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Multiple attenuation in the Radon domain

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 ${\bf m}$, we estimate the data ${\bf d}$via the operator ${\bf L}$ (the HRT) as follows:  
 \begin{displaymath}
\bold{d}=\bold{Lm}
.\end{displaymath} (1)
The unknown model ${\bf m}$ can be estimated in a least-squares sense. We then minimize the objective function  
 \begin{displaymath}
g({\bf m}) = \Vert{\bf Lm-d}\Vert^2.\end{displaymath} (2)
To increase the separability of the multiples and the primaries in the Radon domain ${\bf m}$, 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  
 \begin{displaymath}
g({\bf m}) = \Vert{\bf Lm-d}\Vert^2 + \epsilon^2 \sum_{i=1}^n ln( b +m_i^2),\end{displaymath} (3)
where n is the size of the model space and ${\epsilon}$ and b two constants to be chosen a-priori: ${\epsilon}$ 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 ${\bf m}$ is
\begin{displaymath}
{\bf \hat{m}} =
\left [
 {\bf L'L}+\epsilon^2 {\bf diag}(1/(b+m_i^2))
\right ]^{-1}{\bf L'd},\end{displaymath} (4)
where ${\bf diag}$ defines a diagonal operator. Because the model or the data space can be large, I estimate ${\bf m}$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 $g({\bf m})$. This method has proven efficient for the minimization of the Huber function Guitton and Symes (1999).

From the estimated model ${\bf m}$, 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 ${\bf L}$, 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 up previous print clean
Next: Multiple attenuation with the Up: Theory of multiple attenuation Previous: Theory of multiple attenuation
Stanford Exploration Project
7/8/2003