For this 3-D example, I estimate non-stationary 3-D filters. These filters are constant within a patch. For this dataset, I have a total of 90 patches with ten patches in the time axis and three in the crossline and inline directions. More details on the non-stationary filters can be found in Crawley (2000) and Claerbout and Fomel (2002) . The first step consists in estimating the noise and signal non-stationary PEFs. The noise in Figure is mainly ground-roll. A good noise model can be then obtained by low-passing the data and a signal model by high-passing the data.

Figures and show the noise attenuation results with and without residual weighting, respectively. However, a mask has been applied for the PEF estimation in Figure to not include the missing traces. Figure shows better results in the lower part (below 0.42 s.) of the section where some low-frequency noise is still visible. The difference is particularly obvious in the time section where a channel is clearly visible in Figure . One problem with the input data is that the amplitude decreases with time, especially for the noise (Figure ). Therefore, during the signal and noise PEFs estimation steps, most of the solver efforts are directed toward the PEFs where the noise and signal is the strongest, i.e, the upper part. Thus, we end-up with ``good'' PEFs in the top, and ``bad'' PEFs in the bottom where the noise separation is the less efficient. Figure displays the estimated PEFs for the noise and signal with and without residual weight. Note that in Figures a and c the coefficients (especially the first two) vary a lot with the patch number. They almost go to zero for the PEFs at the bottom. On the contrary, in Figures b and d, the PEFs coefficients are much much uniform across time and offset, as expected with the weighted PEF estimation technique.

Figure 7

Figure 8

Figure 9

patch-geom
Geometry of the patches. The
numbers correspond to the vertical axis of Figure .
Figure 10 |

Figure 11

Figure 12

To finish with the 3-D data example, I show in Figure the noise attenuation result when the noise and signal models are weighted prior to the PEF estimation. In that case, the residual is not weighted, but only the data are, i.e., equation (5). Note that the same inversion parameters ( in equation (8), number of iterations, patch geometry, PEF sizes) are used for both cases. This result is also satisfying but the signal is not as well preserved as it is in Figure . I show a difference plot between the two results in Figure . Looking closely at the time slices (upper panel) of Figures and , we see at large black area between 6000 and 4000 meters in the crossline direction and between 2670 and 3670 meters in the inline direction. We can see the same black shape in the difference plot of Figure , thus demonstrating that more noise has been subtracted in Figure . The same conclusion holds for the white area above the black shape in Figures and . These differences prove that weighting the residual is the correct way of handling amplitude problems with seismic data.

Figure 13

Figure 14

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