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 *v*_{p}/*v*_{s} 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.

12/14/2006