Discussion

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.