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This section describes the kinematic equation that transforms a
subsurface offsetdomain CIG to an openingangledomain CIG,
for the convertedmode case.
The derivation follows the wellknown equations for apparent slowness
in a constantvelocity medium in the neighborhood of the reflection/conversion
point. Our derivation is consistent with those presented by
Fomel (1996);Sava and Fomel (2000); and
Biondi (2005).
The expressions for the partial derivatives of the total
traveltime with respect to the image point coordinates are
as follows Rosales and Rickett (2001a):
 

 
 (1) 
Where S_{s} and S_{r} are the slowness (inverse of velocity) at the
source and receiver locations.
Figure illustrates all the angles in this discussion. The angle is the
direction of the wave propagation for the source, and the angle is the
direction of the wave propagation for the receiver.
Through these set of equations, we obtain:
 

 (2) 
We define two angles, and , to relate and
as follows:
 
(3) 
angles
Figure 1 Angle definition for the kinematic
equation of converted mode ADCIGs

 
The meaning of the angles and will become clear later
in the paper; for now, we will refer to as the fullaperture angle.
Through the change of angles presented on equation (3),
and by following basic trigonometric identities,
we can rewrite equations (2) as follows:
 

 (4) 
where,
 
(5) 
and is the velocity ratio, as for example the PtoS velocity ratio.
This leads to quadratic equations for
and as follows:
 

 (6) 
Each equation has two solutions, which are:
 

 (7) 
The first of equation (7) provides the transformation from
subsurface offsetdomain CIG into angledomain CIG for the
convertedmode case.
This theory is valid under the assumption of constant velocity. However, it
remains valid in a differential sense in an arbitraryvelocity medium, by
considering that is the subsurface half offset. Therefore, the limitation of
constant velocity is on the neighborhood of the image. For , it is important
to consider that every point of the image is related to a point on the velocity model with the same coordinates.
In order to implement this equation, we observe that this can be done by an slantstack
transformation as presented on
Figure . Note that
the contribution along the midpoints is a correction factor needed in order to perform
the transformation. This allows us to do the transformation from SODCIGs to ADCIGs
including the lateral and vertical variations of .
sketch
Figure 2 Slant stack angle transformation from SODCIGs to ADCIGs.
This transformation allows lateral and vertical variation of . 
 