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.
Downward continuation is a product relationship in both the -domain and the kx-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 x2+z2 - v2 t2. 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 x.
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.