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The case of the *N*^{th} partial normal moveout that is not a truncated
Taylor series is easier, so let us examine it first.
Define now the *N*^{th} partial normal moveout by
| |
(5) |

We can see that the truncated Taylor series of equation (5)
is equation (2).
Now we will prove that
| |
(6) |

holds exactly for all N.
Let us denote the application of the *N*^{th} partial normal moveout *i* times by
| |
(7) |

It holds true that
| |
(8) |

The superposition of *N*^{t}*h* partial normal moveouts may be written
| |
(9) |

for and .Let us rewrite this equation into a system of equations:
| |
(10) |

| (11) |

| (12) |

| (13) |

By summing up all the equations we get
| |
(14) |

which is the desired result.

** Next:** PROOF FOR 15-DEGREE PARTIAL
** Up:** Jedlicka: Cascaded normal moveout
** Previous:** 15-DEGREE PARTIAL NORMAL MOVEOUT
Stanford Exploration Project

1/13/1998