** Next:** Conclusion
** Up:** Shan and Biondi: Residual
** Previous:** Amplitude preserving one-way wave

In this section, we demonstrate that the first order approximation of WKBJ is the same as the amplitude
preserving one-way wave equation for *v*=*v*(*z*).
The one-way wave equation with the WKBJ amplitude correction is

| |
(7) |

The WKBJ amplitude correction term can be rewritten as
| |
(8) |

Then in equation (8) can be linearized to , so we have
| |
(9) |

From
and
we have
| |
(10) |

Because
| |
(11) |

and from the dispersion relation , we have
| |
(12) |

and
| |
(13) |

From equation (8) to equation (13), we have
| |
(14) |

So equation (7) can be rewritten as
| |
(15) |

| (16) |

Comparing the amplitude preserving one-way wave equation (equation (4))
with first order approximation of WKBJ (equation (16)),
we find they are same.
So we demonstrate theoretically that the amplitude preserving one-way wave equation is equivalent to the first
order approximation of WKBJ.

** Next:** Conclusion
** Up:** Shan and Biondi: Residual
** Previous:** Amplitude preserving one-way wave
Stanford Exploration Project

10/14/2003