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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 represent the original CMP gather, its velocity
transform, and *N* the velocity transform operator

| |
(1) |

Then, we want to find the unknown model that minimizes
the objective function
| |
(2) |

where 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 that minimizes
we need to solve
leads to
| |
(3) |

| |

| |

which is a system of linear equations to be solved for the unknown
, where the data is the windowed velocity
transform of the original data. The system can be over- or under determined,
depending on how the operators and are defined, but it will
usually be large enough, since is typically 10^{5} points long. Of
course is one of the solutions of
equation 4.
However the window operator ``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 , the nearest neighbor normal moveout-stack. The
advantages of choosing this operator are its computational speed and
the simplicity of its transpose , which is
needed in the conjugate algorithm (Claerbout, 1991).

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** Up:** Cunha and Claerbout: Multiple
** Previous:** Introduction
Stanford Exploration Project

12/18/1997