In an earlier paper I described a method for calculating
mono-frequency Green's functions by using one-way equations in a
polar coordinate frame. This method has the advantage that high
dips can be correctly modeled using a paraxial operator. I use a
extrapolation operator which will accurately model waves
propagating at up to to the *polar* coordinate grid.
These waves may be traveling at much higher angles to the the *
rectangular* coordinate frame and may even be propagating at
angles greater than (overturned waves).

The 2-D polar coordinate wave equation,

can be approximated by a paraxial equation in the frequency domain, where,
This equation can be used for outward extrapolation of a single
frequency from some initial radius *r _{0}*, to give a solution on the
whole plane. The initial solution at

Figure shows the amplitude and unwrapped phase for a polar coordinate Green's function in a medium with velocity that is a linear function of depth. Figure shows the same Green's function mapped back to rectangular coordinates.

Figure 1

Figure 2

All of these calculations are done in parallel for a number of frequencies. If all the frequencies are calculated we then have a complete Green's function. However, calculating all the frequencies can be very expensive so I only calculate a limited number. The missing frequencies must then be estimated from the ones calculated.

11/16/1997