Geoscience Reference
In-Depth Information
Chapter 19
Dissipation
As when the massy substance of the
Earth quivers.
in response to a higher tectonic stress gives
rise to a solid-state viscosity. Seismic waves also
attenuate due to macroscopic phenomena, such
as scattering and interactions between fluid, or
molten, parts of the interior and the solid matrix.
Anelasticty causes the elastic moduli to vary with
frequency;
elastic constants
are not constant. The
equations in this section can be used to correct
seismic velocities for temperature effects due to
anelasticity
. These are different from the
anhar-
monic effects
discussed in other chapters.
Marlowe
Real materials are not perfectly elastic. Solids
creep when a high stress is applied, and the strain
is a function of time. These phenomena are mani-
festations of anelasticity. The attenuation of seis-
mic waves with distance and of normal modes
with time are other examples of anelastic behav-
ior. Generally, the response of a solid to a stress
can be split into an elastic or instantaneous part
and an anelastic or time-dependent part. The
anelastic part contains information about tem-
perature, stress and the defect nature of the solid.
In principle, the attenuation of seismic waves
can tell us about such things as small-scale het-
erogeneity, melt content, dislocation density and
defect mobility. These, in turn, are controlled
by temperature, pressure, stress, history and the
nature of the defects. If these parameters can
be estimated from seismology, they can be used
to estimate other anelastic properties such as
viscosity.
For example, the dislocation density of a crys-
talline solid is a function of the non-hydrostatic
stress. These dislocations respond to an applied
oscillatory stress, such as a seismic wave, but
they are out of phase because of the finite dif-
fusion time of the atoms around the dislocation.
The dependence of attenuation on frequency can
yield information about the dislocations. The
longer-term motions of these same dislocations
Seismic-wave attenuation
The travel time or velocity of a seismic wave
provides an incomplete description of the mate-
rial it has propagated through. The amplitude
and frequency of the wave provide some more
information. Seismic waves attenuate or decay as
they propagate. The rate of attenuation contains
information about the anelastic properties of the
propagation medium.
A propagating wave can be written
A
=
A
0
exp i(
ω
t
−
κ
x
)
where
A
is the amplitude,
the
wave number,
t
the travel time,
x
the distance and
c
ω
the frequency,
κ
the phase velocity. If spatial attenuation
occurs, then
=
ω/κ
κ
is complex. The imaginary part of
κ
∗
is called the spatial attenuation coefficient.
The elastic moduli, M, are now also complex:
κ
,
M
=
M
+
i
M
∗
