scholarly journals Wave-equation migration in generalized coordinate systems

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. Z79-Z80
Author(s):  
Jeffrey Shragge
Keyword(s):  
Wave Equation ◽  
Coordinate Systems ◽  
Generalized Coordinate

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.

An affine generalized optimal scheme with improved free-surface expression using adaptive strategy for frequency-domain elastic wave equation

Geophysics ◽  
2022 ◽  
pp. 1-71
Author(s):  
Shu-Li Dong ◽  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Free Surface ◽  
Elastic Wave ◽  
Rotation Angle ◽  
Surface Expression ◽  
Coordinate Systems ◽  
Elastic Wave Equation ◽  
Point Scheme ◽  
Sampling Intervals ◽  
Grid Points

Effective frequency-domain numerical schemes were central for forward modeling and inversion of the elastic wave equation. The rotated optimal nine-point scheme was a highly used finite-difference numerical scheme. This scheme made a weighted average of the derivative terms of the elastic wave equations in the original and the rotated coordinate systems. In comparison with the classical nine-point scheme, it could simulate S-waves better and had higher accuracy at nearly the same computational cost. Nevertheless, this scheme limited the rotation angle to 45°; thus, the grid sampling intervals in the x- and z-directions needed to be equal. Otherwise, the grid points would not lie on the axes, which dramatically complicates the scheme. Affine coordinate systems did not constrain axes to be perpendicular to each other, providing enhanced flexibility. Based on the affine coordinate transformations, we developed a new affine generalized optimal nine-point scheme. At the free surface, we applied the improved free-surface expression with an adaptive parameter-modified strategy. The new optimal scheme had no restriction that the rotation angle must be 45°. Dispersion analysis found that our scheme could effectively reduce the required number of grid points per shear wavelength for equal and unequal sampling intervals compared to the classical nine-point scheme. Moreover, this reduction improved with the increase of Poisson’s ratio. Three numerical examples demonstrated that our scheme could provide more accurate results than the classical nine-point scheme in terms of the internal and the free-surface grid points.

Riemannian wavefield extrapolation: Nonorthogonal coordinate systems

Geophysics ◽  
2008 ◽  
Vol 73 (2) ◽  
pp. T11-T21 ◽  
Author(s):  
Jeffrey Chilver Shragge
Keyword(s):  
Wave Propagation ◽  
Spherical Geometry ◽  
Coordinate Systems ◽  
Propagation Theory ◽  
Mesh Smoothing ◽  
Computational Overhead ◽  
Wavefield Extrapolation ◽  
2D And 3D ◽  
Smoothing Procedure ◽  
Generalized Coordinate

Riemannian wavefield extrapolation (RWE) is used to model one-way wave propagation on generalized coordinate meshes. Previous RWE implementations assume that coordinate systems are defined by either orthogonal or semiorthogonal geometry. This restriction leads to situations where coordinate meshes suffer from problematic bunching and singularities. Nonorthogonal RWE is a procedure that avoids many of these problems by posing wavefield extrapolation on smooth, but generally nonorthogonal and singularity-free, coordinate meshes. The resulting extrapolation operators include additional terms that describe nonorthogonal propagation. These extra degrees of complexity, however, are offset by smoother coefficients that are more accurately implemented in one-way extrapolation operators. Remaining coordinate mesh singularities are then eliminated using a differential mesh smoothing procedure. Analytic extrapolation examples and the numerical calculation of 2D and 3D Green’s functions for cylindrical and near-spherical geometry validate the nonorthogonal RWE propagation theory. Results from 2D benchmark testing suggest that the computational overhead associated with the RWE approach is roughly 35% greater than Cartesian-based extrapolation.

The wave equation, O(2, 2), and separation of variables on hyperboloids

1978 ◽  
Vol 79 (3-4) ◽  
pp. 227-256 ◽  
Author(s):  
E. G. Kalnins ◽  
W. Miller
Keyword(s):  
Wave Equation ◽  
Harmonic Analysis ◽  
Separation Of Variables ◽  
Beltrami Operator ◽  
Eigenvalue Equation ◽  
Coordinate Systems ◽  
Group Manifold ◽  
Orthogonal Systems

SynopsisWe classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid = 1, finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.

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