REFERENCES

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.

A This appendix discusses how to estimate the coefficients a^ee_nxny, a^oe_nxny, a^eo_nxny and a^oo_nxny in equations (17)-(20).

We begin with equation (17), as the other three equations are silimiar. Let $\Delta k_x$ and $\Delta k_y$ 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

$$F^{ee}(k_x,k_y)\approx \sum_{n_x,n_y}a^{ee}_{n_xn_y}\cos(n_x\Delta xk_x)\cos(n_y\Delta yk_y),$$

for $k_x\in [0,k_x^{{Nyquist}}]$ and $k_y\in [0,k_y^{{Nyquist}}]$, where k_x^Nyquist is the Nyquist wavenumber $\frac{\pi}{\Delta x}$ and k_x^Nyquist is the Nyquist wavenumber $\frac{\pi}{\Delta y}$. Let M_x be $k_x^{{Nyquist}}/\Delta k_x$ and M_y be $k_y^{{Nyquist}}/\Delta k_y$. The coefficients can be estimated by the following fitting goals:  

$${\bf W}({\bf A^{ee}a^{ee}}-{\bf f^{ee}}) \approx {\bf 0},$$ (30)

where

$${\bf a^{ee}}=\left(a^{ee}_{00},a^{ee}_{10},\cdots,a^{ee}_{n_... ...x,n_y} ,a^{ee}_{n_x+1,n_y},\cdots, a^{ee}_{N_x,N_y}\right )^T.$$

${\bf A^{ee}}$ is a matrix with the elements

$$A^{ee}_{mn}=\cos(m_xn_x\Delta k_x \Delta x)\cos(m_yn_y\Delta k_y \Delta y),$$

where m=m_y(M_x+1)+m_x and n=n_y(N_x+1)+n_x. ${\bf f^{ee}}$ is a vector as follows

$${\bf f^{ee}}=\left( f^{ee}_{00},f^{ee}_{10},\cdots,f^{ee}_{m... ...x,m_y} ,f^{ee}_{m_x+1,m_y},\cdots, f^{ee}_{M_x,M_y} \right)^T,$$

where $f^{ee}_{m_x,m_y}=F^{ee}(m_x\Delta k_x,m_y\Delta k_y)$.${\bf W}$ is a diagonal matrix with the weights for the wavenumbers k_x,k_y. High weights are assigned to the wavenumbers of interest. The wavenumbers, such as the evacent energy, are not of interest and are assigned low weights. Given the same weight matrix ${\bf W}$, the matrix ${\bf WA^{ee}}$ are same though f^ee changes with the function F^ee(k_x,k_y). Therefore, QR decomposition is a good way to solve the fitting goal (30). First, we run QR decomposition on matrix ${\bf{WA}}$: ${\bf WA=QR}$, where ${\bf Q}$ is an orthogonal matrix and ${\bf R}$ is an upper triangular matrix. We write down the matrixes ${\bf Q}$ and ${\bf R}$. Given the function F^ee(k_x,k_y), we caluculate ${\bf f^{ee}}$. The solution of fitting goal (30) ${\bf a^{ee}}$ is given by

$${\bf a^{ee}}={\bf R}^{-1}{\bf Q}^T{\bf Wf^{ee}}.$$ (31)

For equation (18), we have the following fitting goal:  

$${\bf W}({\bf A^{oe}a^{oe}}-{\bf f^{oe}}) \approx {\bf 0},$$ (32)

where

$${\bf a^{oe}}=\left(a^{oe}_{00},a^{oe}_{10},\cdots,a^{oe}_{n_... ...x,n_y} ,a^{oe}_{n_x+1,n_y},\cdots, a^{oe}_{N_x,N_y}\right )^T,$$

and ${\bf f^{oe}}$ in equation (32) is

$${\bf f^{oe}}=\left( f^{oe}_{00},f^{oe}_{10},\cdots,f^{oe}_{m... ...x,m_y} ,f^{oe}_{m_x+1,m_y},\cdots, f^{oe}_{M_x,M_y} \right)^T,$$

