Mono-frequency Green's functions

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 $45^\circ$ extrapolation operator which will accurately model waves propagating at up to $45^\circ$ 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 $90^\circ$ (overturned waves).

The 2-D polar coordinate wave equation,

$$\frac{\partial^2 P }{ \partial r^2 } + \frac{1}{r}\frac{\par... ...eta^2} = \frac {1}{v^2} \frac{\partial^2 P }{ \partial t^2 }$$

can be approximated by a paraxial equation in the frequency domain,

$$\frac{\partial P} {\partial r } =\left( \frac{-1}{r^2} + i \... ...alpha}{4r^2} \frac{\partial^2}{\partial \theta^2 } }\right] P$$

where,

$$\alpha = \sqrt{ \frac{\omega^2}{v^2} - \frac{1}{4r^2} }.$$

This equation can be used for outward extrapolation of a single frequency from some initial radius r₀, to give a solution on the whole $(r,\theta)$ 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, $A(r,\theta,\omega)$ and $\phi_w(r,\theta,\omega)$. Starting from the known phase at r=r₀ I perform a phase unwrapping to obtain the unwrapped phase $\phi(r,\theta,\omega)$. 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.

 

polar-greens
Figure 1 Mono-frequency Green's function in polar coordinates. The medium has a velocity gradient as a function of depth, v0=6000m/s dv/dz = 15 (m/s)/m. Frequency is 50Hz. Left frame is amplitude, right frame is unwrapped phase.

 

rect-greens
Figure 2 Mono-frequency Green's function in rectangular coordinates. The medium has a velocity gradient as a function of depth, v0=6000m/s dv/dz = 15 (m/s)/m. Frequency is 50Hz. Left frame is amplitude, right frame is unwrapped phase.

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.