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First, I consider that any seismic data is the sum of signal and
noise as follows:
| |
(1) |

where are the seismic data, the signal we want to
preserve and the noise we wish to attenuate.
In the multiple elimination problem, the noise is the multiples
and the signal the primaries.
Now, assuming that we know the multidimensional PEFs and for the noise and signal components, respectively, we have

| |
(2) |

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

We can easily eliminate in the last equation of the fitting goal
(3) by convolving with . Doing so, we
end up with the following fitting goals:
| |
(4) |

For some field data, it might be useful to add a masking operator on
the data and signal residual and in order to perform
the noise attenuation. It happens for example when the noise
appears after a certain time or offset. I call this masking
operator and I weight the fitting goals in equation (5) as
follows:
| |
(5) |

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

where is a constant to be chosen a-priori that relates to
the signal/noise ratio. The
least-squares inverse for becomes
| |
(7) |

where (') stands for the adjoint. Note that since is a diagonal
operator of zeros and ones, we have . It is interesting to note that
is the inverse spectrum of the noise and
is the inverse spectrum of the signal where we perform the
attenuation. Soubaras (1994) uses a
very similar approach for random noise and more recently
for coherent noise attenuation Soubaras (2001) with F-X PEFs.
Because the size of the data space can be quite large, we estimate
iteratively with a conjugate-gradient method.
In the next section, I describe the PEFs estimation method I use to
compute and needed in equation (7).

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** Up:** Theory of multiple attenuation
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Stanford Exploration Project

7/8/2003