Wide-angle FD and FFD migration using complex Padé approximations

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. S215-S220 ◽  
Author(s):  
Daniela Amazonas ◽  
Jessé C. Costa ◽  
Jörg Schleicher ◽  
Reynam Pestana
Keyword(s):  
Finite Difference ◽  
Special Treatment ◽  
Wide Angle ◽  
Model Data ◽  
Padé Approximations ◽  
Data Set ◽  
Domain Boundaries ◽  
Pade Approximations ◽  
Impulsive Response ◽  
The One

Seismic migration by downward continuation using the one-way wave-equation approximations has two shortcomings: imaging steep-dip reflectors and handling evanescent waves. Complex Padé approximations allow a better treatment of evanescent modes, stabilizing finite-difference migration without requiring special treatment for the migration domain boundaries. Imaging of steep-dip reflectors can be improved using several terms in the Padé expansion. We discuss the implementation and evaluation of wide-angle complex Padé approximations for finite-difference and Fourier finite-difference migration methods. The dispersion relation and the impulsive response of the migration operator provide criteria to select the number of terms and coefficients in the Padé expansion. This ensures stability for a prescribed maximum propagation direction. The implementations are validated on the Marmousi model data set and SEG/EAGE salt model data.

Optimization of the parameters in complex Padé Fourier finite-difference migration

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. S259-S269 ◽  
Author(s):  
Marco Salcedo ◽  
Amélia Novais ◽  
Jörg Schleicher ◽  
Jessé C. Costa
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Rotation Angle ◽  
Computational Cost ◽  
Wide Angle ◽  
Model Data ◽  
Data Set ◽  
Dependent Parameter ◽  
The One

Complex Padé Fourier finite-difference migration is a stable one-way wave-equation technique that allows for better treatment of evanescent modes than its real counterpart, in this way producing fewer artifacts. As for real Fourier finite-difference (FFD) migration, its parameters can be optimized to improve the imaging of steeply dipping reflectors. The dip limitation of the FFD operator depends on the variation of the velocity field. We have developed a wide-angle approximation for the one-way continuation operator by means of optimization of the Padé coefficients and the most important velocity-dependent parameter. We have evaluated the achieved quality of the approximate dispersion relation in dependence on the chosen function of the ratio between the model and reference velocities under consideration of the number of terms in the Padé approximation and the branch-cut rotation angle. The optimized parameters are chosen based on the migration results and the computational cost. We found that by using the optimized parameters, a one-term expansion achieves the highest dip angles. The implementations were validated on the Marmousi data set and SEG/EAGE salt model data.

Optimum split-step Fourier 3D depth migration: Developments and practical aspects

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. S167-S175 ◽  
Author(s):  
Jianfeng Zhang ◽  
Linong Liu
Keyword(s):  
Finite Difference ◽  
Heterogeneous Media ◽  
Fourier Transforms ◽  
Computational Cost ◽  
Fourier Method ◽  
Approximate Algorithm ◽  
Wide Angle ◽  
Data Set ◽  
Depth Migration ◽  
Varying Coefficients

We present an efficient scheme for depth extrapolation of wide-angle 3D wavefields in laterally heterogeneous media. The scheme improves the so-called optimum split-step Fourier method by introducing a frequency-independent cascaded operator with spatially varying coefficients. The developments improve the approximation of the optimum split-step Fourier cascaded operator to the exact phase-shift operator of a varying velocity in the presence of strong lateral velocity variations, and they naturally lead to frequency-dependent varying-step depth extrapolations that reduce computational cost significantly. The resulting scheme can be implemented alternatively in spatial and wavenumber domains using fast Fourier transforms (FFTs). The accuracy of the first-order approximate algorithm is similar to that of the second-order optimum split-step Fourier method in modeling wide-angle propagation through strong, laterally varying media. Similar to the optimum split-step Fourier method, the scheme is superior to methods such as the generalized screen and Fourier finite difference. We demonstrate the scheme’s accuracy by comparing it with 3D two-way finite-difference modeling. Comparisons with the 3D prestack Kirchhoff depth migration of a real 3D data set demonstrate the practical application of the proposed method.

Wide Angle FD and FFD Migration using Complex Padé Approximations

2007 ◽  
Author(s):  
D. Amazonas ◽  
J. Costa ◽  
J. Schleicher ◽  
R. Pestana
Keyword(s):  
Wide Angle ◽  
Padé Approximations ◽  
Pade Approximations

Wide Angle FD and FFD Migration using Complex Padé Approximations

2007 ◽  
Author(s):  
D. Amazonas ◽  
J. Costa ◽  
J. Schleicher ◽  
R. Pestana
Keyword(s):  
Wide Angle ◽  
Padé Approximations ◽  
Pade Approximations

Wide‐angle FD and FFD migration using complex Padé approximations

2007 ◽  
Author(s):  
Daniela Amazonas ◽  
Jessé Costa ◽  
Jörg Schleicher ◽  
Reynam Pestana
Keyword(s):  
Wide Angle ◽  
Padé Approximations ◽  
Pade Approximations

Nucleation of Disorder in Fe3Al Alloys

1967 ◽  
Vol 25 ◽  
pp. 312-313
Author(s):  
P. R. Swann ◽  
W. R. Duff ◽  
R. M. Fisher
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
Annealing Temperature ◽  
Antiphase Domain ◽  
Effect Of Temperature ◽  
Domain Structures ◽  
Transmission Electron ◽  
Domain Boundaries ◽  
Higher Temperature ◽  
The One

Recently we have investigated the phase equilibria and antiphase domain structures of Fe-Al alloys containing from 18 to 50 at.% Al by transmission electron microscopy and Mössbauer techniques. This study has revealed that none of the published phase diagrams are correct, although the one proposed by Rimlinger agrees most closely with our results to be published separately. In this paper observations by transmission electron microscopy relating to the nucleation of disorder in Fe-24% Al will be described. Figure 1 shows the structure after heating this alloy to 776.6°C and quenching. The white areas are B2 micro-domains corresponding to regions of disorder which form at the annealing temperature and re-order during the quench. By examining specimens heated in a temperature gradient of 2°C/cm it is possible to determine the effect of temperature on the disordering reaction very precisely. It was found that disorder begins at existing antiphase domain boundaries but that at a slightly higher temperature (1°C) it also occurs by homogeneous nucleation within the domains. A small (∼ .01°C) further increase in temperature caused these micro-domains to completely fill the specimen.

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