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## A 3-D land data example

I illustrate the weighted PEF estimation on a near-offset section from a 3-D land survey dataset. Figure displays the 3-D data. The noise comes in different flavors. First, we have missing traces, second, we have large amplitude differences from trace to trace (common with land data), and then we have surface waves with strong amplitude variations along the time axis. I first design a weighting function that takes into account the missing traces and correct for the amplitude problems (Figure ). This weighting function is going to be utilized for the PEF estimation. With the time domain formulation, there is no limit in the complexity of the weighting operator since it can mix zero traces with weights of any kind. This would be much more difficult, if not impossible, in the Fourier domain.

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.

data3d
Figure 7
A near-offset section of a 3-D land survey. Some signal is visible near 0.42 s. This section is contaminated with ground-roll. The amplitude varies across time and offset with missing traces as well.

data3dweight
Figure 8
Weighting function for the non-stationary PEFs estimation. Whiter means larger weight. The weight is zero where traces are missing.

pef.comp
Figure 9
PEFs estimated for each patch for the noise and signal. The vertical axis represents the PEF coefficients (the leading one coefficient is omitted) and the horizontal axis represents a patch number (Figure ). (a) Noise PEFs with unweighted residual. (b) Noise PEFs with weighted residual. Note that the noise PEFs are very close to a 1-D second derivative (1,-2,1). (c) Signal PEFs with unweighted residual. (d) Signal PEFs with weighted residual.

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

separ-ns-3d
Figure 11
Signal estimated without weight for the signal and noise PEFs estimation. Some noise remains below 0.42 s.

separ-ns-weight-AGC-3d
Figure 12
Signal estimated with weight for the signal and noise PEFs estimation. The noise is well attenuated. The time slice displays the channel more clearly than in Figure .

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.

separ-ns-AGC-3d
Figure 13
Signal estimated with a weighted noise and signal model for the PEFs estimation.

comp-3d
Figure 14
Difference between Figure and Figure . The signal is better preserved with the weighted residual than with the weighted noise and signal models.

Next: conclusion Up: Signal-noise separation with weighted Previous: A 2-D example
Stanford Exploration Project
7/8/2003