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symmetry. The fifth elastic constant,
F
, requires
information from another direction of propaga-
tion. PH, SH are waves propagating and polarized
in the horizontal direction and PV, SV are waves
propagating in the vertical direction. In the ver-
tical direction SH
8.0
SV; the two shear waves
travel with the same velocity, and this velocity
is the same as SV waves traveling in the horizon-
tal direction. There is no azimuthal variation of
velocity in the horizontal, or symmetry, plane.
Love waves are composed of SH motions, and
Rayleigh waves are a combination of P and SV
motions. In isotropic material Love waves and
Rayleigh waves require only two elastic constants
to describe their velocity. In general, more than
two elastic constants at each depth are requi-
red to satisfy seismic surface-wave data, even
when the azimuthal variation is averaged out,
and complex vertical variations are allowed.
The upper mantle exhibits what is known
as polarization anisotropy. In general, four elas-
tic constants are required to describe Rayleigh-
wave propagation in a homogenous transversely
or equivalent transversely isotropic mantle.
Transverse isotropy, although a special case
of anisotropy, has quite general applicability in
geophysical problems. This kind of anisotropy is
exhibited by laminated or layered solids, solids
containing oriented cracks or melt zones, peri-
dotite massifs, harzburgite bodies, the oceanic
upper mantle and floating ice sheets. A mantle
containing small-scale layering, sills or randomly
oriented dikes will also appear to be macro-
scopically transversely isotropic. Since seismic
waves have wavelengths of tens to hundreds of
kilometers, the scale of the layering can actu-
ally be quite large. If flow in the upper man-
tle is mainly horizontal, then the evidence from
fabrics of peridotite nodules and massifs sug-
gests that the average vertical velocity is less
than the average horizontal velocity, and horizon-
tally propagating SH-waves will travel faster than
SV-waves. In regions of upwelling and subduc-
tion, the slow direction may not be vertical, but if
these regions are randomly oriented, the average
Earth will still display the spherical equivalent
of transverse isotropy. Since the upper mantle is
composed primarily of the very anisotropic crys-
=
1.1
0.90
7.9
7.8
4.4
0.9
4.3
4.2
1.1
4.1
0
60
120
180
Angle of incidence (degrees)
Fig. 20.4
P and S velocities as a function of angle of
incidence relative to the symmetry plane and an anisotropic
parameter, which varies from 0.9 to 1.1 at intervals of 0.05.
Parameters are
V
PV
=
7.752,
V
PH
=
7.994,
V
SV
=
4.343, all in
km/s (after Dziewonski and Anderson, 1981).
tals olivine and pyroxene, and since these crys-
tals tend to align themselves in response to flow
and non-hydrostatic stresses, it is likely that the
upper mantle is anisotropic to the propagation
of elastic waves. Although the preferred orien-
tation in the horizontal plane can be averaged
out by determining the velocity in many direc-
tions or over many plates with different motion
vectors, the vertical still remains a unique
direction. It can be shown that if the azimuthally
varying elastic velocities are replaced by the hor-
izontal averages, then many problems in seismic
wave propagation in more general anisotropic
media can be reduced to the problem of trans-
verse isotropy.
