(10) |

where *h* is the half-offset and *x* is the surface midpoint-location.
By introducing the ratio, , and following
Huub Den Rooijen's 1991 derivations, as well as work by Xu et al. (2001), I
have the following standard form of elliptical equation:

(11) |

This equation introduces two more terms that were absent in equation , namely
*t*_{n}, the NMO-corrected time; and, *D*, this term represents the
lateral shift responsible for the CMP to CCP/CRP correction.
The main difference among the existing PS-DMO operators is in
the computation and correction from CMP to CCP (*D*).
Throughout this thesis I use the definition
presented by Xu et al. (2001):

(12) |

The PS-DMO smile equation is presented as follows:

(13) |

where

(14) | ||

(15) |

12/14/2006