As shown in Figure , the recorded pressure field can be decomposed into the downgoing and upcoming wavefields at the cable depth

(1) |

and, assuming a perfect reflection at the water's surface,

Let be the extrapolation operator for upcoming waves

then

and since we get

(2) |

where

(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 -*k*_{x} domain, the downward extrapolation operator
for upcoming waves can be expressed by

which can be substituted into the Fourier transform of equation (2) to obtain

(4) |

where

(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 -*k*_{x} plane
we can neglect its contribution for the final, downward continued
wavefield.

1/13/1998