Nonlinear scheme

The first few iterations of the CG algorithm lower the energy in the regions where the semblance of the original data was high, but the remaining problem is that the subsequent iterations still focus to much attention in the same places because the weighting function stays the same. To overcome this restriction we need a variable weighting operator that is reevaluated after a given number of steps of the CG solver, resulting in the reformulation of equation (4) as an iterative linearization of a nonlinear problem. Each external step i of the CG solver will work on the residuals $\bf d^i$ defined as  

$$\Delta {\bf d^i} =-{\bf W J^{i-1} ( v - N \: \hat{g}^{i-1})},$$ (5)

where

$${\bf J^{i-1}} = \mbox{semblance}\{ {\bf g^{i-1} = g + \hat{g}^{i-1}} \}.$$

Figure  show the results of the linear and nonlinear semblance weighting schemes. Although with less artifacts in the shallow reflections, and with less deterioration in the primaries (notably the near offset of the primary at 1.6 seconds) relative to the non-weighted algorithm, they fail to eliminate the set of reverberations in the shallow part of the gather. Furthermore, at the near offset traces the multiples are not suppressed, as is usually the case in velocity-domain-based methods.

 

multw01
Figure 3 Results of (a) the multiple suppression using the linear pre-weighted scheme described by equation (4) and (b) multiple suppression using the nonlinear preweighted scheme described by equation (5). In both cases the synthetic dataset of Figure  is used as input.