Interpolation of Green's functions to all frequencies

The simplest possible interpolation scheme would be to assume that the Green's function could be modeled as a single, non-dispersive, zero phase event. It would then be characterized by an amplitude, A(x,z) and frequency independent traveltime $\tau(x,z)$. To estimate these two parameters we only need the Green's function calculated at one frequency,$G(x,z,\omega_0) = A(x,z,\omega_0)e^{i\phi(x,z,\omega_0)}$. The amplitude is assumed to be the same for all frequencies and the traveltime is simply given by the unwrapped phase divided by the frequency, $\tau(x,z) = \phi(x,z,\omega_0)/\omega_0$. The Green's function for all frequencies is then given by, $G(x,z,\omega) = A(x,z) e^{i \omega \tau(x,z) }$. This simple scheme has little advantage over eikonal equation based methods. It uses the same simple model for the Green's function, it does have the advantage of using a calculation in the seismic data bandwidth (rather than at infinite frequency) but it has the disadvantage that it breaks down rapidly in the presence of multiple events.

The next simplest method will also fail in the presence of multiple arrivals but it provides a more general model of the event. This model assumes that both the amplitude and phase are linear functions of frequency over some bandwidth $\omega_0 \leq \omega \leq \omega_1$. Within this band the Green's function can be modeled by an initial amplitude and phase, and an amplitude and phase gradient. Two Green's functions at frequencies $\omega_0$ and $\omega_1$ are required to fit this model,

$$\begin{eqnarray} A_0(x,z) = A(x,z,\omega_0) \\ \delta_\omega A(x,z) = ( A(x,z,\o... ...( \phi(x,z,\omega_1) - \phi(x,z,\omega_0) )/ (\omega_1-\omega_0 ) \end{eqnarray}$$ (1) (2) (3) (4)

The Greens function at any frequency within this band can then be calculated by,

$$\begin{eqnarray} G(x,z,\omega) = A(x,z,\omega) e^{ i\phi(x,z,\omega) } \\ A(x,z... ...mega) = \phi_0(x,z) + \delta_\omega \phi(x,z) ( \omega -\omega_0 )\end{eqnarray}$$ (5) (6) (7)

This model allows the event to have a constant phase shift (as expected for overturned waves ) and for amplitude to vary as a function of frequency. If this model is used piecewise within the seismic bandwidth it can also be used to model a weakly dispersive event.

Figure  shows a time slice through the Greens' function for the same velocity gradient medium used earlier. Three mono-frequency Green's functions were calculated and used to estimate all the frequencies. The time slices were generated by applying an inverse FFT. The expanding wavefront has overturned waves near the top of the section. Figure  shows a depth slice through the Green's function. The phase changes on the limbs of the hyperbola correspond to the point at which the waves overturn. It is encouraging that all the major features expected in this Green's function have been recovered from an extrapolation of only three frequencies.

 

tsl-greens
Figure 3 Time slice through the full Green's function in a V(z) medium. Note the overturned waves near the top of the panel.

 

zsl-greens
Figure 4 Depth slice through the full Green's function in a V(z) medium. Note the phase change on the limbs of the hyperbola.

When there are multiple arrivals at a particular location both of these simple interpolation methods will fail. In this situation an interpolation scheme based on a multiple event model must be used. This is a subject for future research, I will not discuss it further in this paper.