![]() |
![]() |
![]() |
![]() | Downward continuation | ![]() |
![]() |
The basic downward continuation equation for upcoming waves in Fourier space
follows from equation (7.7) by eliminating by using
equation (7.12).
For analysis of real seismic data
we introduce a minus sign because
equation (7.13) refers to downgoing waves
and observed data is made from up-coming waves.
![]() |
---|
inout
Figure 5. Downward continuation of a downgoing wavefield. |
![]() ![]() |
Downward continuation is a product relationship
in both the -domain and the
-domain.
Thus it is a convolution in both time and
.
What does the filter look like in the time and space domain?
It turns out like a cone, that is,
it is roughly an impulse function
of
.
More precisely, it is the Huygens secondary wave source
that was exemplified by ocean waves entering a gap through a storm barrier.
Adding up the response of multiple gaps in
the barrier would be convolution over
.
A nuisance of using Fourier transforms in migration and modeling is that spaces become periodic. This is demonstrated in Figure 7.6. Anywhere an event exits the frame at a side, top, or bottom boundary, the event immediately emerges on the opposite side. In practice, the unwelcome effect of periodicity is generally ameliorated by padding zeros around the data and the model.
![]() |
---|
diag
Figure 6. A reflectivity model on the left and synthetic data using a Fourier method on the right. |
![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | Downward continuation | ![]() |
![]() |