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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

| |
(1) |

where *z*_{0} is the zero-angle depth, is the scattering angle,
*q* is a curvature parameter,
and 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
| |
(2) |

| (3) |

At first order, we can assume that
, 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 .
**sart
**

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.

Equation (2) can be rewritten as

| |
(4) |

where is the image in the angle domain,
is the image in the Radon domain,
is the forward RT operator.
Our goal now is to find the vector that best
synthesizes in a least-squares sense
the data *via* the operator .Therefore, we want to minimize the objective function:
| |
(5) |

We also add a regularization term that enforces sparseness in the
model space . High resolution can be
obtained by imposing a Cauchy distribution in the model space
Sacchi and Ulrych (1995):
| |
(6) |

where *n* is the size of the model space, and *b*
two constants 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
| |
(7) |

where defines a diagonal operator.
Because the model or data space can be large, we
estimate 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 .
From the estimated model , 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 , and subtract them from
the input data to obtain multiple-free gathers.

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** Up:** Sava and Guitton: Multiple
** Previous:** Angle transform
Stanford Exploration Project

7/8/2003