Geoscience Reference
In-Depth Information
The specific quality factor, a convenient dimen-
sionless measure of dissipation, is
due to
scattering
.[
seismic wave scattering
Monte Carlo
]
A variety of physical processes contribute to
attenuation in a crystalline material: motions
of point defects, dislocations, grain bound-
aries and so on. These processes all involve
a high-frequency, or unrelaxed, modulus and
a low-frequency, or relaxed, modulus. At suffi-
ciently high frequencies, or low temperatures,
the defects, which are characterized by a time
constant, do not have time to contribute, and
the body behaves as a perfectly elastic body.
Attenuation is low and
Q
, the seismic quality
factor, is high in the high-frequency limit. At
very low frequencies, or high temperature, the
defects have plenty of time to respond to the
applied force and they contribute an additional
strain. Because the stress cycle time is long com-
pared with the response time of the defect,
stress and strain are in phase and again
Q
is
high. Because of the additional relaxed strain,
however, the modulus is low and the relaxed
seismic velocity is low. When the frequency is
comparable to the characteristic time of the
defect, attenuation reaches a maximum, and the
wave velocity changes rapidly with frequency.
Similar effects are seen in porous or partially
molten
Q
−
1
=
M
∗
/
M
This is related to the energy dissipated per cycle.
Since the phase velocity
k
=
M
=
/ρ
c
it follows that
=
2
k
∗
M
∗
M
Q
−
1
k
=
for
Q
1
In general, all the elastic moduli are complex,
andeachwavetypehasitsown
Q
and velocity,
both frequency dependent. For an isotropic solid
the imaginary parts of the bulk modulus and
rigidity are denoted as
K
∗
and
G
∗
. Most mecha-
nisms of seismic-wave absorption affect the rigid-
ity more than the bulk modulus, and shear waves
more than compressional waves.
Frequency dependence of attenuation
In a perfectly elastic homogenous body, the
elastic wave velocities are independent of fre-
quency. Variations with temperature, and pres-
sure, or volume, are controlled by anharmonic-
ity. In an imperfectly elastic, or anelastic, body
the velocities are dispersive; they depend on
frequency, and this introduces another mecha-
nism for changing moduli, and seismic veloci-
ties with temperature. This is important when
comparing seismic data taken at different fre-
quencies or when comparing seismic and lab-
oratory data. When long-period seismic waves
startedtobeusedinseismology,itwasnoted
that the free oscillation, or normal mode, mod-
els, differed from the classical body wave-models,
which were based on short-period seismic waves.
This is the
body-wave-normal-mode dis-
crepancy
. The discrepancy was resolved when
it was realized that in a real solid, as opposed
to an ideally elastic one, the elastic moduli
were functions of frequency. One has to allow
for this when using body waves, surface waves
and normal modes in the inversion for veloc-
ity vs. depth. The absorption, or dissipation, of
energy, and the frequency dependence of seis-
mic velocity, can be due to
intrinsic anelasticity
,or
solids;
the
elastic
moduli
depend
on
frequency.
These characteristics are embodied in the
standard linear solid
, which is composed
of an elastic spring and a dashpot (or viscous
element) arranged in a parallel circuit, which
is then attached to another spring. At high fre-
quencies the second, or series, spring responds
to the load, and this spring constant is the effec-
tive modulus that controls the total extension.
At low frequencies the other spring and dashpot
both extend, with a time constant characteristic
of the dashpot, the total extension is greater, and
the effective modulus is therefore lower. This sys-
tem is sometimes described as a
viscoelastic solid
.
The temperature dependence of the spring con-
stant, or modulus, represents the anharmonic
contribution to the temperature dependence of
the overall modulus of the system. The tempera-
ture dependence of the viscosity of the dashpot
introduces another term -- the
anelastic term
-- in
