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The Euler-Lagrange equation can be used to find the absolute time (*t*(*x*,*y*)) that minimizes (Fomel, 2002, personal communication)
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(16) |

where *p*_{x} is the dip in the *x* direction and *p*_{y} is the dip in the *y* direction.
This can be simplified for the 2-D case to find the absolute time (*t*(*x*)) that minimizes

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(17) |

The Euler-Lagrange equation Farlow (1993) is used to find the function (*t*(*x*)) that minimizes the equation of this form

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(18) |

where the unknown *t* is a function of *x*.
The Euler-Lagrange equation is
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(19) |

To apply this equation I find

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(20) |

and
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(21) |

Substituting equations ( 20) and ( 21) into equation ( 19) and simplifying, I get:
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(22) |

It is straight forward to extend to the 3-D case.

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Stanford Exploration Project

7/8/2003