Prestack wave-equation depth migration in elliptical coordinates

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. S169-S175 ◽  
Author(s):  
Jeff Shragge ◽  
Guojian Shan
Keyword(s):  
Coordinate Systems ◽  
Data Set ◽  
Depth Migration ◽  
Prestack Migration ◽  
Oblate Spheroidal ◽  
2D Imaging ◽  
Elliptical Coordinates ◽  
The One ◽  
Wavefield Extrapolation ◽  
Root Operator

We extend Riemannian wavefield extrapolation (RWE) to prestack migration using 2D elliptical-coordinate systems. The corresponding 2D elliptical extrapolation wavenumber introduces only an isotropic slowness model stretch to the single-square-root operator. This enables the use of existing Cartesian finite-difference extrapolators for propagating wavefields on elliptical meshes. A poststack migration example illustrates advantages of elliptical coordinates for imaging turning waves. A 2D imaging test using a velocity-benchmark data set demonstrates that the RWE prestack migration algorithm generates high-quality prestack migration images that are more accurate than those generated by Cartesian operators of the equivalent accuracy. Even in situations in which RWE geometries are used, a high-order implementation of the one-way extrapolator operator is required for accurate propagation and imaging. Elliptical-cylindrical and oblate-spheroidal geometries are potential extensions of the analytical approach to 3D RWE-coordinate systems.

3D plane-wave migration in tilted coordinates: A field data example

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA199-WCA209 ◽  
Author(s):  
Guojian Shan ◽  
Robert Clapp ◽  
Biondo Biondi
Keyword(s):  
Plane Wave ◽  
Field Data ◽  
Synthetic Data ◽  
Data Set ◽  
Wave Source ◽  
Surface Data ◽  
The One ◽  
Wave Migration ◽  
Wavefield Extrapolation ◽  
Slant Stacking

We have extended isotropic plane-wave migration in tilted coordinates to 3D anisotropic media and applied it on a Gulf of Mexico data set. Recorded surface data are transformed to plane-wave data by slant-stack processing in inline and crossline directions. The source plane wave and its corresponding slant-stacked data are extrapolated into the subsurface within a tilted coordinate system whose direction depends on the propagation direction of the plane wave. Images are generated by crosscorrelating these two wavefields. The shot sampling is sparse in the crossline direction, and the source generated by slant stacking is not really a plane-wave source but a phase-encoded source. We have discovered that phase-encoded source migration in tilted coordinates can image steep reflectors, using 2D synthetic data set examples. The field data example shows that 3D plane-wave migration in tilted coordinates can image steeply dipping salt flanks and faults, even though the one-way wave-equation operator is used for wavefield extrapolation.

Inline delayed-shot migration in tilted elliptical-cylindrical coordinates

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S187-S197 ◽  
Author(s):  
Jeffrey Shragge ◽  
Guojian Shan
Keyword(s):  
Impulse Response ◽  
Seismic Data ◽  
Finite Difference Methods ◽  
Computational Cost ◽  
Coordinate Systems ◽  
Geologic Structure ◽  
Data Set ◽  
Implicit Finite Difference ◽  
Wavefield Extrapolation ◽  
Generalized Coordinate

Riemannian wavefield extrapolation, a one-way wave-equation method for propagating seismic data on generalized coordinate systems, is extended to inline delayed-shot migration using 3D tilted elliptical-cylindrical (TEC) coordinate meshes. Compared to Cartesian geometries, TEC coordinates are more conformal to the shape of inline delayed-source impulse response, which allows the bulk of wavefield energy to propagate at angles lower to the extrapolation axis, thus improving global propagation accuracy. When inline coordinate tilt angles are well matched to the inline source ray parameters, the TEC coordinate extension affords accurate propagation of both steep-dip and turning-wave components important for successfully imaging complex geologic structure. Wavefield extrapolation in TEC coordinates is no more complicated than propagation in elliptically anisotropic media and can be handled by existing implicit finite-difference methods. Impulse response tests illustrate the phase accuracy of the method and show that the approach is free of numerical anisotropy. Migration tests from a realistic 3D wide-azimuth synthetic derived from a field Gulf of Mexico data set demonstrate the imaging advantages afforded by the technique, including the improved imaging of steeply dipping salt flanks at a reduced computational cost.

