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 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 modeled data with (Figure 2a) and without (Figure 2b) free surface is used. Because the noise and signal PEFs are estimated from almost the exact answer, only 2D filters are estimated. 3D filters can help if the primaries and multiples are correlated in time and offset but uncorrelated across shot position. With 2D filters, the attenuation is performed one shot gather at a time. Figure 5a displays the estimated primaries and Figure 5b the difference with the true primaries (Figure 2b). The bias introduced by the attenuation method is very small. 3D filters would have probably given better results where the difference between Figure 5a and 2b is the strongest (e.g., near 60 kft).

 

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Figure 5 Two constant offset panels at h=1125 m. for the estimated primaries (a) and the difference with the true primaries (b). The true primaries and multiples are used to estimate the PEFs.

Looking now at the same estimated primaries after migration in Figure 6a, we see again that the attenuation gives an excellent result with almost no bias. Some energy is visible in the difference plot in Figure 6b between Figures 6a and 4b where no multiples are actually present. These artifacts come from the fact that the modeled data with and without free surface condition are not perfectly similar where the primaries are located.

 

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Figure 6 (a) Migration result after multiple attenuation when the true primaries and multiples are used to estimate the PEFs. (b) Difference between (a) and Figure 6b. The estimated primaries are almost exact.

From these results it appears that the quality of the multiple attenuation depends essentially on the filters. If the primaries and multiples are known, the primaries with almost no bias are recoverable. 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 be estimated with the auto-convolutional process of the Delft approach. For the primaries, the next section shows that the Spitz approximation gives a very good model if 3D filters are used.