More complicated coordinate frames

The principal advantage of polar coordinates is that the computational grid is a better match to the wavefront than a rectangular grid. Because of this we are able to obtain high dip (in the rectangular frame) accuracy with a $45^\circ$ extrapolator. This principle could be carried further.

In a v(z) medium we could use a grid that followed v(z) raypaths, instead of straight radial lines, and had a second coordinate that was along contours of equal traveltime. In this medium, if we had chosen the correct velocity, the wavefront would always conform to the grid and a low angle extrapolator could be used. There are disadvantages to this coordinate system, the wave equation becomes even more complicated, and the derivatives in ``r'' and ``$\theta$'' are not decoupled as they are in the polar system. The tradeoff to be made is between having a more complicated wave equation and being able to use a lower accuracy paraxial approximation.

In a general velocity medium we could use a local coordinate system that followed all the rays and then use a zero degree extrapolator to propagate the energy. This would be something very close to the method of Gaussian beams Cervený et al. (1982). One disadvantage of Gaussian beams is that the energy always propagates along the asymptotic ray. In contrast when using a paraxial wave equation, whatever coordinate system you choose, you allow energy to propagate at an angle to the coordinate system. In a geometrically dispersive medium the asymptotic ray may not be the true ``ray'' at low frequencies Biondi (1992). Using of a paraxial wave equation allows the energy to take different paths at different frequencies.

Another consideration when using Gaussian beams is that you must trace the rays for all the events that you wish to contribute to your model. If you which the model to contain reflection or diffraction energy you must trace rays corresponding to those wavefronts. This could be regarded as an advantage, at least you get to choose which events are contained in your model. However the polar coordinate system requires no ray tracing at all and for many velocity models a $45^\circ$ wave equation on the polar grid will propagate all events of interest accurately.

In general I feel that the greatest jump in quality comes from the change from rectangular coordinates to polar coordinates. Further refinements may be more effort than they are worth. However, this is obviously a subject worthy of further investigation.