Prestack depth migration of an Alberta Foothills data set—The Husky experience

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 392-398 ◽  
Author(s):  
W.-J. Wu ◽  
L. Lines ◽  
A. Burton ◽  
H.-X. Lu ◽  
J. Zhu ◽  
...  
Keyword(s):  
Velocity Analysis ◽  
Reverse Time ◽  
Prestack Depth Migration ◽  
Data Set ◽  
Depth Migration ◽  
Geological Interpretation ◽  
Prestack Migration ◽  
Velocity Models ◽  
Depth Images ◽  
Migration Velocity Analysis

We produce depth images for an Alberta Foothills line by iteratively using a number of migration and velocity analysis techniques. In imaging steeply dipping layers of a foothills data set, it is apparent that thrust belt geology can violate the conventional assumptions of elevation datum corrections and common midpoint (CMP) stacking. To circumvent these problems, we use migration from topography in which we perform prestack depth migration on the data using correct source and receiver elevations. Migration from topography produces enhanced images of steep shallow reflectors when compared to conventional processing. In addition to migration from topography, we couple prestack depth migration with the continuous adjustment of velocity depth models. A number of criteria are used in doing this. These criteria require that our velocity estimates produce a focused image and that migrated depths in common image gathers be independent of source‐receiver offset. Velocity models are estimated by a series of iterative and interpretive steps involving prestack migration velocity analysis and structural interpretation. Overlays of velocity models on depth migrations should generally show consistency between velocity boundaries and reflection depths. Our preferred seismic depth section has been produced by using prestack reverse‐time depth migration coupled with careful geological interpretation.

3-D prestack Kirchhoff depth migration: From prototype to production in a massively parallel processor environment

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 546-556 ◽  
Author(s):  
Herman Chang ◽  
John P. VanDyke ◽  
Marcelo Solano ◽  
George A. McMechan ◽  
Duryodhan Epili
Keyword(s):  
Model Building ◽  
Velocity Model ◽  
Production Scale ◽  
Data Set ◽  
Depth Migration ◽  
Prestack Migration ◽  
Prototype Development ◽  
Time Migration ◽  
Intel Paragon ◽  
Kirchhoff Depth Migration

Portable, production‐scale 3-D prestack Kirchhoff depth migration software capable of full‐volume imaging has been successfully implemented and applied to a six‐million trace (46.9 Gbyte) marine data set from a salt/subsalt play in the Gulf of Mexico. Velocity model building and updates use an image‐driven strategy and were performed in a Sun Sparc environment. Images obtained by 3-D prestack migration after three velocity iterations are substantially better focused and reveal drilling targets that were not visible in images obtained from conventional 3-D poststack time migration. Amplitudes are well preserved, so anomalies associated with known reservoirs conform to the petrophysical predictions. Prototype development was on an 8-node Intel iPSC860 computer; the production version was run on an 1824-node Intel Paragon computer. The code has been successfully ported to CRAY (T3D) and Unix workstation (PVM) environments.

Seismic depth imaging with the Gabor transform

Geophysics ◽  
2008 ◽  
Vol 73 (3) ◽  
pp. S91-S97 ◽  
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.

Azimuth moveout for 3-D prestack imaging

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 574-588 ◽  
Author(s):  
Biondo Biondi ◽  
Sergey Fomel ◽  
Nizar Chemingui
Keyword(s):  
Integral Operator ◽  
Impulse Response ◽  
Constant Velocity ◽  
Computational Cost ◽  
Data Set ◽  
Depth Migration ◽  
Prestack Migration ◽  
Migration Operator ◽  
Time Space ◽  
Strong Curvature

