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# Wavefield extrapolation operator

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 kx and ky. However, we can decompose the function F(kx,ky) into either even or odd functions of kx and ky, and approximate the even parts with cosine functions and the odd parts with sine functions.

We can decompose the function F(kx,ky) into even and odd parts for the axis kx by
 (11) (12)
We can decompose the operator Fe and Fo into odd or even parts for the axis ky by
 (13) (14) (15) (16)
The function Fee(kx,ky) is an even function of both kx and ky, so it can be approximated by
 (17)
The function Foe(kx,ky) is an even function of ky and an odd function of kx, so it can be approximated by
 (18)
The function Feo(kx,ky) is an even function of kx and an odd function of ky, so it can be approximated by
 (19)
The function Foo(kx,ky) is an odd function of both kx,ky, so it can be approximated by
 (20)
Coefficients aeenxny, aoenxny, aeonxny and aoonxny can be estimated by the weighted least-square 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 aeenxny, aoenxny, aeonxny and aoonxny in detail.

The original operator F(kx,ky) can be obtained from Fee(kx,ky), Feo(kx,ky), Foe(kx,ky) and Foo(kx,ky) by

 F(kx,ky)=Fee(kx,ky)+Feo(kx,ky)+Foe(kx,ky)+Foo(kx,ky). (21)

Appendix B derives the inverse Fourier transform of the functions Fee(kx,ky), Feo(kx,ky), Foe(kx,ky) and Foo(kx,ky), and obtains the inverse Fourier transform of the function F(kx,ky) as follows:
 (22)
where

and cnx,ny is as follows:
 (23) (24) (25) (26) (27)
Let Pz(x,y) be the inverse Fourier transform of Pz(kx,ky). 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.

Next: filter length, cost, and Up: Shan and Biondi: 3D Previous: 3D dispersion relation in
Stanford Exploration Project
5/3/2005