GENERAL FORMULATION OF THE METHOD

Multiple suppression in the velocity domain can be formulated as an optimization problem. The aim is to find some ``model" in the space-time domain, which, when added to the original CMP gather, minimizes the power within a pre-specified window in the velocity-time domain. The precise mathematical formulation follows.

Let $\bf g$ represent the original CMP gather, $\bf v$ its velocity transform, and N the velocity transform operator  

$${\bf v} = {\bf N g}.$$ (1)

Then, we want to find the unknown model $\bf \hat{g}$ that minimizes the objective function  

$${\cal O} = \vert\vert {\bf W N (g + \hat{g})} \vert\vert^2,$$ (2)

where $\bf W$ is a windowing operator that selects the region of the velocity spectrum in which the energy of the multiples is dominant. To find the solution $\bf \hat{g}$ that minimizes ${\cal O}$ we need to solve

$${d {\cal O} \over d {\bf \hat{g}}} = 0$$

leads to

$$\begin{eqnarray} -{\bf W N g} & = & {\bf W N \hat{g}} \\ \nonumber -{\bf W v} & = & {\bf W N \hat{g}} \\ {\bf d} & = & {\bf A m}, \nonumber\end{eqnarray}$$ (3)

which is a system of linear equations to be solved for the unknown ${\bf m} = {\bf \hat{g}}$, where the data $\bf d$ is the windowed velocity transform of the original data. The system can be over- or under determined, depending on how the operators $\bf W$ and $\bf N$ are defined, but it will usually be large enough, since $\bf m$ is typically 10⁵ points long. Of course ${\bf \hat{g}}= -{\bf g}$ is one of the solutions of equation 4. However the window operator $\bf W$ ``transfer" the events that map outside the window (which are predominantly primaries) to the null space. The use of conjugate gradients (CG) seems to be a suitable choice for solving this problem especially because a few iterations would not only be enough but would also contribute to the stabilization of the process, when noise is present. We have chosen to use a very simple form for the velocity transform operator $\bf N$, the nearest neighbor normal moveout-stack. The advantages of choosing this operator are its computational speed and the simplicity of its transpose ${\bf N^{\prime}}$, which is needed in the conjugate algorithm (Claerbout, 1991).