Playing Devil's Advocate: What do the multiples add?  

LSJIMP seeks to exploit another type of multiplicity in the data, that between multiples and primaries. I claimed in Chapter that by adding the model regularization which differences between images (section ), we expect that information from the multiples can fill illumination holes or missing trace and also lead to better discrimination between signal and noise. The veracity of this claim is central to the labeling of LSJIMP as a ``joint imaging'' algorithm. If false, then we conclude that the multiples add nothing to the inversion. I ran a simple test to determine what, if anything, the multiples add to the LSJIMP inversion, I ``turn off'' the regularization which differences across images by setting $\epsilon_2=0$ in equation (). Figures - show the results of this test.

Figure shows the stack of the estimated primaries, $\bold m_0$, with $\epsilon_2=0$, and can be compared directly with Figure . Differences are apparent, although subtle. Generally, we notice a loss of coherency in the estimated multiples (difference panel).

 

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Figure 25 Top: raw data stack. Center: estimated LSJIMP primaries stack, with $\epsilon_2=0$. Bottom: difference panel (estimated multiples) stack. Compare with Figure .

More revealing are Figures and , which show a zoomed view of two regions of Figure , and are directly comparable to Figures and , respectively. Comparing Figure to Figure , we again note a general decrease in estimated multiple coherency when $\epsilon_2=0$. We also can see that in regions where multiples overlap primaries, like at 3.7 seconds/1200 meters, setting $\epsilon_2=0$ leads to some losses in primary energy. Comparing Figure to Figure , we see that setting $\epsilon_2=0$ leads to a generally worse result. Less multiple energy is removed, particularly from some of the salt-related multiples, like TSPLWB and BSPLWB, and again, the subtracted energy is less coherent.

 

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Figure 26 Zoom on Figure , from the sedimentary basin section of the data. Top: raw data stack. Center: estimated LSJIMP primaries stack, with $\epsilon_2=0$. Bottom: difference panel (estimated multiples) stack. Compare with Figure .

 

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Figure 27 Zoom on Figure , from the subsalt section of the data. Top: raw data stack. Center: estimated LSJIMP primaries stack, with $\epsilon_2=0$. Bottom: difference panel (estimated multiples) stack. Compare with Figure .

Finally, Figure compares the result of setting $\epsilon_2=0$ in the prestack sense, at CMP 55 of 750. The left-hand panels compare the estimated primaries with $\epsilon_2=1.0$ and $\epsilon_2=0$, while the right-hand panels compare (after NMO) the data residuals for $\epsilon_2=1.0$ and $\epsilon_2=0$. The panels are split in half vertically and clipped at a different value, labeled on the plot, for display purposes. Comparing the estimated primaries, we see from the small oval that where multiples and primaries overlap, setting $\epsilon_2=0$ reduces the quality of the separation. Primaries are less coherent with offset, and the primary panel contains some energy corresponding to the seabed pure multiple. From the larger oval, notice that for the strongest multiples, setting $\epsilon_2=0$leads to poorer separation. Comparing the data residuals, we see from the top pair of ovals that if $\epsilon_2\gt$, we generally somewhat damage the primaries, which we expect if we have velocity errors, mis-alignment between imaged primaries and multiples, or incorrect reflection coefficient. This issue was discussed earlier, in section . However, we also note from the lower pair of ovals, that setting $\epsilon_2=0$ seems to have reduced our ability to accurately model the important multiples.

 

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Figure 28 Left-hand panels: Estimated LSJIMP primaries at CMP 55 of 750, with $\epsilon_2=1.0$ and $\epsilon_2=0.0$. Right-hand panels: Weighted data residuals at CMP 55 (after NMO) with $\epsilon_2=1.0$ and $\epsilon_2=0.0$. Left-hand and right-hand panels split in half along time axis and clipped independently for display clarity.