Estimating biases

A bias is a processing footprint left by the multiple attenuation technique, e.g., edge effects from the non-stationary PEFs. In an ideal but unrealistic case, a model for both the primaries and the multiples might be available. In this case, a bias is also any difference between the true primaries and the estimated primaries after attenuation of the multiples. In this section I demonstrate that the bias is minimum with the pattern-based approach.

For the pattern model of the primaries, the answer, i.e., the data modeled without the free surface condition, is used. For the multiples, the difference between the FS (Figure a) and NFS (Figure b) datasets is used. Because the noise and signal PEFs are estimated from accurate pattern models, only 2-D filters are estimated. 3-D filters can help if the primaries and multiples are correlated in time and offset but uncorrelated across shot position. With 2-D filters, the attenuation is performed one shot gather at a time. Figure a displays the estimated primaries and Figure b the difference with the true primaries (Figure b). The bias introduced by the attenuation method is very small. 3-D filters would have given better results where the difference between Figure a and b is the strongest (e.g., near 20 km).

Looking now at the same estimated primaries after migration in Figure a, we see again that the attenuation gives a very good result with little bias. Some energy is visible in the difference plot in Figure b where no multiples are actually present, however. These artifacts have two origins. First, below 4,000 m, some primaries are affected by the multiple attenuation process, especially at far offset where primaries and multiples overlap. Second, above 4,000 m, the amplitude of the reflections for the sea floor and the top of salt are slightly different between the FS and NFS datasets. These differences are migrating at the reflector positions in Figure b but with a very small energy, however.

From these results it appears that the quality of the multiple attenuation depends essentially on the filters. If accurate models for both the primaries and multiples are available, the primaries are recoverable with their true amplitude. Therefore, we should always try to find the best models for the signal and the noise. In practice, a very accurate model of the multiples can often be estimated with the auto-convolutional process of the Delft approach Verschuur et al. (1992). For the primaries, the next section shows that the Spitz approximation gives a very good model if 3-D filters are used for multiple removal.