(12) | ||

(13) |

The basic downward continuation equation for upcoming waves in Fourier space
follows from equation (7) by eliminating *p* by using
equation (12).
For analysis of real seismic data
we introduce a minus sign because
equation (13) refers to downgoing waves
and observed data is made from up-coming waves.

(14) |

Figure 5

Downward continuation is a product relationship
in both the -domain and the *k*_{x}-domain.
Thus it is a convolution in both time and *x*.
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 *x ^{2}*+

A nuisance of using Fourier transforms in migration and modeling is that spaces become periodic. This is demonstrated in Figure 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.

Figure 6

12/26/2000