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Multiple attenuation with a pattern-based approach

Pattern-based techniques have been recently proposed as a way of removing coherent noise Brown and Clapp (2000); Guitton et al. (2001a); Guitton and Cambois (1998); Manin and Spitz (1995); Soubaras (2001). The philosophy behind these methods is that the noise and the signal have different moveouts and/or amplitude behaviors that can be utilized to discriminate and separate them. The patterns can be reliably estimated with prediction-error filters Claerbout and Fomel (2001). In this section, I describe my implementation of a pattern-based technique.

Guitton (2003) describes in detail the pattern-based method used in this paper. The main idea is that the patterns of both the noise ${\bf n}$ and the signal ${\bf s}$ can be estimated with time domain non-stationary prediction-error filters. Thanks to the helical boundary conditions Claerbout (1998); Mersereau and Dudgeon (1974), these filters can have any dimension.

The two important components of the proposed method are the noise PEFs ${\bf N}$ and the signal PEFs ${\bf S}$. Then, in Guitton (2003) I show that the signal can be estimated by minimizing the objective function  
g({\bf s})=\Vert{\bf M(Ns - Nd)}\Vert^2+\epsilon^2\Vert{\bf MSs}\Vert^2\end{displaymath} (7)
where ${\bf M}$ is a masking operator that preserves the signal where no multiples are present and ${\bf d}$ is the data vector (primaries plus multiples). As demonstrated by Abma (1995), this approach is similar to Wiener filtering. Therefore, the noise and signal are separated according to their multivariate spectra that we approximate with non-stationary PEFs. One important assumption made is that the noise and signal are uncorrelated. If they are correlated, then complications arise like estimating ${\epsilon}$ accurately or computing the cross-sprectrum between the noise and the signal.

To estimate the noise and signal PEFs, we need to derive a model of the noise and a model of the signal. For the multiple attenuation problem, the model given by the Delft approach is the best choice. This choice is not restrictive because we could use other techniques to estimate a model of the multiples. For instance, Haines et al. (2003) show that a Radon transform can give a satisfying model of the multiples while improving on the Radon demultiple result. Another important assumption made with the pattern-based approach is that the model has accurate kinematics and amplitudes. It is well established that with one iteration only, the Delft approach does not properly model amplitudes for high-order multiples Guitton et al. (2001b); Wang and Levin (1994). Guitton (2003) shows that 3-D filters cope better with modeling inaccuracies than 2-D filters. Once the multiple model is computed and the noise PEFs ${\bf N}$ are estimated, I convolve ${\bf N}$ with the data ${\bf d}$ to obtain a signal model from which I estimate the signal PEFs ${\bf S}$ (Spitz, 2001, personal communication).

The proposed pattern-based approach works very well. I show in Guitton (2003) a very detailed comparison of this technique for 2-D and 3-D filters. In the next section, I compare the three multiple attenuation techniques, i.e., Radon, Delft and the pattern-based approach on a Gulf of Mexico dataset provided by WesternGeco. This example proves that the best multiple attenuation result is obtained with the pattern-based approach and 3-D filters.

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