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When the medium is TI and no information is available
about *SV*-wave phase velocities,
it is still possible to obtain four
elastic constants from *P*-wave phase velocities alone near both
axes. This is done by solving the system of
equations (4), (5), (13), and (14),
which yields:

| |
(1) |

| (2) |

| (3) |

| (4) |

This set of equations forms the *P*-wave double elliptic
approximation of the elastic constants in a TI medium.
In this approximation, as well as in the previous ones,
the assumption of transverse isotropy is crucial. When the medium
is isotropic (*W*_{P,x} = *W*_{P,z} = *W*_{P,xnmo} = *W*_{P,znmo})
there is no way to calculate *W*_{44} (the shear moduli)
from *P*-wave
phase velocities alone because equation (24c) is indeterminate.
When the medium is weakly anisotropic the
estimation of *W*_{44} using this approximation may still be unreliable
because both the numerator and the denominator in (24c) are close to zero.
We will see later in the
examples that even when the medium is moderately
anisotropic this approximation
breaks down
quickly for phase angles not close to the axes, unlike the previous
vertical and horizontal double elliptic approximations (equations (22) and
(23), respectively) that have a wider range a validity.

If only *SV*-wave phase velocities near the axes
are available, it is not possible
to obtain the corresponding elastic constants because the system
of equations (7), (8), (16), and (17) is underdetermined.

** Next:** OBTAINING THE PHASE VELOCITIES
** Up:** INVERSE MAPPING
** Previous:** Using P- and SV-wave
Stanford Exploration Project

11/17/1997