- Gazdag, J. and P. Sguazzero, 1984, Migration of seismic data by phase-shift plus interpolation: Geophysics,
**69**, 124-131. - Guggenheimer, H., 1977, Differential Geometry: Dover Publications, Inc., New York.
- Sava, P. and S. Fomel, 2005, Riemannian wavefield extrapolation: Geophysics,
**70**, T45-T56. - Shragge, J. and P. Sava, 2004, Incorporating topography into wave-equation imaging through conformal mapping: SEP-
**117**, 27-42. - Shragge, J., 2006a, Generalized riemannian wavefield extrapolation: SEP-
**124**. - Shragge, J., 2006b, Structured mesh generation using differential methods: SEP-
**124**. - Stoffa, P. L., J. T. Fokkema, R. M. de Luna Freire, and W. P. Kessinger, 1990, Split-step Fourier migration: Geophysics,
**55**, no. 04, 410-421.

A The extrapolation wavenumber developed in equation 14 is appropriate for any non-orthogonal Riemannian geometry. However, there are a number of situations where symmetry or partial orthogonality are present. Moreover, one may wish to make a kinematic approximation where all of the imaginary components of the wavenumber are ignored. These situations are discussed in this Appendix.

**3D Semi-orthogonal Coordinate Systems** - Semi-orthogonal
coordinate systems occur where one coordinate is orthogonal to the
other two coordinates Sava and Fomel (2005). In these cases the
*m ^{13}* and

(35) |

(36) |

**3-D Kinematic Coordinate Systems** - Wave-equation migration
amplitudes are generally inexact in laterally variant media - even in
a Cartesian based system. Hence, one beneficial approximation that
reduces computational cost is to consider only the kinematic terms in
equation 14. This approximate generates the following
extrapolation wavenumber,

(37) |

(38) |

**3-D Kinematic Semi-orthogonal coordinate systems** - Combining the
two above restrictions yields the following extrapolation wavenumber,

(39) |

(40) |

**2-D Non-orthogonal coordinate systems** - Two-dimensional
situations are handled by identifying . Hence,
all derivatives in the associated metric tensor *g*^{ij} with respect
coordinate are identically zero. Hence, a 2-D non-orthogonal
coordinate system can be represented by

(41) |

(42) |

**2-D Non-orthogonal Kinematic Coordinate Systems** -
Two-dimensional kinematic situations are handled through identity
. Again, all derivatives in the associated metric tensor
*g*^{ij} with respect coordinate are identically zero, and
the 2-D non-orthogonal kinematic extrapolation wavenumber is

(43) |

(44) |

**2-D Orthogonal Coordinate Systems** - Two-dimensional situations
are handled with . Accordingly, all derivatives in the
associated
metric tensor *g*^{ij} with respect coordinate are identically
zero, and the 2-D non-orthogonal coordinate system is represented by

(45) |

(46) |

**2-D Orthogonal Kinematic Coordinate Systems** - The above two
approximations can be combined to yield the following extrapolation
wavenumber for 2-D orthogonal kinematic coordinate systems,

(47) |

(48) |

4/5/2006