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Finite differences

We denote the source wavefield by $q(x,\omega)=FFT\left[ {\mbox{Shot}\left( {\vec x,t} \right)} \right]_{\left( \omega \right)}$. Equation (13) can be written:
\begin{displaymath}
p_D =
\frac{1}{2i}\left(\frac{\omega^2}{v^2}+\frac{\partial^2}{\partial x^2}\right)^{-\frac{1}{2}}q.\end{displaymath} (15)
We then take a Taylor expansion of $\Lambda^{-1}$ by considering kx small compared to $\frac{\omega}{v}$: 
 \begin{displaymath}
p_D \simeq
\frac{iv}{2\omega}
\left(
-1+\frac{v^2}{2\omega^2}
\frac{\partial^2}{\partial x^2}
\right)q.\end{displaymath} (16)
Explicit finite differences are used to obtain a numerical solution:
\begin{eqnarray}
p_D^x 
&=&
\frac{iv}{2\omega}
\left[
-q^x+\frac{v^2}{2\omega^2}...
 ...lta_3q^{x-1}-\left(1+2\Delta_3\right)q^{x}+\Delta_3q^{x+1}\right],\end{eqnarray} (17)
(18)
where $\Delta_3=\frac{1}{2}\left(\frac{v}{\omega \Delta x}\right)^2$.
next up previous print clean
Next: Mixed domain Up: Implementing the operator Previous: Implementing the operator
Stanford Exploration Project
7/8/2003