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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, *k*_{z} is not an even function
of *k*_{x}, 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*(*k*_{x}) is not an even function, but
can be divided *F*(*k*_{x}) into two parts: even function *F*^{e}(*k*_{x}) and odd function *F*^{o}(*k*_{x}),
| |
(8) |

| (9) |

To design the explicit correction operator, we specify *F*^{e}(*k*_{x}) in the form

| |
(10) |

and *F*^{o}(*k*_{x}) in the form
| |
(11) |

where *a*_{n},*b*_{n} 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 *k*_{x}.
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 *c*_{0}=*a*_{0}, 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