We introduce a new partial prestack‐migration operator called “azimuth moveout” (AMO) that rotates the azimuth and modifies the offset of 3-D prestack data. Followed by partial stacking, AMO can reduce the computational cost of 3-D prestack imaging. We have successfully applied AMO to the partial stacking of a 3-D marine data set over a range of offsets and azimuths. When AMO is included in the partial‐stacking procedure, high‐frequency steeply dipping energy is better preserved than when conventional partial‐stacking methodologies are used. Because the test data set requires 3-D prestack depth migration to handle strong lateral variations in velocity, the results of our tests support the applicability of AMO to prestack depth‐imaging problems. AMO is a partial prestack‐migration operator defined by chaining a 3-D prestack imaging operator with a 3-D prestack modeling operator. The analytical expression for the AMO impulse response is derived by chaining constant‐velocity DMO with its inverse. Equivalently, it can be derived by chaining constant‐velocity prestack migration and modeling. Because 3-D prestack data are typically irregularly sampled in the surface coordinates, AMO is naturally applied as an integral operator in the time‐space domain. The AMO impulse response is a skewed saddle surface in the time‐midpoint space. Its shape depends on the amount of azimuth rotation and offset continuation to be applied to the data. The shape of the AMO saddle is velocity independent, whereas its spatial aperture is dependent on the minimum velocity. When the azimuth rotation is small (⩽20°), the AMO impulse response is compact, and its application as an integral operator is inexpensive. Implementing AMO as an integral operator is not straightforward because the AMO saddle may have a strong curvature when it is expressed in the midpoint coordinates. An appropriate transformation of the midpoint axes to regularize the AMO saddle leads to an effective implementation.

One-way wave-equation migration in log-polar coordinates

Geophysics ◽  
2013 ◽  
Vol 78 (2) ◽  
pp. S59-S67 ◽  
Author(s):  
Diako Hariri Naghadeh ◽  
Mohamad Ali Riahi
Keyword(s):  
Wave Equation ◽  
Polar Coordinates ◽  
Finite Difference Schemes ◽  
Coordinate Systems ◽  
Low Frequencies ◽  
Wave Propagation Direction ◽  
The One ◽  
Wavefield Extrapolation ◽  
High Order Finite Difference ◽  
Extrapolation Operator

We obtained acoustic wave and wavefield extrapolation equations in log-polar coordinates (LPCs) and tried to enhance the imaging. To achieve this goal, it was necessary to decrease the angle between the wavefield extrapolation axis and wave propagation direction in the one-way wave-equation migration (WEM). If we were unable to carry it out, more reflection wave energy would be lost in the migration process. It was concluded that the wavefield extrapolation operator in LPCs at low frequencies has a large wavelike region, and at high frequencies, it can mute the evanescent energy. In these coordinate systems, an extrapolation operator can readily lend itself to high-order finite-difference schemes; therefore, even with the use of inexpensive operators, WEM in LPCs can clearly image varied (horizontal and vertical) events in complex geologic structures using wide-angle and turning waves. In these coordinates, we did not encounter any problems with reflections from opposing dips. Dispersion played important roles not only as a filter operator but also as a gain function. Prestack and poststack migration results were obtained with extrapolation methods in different coordinate systems, and it was concluded that migration in LPCs can image steeply dipping events in a much better way when compared with other methods.

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.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

Intuitionistic fuzzy two-factor variance analysis of movie ticket sales

2021 ◽  
pp. 1-11
Author(s):  
Velichka Traneva ◽  
Stoyan Tranev
Keyword(s):  
Comparative Analysis ◽  
Real Data ◽  
Intuitionistic Fuzzy Sets ◽  
Real Numbers ◽  
Data Set ◽  
Important Method ◽  
Ticket Sales ◽  
Intuitionistic Fuzzy ◽  
The One ◽  
First Time

Analysis of variance (ANOVA) is an important method in data analysis, which was developed by Fisher. There are situations when there is impreciseness in data In order to analyze such data, the aim of this paper is to introduce for the first time an intuitionistic fuzzy two-factor ANOVA (2-D IFANOVA) without replication as an extension of the classical ANOVA and the one-way IFANOVA for a case where the data are intuitionistic fuzzy rather than real numbers. The proposed approach employs the apparatus of intuitionistic fuzzy sets (IFSs) and index matrices (IMs). The paper also analyzes a unique set of data on daily ticket sales for a year in a multiplex of Cinema City Bulgaria, part of Cineworld PLC Group, applying the two-factor ANOVA and the proposed 2-D IFANOVA to study the influence of “ season ” and “ ticket price ” factors. A comparative analysis of the results, obtained after the application of ANOVA and 2-D IFANOVA over the real data set, is also presented.

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