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,

$$k_\xi_3= i a_3 \pm \left[ a_4^2 \omega^2 - a_5^2 k_\xi_1... ... a_8 k_\xi_1 + i a_9 k_\xi_2 - a^2_{10} \right]^{\frac{1}{2}},$$ (35)

where,

$$\mathbf{a} = \left[ 0 \;\;\; 0 \;\;\; \frac{ n_3}{2 m^{33}}... ...ac{n_2}{m^{33}} \;\;\; \frac{n_3}{m^{33}} \right]^{\mathbf{T}}.$$ (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,

$$k_\xi_3= a_1 k_\xi_1+ a_2 k_\xi_2 \pm \left[ a_4^2 \omega^2... ...\xi_2^2 - a_7 k_\xi_1k_\xi_2 - a_{10}^2 \right]^{\frac{1}{2}},$$ (37)

where,

$$\mathbf{a} = \left[ -\frac{ g^{13} }{ g^{33} }\;\;\; -\frac... ...;\;\; 0\;\;\; 0\;\;\; \frac{n_3}{m^{33}} \right]^{\mathbf{T}}.$$ (38)

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

$$k_\xi_3= \pm \left[ a_4^2 \omega^2 - a_5^2 k_\xi_1^2 - a_... ...\xi_2^2 - a_7 k_\xi_1k_\xi_2 - a_{10}^2 \right]^{\frac{1}{2}},$$ (39)

where,

$$\mathbf{a} = \left[ 0 \;\;\; 0 \;\;\; 0 \;\;\; \frac{ \ss }... ...;\; 0 \;\;\; 0 \;\;\; \frac{n_3}{m^{33}} \right]^{\mathbf{T}}.$$ (40)

Note that the expression $\Phi=1$, and that components of the metric tensor are significantly simplified.

2-D Non-orthogonal coordinate systems - Two-dimensional situations are handled by identifying $\xi_2=0$. Hence, all derivatives in the associated metric tensor g^ij with respect coordinate $\xi_2$ are identically zero. Hence, a 2-D non-orthogonal coordinate system can be represented by  

$$k_\xi_3= a_1 k_\xi_1+ i a_3 \pm \left[ a_4^2 \omega^2 - a_5^2 k_\xi_1^2 + a_8 i k_\xi_1 - a_{10}^2 \right]^{\frac{1}{2}},$$ (41)

where,

$$\mathbf{a} = \left[ -\frac{ g^{13} }{ g^{33} }\;\;\; 0 \;\;... ...^{33}} \;\;\; 0 \;\;\; \frac{n_3}{m^{33}} \right]^{\mathbf{T}}.$$ (42)

2-D Non-orthogonal Kinematic Coordinate Systems - Two-dimensional kinematic situations are handled through identity $\xi_2=0$. Again, all derivatives in the associated metric tensor g^ij with respect coordinate $\xi_2$ are identically zero, and the 2-D non-orthogonal kinematic extrapolation wavenumber is

$$k_\xi_3= a_1 k_\xi_1 \pm \left[ a_4^2 \omega^2 - a_5^2 k_\xi_1^2 - a_{10}^2 \right]^{\frac{1}{2}},$$ (43)

where,

$$\mathbf{a} = \left[ -\frac{ g^{13} }{ g^{33} } \;\;\; 0 \;\... ...\;\; 0 \;\;\; 0 \;\;\; \frac{n_3}{m^{33}} \right]^{\mathbf{T}}.$$ (44)

2-D Orthogonal Coordinate Systems - Two-dimensional situations are handled with $\xi_2=g_{13}=0$. Accordingly, all derivatives in the associated metric tensor g^ij with respect coordinate $\xi_2$ are identically zero, and the 2-D non-orthogonal coordinate system is represented by

$$k_\xi_3= i a_3 \pm \left[ a_4^2 \omega^2 - a_5^2 k_\xi_1^2 + i a_8 k_\xi_1 - a_{10}^2 \right]^{\frac{1}{2}},$$ (45)

where,

$$\mathbf{a} = \left[ 0 \;\;\; 0 \;\;\; \frac{ n_3 }{ 2 m^{33... ...^{33}} \;\;\; 0 \;\;\; \frac{n_3}{m^{33}} \right]^{\mathbf{T}}.$$ (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,

$$k_\xi_3= \pm \left[ a_4^2 \omega^2 - a_5^2 k_\xi_1^2 - a_{10}^2 \right]^{\frac{1}{2}},$$ (47)

where,

$$\mathbf{a} = \left[ 0 \;\;\; 0 \;\;\; 0 \;\;\; \frac{ \ss }... ...\;\; 0 \;\;\; 0 \;\;\; \frac{n_3}{m^{33}} \right]^{\mathbf{T}}.$$ (48)