On anelliptic approximations for velocities in transversally isotropic media |
Wavefront propagation in the general anisotropic media can be described with the anisotropic eikonal equation
In the case of VTI media, the three modes of elastic wave propagation
(
,
, and
) have the following well-known explicit
expressions for the phase velocities (Gassmann, 1964):
The group velocity describes the propagation of individual ray trajectories . It can be determined from the phase velocity using the general expression
The group velocity has a particularly simple form in the case of elliptic anisotropy. Specifically, the phase velocity squared has the quadratic form
The situation is more complicated in the anelliptic case. Figure 1 shows the and phase velocity profiles in a transversely isotropic material - Greenhorn shale (Jones and Wang, 1981), which has the parameters kms , kms , kms , and kms . Figure 2 shows the corresponding group velocity profiles. The non-convexity of the phase velocity causes a multi-valued (triplicated) group velocity profile. The shapes of all the surfaces are clearly anelliptic.
exph
Figure 1. Phase velocity profiles for (outer curve) and (inner curve) waves in a transversely isotropic material (Greenhorn shale). |
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exgr
Figure 2. Group velocity profiles for (outer curve) and (inner curve) waves in a transversely isotropic material (Greenhorn shale). |
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A simple model of anellipticity is suggested by the Muir approximation (Dellinger et al., 1993; Muir and Dellinger, 1985), reviewed in the next section.
On anelliptic approximations for velocities in transversally isotropic media |