The PEFs used here are time-space domain non-stationary filters to cope with the variability of seismic data with time, offset and shot position. Implementing non-stationary filters is not an easy task. A possible solution is to break-up the dataset into patches and estimate a filter for each patch. However, reassembling these patches create edge effects in the overlapping zones. Alternatively, non-stationary convolution or combination can be used to estimate one filter per data point Margrave (1998) or more realistically, per micro-patch Crawley (2000). A micro-patch is made of a succession of data points that share the same filter. With this technique, the filters can vary smoothly across the dataset while leaving almost no edge effect.

A detailed description of the non-stationary filters is given in Appendix . Here I present the most important steps of the filter estimation procedure only. Calling the non-stationary combination matrix Margrave (1998) with the data vector from which we want to estimate the filters and the unknown PEFs coefficients, one way to estimate the PEFs is to minimize the length of residual vectors Claerbout and Fomel (2002):

(45) |

(46) |

If one filter is used per data point, the matrix of unknown coefficients can be enormous, making the problem very underdetermined. This difficulty can be overcome in two ways. Firstly, the filter is kept constant inside a micro-patch. Secondly, a smoothing operator is introduced to penalize strong variations between neighboring filters. Both strategies are considered here. Introducing the Laplacian operator , equation () is augmented as follows:

(47) |

(48) |

The amplitudes of seismic data vary across offset, shot and time. Large amplitude variations can be troublesome with least-squares inversion because they tend to bias the final result Claerbout (1992) by ignoring or over-fitting some areas of the data. Therefore, it is important to make sure that these amplitude variations do not affect our processing. One solution is to apply a weight to the data prior to the inversion like Automatic Gain Control (AGC) or a geometrical spreading correction. Alternatively, a weight can be incorporated inside the fitting goals in equation ():

(49) |

(50) |

Figure 1

filters (Figure ) and demonstrate that 3-D filters lead to the best noise attenuation results. When 2-D filters are used, the multiple attenuation is performed on one shot gather at a time. When 3-D filters are used, the multiple attenuation is performed on one macro-gather at a time. A macro-gather is a cube made of adjacent shots with all the offsets and time samples. There is an overlapping zone of five shot gathers between neighboring macro-patches. When the multiple attenuation is done, the macro-gathers are reassembled to form the final result. The next section describes how to choose the model in equation () when PEFs for multiples and primaries are estimated.

5/5/2005