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# Derivation of the PS-DMO and PS-AMO operators

This appendix presents the extension to 3-D of the PS-DMO operator introduced in Chapter 2, followed by the derivation of the PS-AMO operator presented in Chapter 4.

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:

 (87)

where

 (88)

 (89)

and

 (90)

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:

 (91)

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

 (92)

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:

 (93)

Figure  shows that and are parallel, as well as and . Therefore, we can rewrite equation as

 (94)

Both and can be expressed in terms of the final midpoint position, , by using the rule of sines in the triangle (,,), in Figure , as

 (95) (96)

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.

Next: PS-CAM theoretical impulse response Up: Imaging of converted-wave Ocean-bottom Previous: Tangent to the impulse
Stanford Exploration Project
12/14/2006