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The acoustic wave-equation for wavefield, , in a generalized
Riemannian space is
| |
(19) |

where the is frequency, is the propagation slowness, and
is the Laplacian operator
| |
(20) |

Substituting equation 20 into 19 generates
a Helmholtz equation appropriate for propagating waves through a 3D
space
| |
(21) |

Expanding the derivative terms and multiplying through by yields
| |
(22) |

Defining *n*_{j} as
| |
(23) |

leads to a more compact notation of equation 22
| |
(24) |

Developing a wave-equation dispersion relation is achieved by
replacing the partial differential operators acting on wavefield
with their Fourier domain duals
| |
(25) |

where is the Fourier domain dual of differential operator
. Equation 25 represents
the dispersion relationship for wavefield propagation on a generalized
3-D Riemannian space.

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Stanford Exploration Project

1/16/2007