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Analytical formulation

Denote the surface position vector by
\begin{displaymath}
\vec x = (x_1 ,x_2 ),\end{displaymath} (1)
the temporal frequency by $\omega$, and the velocity by
\begin{displaymath}
v = v\left( {\vec x,z}\right).\end{displaymath} (2)
We define the operator $\Lambda$ as
\begin{displaymath}
\Lambda = i\sqrt {\frac{{\omega ^2 }}{{v^2 }} + \frac{{\part...
 ...{\partial x_1^2 }} + \frac{{\partial ^2 }}{{\partial x_2^2 }}},\end{displaymath} (3)
and the operator $\Gamma$ as
\begin{displaymath}
\Gamma = \frac{1}{{2v}}\frac{{\partial v}}{{\partial z}}\fra...
 ...x_1^2 }} + \frac{{\partial ^2 }}{{\partial x_2^2 }}} \right)}}.\end{displaymath} (4)
Without the amplitude correction, shot profile migration proceeds by computing the upward going and the downward going wavefields at all depths:
\begin{eqnarray}
\left( {\frac{\partial }{{\partial z}} - \Lambda } \right)U &=&...
 ...\left( {\frac{\partial }{{\partial z}} + \Lambda } \right)D &=& 0.\end{eqnarray} (5)
(6)
To solve these equations, we need boundary conditions (inputs into the downward continuation). The recorded shot gather is taken as the upper boundary condition for the upward going wavefield, and a Dirac, approximated by a small wavelet, is taken as the upper boundary condition for the downward going wavefield:
\begin{eqnarray}
U\left( {\vec x,z = 0,\omega } \right) &=& FFT\left[ {\mbox{Sho...
 ...ec x - \vec x_{shot} ,t} \right)} \right]_{\left( \omega \right)} \end{eqnarray} (7)
(8)
The reflectivity image is produced using the imaging condition:  
 \begin{displaymath}
R\left( {\vec x,z} \right) = \sum\limits_\omega {\frac{{U\le...
 ... x,z,\omega } \right)}}{{D\left( {\vec x,z,\omega } \right)}}}.\end{displaymath} (9)
According to Zhang et al. (2003a), to improve the amplitude and phase of the image R, we only need to add the $\Gamma$ operator into the downward continuation equations:
      \begin{eqnarray}
\left( {\frac{\partial }{{\partial z}} - \Lambda - \Gamma }
\ri...
 ...rac{\partial }{{\partial z}} + \Lambda - \Gamma } \right)p_D &=& 0\end{eqnarray} (10)
(11)
and while taking the same boundary conditions for the upward going wavefield, to apply a correction to the downward going wavefield boundary condition:
   \begin{eqnarray}
p_U \left( {\vec x,z = 0,\omega } \right) &=& FFT\left[ {\mbox{...
 ...x{Shot}\left( {\vec x,t} \right)} \right]_{\left( \omega \right)}.\end{eqnarray} (12)
(13)
The imaging condition remains the same:
\begin{displaymath}
R\left( {\vec x,z} \right) = \sum\limits_\omega {\frac{{p_U\...
 ...,z,\omega } \right)}}{{p_D\left( {\vec x,z,\omega } \right)}}}.\end{displaymath} (14)
The only differences between the amplitude-preserving algorithm and the traditional one are the use of the $\Gamma$ operator in downward continuation and the application of the $\Lambda^{-1}$operator to the shot boundary condition. In the following sections, we will describe implementations of the two operators in the $\omega-x$ domain (finite-difference), as well as in the mixed domain. We will also show 2-D synthetic examples of applying the operators.
next up previous print clean
Next: Implementing the operator Up: Vlad et al.: Improving Previous: Introduction
Stanford Exploration Project
7/8/2003