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In order to account for the apex-shift of the diffracted
multiples (*h*), we define the forward and adjoint Radon transforms as a
modified version of the ``tangent-squared'' Radon transform introduced
by Biondi and Symes 2003. We define the
transformation from data space (ADCIGs) to model space
(Radon-transformed domain) as:
and from model space to data space as
where *z* is depth in the data space,
is the aperture angle, *z*' is the depth in the model space,
*q* is the moveout curvature
and *h* is the lateral apex shift. In this way, we transform the
two-dimensional data space of ADCIGs, , into a three-dimensional
model space, *m*(*z*',*q*,*h*).
In the ideal case, primaries would be perfectly horizontal in the ADCIGs
and would thus map in the model space to the zero-curvature (*q*=0) plane,
*i*.*e*., a plane of dimensions depth and apex-shift distance (*h*,*z*').
Specularly-reflected multiples would map to the zero apex-shift distance
(*h*=0) plane, *i*.*e*., a plane of dimensions depth and curvature (*q*,*z*').
Diffracted multiples would map elsewhere in the cube depending on
their curvature and apex-shift distance.

** Next:** Sparsity Constraint
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Stanford Exploration Project

5/23/2004