(1) |

(2) |

For VTI media, the true dispersion relation requires solving a quartic equation numerically Shan and Biondi (2005). With the assumption that the S-wave velocity is much smaller than the P-wave velocity, the dispersion relation for VTI media can be obtained analytically and represented as follows:

(3) |

(4) |

As in the isotropic case, the coefficients and can also be obtained by least-squares optimization. The advantage of least-squares approximation is that I do not have to derive an explicit approximated expression for the dispersion relation analytically. This is especially useful for anisotropic media. For VTI media, I can use the true dispersion relation, and no assumption of small S-wave velocity and weak anisotropy is necessary.

Generally, the Padé approximation suggests that
if the function , then *S*_{z}(*S*_{r}) can be approximated by a rational function
*R*_{n,m}(*S*_{r}):

(5) |

kz1
Dispersion relation: curve A is the true dispersion relation; B is the aprroximate
dispersion relation by Tayor-series analysis; C is the approximate dispersion relation by optimization.
Figure 1 |

err1
Relative dispersion error: curve D is the relative dispersion error of the approximation by Taylor-series
analysis; E is the relative dispersion error of the approximation by optimization.
Figure 2 |

We can obtain the coefficients *a*_{i} and *b*_{i} by solving the following optimization problem:

(6) |

(7) |

(8) |

(9) |

The dispersion error of approximation (9) is given by

(10) |

For the second-order approximation (*m*=1,*n*=1), Figure
shows the true and approximated dispersion relation, given and .
In Figure , curve A is the true dispersion relation curve. B is the approximated dispersion
suggested by Ristow and Ruhl (1997), in which and . C is the approximated dispersion
relation by the least-squares optimization, in which and .
The dispersion relation by optimization (C) approximates the true dispersion relation better than
the approximation using Taylor-series analysis and the weak anisotropy assumption.

Figure shows the relative dispersion error. D is the relative dispersion error of the approximation using the Taylor-series analysis. E is the relative dispersion error of the optimized one-way wave operator. Figure shows that opimization greatly improves the dispersion relation. If we accept a one-percent dispersion error, the optimized one-way wave-equation is accurate to while the approximation using Taylor-series analysis is accurate to only .

Figure 3

4/5/2006