On anelliptic approximations for qP velocities in TI and orthorhombic media |

Similar to the derivations by Fomel (2004), the Muir-Dellinger approximations (Muir and Dellinger, 1985; Dellinger et al., 1993) serve as the starting point of our derivation. The Muir-Dellinger phase-velocity approximation is of the following form:

where is the anelliptic parameter ( in case of elliptical anisotropy), denotes the horizontal ( ) velocity squared, denotes the vertical ( ) velocity squared, and describes the elliptical part of the velocity and is defined by

(13) |

The group-velocity approximation takes a similar form, but with symmetric changes in the coefficients and variables,

where , , is group angle (from vertical), denotes the horizontal slowness squared, denotes the vertical slowness squared, , and describes the elliptical part of the slowness and is defined by

(15) |

As suggested by Muir and Dellinger (1985), the parameter can be found by fitting the phase-velocity curvature around either the vertical axis ( ) or the horizontal axis ( ). The explicit expressions of fitting in those two cases are given in equations 1 and 2. If we define in equation 14 by fitting the group velocity curvature around either or , we find that

(16) |

Extending this idea, Dellinger et al. (1993) proposed four-parameter approximations for phase and group velocites using both and .

On anelliptic approximations for qP velocities in TI and orthorhombic media |

2017-04-14