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Multiple attenuation with Radon transforms (RT) are popular and robust
methods Foster and Mosher (1992). These techniques use the
moveout discrepancy between primaries and multiples in order to separate them.
In the method used here, I sort the data into Common Mid Point (CMP)
gathers and remap them with a hyperbolic Radon transform (HRT).
The estimation of the Radon domain can be cast as a linear
transformation Thorson and Claerbout (1985) where, given a model
, we estimate the data *via* the operator (the HRT) as follows:
| |
(1) |

The unknown model can be estimated in a least-squares
sense. We then minimize the objective function
| |
(2) |

To increase the separability of the multiples and the primaries in the
Radon domain , I add a regularization term in equation
(2) that will enforce sparseness in the model space.
The regularization term is a Cauchy function
Sacchi and Ulrych (1995). Thus, we have to minimize the new objective
function
| |
(3) |

where *n* is the size of the model space and and *b*
two constants to be chosen *a-priori*: controls the amount of
sparseness in the model space and *b* relates to the minimum value below which
everything in the Radon domain should be zeroed.
The least-squares inverse of is
| |
(4) |

where defines a diagonal operator.
Because the model or the data space can be large, I estimate iteratively. Note that the objective function is non-linear because
of the logarithm in the regularization term.
Therefore, I use a quasi-Newton method called limited-memory BFGS
Broyden (1969); Fletcher (1970); Goldfarb (1970); Nocedal (1980); Shanno (1970)
to find the minimum of . This method has proven
efficient for the minimization of the Huber function Guitton and Symes (1999).
From the estimated model , I separate the multiples from the
primaries in the Radon domain. In the following examples, the muting function is
identical for each CMP gather.
We transform back the multiples in the image
space by applying , and subtract them from
the input data to obtain multiple-free gathers.

Multiple attenuation with Radon transforms is widely used in the
industry. For complex geology, however, Radon transforms are not
optimal because the moveout discrepancies can be too small or the
moveout of both multiples and primaries can be quite distorted
Matson et al. (1999). Therefore, more sophisticated multiple
attenuation techniques are needed. Two of these techniques are the
Delft approach Verschuur et al. (1992) and the inverse-scattering
method Weglein et al. (1997). In the next section, I describe my
implementation of the Delft approach for multiple attenuation.

** Next:** Multiple attenuation with the
** Up:** Theory of multiple attenuation
** Previous:** Theory of multiple attenuation
Stanford Exploration Project

7/8/2003