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The phase velocity expression
for *P*- and *SV*-waves in TI
media is (Auld, 1990)

| |
(1) |

| |

where is the phase velocity squared and
is the phase angle from the vertical.
*W*_{ij} is the (*ij*)^{th}
elastic modulus divided by density, with units of
velocity squared; I refer to the quantity *W*_{ij} as an ``elastic constant''
in the remainder of this paper.
The ``+'' sign in front of the square root corresponds to
*P*-waves and the ``-'' to *SV*-waves.
For *SH*-waves, the expression for the phase velocity is
(Auld, 1990)
| |
(2) |

Near the vertical and horizontal axes equation (1a) is elliptical.
The velocities that describe the corresponding ellipses are called
elliptical velocities. In the next two subsections, I rederive
equations contained in previous papers (Levin, 1979; Levin, 1980;
Muir, 1990b) that are needed
to calculate these elliptical velocities from the elastic constants. The
expressions that result
are used later in the paper to solve the inverse mapping
when
the data have narrow aperture.

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** Up:** FORWARD MAPPING
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Stanford Exploration Project

11/17/1997