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# Phase and group angles and velocities

In anisotropic wave propagation the phase angles and velocities are different from the group angles and velocities. In this section I briefly review the concepts of phase and group angles and velocities and the relationships between these physical quantities. I also define the particular approximation to a VTI medium that I use in the numerical examples.

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 VV, VH, VN, 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 SV, SH, SN, are respectively the vertical slowness, the horizontal slowness and the NMO slowness.

The numerical results obtained by modeling and migrating synthetic seismic data and by migrating the real 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 kx and kz 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).

Next: Angle Gathers by anisotropic Up: Biondi: Anisotropic ADCIGs Previous: Biondi: Anisotropic ADCIGs
Stanford Exploration Project
11/1/2005