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When the noise and signal PEFs have been determined, the task of multiple
attenuation follows. First consider that the seismic data
are the sum of signal (primaries) and noise (multiples):
| |
(58) |

where is the signal we want to
preserve and the noise we wish to attenuate.
By definition of the signal and noise PEFs,
the following relationships hold:

| |
(59) |

Equations () and () can
be combined to solve a constrained problem to separate signal from
noise as follows:
| |
(60) |

The scalar is related to the S/N ratio. In practice,
however, is estimated by trial and error. Replacing by in equation () yields
| |
(61) |

Sometimes it is useful to add a masking operator for
the noise and signal residuals and when performing
the noise attenuation. For example, in areas of the data
where no multiples are present, the signal should be preserved. For
instance, a mute zone can be taken into account very easily.
Calling this masking operator, the fitting goals
in equation () are weighted as follows:
| |
(62) |

Solving for in a least-squares sense leads to the objective
function
| |
(63) |

It is interesting to look at the least-squares inverse for :
| |
(64) |

where (') stands for the adjoint. Because is a diagonal
matrix of zeros and ones, . In practice, all computations
are done in the time domain. In the Fourier domain, equation
() demonstrates that the least-squares estimate of is obtained by combining the spectra of both noise and signal.
Abma (1995) shows that equation () is
similar to Wiener filtering for random noise attenuation.
Soubaras (1994) uses a very similar approach for random noise
attenuation and more recently for coherent noise attenuation
Soubaras (2001).
Because the size of the data space can be quite large,
is estimated iteratively with the conjugate-gradient method.
Therefore, multiple attenuation with prediction-error filters (in 2-D or 3-D) is
two-step process where (1) noise and signal PEFs are estimated and
(2) signal and noise are separated according to their multidimensional
spectra. The next section illustrates this technique with the
Sigsbee2B dataset.

** Next:** Multiple attenuation with the
** Up:** Multiple attenuation: Theory and
** Previous:** Signal and noise PEFs
Stanford Exploration Project

5/5/2005