A trace with input offset vector and midpoint position is first transformed to its corresponding CCP position and zero offset. By defining the new offset and azimuth position and by applying inverse PS-DMO, we transform the data to a new CCP position and its corresponding CMP position.
Here, we follow the same procedure as Biondi et al. (1998); Fomel and Biondi (1995) for the derivation of the PS-AMO operator.
First, we refer to equations (1) and (2) in order to understand the relationship between CMP and CCP for the 3D case. We rewrite equation (2) as
where is the angle between the midpoint vector () and the transformation vector ().
We can then rewrite equation (1) as
is an extension of and lies in the CCP space. Figure 1 shows both and in the same plane. Since the vectors are parallel, the angle between and is the same as the angle between and .If the coordinate system is aligned with the midpoint coordinates, then the angle is the same as the azimuth (). changes after and before PS-AMO. This variation is responsible for the event movement along the common conversion point.
Figure 1 Definition of offset vector and transformation vector , before and after PS-AMO
Figure 2 shows how event movement along CCP changes with depth. This is due to the dependence of with respect to vp, and tn. This variance with depth will persist even in a constant velocity media. Figure 2 also illustrates that the time after PS-AMO (t2) has a new and , therefore, a new CCP position.
Figure 2 Comparison between the CMP and CCP position in the PS-AMO operator
Continuing with the procedure presented by Fomel and Biondi (1995) to obtain the PS-AMO operator, we cascade PS-DMO [equation (5)] with its inverse. Figure 3 shows a scheme of the PS-AMO transformation. A trace with input offset vector and midpoint at the origin is transformed into equivalent data with output offset vector and midpoint position . The data is first transformed to its corresponding CCP position and . Subsequently, the inverse PS-DMO repositions the data to a new midpoint position with a new offset vector .
Figure 3 CMP-CCP plane, PS-AMO geometrical interpretation.
The new trace position is defined by
Both and can be expressed as terms of the final midpoint position by using the rule of sines in the triangle (,,)in Figure 3 as
The final expression takes the form of
This expression represents the azimuth rotation in both the CCP domain and the CMP domain.