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DERIVING AMO AS A CASCADE OF DMO AND INVERSE DMO

IN THE TIME-SPACE DOMAIN

In this appendix, we present an alternative derivation of the AMO operator. The entire derivation is carried out in the time-space domain. It applies the idea of cascading DMO and inverse DMO, developed in appendix A, but uses the integral formulation of DMO Deregowski and Rocca (1981); Deregowski (1986); Hale (1991) in place of the frequency-domain DMO.

Let $P_1\left({\bf m}_1,t_1;{\bf h}_{1}\right)$ be the input of an AMO operator (common-azimuth and common-offset seismic reflection data after normal moveout correction) and $P_2\left({\bf m}_2,t_2;{\bf h}_{2}\right)$ be the output. Then the three-dimensional AMO operator takes the following general form:  
 \begin{displaymath}
P_2\left({\bf m}_2,t_2;{\bf h}_{2}\right) = 
\left\vert{\bf ...
 ... h}_{1},{\bf h}_{2}\right);\,
{\bf h}_{1}\right)\,d{\bf m}_1\;,\end{displaymath} (39)
where $\left\vert{\bf D}_{t_2}\right\vert$ is the differentiation operator (equivalent to multiplication by $\vert\omega_2\vert$ in the frequency domain), ${\bf \Delta m}= {\bf m}_2-{\bf m}_1$ is the difference vector between the input and the output midpoints, $t_2\,\sigma_{12}$ is the summation path, and w12 is the weighting function.

To derive (40) in the time-space domain we cascade an integral DMO operator of the form  
 \begin{displaymath}
P_0\left({\bf m}_0,t_0;{\bf 0}\right)={\bf D}_{-t_0}^{1/2}\,...
 ...elta m}_{10},{\bf h}_{1}\right);
{\bf h}_{1}\right)\,d\hat{x}_1\end{displaymath} (40)
with an inverse DMO of the form  
 \begin{displaymath}
P_2\left({\bf m}_2,t_2;{\bf h}_{2}\right)={\bf D}_{t_2}^{1/2...
 ...Delta m}_{02},{\bf h}_{2}\right);
{\bf 0}\right)\,d\hat{x}_0\;.\end{displaymath} (41)
Where $t_0\, \sigma_{10}$ and $t_2\,\sigma_{02}$ are the summation paths of the DMO and inverse DMO operators Deregowski and Rocca (1981):  
 \begin{displaymath}
\sigma_{10}({\bf \Delta m},{\bf h}_{1})=
\frac{h_{1}}{\sqrt{...
 ...a m}, {\bf h}_{2}})=
\frac{\sqrt{h_{2}^2-\Delta m^2}}{h_{2}}\;;\end{displaymath} (42)
w10 and w02 are the corresponding weighting functions (amplitudes of impulse responses); $\hat{x}_1$ is the component of ${\bf m}_1$ along the ${\bf h}_{1}$ azimuth; $\hat{x}_0$ is the component of ${\bf m}_0$ along the ${\bf h}_{2}$ azimuth; and ${\bf \Delta m}_{10} = {\bf m}_0- {\bf m}_1$, ${\bf \Delta m}_{02} = {\bf m}_2- {\bf m}_0$. ${\bf D}_t^{1/2}$ stands for the operator of half-order differentiation (equivalent to the $(i \omega)^{1/2}$ multiplication in Fourier domain).

Both DMO and inverse DMO operate as 2-D operators on 3-D seismic data, because their apertures are defined on a line. This implies that for a given input midpoint ${\bf m}_1$, the corresponding location of ${\bf m}_0$must belong to the line going through ${\bf m}_1$, with the azimuth $\theta_{1}$ defined by the input offset ${\bf h}_{1}$. Similarly, ${\bf m}_0$must be on the line going through ${\bf m}_2$ with the azimuth $\theta_{2}$of ${\bf h}_{2}$.These geometrical considerations lead us to the following conclusion: For a given pair of input and output midpoints ${\bf m}_1$ and ${\bf m}_2$ of the AMO operator, the corresponding midpoint ${\bf m}_0$ on the intermediate zero-offset gather is determined by the intersection of two lines drawn through ${\bf m}_1$ and ${\bf m}_2$ in the offset directions. Applying the geometric connection among the three midpoints, we can find the cascade of the DMO and inverse DMO operators in one step. For this purpose, it is sufficient to notice that the angles in the triangle, formed by the midpoints ${\bf m}_1$,${\bf m}_0$, and ${\bf m}_2$, satisfy the law of sines:  
 \begin{displaymath}
\left\vert {\Delta m\over \ {\sin \Delta \theta}} \right\ver...
 ...m_{02} \over \ {\sin (\theta_1-\Delta \varphi)}} \right\vert\;.\end{displaymath} (43)

Substituting equation (41) into (42), taking into account (44), and neglecting the low-order asymptotic terms, produces the 3-D integral AMO operator (40), where

\begin{displaymath}
\sigma_{12}\left({\bf \Delta m},{\bf h}_{1},{\bf h}_{2}\righ...
 ...h_{2}^2 - \Delta m_{02}^2} \over
{h_{1}^2 - \Delta m_{10}^2}}}=\end{displaymath}

 
 \begin{displaymath}
\frac{h_{1}}{h_{2}}\sqrt{
\frac
{h_{2}^2\sin^2\Delta \theta-...
 ...n^2\Delta \theta-\Delta m^2\sin^2(\theta_2-\Delta \varphi)}}\;,\end{displaymath} (44)
 
 \begin{displaymath}
w_{12}\left({\bf \Delta m},{\bf h}_{1},{\bf h}_{2},t_2\right...
 ... {\bf h}_{2}\right)
 \right)\,
} \over {
\sin{\Delta \theta}}}.\end{displaymath} (45)
Equation (45) is the reciprocal of, and thus equivalent to equation (1) in the main text. The factor $\sin{\Delta \theta}$ in the denominator of the equation (46) appears as the result of the midpoint-coordinate transformation $d{\bf m}_1= d\hat{x}_0\,d\hat{x}_0\,\sin{\Delta \theta}$.

The time-and-space analogue of amplitude-preserving DMO Black et al. (1993) has the weighting function  
 \begin{displaymath}
w_{10}\left({\bf \Delta m}_{10},{\bf h}_{1},t_0\right) = 
\s...
 ..._{10}^2} \over
{h_{1}\,
\left(h_{1}^2-\Delta m_{10}^2 \right)}}\end{displaymath} (46)
while its asymptotic inverse has the weighting function  
 \begin{displaymath}
w_{02}\left({\bf \Delta m}_{02},{\bf h}_{2},t_2\right) = 
\s...
 ...2\,\pi}\,
{h_{2}\over
\left(h_{2}^2-\Delta m_{02}^2 \right)}\;.\end{displaymath} (47)
Inserting (47) and (48) into (46), and using the equality $\sqrt{{\bf D}_{-t_2} t_2}=\sqrt{{\bf D}_{-t_0} t_0}$,similarly to appendix A, yields
   \begin{eqnarray}
\lefteqn{w_{12}\left({\bf \Delta m},{\bf h}_{1},{\bf h}_{2},t_2...
 ...-\Delta \varphi)}
\over
{h_{2}^2\sin^2\Delta \theta}}}
\right)}\;,\end{eqnarray}
(48)
which is equivalent to equation (4) in the main text.



 
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Next: 2-D AMO operator Up: Biondi, Fomel & Chemingui: Previous: 2-D AMO operator
Stanford Exploration Project
11/11/1997