The first test, shown in Figure 3, was performed on a CMP gather from a 2-D dataset acquired by WesternGeco in the Gulf of Mexico. Characterized by strong water-bottom and top-of-salt multiple reflections, this dataset was the focus of the 1997 SEG multiple attenuation workshop. To model the signal and noise, I adopted a simple approach: given a random zero-offset section and a stacking velocity, apply inverse NMO and inverse NMO for first-order water-bottom multiples Brown (2002) to create a signal and noise models, respectively.
The panels on the top row of Figure 3 have been NMO-corrected with the stacking velocity, to facilitate comparison. Estimated primary reflections should be flat. We can see from the estimated signal and noise panels that my approach has produced an excellent separation result. Many totally obscured primaries now appear from beneath the multiple train. To dealias the data, I applied NMO with water velocity (4900 ft/sec). The effects of this step can be seen in the panels on the prior models and estimated dips panels of Figure 3. In this case, I fixed the signal slope, not the noise slope.
The method performed less impressively at near offsets, where the signal and noise dips both tend to 0. In these regions, the inversion simply ``splits the difference,'' according to the parameter in equation (17).
I conducted a second multiple separation test on CMP gather taken from the ``Mobil AVO'' dataset Lumley et al. (1994). The gather is characterized by a strong train of first order water-bottom multiples, as well as strong primary events partially concealed under the noise. These deep primaries provide an excellent benchmark to test the signal preservation characteristics of my method.
The separation results are shown in Figure 4. As with the previous example (Figure 3), inverse NMO was used to create the prior models, and the prior signal slope was fixed in the slope estimation step. Again, the separation results are quite impressive. The embedded primary events appear to be perfectly preserved, and most of the reverberations are segregated to the noise panel. We expect the same near-offset behavior as the previous example, though the results in this region look plausible. Like before, the data were dealiased with an NMO-correction of 2.0 km/sec.
My final test was conducted on a 2-D receiver line extracted from a 3-D shot gather acquired by Saudi Aramco. While the ground roll looks impossibly strong to conceal any extractable information, there is indeed no shortage of primaries under the noise cone.
The signal and noise are approximately, but not perfectly, separable in temporal frequency Brown et al. (1999). To obtain an approximate noise model, I applied a lowpass filter with a cutoff of 10 Hz. Conversely, I obtained a signal model by applying a highpass filter with a 35 Hz cutoff. Unlike the previous two examples, I treated the noise slope as fixed in the slope estimation step. In this case, the noise is simpler than the signal. We have confidence in the noise slope; everything else is treated as signal.
While the results are not as impressive, they are good nontheless. It is not difficult to find many coherent primary events that have been unmasked from under the strong ground roll. Notice that near zero offset, some noise has leaked into the signal model. Although the data were again dealiased with an NMO correction (decreasing velocity), the noise is still spatially aliased at far offsets, a fact confirmed by a look at the estimated noise slope. Th separation results are visibly compromised in those regions. Furthermore, a persistent ``ringing'' is present around zero offset. Fomel (2000, 2001a,b) solved the problem by supplementing the noise decorrelation filter [ in equation (17)] with a 3-point notch filter.