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An instructive example is a coordinate system formed by a shearing
action on a Cartesian mesh (see figure ). A sheared
Cartesian coordinate system is defined by,
 
(21) 
where is the shear angle of the coordinate system
( is Cartesian).
2Dexamp
Figure 1 2D sheared Cartesian
coordinates. Left panel: Physical domain represented by sheared
Cartesian coordinates defined by ; Right panel:
GRWE domain chosen to be the unit square . 
 
This system reduces to a more workable set of two equations,
 
(22) 
that has a metric tensor g_{ij} given by,
 
(23) 
with a determinant and an associated
metric tensor g^{ij} given by,
 
(24) 
Note that because the tensor in equation 24 is
coordinate invariant, equation 10 simplifies to,
 
(25) 
and generates the following dispersion relation,
 
(26) 
Expanding out these terms leads to an expression for wavenumber ,
 
(27) 
Substituting the values of the associated metric tensor in
equation 24 into equation 27 yields,
 
(28) 
A numerical test using a Cartesian coordinate system sheared at
25 from vertical is shown in figure .
The background velocity model is 1500 ms^{1} and the zerooffset
data consist of 4 flat planewaves t=0.2, 0.4, 0.6 and 0.8 s.
As expected, the zerooffset migration results image reflectors at
depths z=300, 600, 900, and 1200 m. Note that the propagation has
created boundary artifacts: those on the left are reflections due to a
truncated coordinate system while those on the right are hyperbolic
diffractions caused by truncated planewaves.
Rays0
Figure 2 Sheared Cartesian coordinate
system test. Coordinate system shear angle and velocity are
and 1500 ms^{1}, respectively. Zerooffset
data consist of 4 flat planewave impulses at t=0.2, 0.4, 0.6 and
0.8 s that are correctly imaged at depths z=300, 600, 900, and
1200 m.

 
Next: Example 2  Polar
Up: Shragge: GRWE
Previous: SplitStep Fourier Approximation
Stanford Exploration Project
4/5/2006