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One question naturally arising when using RWE propagation in
prestack migration is how does one obtain the optimal trade-off
between: i) incorporating wave-propagation effects directly in a more
dynamic coordinate system (e.g. through a ray-traced coordinate
system); and ii) using higher-order extrapolators in coordinate
systems not strictly conformal to the wavefield propagation direction.
Based on our experience, we argue that a parametric coordinate system
(such as tilted Cartesian or elliptic meshes) offers the advantage of
being able to develop analytic extrapolation operators that readily
lend themselves to accurate, high-order, finite-difference schemes.
Importantly, while coordinate systems based on ray-tracing better
conform to the direction of wavefield extrapolation, numerically
generated meshes do not lend themselves as easily to high-order
extrapolators due to the greater number, and spatially variability of,
the mixed-domain coefficients.
A second question worth addressing is how can the elliptic coordinate
approach be extended to 3D prestack shot-profile migration. Appendix A presents
two candidate coordinate systems. The elliptic cylindrical system
extends the 2D elliptic by constant factor in the third dimension and
realizes a fairly basic expression for the extrapolation wavenumber.
This coordinate system, though, does not propagate overturning waves
in the cross-line direction. The second coordinate system, oblate
spheroidal, incorporates more spherical geometry and enables
overturning wave propagation in the cross-line direction, but yields a
more complicated extrapolation wavenumber. As
expected, both of these coordinate systems reduce to the above 2D
elliptic expression for a zero cross-line wavenumber.

** Next:** Concluding Remarks
** Up:** 2D synthetic tests
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Stanford Exploration Project

5/6/2007