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The anelliptic approximations are useful in their own right for
fitting measured phase- and group-velocity surfaces, but it is also
useful to know how to relate these approximations to a real anisotropic
symmetry system. Because the approximations are two-dimensional, the
appropriate symmetry system is transverse isotropy (TI).
The equations for TI media are most easily written in terms of
phase-velocity squared, suggesting we follow the notation of
equation ().
Let be the phase-velocity squared measured at
phase angle from the vertical, and
let and .Express the elastic constants in units of
phase-velocity squared as well, so .Then we have for the TI SH wavetype

| |
(17) |

Equation () is linear because SH waves
in TI media are exactly elliptically anisotropic; by matching coefficients
in equations () and () we find
the required anelliptic parameters:

| |
(18) |

The TI *q*P-*q*SV wavetype is rather more complicated:

| |
(19) |

where the + sign is for *q*P, - for *q*SV.
We can find *W*_{z} or *W*_{x} by substituting *S*=0
or *S*=1, respectively, into equation ().
To find or
, however,
we first have to fit a paraxial ellipse
about the *z* or *x* axes, respectively.
We have already seen that elliptical anisotropy is linear in these coordinates.
To find the equation for the paraxial ellipse about
the *z* axis we therefore linearize about *S*=0, obtaining
| |
(20) |

Next, remembering that is the horizontal velocity for the paraxial approximation
about the *z* axis, we set *S*=1 (horizontal propagation)
in equation (), obtaining
| |
(21) |

Similarly,
| |
(22) |

For the TI *q*P wavetype we obtain:

| |
(23) |

For the TI *q*SV wavetype we obtain:
| |
(24) |

** Next:** Approximating TI dispersion relations
** Up:** Dellinger, Muir, & Karrenbach:
** Previous:** THE SECOND ANELLIPTIC APPROXIMATION
Stanford Exploration Project

11/17/1997