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Expanding equation (1a) around and neglecting
terms in , results in:

| |
(2) |

| |

Choosing the positive root yields the
*P*-wave phase velocity near the vertical axis, as follows:

| |
(3) |

where
, ,
| |
(4) |

and
| |
(5) |

*W*_{P,z} is the vertical *P*-wave phase velocity squared and *W*_{P,xnmo}
is the horizontal normal moveout (*NMO*) phase velocity squared.
Choosing the negative root in equation
(2) yields *SV*-wave phase velocities near the vertical axis,
as follows:

| |
(6) |

where
| |
(7) |

and
| |
(8) |

The previous expressions for the *NMO*
velocities agree with the results
of Thomsen (1986) and Vernik and Nur (1992).

The expression for *SH*-wave phase velocities near the
vertical axis is

| |
(9) |

where
| |
(10) |

and
| |
(11) |

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Stanford Exploration Project

11/17/1997