Next: About this document ...
Up: Shragge: GRWE
Previous: Acknowledgements
- Gazdag, J. and P. Sguazzero, 1984, Migration of seismic data by phase-shift plus interpolation: Geophysics, 69, 124-131.
-
- Guggenheimer, H., 1977, Differential Geometry: Dover Publications, Inc., New York.
-
- Sava, P. and S. Fomel, 2005, Riemannian wavefield extrapolation: Geophysics, 70, T45-T56.
-
- Shragge, J. and P. Sava, 2004, Incorporating topography into wave-equation imaging through conformal mapping: SEP-117, 27-42.
-
- Shragge, J., 2006a, Generalized riemannian wavefield extrapolation: SEP-124.
-
- Shragge, J., 2006b, Structured mesh generation using differential methods: SEP-124.
-
- Stoffa, P. L., J. T. Fokkema, R. M. de Luna Freire, and W. P. Kessinger, 1990, Split-step Fourier migration: Geophysics, 55, no. 04, 410-421.
-
A
The extrapolation wavenumber developed in equation 14
is appropriate for any non-orthogonal Riemannian geometry. However,
there are a number of situations where symmetry or partial
orthogonality are present. Moreover, one may wish to make a kinematic
approximation where all of the imaginary components of the wavenumber
are ignored. These situations are discussed in this Appendix.
3D Semi-orthogonal Coordinate Systems - Semi-orthogonal
coordinate systems occur where one coordinate is orthogonal to the
other two coordinates Sava and Fomel (2005). In these cases the
m13 and m23 components of the weighted metric tensor are
identically zero, which leads to the following extrapolation
wavenumber,
| |
(35) |
where,
| |
(36) |
which are the coefficients recovered by Sava and Fomel (2005).
3-D Kinematic Coordinate Systems - Wave-equation migration
amplitudes are generally inexact in laterally variant media - even in
a Cartesian based system. Hence, one beneficial approximation that
reduces computational cost is to consider only the kinematic terms in
equation 14. This approximate generates the following
extrapolation wavenumber,
| |
(37) |
where,
| |
(38) |
3-D Kinematic Semi-orthogonal coordinate systems - Combining the
two above restrictions yields the following extrapolation wavenumber,
| |
(39) |
where,
| |
(40) |
Note that the expression , and that components of the metric tensor are
significantly simplified.
2-D Non-orthogonal coordinate systems - Two-dimensional
situations are handled by identifying . Hence,
all derivatives in the associated metric tensor gij with respect
coordinate are identically zero. Hence, a 2-D non-orthogonal
coordinate system can be represented by
| |
(41) |
where,
| |
(42) |
2-D Non-orthogonal Kinematic Coordinate Systems -
Two-dimensional kinematic situations are handled through identity
. Again, all derivatives in the associated metric tensor
gij with respect coordinate are identically zero, and
the 2-D non-orthogonal kinematic extrapolation wavenumber is
| |
(43) |
where,
| |
(44) |
2-D Orthogonal Coordinate Systems - Two-dimensional situations
are handled with . Accordingly, all derivatives in the
associated
metric tensor gij with respect coordinate are identically
zero, and the 2-D non-orthogonal coordinate system is represented by
| |
(45) |
where,
| |
(46) |
2-D Orthogonal Kinematic Coordinate Systems - The above two
approximations can be combined to yield the following extrapolation
wavenumber for 2-D orthogonal kinematic coordinate systems,
| |
(47) |
where,
| |
(48) |
Next: About this document ...
Up: Shragge: GRWE
Previous: Acknowledgements
Stanford Exploration Project
4/5/2006