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:

 

$$\frac{t_0^2}{t_n^2} + \frac{\Vert {\bf y} \Vert^2}{\Vert {\bf H} \Vert^2} =1,$$ (87)

where

 

$$\Vert {\bf y}\Vert^2 = \Vert {\bf x} + {\bf D}\Vert^2,$$ (88)

 

$${\bf H} = \frac{2\sqrt{\gamma}}{1+\gamma} {\bf h} = \alpha {\bf h},$$ (89)

and

 

$${\bf D}=\left[ 1 + \frac{4\gamma \Vert {\bf h}\Vert ^2}{v_p^... ...Vert {\bf h} \Vert^2}\right] \frac{1-\gamma}{1+\gamma} {\bf h}.$$ (90)

Here, $\bf x$ is the midpoint position vector, $\bf {h}$ is the offset vector, $\bf {D}$ is the transformation vector responsible for the CMP to CRP correction, and $\gamma$ is the v_p/v_s ratio.

I use the 3-D PS-DMO operator to derive the PS-AMO operator. Since the vectors $\bf x$ and $\bf {D}$ are collinear, and from substituing equation  into equation , we obtain the first of two time shifts corresponding to the PS-AMO transformation:

 

$$t_0^2=t_1^2 \left ( \frac{\Vert{\bf H_1}\Vert^2 - \vert\vert... ...}} + {\bf D_{10}}\vert\vert^2}{\Vert{\bf H_1}\Vert^2} \right ),$$ (91)

where ${\bf x_{10}}$ corresponds to the intermediate transformation to zero-offset. The vector, ${\bf x_{10}} + {\bf D_{10}}$, 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, ${\bf x_{10}}$, to the final trace position, is

 

$$t_2^2=t_0^2 \left ( \frac{\Vert{\bf H_2}\Vert^2}{\Vert{\bf H... ... - \vert\vert{\bf x_{02}} + {\bf D_{02}}\vert\vert^2} \right ),$$ (92)

where the vector (${\bf x_{02}} + {\bf D_{02}}$) corresponds to the transformation from zero-offset to the final CMP position. The transformation vectors, ($\bf D_{10}$ and $\bf D_{02}$, both comes from equation with the offset vector, $\bf {h}$, equals to the input offset and the output offset, respectively.

Finally, combining equations and I obtain the expression for the PS-AMO operator:

 

$$t_2^2=t_1^2 \frac{\Vert{\bf H_{2}}\Vert^2}{\Vert{\bf H_{1}}\... ...{2}}\Vert^2 - \Vert{\bf x_{02}} +{\bf D_{02}}\Vert^2 } \right).$$ (93)

Figure  shows that ${\bf x_{10}}$ and $\bf D_{10}$ are parallel, as well as $\bf x_{02}$ and $\bf D_{02}$. Therefore, we can rewrite equation as

 

$$t_2^2=t_1^2 \frac{\Vert{\bf H_{2}}\Vert^2}{\Vert{\bf H_{1}}\... ...t{\bf D_{02}}\Vert^2 - 2{\bf x_{02}}\cdot{\bf D_{02}}} \right).$$ (94)

Both ${\bf x_{10}}$ and $\bf x_{02}$ can be expressed in terms of the final midpoint position, $\bf x$, by using the rule of sines in the triangle ($\bf x$,${\bf x_{10}}$,$\bf x_{02}$), in Figure , as

$$\begin{eqnarray} \bf {x}_{10} & = & \bf {x} \frac{\sin(\theta_2 - \Delta\phi)}{\... ...x} \frac{\sin(\theta_1 - \Delta\phi)}{\sin(\theta_1 - \theta_2)} .\end{eqnarray}$$ (95) (96)

By introducing equations and into equation and by replacing $\bf H_1$ and $\bf H_2$ for their definition on equation , I obtain the final expression for the PS-AMO operator, that is equation in Chapter 4.