** Next:** Impulse responses
** Up:** Shan: Implicit migration for
** Previous:** Optimized one-way wave equation

In the approximated dispersion relation (6), replacing *S*_{z} and *S*_{x} by the partial differential operators
and , we obtain a partial differential equation as follows:
| |
(7) |

Equation (7) can be solved by cascading as follows:
| |
(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:
and
we can derive the following finite difference equation:
| |
(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