where $f^{oe}_{m_x,m_y}=F^{oe}(m_x\Delta k_x,m_y\Delta k_y)$. ${\bf A^{oe}}$ in equation (32) is a matrix with the elements

$$A^{oe}_{mn}=\sin(m_xn_x\Delta k_x \Delta x)\cos(m_yn_y\Delta k_y \Delta y),$$

where m=m_y(M_x+1)+m_x and n=n_y(N_x+1)+n_x. For equation (19), we have the following fitting goal:  

$${\bf W}({\bf A^{eo}a^{eo}}-{\bf f^{eo}}) \approx {\bf 0},$$ (33)

where

$${\bf a^{eo}}=\left(a^{eo}_{00},a^{eo}_{10},\cdots,a^{eo}_{n_... ...x,n_y} ,a^{eo}_{n_x+1,n_y},\cdots, a^{eo}_{N_x,N_y}\right )^T,$$

and ${\bf f^{eo}}$ in equation (32) is

$${\bf f^{eo}}=\left( f^{eo}_{00},f^{eo}_{10},\cdots,f^{eo}_{m... ...x,m_y} ,f^{eo}_{m_x+1,m_y},\cdots, f^{eo}_{M_x,M_y} \right)^T,$$

where $f^{eo}_{m_x,m_y}=F^{eo}(m_x\Delta k_x,m_y\Delta k_y)$. ${\bf A^{eo}}$ in equation (33) is a matrix with the elements

$$A^{eo}_{mn}=\cos(m_xn_x\Delta k_x \Delta x)\sin(m_yn_y\Delta k_y \Delta y),$$

where m=m_y(M_x+1)+m_x and n=n_y(N_x+1)+n_x. For equation (20), we have the following fitting goal:  

$${\bf W}({\bf A^{oo}a^{oo}}-{\bf f^{oo}}) \approx {\bf 0},$$ (34)

where

$${\bf a^{oo}}=\left(a^{oo}_{00},a^{oo}_{10},\cdots,a^{oo}_{n_... ...x,n_y} ,a^{oo}_{n_x+1,n_y},\cdots, a^{oo}_{N_x,N_y}\right )^T,$$

and ${\bf f^{oo}}$ in equation (32) is

$${\bf f^{oo}}=\left( f^{oo}_{00},f^{oo}_{10},\cdots,f^{oo}_{m... ...x,m_y} ,f^{oo}_{m_x+1,m_y},\cdots, f^{oo}_{M_x,M_y} \right)^T,$$

where $f^{oo}_{m_x,m_y}=F^{oo}(m_x\Delta k_x,m_y\Delta k_y)$. ${\bf A^{oo}}$ in equation (34) is a matrix with the elements

$$A^{oo}_{mn}=\sin(m_xn_x\Delta k_x \Delta x)\sin(m_yn_y\Delta k_y \Delta y),$$

where m=m_y(M_x+1)+m_x and n=n_y(N_x+1)+n_x. Fitting goals (32), (33), and (34) can be solved in the same way as equation (30). The solution of fitting goal (32), (33), (34) are given by

$$\begin{eqnarray} {\bf a^{oe}}&=&({\bf R^{oe}})^{-1}({\bf Q^{oe}})^T{\bf Wf^{oe}}... ... {\bf a^{oo}}&=&({\bf R^{oo}})^{-1}({\bf Q^{oo}})^T{\bf Wf^{oo}},\end{eqnarray}$$ (35) (36) (37)

where ${\bf Q^{oe}}$, ${\bf R^{oe}}$, ${\bf Q^{eo}}$, ${\bf R^{eo}}$ and ${\bf Q^{oo}}$, ${\bf R^{oo}}$ are the QR decomposition result of the matrixs ${\bf WA^{oe}}$, ${\bf WA^{eo}}$ and ${\bf WA^{oo}}$, respectively. B In this appendix, we derive the inverse Fourier transform of the correction operator F(k_x,k_y).

