Next: Impulse responses
Up: Shan: Implicit migration for
Previous: Optimized one-way wave equation
In the approximated dispersion relation (6), replacing Sz and Sx by the partial differential operators
and
, we obtain a partial differential equation as follows:
| ![\begin{displaymath}
\frac{\partial }{\partial z}P=i\frac{\omega}{V_{P0}}\left( S...
...{V_{P0}^2}{\omega^2}\frac{\partial^2}{\partial x^2} } \right)P.\end{displaymath}](img24.gif) |
(7) |
Equation (7) can be solved by cascading as follows:
| ![\begin{eqnarray}
\frac{\partial }{\partial z}P&=&i\frac{\omega}{V_{P0}}S_{z0}P,
...
...rac{V_{P0}^2}{\omega^2}\frac{\partial^2}{\partial x^2} } \right)P.\end{eqnarray}](img25.gif) |
(8) |
| (9) |
| (10) |
Equation (8) can be solved by a phase-shift in the space domain.
Let
, where
and
are the grid size of finite-difference scheme.
In equation (9), replacing the partial differential operators by the finite-difference operators as follows:
![\begin{displaymath}
\frac{\partial }{\partial x}P(\omega,n\Delta z,i\Delta x)\approx\delta_xP^n_i= \frac{P_{i+1}^n-P_{i-1}^n}{2\Delta x}\end{displaymath}](img29.gif)
and
![\begin{displaymath}
\frac{\partial^2 }{\partial x^2}P(\omega,n\Delta z,i\Delta x...
... \delta_x^2P^n_i=\frac{P_{i+1}^n-2P_i^n+P_{i-1}^n}{\Delta x^2},\end{displaymath}](img30.gif)
we can derive the following finite difference equation:
| ![\begin{displaymath}
\left(1+(\frac{ia_1\Delta z}{2}\frac{V_{P0}}{\omega}-b_1\fra...
...ega^2})\delta_x^2-\frac{c_1\Delta z}{2}\delta_x \right) P_i^{n}\end{displaymath}](img31.gif) |
(11) |
Fourier analysis shows that the finite-difference scheme (11) is stable. Its computational cost is
almost same as that of the finite-difference scheme for isotropic media. Equation (10) can be solved similarly.
Next: Impulse responses
Up: Shan: Implicit migration for
Previous: Optimized one-way wave equation
Stanford Exploration Project
1/16/2007