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Visual inspection of Figure 3 motivates the two forms of
regularization utilized in this paper. Find any first-order multiple on the section marked
``NMO for Primaries''. Notice that the corresponding event on the first- and second-order
pseudo-primary panels, originally second- and third-order multiples, respectively, all have
a different moveout. In fact, the only events which are kinematically consistent across all
offsets are the flattened primary and pseudo-primaries. The other events, all crosstalk, are
inconsistent between panels. Therefore, the first regularization
operator seeks to penalize the difference between the , at fixed . To account
for the dissimilarity of the AVO of primaries and multiples, this difference is taken at
different offsets, as defined in equation (4). Written in the form of a model
residual vector, this difference is:
| |
(8) |

The third index in equation (9), *i*, ranges from 0 to *n*_{p}-1, where
*n*_{p} is the highest order multiple modeled in the inversion [see equation (6)].

The second form of regularization used in this paper is the more obvious of the two:
a difference operator along offset. This difference exploits the fact that all
non-primaries are not flat after NMO. Again, we can write this difference in the form
of a model residual vector:

| |
(9) |

The second regularization is applied to all the . A similar approach is used
by Prucha et al. (2001) to regularize prestack depth migration in the angle
domain.

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** Up:** methodology
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Stanford Exploration Project

6/10/2002