The 2-D PS-DMO operator in equations and , from Chapter 2, extend to 3-D by replacing the offset and midpoint coordinates for the offset and midpoint vectors respectively. This extension gives the 3-D expression for the PS-DMO operator:
Here, is the midpoint position vector, is the offset vector, is the transformation vector responsible for the CMP to CRP correction, and is the vp/vs ratio.
I use the 3-D PS-DMO operator to derive the PS-AMO operator. Since the vectors and are collinear, and from substituing equation into equation , we obtain the first of two time shifts corresponding to the PS-AMO transformation:
where corresponds to the intermediate transformation to zero-offset. The vector, , relates to the transformation from CMP to CRP, which is an intrinsic property of PS-DMO operator.
The second time shift for the PS-AMO operator corresponds to the transformation from the intermediate zero-offset position, , to the final trace position, is
where the vector () corresponds to the transformation from zero-offset to the final CMP position. The transformation vectors, ( and , both comes from equation with the offset vector, , equals to the input offset and the output offset, respectively.
Finally, combining equations and I obtain the expression for the PS-AMO operator:
Figure shows that and are parallel, as well as and . Therefore, we can rewrite equation as
Both and can be expressed in terms of the final midpoint position, , by using the rule of sines in the triangle (,,), in Figure , as
By introducing equations and into equation and by replacing and for their definition on equation , I obtain the final expression for the PS-AMO operator, that is equation in Chapter 4.