It is well known that the inverse Fourier transform of the function $\cos{n_x\Delta xk_x}$, $\sin{n_x\Delta xk_x}$, $\cos{n_y\Delta yk_y}$, $\sin{n_y\Delta yk_y}$ are:

$$\begin{eqnarray} {\mathcal{F}}^{-1}\{\cos(n_x\Delta x k_x)\}&=&\frac{1}{2}(\delt... ...) \}&=& \frac{1}{2i}(\delta(y-n_y\Delta y)-\delta(y+n_y\Delta y)).\end{eqnarray}$$ (38) (39) (40) (41)

Let $\delta_{\pm n_x}=\delta(x\pm n_x\Delta x)$, $\delta_{\pm n_y}=\delta(y \pm n_y\Delta y)$ and $\delta_{\pm n_x,\pm n_y}=\delta({x\pm n_x\Delta x,y\pm n_y\Delta y})$.The inverse Fourier transform of the function $\cos({n_x\Delta xk_x})\cos({n_y\Delta yk_y})$ is :

$$\begin{eqnarray} {\mathcal{F}}^{-1}\{\cos{(n_x\Delta xk_x)}\cos(n_y\Delta y k_y)... ..._{-n_x,+n_y}+\delta_{+n_x,-n_y}+\delta_{+n_x,+n_y}).\ \ \ \ \ \ \ \end{eqnarray}$$ (42) (43)

Similarly, the inverse Fourier transform of the functions $\cos({n_x\Delta xk_x})\sin({n_y\Delta yk_y})$,
$\sin({n_x\Delta xk_x})\cos({n_y\Delta yk_y})$ and $\sin({n_x\Delta xk_x})\sin({n_y\Delta yk_y})$ are :

$$\begin{eqnarray} {\mathcal{F}}^{-1}\{\cos{(n_x\Delta xk_x)}\sin(n_y\Delta y k_y)... ...n_y}-\delta_{+n_x,-n_y}+\delta_{+n_x,+n_y}).\ \ \ \ \ \ \ \ \ \ \ \end{eqnarray}$$ (44) (45) (46)

Therefore the inverse Fourier transform of the functions F^ee(k_x,k_y), F^oe(k_x,k_y), F^eo(k_x,k_y) and F^oo(k_x,k_y) are:

$$\begin{eqnarray} {\mathcal{F}}^{-1}\{ F^{ee}(k_x,k_y)\}&=&\sum_{n_x,n_y}\frac{1}... ...x,+n_y}-\delta_{+n_x,-n_y}+\delta_{+n_x,+n_y}). \ \ \ \ \ \ \ \ \ \end{eqnarray}$$ (47) (48) (49) (50)

The correction operator F(k_x,k_y) is the sum of F^ee(k_x,k_y), F^oe(k_x,k_y), F^eo(k_x,k_y) and F^oo(k_x,k_y). Therefore the inverse Fourier transform of the correction operator is:

$$\begin{eqnarray} {\mathcal{F}}^{-1}\{ F(k_x,k_y)\}&=&{\mathcal{F}}^{-1}\{ F^{ee}... ...-n_y} +\delta_{+n_x,+n_y} c_{+n_x,+n_y})\ \ \ \ \ \ \ \ \ \ \ \ \ \end{eqnarray}$$ (51) (52)

where

$$\begin{eqnarray} c_{-n_x,-n_y}&=&[(a^{ee}_{n_x,n_y}-a^{oo}_{n_x,n_y})-i(a^{eo}_{... ...{n_x,n_y}-a^{oo}_{n_x,n_y})+i(a^{eo}_{n_x,n_y}+a^{oe}_{n_x,n_y})].\end{eqnarray}$$ (53) (54) (55) (56)