Examples with synthetic data

As a first example, we will consider the synthetic dataset shown in panel (a) of Figure . There are two primaries (black) and four multiples (white). The traveltimes of both primaries and multiples were computed analytically from a three-layer model: water layer, a sedimentary layer and a half space. The estimates of the multiples (b) and primaries (c) were computed by adding 40% of the primaries to the multiples and 40% of the multiples to the primaries, respectively. The goal is to simulate a situation in which the kinematics of the estimates of primaries and multiples are both correct but there is strong cross-talk (leakage) which may happen, for example, after the multiples are estimated via filtering in the Radon domain.

 

syn1_estimates1
Figure 1 Synthetic CMP gather (a) showing two primaries (black) and four multiples from a three horizontal layer model. The initial estimates of multiples (b) and primaries (c) are contaminated with 40% cross-talk.

 

syn1_matched_muls
Figure 2 Matched estimates of multiples after one (a), three (b) and five (c) non-linear iterations of the algorithm.

 

syn1_matched_prims
Figure 3 Matched estimates of primaries after one (a), two (b) and three (c) non-linear iterations of the algorithm.

Figure  shows the estimated multiples after one, two and three non-linear iterations of the algorithm. The corresponding results for the estimated primaries are shown in Figure . In both figures we see that the cross-talk is substantially reduced after the first non-linear iteration and is completely eliminated after the third. Notice the hole in the top multiple and the bottom primary in the final estimates. This is actually present in the data (panel (a) in Figure ) and is an artifact because both primaries and multiples were modeled with the same amplitude and opposite polarity. Consider now the more realistic situation of kinematic and offset-dependent amplitude errors and noise as shown in Figure . The multiple and primary estimates were obtained via migration-demigration as described in a previous report (). Clearly these are imperfect estimates with cross-talk on primaries and multiples and other noises. Panel (a) of Figures  and  show the results after one iteration of the non-linear inversion whereas panels (b) and (c) of the same figures show the results after three and five non-linear iterations respectively. There is still some localized cross-talk from the multiples into the primaries, but given how imperfect the initial estimates were, the result is encouraging.

 

syn2_estimates1
Figure 4 Original CMP gather (a), initial estimate of multiples (b) and primaries (c).

 

syn2_matched_muls
Figure 5 Matched estimates of multiples after one (a), three (b) and five (c) non-linear iterations of the algorithm.

 

syn2_matched_prims
Figure 6 Matched estimates of primaries after one (a), three (b) and five (c) non-linear iterations of the algorithm.

The next example uses the well-known Sigsbee model to illustrate the method in the image space on an angle stack. The estimate of the multiples was computed with an image space version of SRME (). Figure  shows the modeled data after shot profile migration (a) and the estimated multiples (b). Both panels are plotted at the exact same clip value. Notice that the estimate of the multiples is accurate only in kinematics, not in amplitudes or frequency content.

 

sgsb_estimates1
Figure 7 Migrated angle stack of the Sigsbee model (a), initial estimates of multiples (b).

 

sgsb_matched_prims
Figure 8 Estimated primaries after one (a), two (b) and three (c) iterations of the non-linear inversion.

 

sgsb_matched_muls
Figure 9 Estimated multiples after one (a), two (b) and three (c) iterations of the non-linear inversion.

In contrast with the previous examples, in this case we do not have an independent initial estimate of the primaries. We could subtract the estimates of the multiples from the data, but the corresponding estimate of the primaries is too distorted and using it actually hurts the chances of matching the multiples to the data (since this example is in the image space, ''data'' means the migrated image with primaries and multiples). Other option is to use the data itself as the initial estimate of the primaries. We found, however, that a better alternative is to do a first iteration setting $\mu=0$, meaning only the multiples need to be matched. Once matched, the multiples are subtracted from the data to get the estimate of the primaries for the next iteration Figures  and  show a close-up view of the matched primaries and multiples, respectively, after one, two and three non-linear iterations. After the first iteration, the most obvious multiples contaminating the estimate of the primaries have been attenuated (compare panels (a) of Figures  and ). The second iteration helps attenuate the multiples further, although is hard to appreciate in these small figures. See, for example the multiple inside the salt and in the bottom right corner of panel (b). The third iteration does not help appreciably. The remaining multiples have too much dip and would require a long filter that could also match the primaries. On the estimate of the multiples, again the first iteration extracts the most significant multiples and the second iteration locally correct the amplitudes. The third iteration actually hurts the estimate of the multiples because the effect of the regularization term becomes significant as the match of both the primaries and the multiples to the data improves. The net result is an estimate of the primaries that is close to the primaries in the original image. The estimate of the multiples, however, is weaker than it should. If for some reason we wanted the multiples, we could subtract the estimated primaries from the original image.