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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 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
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Stanford Exploration Project
11/17/1997