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For a homogeneous medium, the wavefield can be extrapolated by an anisotropic phase shift in the Fourier domain as follows:
 
(9) 
In reality, the velocity and anisotropy parameters change laterally. PSPI, explicit methods, or a combination
of PSPI and explicit correction will remedy this problem. We extrapolate the wavefield by an isotropic operator
with an explicit correction operator as follows:
 
(10) 
where .In VTI media, the correction operator is circularly symmetric.
This allows us to use a 1D algorithm to replace the 2D operator by McClellan transformations
Hale (1991a); McClellan and Chan (1977); McClellan and Parks (1972).
However, tilting the symmetry axis in tilted TI media breaks the circular symmetry.
As a result, we need to design a 2D convolution operator in the Fourier domain for wavefield
extrapolation in 3D tilted TI media.
The correction operator is not symmetric for axes x or y in tilted TI media. This means
is not a even function of k_{x} and k_{y}.
However, we can decompose the function F(k_{x},k_{y}) into either even or odd functions of k_{x} and k_{y}, and approximate
the even parts with cosine functions and the odd parts with sine functions.
We can decompose the function F(k_{x},k_{y}) into even and odd parts for the axis k_{x} by
 
(11) 
 (12) 
We can decompose the operator F^{e} and F^{o} into odd or even parts for the axis k_{y} by
 
(13) 
 (14) 
 (15) 
 (16) 
The function F^{ee}(k_{x},k_{y}) is an even function of both k_{x} and k_{y}, so
it can be approximated by
 
(17) 
The function F^{oe}(k_{x},k_{y}) is an even function of k_{y} and an odd function of k_{x},
so it can be approximated by
 
(18) 
The function F^{eo}(k_{x},k_{y}) is an even function of k_{x} and an odd function of k_{y},
so it can be approximated by
 
(19) 
The function F^{oo}(k_{x},k_{y}) is an odd function of both k_{x},k_{y}, so it can be approximated by
 
(20) 
Coefficients a^{ee}_{nxny}, a^{oe}_{nxny}, a^{eo}_{nxny} and a^{oo}_{nxny}
can be estimated by the weighted leastsquare method Thorbecke (1997),
which can be solved by QR decomposition Baumstein and Anderson (2003); Shan and Biondi (2004b).
Appendix A discusses how to estimate the coefficients a^{ee}_{nxny}, a^{oe}_{nxny}, a^{eo}_{nxny}
and a^{oo}_{nxny} in detail.
The original operator F(k_{x},k_{y}) can be obtained from F^{ee}(k_{x},k_{y}), F^{eo}(k_{x},k_{y}), F^{oe}(k_{x},k_{y}) and F^{oo}(k_{x},k_{y}) by

F(k_{x},k_{y})=F^{ee}(k_{x},k_{y})+F^{eo}(k_{x},k_{y})+F^{oe}(k_{x},k_{y})+F^{oo}(k_{x},k_{y}).

(21) 
Appendix B derives the inverse Fourier transform of the functions F^{ee}(k_{x},k_{y}), F^{eo}(k_{x},k_{y}), F^{oe}(k_{x},k_{y}) and F^{oo}(k_{x},k_{y}), and obtains the inverse Fourier transform of the function F(k_{x},k_{y}) as follows:
 
(22) 
where
and c_{nx,ny} is as follows:
 
(23) 
 (24) 
 (25) 
 (26) 
 (27) 
Let P^{z}(x,y) be the inverse Fourier transform of P^{z}(k_{x},k_{y}). It is well known that
where ``**'' is 2D convolution.
From the Fourier transform theory, we have
 
(28) 
Therefore, we can apply the correction operator on the wavefield in the space domain as follows:
 
(29) 
From the above derivation, we know that the correction operator is designed in the Fourier domain and is implemented as a convolution in the space domain. For
a laterally varying medium, we build a table of the convolution coefficients.
When we run wavefield extrapolation, for each space position, we search for the
corresponding convolution coefficients from that table and convolve the wavefield with these coefficients at that space position.
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Stanford Exploration Project
5/3/2005