- Abramowitz, M., and Stegun, I., 1972, Solutions of quartic equations: New York: Dover., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 17-18.
- Baumstein, A., and Anderson, J., 2003, Wavefield extrapolation in laterally varying VTI media: Soc. of Expl. Geophys., 73rd Ann. Internat. Mtg., 945-948.
- Blacquiere, G., Debeye, H. W. J., Wapenaar, C. P. A., and Berkhout, A. J., 1989, 3D table-driven migration: Geophys. Prosp.,
**37**, no. 08, 925-958. - Ferguson, R. J., and Margrave, G. F., 1998, Depth migration in TI media by nonstationary phase shift: Soc. of Expl. Geophys., 68th Ann. Internat. Mtg, 1831-1834.
- Hale, D., 1991a, 3-D depth migration via McClellan transformations: Geophysics,
**56**, no. 11, 1778-1785. - Hale, D., 1991b, Stable explicit depth extrapolation of seismic wavefields: Geophysics,
**56**, no. 11, 1770-1777. - Holberg, O., 1988, Towards optimum one-way wave propagation: Geophys. Prosp.,
**36**, no. 02, 99-114. - McClellan, J., and Chan, D., 1977, A 2-D FIR filter structure derived from the Chebychev recursion: IEEE Trans. Circuits Syst.,,
**CAS-24**, 372-378. - McClellan, J. H., and Parks, T. W., 1972, Equiripple approximation of fan filters: Geophysics,
**37**, no. 04, 573-583. - Ristow, D., and Ruhl, T., 1997, Migration in transversely isotropic media using implicit operators: Soc. of Expl. Geophys., 67th Ann. Internat. Mtg, 1699-1702.
- Rousseau, J. H. L., 1997, Depth migration in heterogeneous, transversely isotropic media with the phase-shift-plus-interpolation method: Soc. of Expl. Geophys., 67th Ann. Internat. Mtg, 1703-1706.
- Shan, G., and Biondi, B., 2004a, Imaging overturned waves by plane-wave migration in tilted coordinates: 74th Ann. Internat. Mtg., Soc. of Expl. Geophys., Expanded Abstracts, 969-972.
- Shan, G., and Biondi, B., 2004b, Wavefield extrapolation in laterally-varying tilted TI media: SEP-
**117**, 1-10. - Shan, G., and Biondi, B., 2005, Imaging steeply dipping reflectors in TI media by wavefield extrapolation: SEP-
**120**, 63-76. - Thomsen, L., 1986, Weak elastic anisotropy: Geophysics,
**51**, no. 10, 1954-1966. - Thorbecke, J., 1997, Common focus point technology: Delft University of Technology., Ph.D. thesis.
- Tsvankin, I., 1996, P-wave signatures and notation for transversely isotropic media: An overview: Geophysics,
**61**, no. 02, 467-483. - Uzcategui, O., 1995, 2-D depth migration in transversely isotropic media using explicit operators: Geophysics,
**60**, no. 06, 1819-1829. - Zhang, J., Verschuur, D. J., and Wapenaar, C. P. A., 2001a, Depth migration of shot records in heterogeneous, tranversely isotropic media using optimum explicit operators: Geophys. Prosp.,
**49**, no. 03, 287-299. - Zhang, J., Wapenaar, C., and Verschuur, D., 2001b, 3-D depth migration in VTI media with explicit extrapolation operators: Soc. of Expl. Geophys., 71st Ann. Internat. Mtg, 1085-1088.

We begin with equation (17), as the other three equations are silimiar.
Let and be the sampling of the wavenumbers *k*_{x}
and *k*_{y}, respectively. To mimic the behavior of the orginal operator
*F*^{ee},
we need to estimate *a*^{ee}_{nxny}, so that

(30) |

(31) |

(32) |

(33) |

(34) |

(35) | ||

(36) | ||

(37) |

It is well known that the inverse Fourier transform of the function , , , are:

(38) | ||

(39) | ||

(40) | ||

(41) |

(42) | ||

(43) |

and are :

(44) | ||

(45) | ||

(46) |

(47) | ||

(48) | ||

(49) | ||

(50) |

(51) | ||

(52) |

(53) | ||

(54) | ||

(55) | ||

(56) |

5/3/2005