Analysis of higher‐order, finite‐difference schemes in 3-D reverse‐time migration

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 845-856 ◽  
Author(s):  
Wen‐Jing Wu ◽  
Larry R. Lines ◽  
Han‐Xing Lu
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Higher Order ◽  
Second Order ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Scalar Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration

The design and usefulness of practical algorithms for 3-D reverse‐time depth migration is examined, while demonstrating data applications of the algorithm. We evaluate quantitatively the accuracy of the finite‐difference operator from second‐order to eighth‐order for the scalar wave equation by comparing numerical and analytical solutions. The results clearly show the advantage of using higher‐order, finite‐difference schemes, especially from second‐order to fourth‐order for space derivatives. Hence, a finite‐difference method with the accuracy of fourth‐order in space and second‐order in time is applied to 3-D full scalar wave equations in reverse‐time migration. Considerable savings in CPU and memory are obtained by using larger horizontal grid spacings than the vertical spacings. Therefore, we derive dispersion and stability conditions for such unequal grid spacing. The 3-D reverse‐time migration of data from the Hibernia Field shows an image improvement over 2-D migrations, despite the fact that much of the structure was considered to be approximately 2-D. Stable, nondispersive, finite‐difference methods of higher order can allow for tractable and efficient 3-D reverse‐time migration solutions.

scholarly journals Higher order schemes and Richardson extrapolation for singular perturbation problems

1989 ◽  
Vol 39 (1) ◽  
pp. 129-139 ◽  
Author(s):  
Dragoslav Herceg ◽  
Relja Vulanović ◽  
Nenad Petrović
Keyword(s):  
Finite Difference ◽  
Singular Perturbation ◽  
Fourth Order ◽  
Richardson Extrapolation ◽  
Higher Order ◽  
Difference Schemes ◽  
Singular Perturbation Problems ◽  
Finite Difference Schemes ◽  
Perturbation Problems ◽  
Uniform Accuracy

Semilinear singular perturbation problems are solved numerically by using finite–difference schemes on non-equidistant meshes which are dense in the layers. The fourth order uniform accuracy of the Hermitian approximation is improved by the Richardson extrapolation.

Coupled equations for reverse time migration in transversely isotropic media

Geophysics ◽  
2010 ◽  
Vol 75 (1) ◽  
pp. S11-S22 ◽  
Author(s):  
Paul J. Fowler ◽  
Xiang Du ◽  
Robin P. Fletcher
Keyword(s):  
Dispersion Relation ◽  
Fourth Order ◽  
Transversely Isotropic ◽  
Shear Velocity ◽  
Second Order ◽  
Vertical Shear ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Coupled Equations ◽  
Time Migration

Reverse time migration (RTM) images reflectors by using time-extrapolation modeling codes to synthesize source and receiver wavefields in the subsurface. Asymptotic analysis of wave propagation in transversely isotropic (TI) media yields a dispersion relation describing coupled P- and SV-wave modes. This dispersion relation can be converted into a fourth-order scalar partial differential equation (PDE). Increased computational efficiency can be achieved using equivalent coupled second-order PDEs. Analysis of the corresponding dispersion relations as matrix eigenvalue systems allows one to characterize all possible coupled linear second-order systems equivalent to a given linear fourth-order PDE and to determine which ones yield optimally efficient finite-difference implementations. Setting the shear velocity along the axis of symmetry to zero yields a simpler approximate TI wave equation that is more efficient to implement. This simpler approximation, however, can become unstable for some plausible combinations of anisotropic parameters. The same eigensystem analysis can be applied using finite vertical shear velocity to obtain solutions that avoid these instability problems.

Wave extrapolation in the spatial wavelet domain with application to poststack reverse‐time migration

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 589-600 ◽  
Author(s):  
Yafei Wu ◽  
George A. McMechan
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Wavelet Domain ◽  
Scalar Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Step ◽  
Scalar Wave Equation ◽  
Time Migration ◽  
Spatial Coordinates

A wavelet transformation is performed over each of the spatial coordinates of the scalar wave equation. This transformed equation is solved directly with a finite‐difference scheme for both homogeneous and smooth inhomogeneous media. Wavefield extrapolation is performed completely in the spatial wavelet domain without transforming back into the space domain at each time step. The wavelet coefficients are extrapolated, rather than the wavefield itself. The numerical solution of the scalar wave equation in the spatial wavelet domain is closely related to the finite‐difference method because of the compact support of the wavelet bases. Poststack reverse‐time migration is implemented as an application. The resolution spaces of the wavelet transform provide a natural framework for multigrid analysis. Migrated images are constructed from various resolution spaces.

Q least-squares reverse time migration based on the first-order viscoacoustic quasidifferential equations

Geophysics ◽  
2021 ◽  
pp. 1-65
Author(s):  
Yingming Qu ◽  
Yixin Wang ◽  
Zhenchun Li ◽  
Chang Liu
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Surface Topography ◽  
Density Perturbation ◽  
Second Order ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
First Order ◽  
Time Migration ◽  
Grid Scheme

Seismic wave attenuation caused by subsurface viscoelasticity reduces the quality of migration and the reliability of interpretation. A variety of Q-compensated migration methods have been developed based on the second-order viscoacoustic quasidifferential equations. However, these second-order wave-equation-based methods are difficult to handle with density perturbation and surface topography. In addition, the staggered grid scheme, which has an advantage over the collocated grid scheme because of its reduced numerical dispersion and enhanced stability, works in first-order wave-equation-based methods. We have developed a Q least-squares reverse time migration method based on the first-order viscoacoustic quasidifferential equations by deriving Q-compensated forward-propagated operators, Q-compensated adjoint operators, and Q-attenuated Born modeling operators. Besides, our method using curvilinear grids is available even when the attenuating medium has surface topography and can conduct Q-compensated migration with density perturbation. The results of numerical tests on two synthetic and a field data sets indicate that our method improves the imaging quality with iterations and produces better imaging results with clearer structures, higher signal-to-noise ratio, higher resolution, and more balanced amplitude by correcting the energy loss and phase distortion caused by Q attenuation. It also suppresses the scattering and diffracted noise caused by the surface topography.

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