Examples with real data

We will now illustrate the method with an Angle-Domain Common-Image Gather (ADCIG) from a seismic line acquired in the Gulf of Mexico. Figure  shows the ADCIG together with the estimates of the multiples and primaries. The estimate of the multiples was obtained with an apex-shifted Radon transform in the image space Alvarez et al. (2004) and the estimate of the primaries was obtained simply by subtracting it from the data. Notice the residual primary energy just below 3000 m in the estimate of the primaries. Note also the residual energy from the multiples in the estimate of the primaries.

 

adcig1_estimates1
Figure 10 ADCIG (a), initial estimate of the multiples (b), and the primaries (c). Note the crosstalk on both panels.

 

adcig1_matched_prims
Figure 11 Estimated primaries after one (a), five (b) and ten (c) non-linear iterations.

Figure  shows the ADCIG after one, five and ten non-linear iterations. The first iteration attenuates the strongest residual multiples (compare panel (a) of Figure  with panel (c) of Figure ).

 

adcig1_matched_muls
Figure 12 Estimated multiples after one (a), five (b) and ten (c) non-linear iterations.

Subsequent iterations further reduce the residual multiples. Also, although hard to see in the hard copy, the primary energy that contaminated the estimate of multiples below 3000 m has been mapped back to the primaries. Figure  shows the corresponding results for the multiples. Notice again that the residual primary energy has been severely attenuated.

As a final example, consider the problem of separating ground-roll from body waves in land data. This a more challenging application of our implementation of the algorithm because the signal has curvature that changes rapidly with both offset and time so to match it we need small filters in relatively small patches. The ground-roll, on the other hand, has little global curvature (although it may have strong local curvature due to aliasing) and matching it is more successful with large filters in large patches. Figure  shows the original shot as well as the initial estimates of the body waves and the ground-roll.

 

shot1_estimates1
Figure 13 Land shot gather with strong ground-roll (a), initial estimate of ground-roll (b), and body waves (c).

 

shot1_matched_bw
Figure 14 Estimate of body waves after one non-linear iteration (a), after 5 non-linear iterations (b) and after 20 non-linear iterations (c). Notice how after the fifth iteration the ground-roll is essentially gone.

The ground-roll estimate was computed simply by high-cut filtering the data to 24 Hz using a Butterworth filter with six poles. We allowed significant energy from the body waves to leak into the estimate of the ground-roll to illustrate the problem described in the previous paragraph. Similarly, the estimate of the body waves was computed by low-cut filtering the data to 18 Hz also with a Butterworth filter with 6 poles. Since we don't want to reduce the low frequency components of the signal too much, we allowed strong ground-roll to leak into the estimate of the body waves. The purpose is to eliminate this ground-roll without hurting the signal and ideally, mapping back some of the body-waves from the estimate of the ground-roll.

 

shot1_matched_gr
Figure 15 Estimate of ground-roll after one non-linear iteration (a), after 5 non-linear iterations (b) and after 20 non-linear iterations (c). Some of the body waves have been removed in panel (c) but much still remains.

Figure  shows the estimate of the body-waves after one, five and 20 non-linear iterations of the proposed algorithm. Even after just the first iteration, most of the ground-roll has been eliminated and after five iterations it is almost gone. For this example we used just two patches in time and one in offset. Figure  shows similar results for the ground-roll. Since the patches were so large, the energy of the leaked body-waves were only slightly attenuated (see the reflector at about 1.7 secs). This energy was mapped back to the estimate of the body-waves.