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# Example 1 - 2-D Sheared Cartesian Coordinates

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 2-D 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 gij given by,
 (23)
with a determinant and an associated metric tensor gij 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 zero-offset data consist of 4 flat plane-waves t=0.2, 0.4, 0.6 and 0.8 s. As expected, the zero-offset 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 plane-waves.

 Rays0 Figure 2 Sheared Cartesian coordinate system test. Coordinate system shear angle and velocity are and 1500 ms-1, respectively. Zero-offset data consist of 4 flat plane-wave 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: Split-Step Fourier Approximation
Stanford Exploration Project
4/5/2006