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First consider that any seismic data are the sum of signal (primaries) and
noise (multiples) as follows:
| |
(1) |

where is the signal we want to
preserve and the noise we wish to attenuate.
Now assuming that the multidimensional PEFs and are known for the noise and signal components, respectively (see the following section for
a complete description of the PEFs estimation step), we have

| |
(2) |

by definition of the PEFs. Equations (1) and (2) can
be combined to solve a constrained problem to separate signal from
noise as follows:
| |
(3) |

The noise can be eliminated in the last equation of the fitting goal
(3) by convolving with . Doing so, equation
(3) becomes:
| |
(4) |

Sometimes it is useful to add a masking operator on
the noise and signal residuals and when performing
the noise attenuation. It happens for example when we want to isolate
and preserve parts of the data where no multiples are present. For
instance, a mute zone can be taken into account very easily.
Calling this masking operator, the fitting goals
in equation (4) are weighted as follows:
| |
(5) |

Solving for in a least-squares sense leads to the objective
function
| |
(6) |

where is a trade-off parameter that relates to
the signal/noise ratio. The
least-squares inverse for becomes
| |
(7) |

where (') stands for the adjoint. Soubaras (1994) uses a
very similar approach for random noise attenuation and more recently
for coherent noise attenuation Soubaras (2001) with F-X PEFs.
Because the size of the data space can be quite large,
is estimated iteratively with a conjugate-gradient method.
In the next section, I describe how both and
are estimated.

** Next:** Filter estimation
** Up:** Theory of multiple attenuation
** Previous:** Theory of multiple attenuation
Stanford Exploration Project

5/23/2004