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# ANELLIPTIC PARAMETERS FOR TI MEDIA

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 qP-qSV wavetype is rather more complicated:

 (19)
where the + sign is for qP, - for qSV. We can find Wz or Wx 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 qP wavetype we obtain:
 (23)

For the TI qSV wavetype we obtain:
 (24)

Next: Approximating TI dispersion relations Up: Dellinger, Muir, & Karrenbach: Previous: THE SECOND ANELLIPTIC APPROXIMATION
Stanford Exploration Project
11/17/1997