Hybrid migration by finite‐difference and phase shift

1992 ◽  
Author(s):  
C. M. Haddow
Keyword(s):  
Phase Shift ◽  
Finite Difference

scholarly journals Verification of seiching processes in a large and deep lake (Trichonis, Greece)

2000 ◽  
Vol 1 (1) ◽  
pp. 79 ◽  
Author(s):  
I. ZACHARIAS
Keyword(s):  
Time Series ◽  
Spectral Analysis ◽  
Phase Shift ◽  
Finite Difference ◽  
Phase Difference ◽  
Experimental Verification ◽  
Computational Analysis ◽  
Power Spectral Analysis ◽  
Deep Lake ◽  
Power Spectral

A computational analysis of the periods and structure of surface seiches of Lake Trichonis in Greece and its experimental verification from three simultaneous water gauge recordings, mounted along the shores in Myrtia, Panetolio and Trichonio is given. The first five theoretical modes are calculated with a finite difference code of tidal equations, which yield the eigenperiodes, co-range and co-tidal lines that are graphically displayed and discussed in detail.Experimental verifications are from recordings taken during spring. Visual observations of the record permit identification of the five lowest order modes, including inter station phase shift. Power spectral analysis of two time series and interstation phase difference and coherence spectra allow the identification of the same five modes. Agreement between the theoretically predicted and the experimentally determined periods was excellent for most of the calculated modes.

Fourier finite‐difference migration

Geophysics ◽  
1994 ◽  
Vol 59 (12) ◽  
pp. 1882-1893 ◽  
Author(s):  
Dietrich Ristow ◽  
Thomas Rühl
Keyword(s):  
Phase Shift ◽  
Finite Difference ◽  
Difference Operator ◽  
Shift Operator ◽  
Velocity Model ◽  
Downward Continuation ◽  
The Other ◽  
Finite Difference Schemes ◽  
Other Hand ◽  
Arbitrary Velocity

Many existing migration schemes cannot simultaneously handle the two most important problems of migration: imaging of steep dips and imaging in media with arbitrary velocity variations in all directions. For example, phase‐shift (ω, k) migration is accurate for nearly all dips but it is limited to very simple velocity functions. On the other hand, finite‐difference schemes based on one‐way wave equations consider arbitrary velocity functions but they attenuate steeply dipping events. We propose a new hybrid migration method, named “Fourier finite‐difference migration,” wherein the downward‐continuation operator is split into two downward‐continuation operators: one operator is a phase‐shift operator for a chosen constant background velocity, and the other operator is an optimized finite‐difference operator for the varying component of the velocity function. If there is no variation of velocity, then only a phase‐shift operator will be applied automatically. On the other hand, if there is a strong variation of velocity, then the phase‐shift component is suppressed and the optimized finite‐difference operator will be fully applied. The cascaded application of phase‐shift and finite‐difference operators shows a better maximum dip‐angle behavior than the split‐step Fourier migration operator. Depending on the macro velocity model, the Fourier finite‐difference migration even shows an improved performance compared to conventional finite‐difference migration with one downward‐continuation step. Finite‐difference migration with two downward‐continuation steps is required to reach the same migration performance, but this is achieved with about 20 percent higher computation costs. The new cascaded operator of the Fourier finite‐difference migration can be applied to arbitrary velocity functions and allows an accurate migration of steeply dipping reflectors in a complex macro velocity model. The dip limitation of the cascaded operator depends on the variation of the velocity field and, hence, is velocity‐adaptive.

A stable one-way wave propagator for VTI media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB3-WB17 ◽  
Author(s):  
Peter M. Bakker
Keyword(s):  
Phase Shift ◽  
Finite Difference ◽  
Vertical Direction ◽  
Transversely Isotropic ◽  
Difference Operators ◽  
Wide Angle ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
Reference Medium

For the purpose of one-way wave-equation imaging, a pseudoscreen propagator is developed for transversely isotropic media with vertical axes of symmetry. The phase shift for propagation through a depth slice is decomposed into three terms: a Gazdag phase shift for propagation in a laterally homogeneous reference medium, a correction for lateral variability of vertical propagation, and a remaining wide-angle term for oblique directions of propagation. Based on rational function approximation for this remaining wide-angle term, a Fourier finite-difference (FFD) approach with four-way splitting is applied. Fourth-order Padé approximation is unsatisfactory in anelliptic media for large propagation angles with respect to the vertical direction. Therefore, a method of coefficient optimization is developed in conjunction with a method of choosing an adequate homogeneous reference medium in a depth slice. By symmetrizing the finite-difference operators, and because of the choice of the optimized coefficients, the propagator is stable in the sense that the least-squares norm of the wavefield, measured for a frequency-depth slice, does not grow with increasing depth of propagation. A small amount of artificial damping is applied to suppress artifacts that appear at the critical angle defined by the velocities in the reference medium and the actual medium. Synthetic examples confirm that good kinematic accuracy is achieved for a wide range of propagation angles (typically up to 60°).

scholarly journals Compensating finite‐difference errors in 3-D migration and modeling

Geophysics ◽  
1991 ◽  
Vol 56 (10) ◽  
pp. 1650-1660 ◽  
Author(s):  
Zhiming Li
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Finite Difference ◽  
Interpolation Method ◽  
Splitting Method ◽  
Three Dimensional ◽  
Lateral Velocity ◽  
Frequency Space ◽  
Difference Grid ◽  
Imaging Results

One‐pass three‐dimensional (3-D) depth migration potentially offers more accurate imaging results than does conventional two‐pass 3-D migration for variable velocity media. Conventional one‐pass 3-D migration, using the method of finite‐difference inline and crossline splitting, however, creates large errors in the image of complex structures. These errors are due to paraxial wave‐equation approximation of the one‐way wave equation, inline‐crossline splitting, and finite‐difference grid dispersion. To compensate for these errors, and still preserve the efficiency of the conventional finite‐difference splitting method, a phase‐correction operator is derived by minimizing the difference between the ideal 3-D migration (or modeling) and the actual, conventional 3-D migration (or modeling). For frequency‐space 3-D finite‐difference migration and modeling, the compensation operator is implemented using either the phase‐shift, or phase‐shift‐plus‐interpolation method, depending on the extent of lateral velocity variations. The compensation operator increases the accuracy of handling steep dips, suppresses the inline and crossline splitting error, and reduces finite‐difference grid dispersions.

Object Wave Determination by Single-Sideband Methods

1973 ◽  
Vol 31 ◽  
pp. 266-267
Author(s):  
Kenneth H. Downing ◽  
Benjamin M. Siegel
Keyword(s):  
Phase Shift ◽  
Carbon Films ◽  
Object Wave ◽  
Electron Wave ◽  
Phase Object ◽  
Heavy Atoms ◽  
Single Sideband ◽  
Thin Carbon ◽  
Additional Phase Shift ◽  
Additional Phase

Under the “weak phase object” approximation, the component of the electron wave scattered by an object is phase shifted by π/2 with respect to the unscattered component. This phase shift has been confirmed for thin carbon films by many experiments dealing with image contrast and the contrast transfer theory. There is also an additional phase shift which is a function of the atomic number of the scattering atom. This shift is negligible for light atoms such as carbon, but becomes significant for heavy atoms as used for stains for biological specimens. The light elements are imaged as phase objects, while those atoms scattering with a larger phase shift may be imaged as amplitude objects. There is a great deal of interest in determining the complete object wave, i.e., both the phase and amplitude components of the electron wave leaving the object.

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