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.

True-amplitude one-way wave equation migration in the mixed domain

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S199-S209 ◽  
Author(s):  
Flor A. Vivas ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Numerical Experiments ◽  
Wave Equations ◽  
Laplacian Operator ◽  
Single Shot ◽  
Lateral Velocity ◽  
Data Set ◽  
Classical Formulation ◽  
Imaging Tool

One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium,the extrapolation of the wavefields with the true-amplitude one-way wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain true-amplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.

Migration of seismic data by phase shift plus interpolation

Geophysics ◽  
1984 ◽  
Vol 49 (2) ◽  
pp. 124-131 ◽  
Author(s):  
Jeno Gazdag ◽  
Piero Sguazzero
Keyword(s):  
Phase Shift ◽  
Wave Field ◽  
Seismic Data ◽  
Three Dimensional ◽  
Reference Wave ◽  
Second Step ◽  
Lateral Velocity ◽  
Wave Fields ◽  
Shift Method ◽  
Phase Shift Method

Under the horizontally layered velocity assumption, migration is defined by a set of independent ordinary differential equations in the wavenumber‐frequency domain. The wave components are extrapolated downward by rotating their phases. This paper shows that one can generalize the concepts of the phase‐shift method to media having lateral velocity variations. The wave extrapolation procedure consists of two steps. In the first step, the wave field is extrapolated by the phase‐shift method using ℓ laterally uniform velocity fields. The intermediate result is ℓ reference wave fields. In the second step, the actual wave field is computed by interpolation from the reference wave fields. The phase shift plus interpolation (PSPI) method is unconditionally stable and lends itself conveniently to migration of three‐dimensional data. The performance of the methods is demonstrated on synthetic examples. The PSPI migration results are then compared with those obtained from a finite‐difference method.

scholarly journals A Comparison of Splitting Techniques for 3D Complex Padé Fourier Finite Difference Migration

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Jessé C. Costa ◽  
Débora Mondini ◽  
Jörg Schleicher ◽  
Amélia Novais
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Computational Cost ◽  
Three Dimensional ◽  
Depth Level ◽  
Numerical Examples ◽  
The Cost ◽  
Comparable Quality ◽  
3D Problem ◽  
Dimensional Wave

Three-dimensional wave-equation migration techniques are still quite expensive because of the huge matrices that need to be inverted. Several techniques have been proposed to reduce this cost by splitting the full 3D problem into a sequence of 2D problems. We compare the performance of splitting techniques for stable 3D Fourier finite-difference (FFD) migration techniques in terms of image quality and computational cost. The FFD methods are complex Padé FFD and FFD plus interpolation, and the compared splitting techniques are two- and four-way splitting as well as alternating four-way splitting, that is, splitting into the coordinate directions at one depth and the diagonal directions at the next depth level. From numerical examples in homogeneous and inhomogeneous media, we conclude that, though theoretically less accurate, alternate four-way splitting yields results of comparable quality as full four-way splitting at the cost of two-way splitting.

An explicit, unconditionally stable, 15‐degree depth migration and modeling algorithm implemented in poststack, directional, and prestack modes

Geophysics ◽  
1991 ◽  
Vol 56 (9) ◽  
pp. 1412-1422
Author(s):  
Alvin K. Benson
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Three Dimensional ◽  
Inhomogeneous Media ◽  
Unconditional Stability ◽  
Data Sets ◽  
Unconditionally Stable ◽  
Depth Migration ◽  
Implicit Schemes ◽  
Modeling Algorithm

An explicit, unconditionally stable, finite‐difference depth migration and modeling algorithm is formulated and implemented for the fifteen‐degree wave equation in poststack, directional (rotational), and prestack modes for inhomogeneous media. It is about two times faster than implicit schemes. The simplicity, unconditional stability, and speed of the algorithm are appealing for numerous applications, especially prestack and three‐dimensional data sets.

MIGRATION BY FOURIER TRANSFORM

Geophysics ◽  
1978 ◽  
Vol 43 (1) ◽  
pp. 23-48 ◽  
Author(s):  
R. H. Stolt
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Scalar Wave ◽  
Conjugate Space ◽  
And Migration

Wave equation migration is known to be simpler in principle when the horizontal coordinate or coordinates are replaced by their Fourier conjugates. Two practical migration schemes utilizing this concept are developed in this paper. One scheme extends the Claerbout finite difference method, greatly reducing dispersion problems usually associated with this method at higher dips and frequencies. The second scheme effects a Fourier transform in both space and time; by using the full scalar wave equation in the conjugate space, the method eliminates (up to the aliasing frequency) dispersion altogether. The second method in particular appears adaptable to three‐dimensional migration and migration before stack.

Prestack migration by split‐step DSR

Geophysics ◽  
1996 ◽  
Vol 61 (5) ◽  
pp. 1412-1416 ◽  
Author(s):  
Alexander Mihai Popovici
Keyword(s):  
Phase Shift ◽  
Square Root ◽  
Variable Velocity ◽  
Step Method ◽  
Lateral Velocity ◽  
Data Set ◽  
Prestack Migration ◽  
Imaging Results ◽  
Velocity Variations ◽  
Migration Equation

The double‐square‐root (DSR) prestack migration equation, though defined for depth variable velocity, can be used to image media with strong velocity variations using a phase‐shift plus interpolation (PSPI) or split‐step correction. The split‐step method is based on applying a phase‐shift correction to the extrapolated wavefield—a correction that attempts to compensate for the lateral velocity variations. I show how to extend DSR prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the Marmousi data set. The split‐step DSR migration is very fast and offers excellent imaging results.

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