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A This Appendix is a step-by-step derivation of the Fourier finite-difference equation, Equation (2), one of the most general forms of the equations describing mixed-domain migration.

I begin with Taylor series approximations of the single square root
equations for the vertical wavenumbers corresponding to the true
slowness (*s*)

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

(27) |

(28) |

(29) |

The equivalent 2^{nd} order equation takes the form:

We can write an analogous form for Equation (29) using a continuous fraction expansion

(30) |

B This appendix is a step-by-step derivation of the generalized screen equation.

I begin with the single square root equation for the true slowness (*s*)

We can replace *k*_{z}_{o} in *k*_{z} to get

(31) |

Next, we can write a Taylor series expansion, assuming a small slowness squared
perturbation *s ^{2}*-

We can make *k*_{z}_{o} explicit for the terms of the sum and obtain

(32) |

(33) |

Because Equation (33) becomes unstable when the denominator of the terms in the summation vanishes, we replace these terms with another Taylor series expansion:

(34) |

We can obtain the split-step equation through a sequence of approximations in
Equation (34): first, we limit the terms of the inner Taylor series to two
(*i*=1,2) and the terms of the outer series to one (*n*=1), therefore

Next, we approximate and , and get

which reduces to the split-step equation9/5/2000