Regularization of the Least-Squares Problem

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 $\bold m_i$, at fixed $\tau$. 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:  

$$\bold r_m^{[1]}(\tau,x,i) = m_i(\tau,h_p) - m_{i+1}(\tau,x).$$ (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:  

$$\bold r_m^{[2]}(\tau,x,i) = m_i(\tau,x) - m_i(\tau,x+\Delta x).$$ (9)

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