Introduction

In the presence of complex geology where multipathing, illumination gaps, and coherent noise are prevalent, the most advanced techniques need to be used for both preprocessing and imaging. For multiple attenuation, Weglein (1999) shows that current technology may be divided into filtering methods, which exploit the periodicity and the separability (move-out discrepancies) of the multiples, and wavefield methods, which first predict, then subtract the multiples Dragoset and MacKay (1993); Verschuur et al. (1992); Weglein et al. (1997).

Traditionally, filtering techniques are chosen because of their robustness and low cost. However, filtering techniques have some limitations when tackling multiples in complex media. For example predictive deconvolution in the ray parameter domain fails when the water bottom is not flat Treitel et al. (1982). Furthermore, numerous authors Bishop et al. (2001); Matson et al. (1999); Paffenholz et al. (2002) have presented cases where wavefield approaches such as surface-related multiple elimination (SRME) attenuate multiples much better than filtering techniques such as radon-based methods.

Wavefield techniques start usually with a prediction step where surface-related multiples are modeled from the data with Lokshtanov (1999); Wiggins (1988) or without any subsurface information. Then the multiples are subtracted from the data. Both aspects of the multiple attenuation procedure are important but most of today's efforts are concentrated on the prediction, not on the subtraction.

With SRME Verschuur et al. (1992), two significant assumptions are usually made for the subtraction. First, it is assumed that the signal has minimum energy, leading to the adaptive subtraction of the multiples with a $\ell^2$ norm. This assumption might not hold where primaries and multiples interfere Spitz (1999), however. Chapter shows that when primaries are much stronger than multiples, the $\ell^1$ norm should be used instead. Second, it is assumed that the multiples are accurately modeled. This point relies on the acquisition or the interpolation/extrapolation of the data to provide the necessary traces for the prediction step. In practice, however, the data are never acquired densely enough and the interpolation schemes are never perfect, especially with sparse acquisition geometries. For instance, 3-D seismic marine data, although well sampled in the inline direction, lack of crossline offsets. Consequently, the prediction is not as accurate as required. Therefore, other subtraction techniques are desirable when adaptive subtraction fails to recover the primaries and when the multiple model is not precise enough. As stated by Berkhout (2004, personal communication), the subtraction step is the weakest component of SRME, and more work needs to be done in this direction.

A new class of multiple attenuation techniques has recently emerged to circumvent some of the limitations of adaptive subtraction and of the modeling. These techniques are called pattern-based because they discriminate primaries from multiples according to their multidimensional spectra Brown and Clapp (2000); Fomel (2002); Guitton and Cambois (1998); Manin and Spitz (1995). This Chapter presents one implementation of a pattern-based approach based on prediction-error filters (PEFs). By construction, PEFs approximate the inverse spectrum of the data from which they are estimated Burg (1975). Therefore, PEFs can serve as proxies for the patterns of both primaries and multiples and used for signal-noise separation Guitton (2002).

This Chapter begins with a description of non-stationary PEFs estimation. The helical boundary conditions Claerbout (1998); Mersereau and Dudgeon (1974) are used to estimate 2-D and 3-D filters. Both noise (multiples) and signal (primaries) PEFs are needed for the attenuation. Because the signal is usually unknown, a method that needs only the noise and the input data to derive a pattern model for the signal is presented. Having estimated the noise and signal PEFs, the multiple subtraction method (pattern-based) that separates primaries from multiples in a least-squares sense is then described.

To illustrate the efficiency of the pattern-based approach, surface-related multiples are attenuated for the Sigsbee2B synthetic dataset. The results of multiple attenuation are analyzed after migration to assess the effects of the proposed technique on the primaries. This example illustrates that the pattern-based approach can lead to a very good elimination of the multiples if an accurate pattern model for both primaries and multiples is available. In addition, this dataset shows that 3-D PEFs preserve the primaries better than 2-D filters.

Then, adaptive and pattern-based subtractions are compared on a synthetic data example provided by BP. This example proves that when multiples and primaries are spatially uncorrelated (i.e., different patterns), PEFs attenuate multiples better than adaptive subtraction. Finally, I show multiple attenuation results on a Gulf of Mexico 2-D line before and after migration. This last example illustrates that the pattern-based method is robust to model inadequacies.