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# Derivation of Snell Resampling Operator

In the following appendix, I derive the Snell resampling operation, equation
(). The graphical basis for the derivation is Figure
. Since the pegleg multiple and primary in the figure have
the same emergence angle, , the stepout, or spatial derivative, of the
traveltime curves of the two events is the same at *x* and *x*_{p}. First we
compute the stepout of the primary event, starting from the standard NMO equation:
| |
(49) |

| (50) |

| (51) |

Using equations () and (), we can similarly compute
the stepout of the corresponding *j*^{th}-order pegleg multiple:
| |
(52) |

Finally, we compute *x*_{p} as a function of *x* by squaring equations
() and (), setting them equal,
and substituting traveltime equations () and ()
for *t*_{m} and *t*_{p}, respectively:
| |
(53) |

| (54) |

| (55) |

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Stanford Exploration Project

5/30/2004