-S249 ◽  
Author(s):  
Xiao-Bi Xie ◽  
Hui Yang
Keyword(s):  
Wave Equation ◽  
Synthetic Data ◽  
Velocity Model ◽  
Migration Velocity ◽  
Data Sets ◽  
Velocity Analysis ◽  
Depth Migration ◽  
Sensitivity Kernel ◽  
Migration Velocity Analysis ◽  
Tomography Method

We have derived a broadband sensitivity kernel that relates the residual moveout (RMO) in prestack depth migration (PSDM) to velocity perturbations in the migration-velocity model. We have compared the kernel with the RMO directly measured from the migration image. The consistency between the sensitivity kernel and the measured sensitivity map validates the theory and the numerical implementation. Based on this broadband sensitivity kernel, we propose a new tomography method for migration-velocity analysis and updating — specifically, for the shot-record PSDM and shot-index common-image gather. As a result, time-consuming angle-domain analysis is not required. We use a fast one-way propagator and multiple forward scattering and single backscattering approximations to calculate the sensitivity kernel. Using synthetic data sets, we can successfully invert velocity perturbations from the migration RMO. This wave-equation-based method naturally incorporates the wave phenomena and is best teamed with the wave-equation migration method for velocity analysis. In addition, the new method maintains the simplicity of the ray-based velocity analysis method, with the more accurate sensitivity kernels replacing the rays.

scholarly journals Wave-equation time migration

Geophysics ◽  
2020 ◽  
pp. 1-58
Author(s):  
Sergey Fomel ◽  
Harpreet Kaur
Keyword(s):  
Wave Equation ◽  
Time Domain ◽  
Field Data ◽  
Model Building ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Time Migration ◽  
Expensive Model ◽  
Approximate Equations ◽  
Ray Coordinates

Time migration, as opposed to depth migration, suffers from two well-known shortcomings: (1)approximate equations are used for computing Green’s functions inside the imaging operator; (2) in case of lateral velocity variations, the transformation between the image ray coordinates andthe Cartesian coordinates is undefined in places where the image rays cross. We show that thefirst limitation can be removed entirely by formulating time migration through wave propagationin image-ray coordinates. The proposed approach constructs a time-migrated image without relyingon any kind of traveltime approximation by formulating an appropriate geometrically accurateacoustic wave equation in the time-migration domain. The advantage of this approach is that thepropagation velocity in image-ray coordinates does not require expensive model building and canbe approximated by quantities that are estimated in conventional time-domain processing. Synthetic and field data examples demonstrate the effectiveness of the proposed approach and show that theproposed imaging workflow leads to a significant uplift in terms of image quality and can bridge thegap between time and depth migrations. The image obtained by the proposed algorithm is correctlyfocused and mapped to depth coordinates it is comparable to the image obtained by depth migration.

Wave-equation global datuming based on the double square root operator

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. U35-U43 ◽  
Author(s):  
Wenge Liu ◽  
Bo Zhao ◽  
Hua-wei Zhou ◽  
Zhenhua He ◽  
Hui Liu ◽  
...  
Keyword(s):  
Wave Equation ◽  
Synthetic Data ◽  
Square Root ◽  
New Wave ◽  
Lateral Velocity ◽  
Near Surface ◽  
Traditional Procedure ◽  
Wavefield Continuation ◽  
Static Corrections ◽  
Root Operator

Current schemes for removing near-surface effects in seismic data processing use either static corrections or wave-equation datuming (WED). In the presence of rough topography and strong lateral velocity variations in the near surface, the WED scheme is the only option available. However, the traditional procedure of WED downward continues the sources and receivers in different domains. A new wave-equation global-datuming method is based on the double-square-root operator, implementing the wavefield continuation in a single domain following the survey sinking concept. This method has fewer approximations and therefore is more robust and convenient than the traditional WED methods. This method is compared with the traditional methods using a synthetic 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.

Migration with the full acoustic wave equation

Geophysics ◽  
1983 ◽  
Vol 48 (6) ◽  
pp. 677-687 ◽  
Author(s):  
Dan D. Kosloff ◽  
Edip Baysal
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Physical Model ◽  
Wave Equations ◽  
Synthetic Data ◽  
Acoustic Wave Equation ◽  
Model Data ◽  
Lateral Velocity ◽  
Alternative Approach

Conventional finite‐difference migration has relied on one‐way wave equations which allow energy to propagate only downward. Although generally reliable, such equations may not give accurate migration when the structures have strong lateral velocity variations or steep dips. The present study examined an alternative approach based on the full acoustic wave equation. The migration algorithm which developed from this equation was tested against synthetic data and against physical model data. The results indicated that such a scheme gives accurate migration for complicated structures.

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.

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