As shown in Figure , the recorded pressure field can be decomposed into the downgoing and upcoming wavefields at the cable depth
(1) |
Let be the extrapolation operator for upcoming waves
then
(2) |
(3) |
The operator defined in equation (3) performs
two distinct processes: the conversion
of the recorded pressure field into the upcoming pressure field, and its
downward continuation to the sea bottom.
In the -kx domain, the downward extrapolation operator for upcoming waves can be expressed by
(4) |
(5) |
It is evident from equations (3) and (5) that the operator has poles at , corresponding to vertical wavelengths of .
For waves with these wavelengths, the downgoing and upcoming fields will cancel each other at the cable depth (assuming perfect reflection at the water's surface), and the result is a null recorded wavefield. Of course no information can be obtained for the upcoming field in this case, but since it represents only a couple of lines in the -kx plane we can neglect its contribution for the final, downward continued wavefield.