Linear scheme

As indicated in the examples, the basic scheme of equation (4) suppress part of the multiples but also interferes with the nearby primaries. Furthermore, the converted waves are suppressed in conjunction with the multiples. The deterioration of the primaries is caused by the fact that some of energy associated with the primaries extends over the windowed region and the operator, as defined in equation (4), acts uniformly over all the window. A possible solution for that is to use a weighting function, designed to privilege the more energetic regions inside the window, which are more likely related to multiples. Equation (4) is replaced in this case by

$$\begin{eqnarray} -{\bf W J N g} & = & {\bf W J N \hat{g}} \\ \nonumber -{\bf W J v} & = & {\bf W J N \hat{g}}, \end{eqnarray}$$ (4)

where $\bf J$ is a diagonal weighting operator chosen to represent an approximation for the noise (multiples) covariance matrix. We have chosen the operator $\bf J$ to be the smoothed semblance velocity spectrum of the original data (Figure -a).