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# Kinematic equations

This section describes the kinematic equation that transforms a subsurface offset-domain CIG to an opening-angle-domain CIG, for the converted-mode case. The derivation follows the well-known equations for apparent slowness in a constant-velocity 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 Ss and Sr 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 full-aperture 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 P-to-S 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 offset-domain CIG into angle-domain CIG for the converted-mode case. This theory is valid under the assumption of constant velocity. However, it remains valid in a differential sense in an arbitrary-velocity 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 slant-stack 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 .