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,
This equation can be used for outward extrapolation of a single frequency from some initial radius r0, to give a solution on the whole plane. The initial solution at r=r0 is the solution for a homegenous medium.I choose to express this solution in terms of amplitude and wrapped phase, and . Starting from the known phase at r=r0 I perform a phase unwrapping to obtain the unwrapped phase . The amplitude and phase are then mapped back to the rectangular coordinate frame using a simple bilinear interpolation.
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.
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.