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# Explicit correction operator

Explicit extrapolation operators have been proved useful in isotropic wavefield extrapolation Hale (1991); Hale (1991); Blacquiere et al. (1989); Holberg (1988); Thorbecke (1997). They are also applied in wavefield extrapolation for TI media Zhang et al. (2001). For an isotropic or VTI medium, the extrapolation operator is symmetric and can be approximated by a cosine function series. For a tilted TI medium, kz is not an even function of kx, and the extrapolation operator is asymmetric. Thus, we need both the sine and cosine function series to approximate the correction operator in the wavenumber domain. In equation (6), F(kx) is not an even function, but can be divided F(kx) into two parts: even function Fe(kx) and odd function Fo(kx),
 (8) (9)

To design the explicit correction operator, we specify Fe(kx) in the form
 (10)
and Fo(kx) in the form
 (11)
where an,bn are complex coefficients. These coefficients can be determined by the following weighted least-squares fitting goals:
 (12)

 (13)
where

is an matrix with elements , , and . is an matrix with elements , , and . is a vector with elements , . is a vector with elements , . is a diagonal matrix with proper weights for the wavenumber kx. One way to solve the fitting goal (12) is to do QR decomposition Golub and Van Loan (1996) of the matrix :, where is an orthogonal matrix and is an upper triangular matrix. Then the coefficient vector is given by
 (14)
We can solve the fitting goal in equation (13) and obtain the coefficient vector in the same way. After we have the coefficient vectors and , we can combine them into the coefficients for the explicit correction operator. From Fourier transform theory, it is well known that the inverse Fourier transform of the function and are:
 (15) (16)
Thus, the inverse Fourier transform of the function is

Therefore, the explicit correction operator is:
 (17)
where c0=a0, and

In 3-D, based on the following trigonometric identity,
 (18)
we can run McClellan transformations Hale (1991); McClellan and Chan (1977); McClellan and Parks (1972) for the cosine terms. Similarly, based on the trigonometric identity:
 (19)
we can design a recursive operator similar to McClellan transformations for the sine terms.

ico
Figure 1
Impulse response of isotropic phase-shift with an anisotropic correction operator.

icoffd
Figure 2
Impulse response of isotropic FFD with an anisotropic correction operator.

phsift
Figure 3
Impulse response of anisotropic phase-shift.

Next: Impulse response tests Up: Shan and Biondi: Anisotropic Previous: Extrapolation operator in laterally
Stanford Exploration Project
10/23/2004