Aliased Data

Ground roll is nearly always spatially aliased, so the relatively unaliased example of Figure 3 is a somewhat unrealistic exception to the practical rule. To inject some realism, we decimated the original 2-D shot gather (Figure 3) by a factor of two in offset, as shown in Figure 8, so that the ground roll is quite aliased. Figure 9 compares the RTT of the decimated data. The results are disappointing. Looking at the v-interpolation without infill panel (top), the human eye can easily interpolate vertically to reconstruct the radial events in RT space. Unfortunately, the v-interpolation panel with infill does not have the desired vertical coherence. In fact, it would seem that the central premise motivating this paper -- that the RTT maps ground roll to zero temporal frequency -- is violated. Figures 10 and 11 are analogous to Figures 6 and 7 -- they are the estimates of signal and noise, respectively. All implementations (v-interpolation with and without infill, and x-interpolation) do an relatively poor job of noise suppression.

A simple way to dealias linear ground roll is to apply a linear moveout (LMO) correction. Figure 12 shows the result of applying a 1.5 km/sec LMO correction to the decimated data of Figure 8. The ground roll is no longer spatially aliased, but the primaries are also no longer ``flat'', as they were originally. As a result, interpolation errors for the x-interpolation RTT will increase. Figure 13 compares the RTT panels for the decimated/LMO'ed data. The ground roll now occupies a higher effective velocity band, and more importantly, is much closer to zero temporal frequency than in Figure 9. The noise suppression achieved (Figure 14) is better than the case in which LMO was not used (Figure 10). As expected, and mentioned above, the x-interpolation RTT leads to severe losses of signal energy, quite a bit more severe than either of the two v-interpolation implementations, as can be seen in Figure 15. Unfortunately, both v-interpolation implementations seem to suffer some small signal losses, which suggests that LMO may actually be ``aliasing'' the primaries by mapping them to low temporal frequency in the RT domain.

 

hectoralias-dat Figure 8 Same 2-D shot gather as Figure 3, only decimated by a factor of two in offset.

 

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Figure 9 Top: v-interpolation without infill. Middle: v-interpolation with infill. Bottom: x-interpolation.

 

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Figure 10 Estimated signal. Top: v-interpolation without infill. Middle: v-interpolation with infill. Bottom: x-interpolation.

 

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Figure 11 Estimated noise. Panels defined as in Figure 10.

 

hectorlmo-dat Figure 12 Decimated 2-D shot gather (Figure 8), after 1.0 km/sec linear moveout correction.

 

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Figure 13 Top: v-interpolation without infill. Middle: v-interpolation with infill. Bottom: x-interpolation.

 

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Figure 14 Estimated signal. Top: v-interpolation without infill. Middle: v-interpolation with infill. Bottom: x-interpolation.

 

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Figure 15 Estimated noise. Panels defined as in Figure 14.