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In anisotropic media
the group angles and velocities
do not coincide with
the phase angles and velocities.
The transformation
from phase velocity to group velocity *V*
is conventionally defined as the following
Tsvankin (2001):
| |
(1) |

where is the phase propagation angle.
The associated transformation from phase angles
to group angles is defined
as:
| |
(2) |

Notice that throughout this paper I use the tilde
symbol to distinguish between phase quantities (with a tilde)
and group quantities (without a tilde).
Dellinger and Muir (1985) propose,
and heuristically motivate,
the following symmetric relations for the inverse transforms:

| |
(3) |

where and *S* are respectively the phase slowness
and the group slowness,
and
| |
(4) |

I use the heuristic relation in equation 4
to derive some of the analytical results presented in this paper.
Furthermore, I use all the above relationships
to compute the kinematic numerical results presented
in this paper.
The numerical results, though not the analytical results,
are also dependent on the choice
of a specific approximation of the anisotropic
phase-velocity function.
I used the following VTI approximation for the phase velocity:

| |
(5) |

where *V*_{V}, *V*_{H}, *V*_{N}, are respectively
the vertical velocity, the horizontal velocity and the
NMO velocity.
Following Fowler (2003),
the corresponding approximation for the group velocity is
the following:
| |
(6) |

where *S*_{V}, *S*_{H}, *S*_{N}, are respectively
the vertical slowness, the horizontal slowness and the
NMO slowness.
The numerical results obtained by modeling and migrating
synthetic seismic data were obtained by source-receiver
depth continuation (upward for modeling and downward for migration)
using the following dispersion relation:

| |
(7) |

where is the temporal frequency,
and *k*_{x} and *k*_{z} are respectively the horizontal and
vertical wavenumbers.
The dispersion relation shown in equation 7
corresponds to the velocity and slowness functions
in equations 5 and 6
Fowler (2003).