## REFERENCES

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 m13 and m23 components of the weighted metric tensor are identically zero, which leads to the following extrapolation wavenumber,
 (35)
where,
 (36)
which are the coefficients recovered by Sava and Fomel (2005).

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)
where,
 (38)

3-D Kinematic Semi-orthogonal coordinate systems - Combining the two above restrictions yields the following extrapolation wavenumber,
 (39)
where,
 (40)
Note that the expression , and that components of the metric tensor are significantly simplified.

2-D Non-orthogonal coordinate systems - Two-dimensional situations are handled by identifying . Hence, all derivatives in the associated metric tensor gij with respect coordinate are identically zero. Hence, a 2-D non-orthogonal coordinate system can be represented by
 (41)
where,
 (42)

2-D Non-orthogonal Kinematic Coordinate Systems - Two-dimensional kinematic situations are handled through identity . Again, all derivatives in the associated metric tensor gij with respect coordinate are identically zero, and the 2-D non-orthogonal kinematic extrapolation wavenumber is
 (43)
where,
 (44)

2-D Orthogonal Coordinate Systems - Two-dimensional situations are handled with . Accordingly, all derivatives in the associated metric tensor gij with respect coordinate are identically zero, and the 2-D non-orthogonal coordinate system is represented by
 (45)
where,
 (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)
where,
 (48)