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Multiple suppression

Multiple attenuation methods using Radon transforms (RT) are popular and robust Foster and Mosher (1992). These techniques use the moveout discrepancy between primaries and multiples for discrimination. Usually, the multiple attenuation is carried out with Common Midpoint (CMP) gathers after NMO correction Kabir and Marfurt (1999). Then, the NMOed data are mapped with a parabolic Radon transform (PRT) into a domain where primaries and multiples are separable.

One desirable property of a Radon transform is that events in the Radon domain be well focused. This property makes the signal/noise separation much easier and decreases the transformation artifacts. These artifacts come essentially from the null-space associated with the RTs Thorson and Claerbout (1985). The RTs can be made sparse in the Fourier domain Hugonnet et al. (2001) or in the time domain Sacchi and Ulrych (1995). The Fourier domain approach has the advantage of allowing fast computation of the Radon panel Kostov (1990). However, the sparse condition developed so far Hugonnet et al. (2001) does not focus the energy in the time axis. Therefore, in our implementation of the RTs, we opted for a time domain formulation with a Cauchy regularization in order to enforce sparseness.

A generic equation for a Radon transform in the angle domain is  
z(q,\gamma)=z_0+q \; g(\gamma),\end{displaymath} (1)
where z0 is the zero-angle depth, $\gamma$ is the scattering angle, q is a curvature parameter, and $g(\gamma)$ is a function that represents the moveout in the CIGs. The modeling equation from the Radon domain to the image domain and its adjoint are
d(z,\gamma) &=& \sum_{z_0}\sum_{q} m(z_0,q) 
 \delta \left[ z_0...
 \delta \left[ z -\left(z_0+q\;g(\gamma)\right) \right].\end{eqnarray} (2)
At first order, we can assume that $g(\gamma)=\gamma^2$, which shows that Equation (1) corresponds to the definition of a parabola. However, for angle-domain common image gathers, Biondi and Symes (2003) demonstrate that a better approximation is $g(\gamma)=tan(\gamma)^2$.

Figure 3
Radon transform of angle-domain CIGs using the parabolic equation (left) and the tangent equation (right). Though still an approximation, the tangent equation focuses better the events in the Radon domain.
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Equation (2) can be rewritten as  
,\end{displaymath} (4)
where ${\bf d}$ is the image in the angle domain, ${\bf m}$ is the image in the Radon domain, ${\bf L}$ is the forward RT operator. Our goal now is to find the vector ${\bf m}$ that best synthesizes in a least-squares sense the data ${\bf d}$ via the operator ${\bf L}$.Therefore, we want to minimize the objective function:  
f({\bf m}) = \Vert{\bf Lm-d}\Vert^2.\end{displaymath} (5)
We also add a regularization term that enforces sparseness in the model space ${\bf m}$. High resolution can be obtained by imposing a Cauchy distribution in the model space Sacchi and Ulrych (1995):  
f({\bf m}) = \Vert{\bf Lm-d}\Vert^2 + \epsilon^2 \sum_{i=1}^n ln( b +m_i^2),\end{displaymath} (6)
where n is the size of the model space, ${\epsilon}$ and b two constants 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
{\bf \hat{m}} = 
\left [
 {\bf L'L}+\epsilon^2 {\bf diag}(1/(b+m_i^2))
\right ]^{-1}{\bf L'd},\end{displaymath} (7)
where ${\bf diag}$ defines a diagonal operator. Because the model or data space can be large, we estimate ${\bf m}$ iteratively. The objective function in Equation (6) is non-linear because the model appears in the definition of the regularization term. Therefore, we use a limited-memory quasi-Newton method Guitton and Symes (1999) to find the minimum of $f({\bf m})$.

From the estimated model ${\bf m}$, we separate the multiples from primaries in the Radon domain, using their distinct q values. We transform the multiples back to the image domain by applying ${\bf L}$, and subtract them from the input data to obtain multiple-free gathers.

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