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Most of the treatment that follows is carried out
either in the -*p* or in the - domain. To simplify
the analysis, avoiding the unnecessary intrusion of convolutions along
the axis, the discussion focuses on the points in space
where the wavefield is measured, that is, at the points (*x*,*z* = *z*_{0}) where
*z*_{0} is the cable depth. At these points the bulk moduli and density
are constant and known: *K*(*x*,*z*_{0})=*K*, and
The pressure field at any position of the space can be represented
as the superposition of an upward propagating wavefield with a
downward propagating wavefield :

| |
(5) |

Continuation of these wavefields in the frequency-horizontal wavenumber
domain (-) is controlled by the following equations:
| |
(6) |

| (7) |

where is the vertical wavenumber, which relates to the
horizontal slowness *p* through the dispersion relation
| |
(8) |

To separate these two wavefields, we must make the following assumptions:

- The water-air reflection coefficient
^{} is

- The water surface is nearly horizontal.

- The cable depth is a smooth function of the receiver position.

Under these assumptions, the upcoming and downgoing fields can be related by
a simple time shift equation in the -*p* domain:
| |
(9) |

where *q* is the vertical slowness. When the source is located
below the cable equation (9) can be applied without
restrictions, and the only event for which it doesn't hold perfectly true,
for cases in which the source is above the cable, is the direct wave.
Substituting equation (9) into equation (5),
both in the -*p* domain, we obtain

| |
(10) |

which results in the following equations for separation of the
two wavefields:
| |
(11) |

| |
(12) |

** Next:** Calculating the pressure gradient
** Up:** THEORETICAL BACKGROUND
** Previous:** Relating the pressure and
Stanford Exploration Project

11/18/1997