As shown in Figure , the recorded pressure field can be decomposed into the downgoing and upcoming wavefields at the cable depth
Let be the extrapolation operator for upcoming waves
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
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.