of 10
Chapter
19
Dissipation
As
when
the
massy
substance
of
the
Earth
quivers.
M a rl owe
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
examp
le,
the
dis
l
ocation
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
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
an
elasticity.
These
are
different
from
the
anhar-
monic
eff
ec
ts
discussed
in
other
chapters.
Seismic-wave
attenuation
The
travel
time
or
ve
lo
city
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(
wt -
KX)
where
A
is
the
amplitude,
w
the
frequency
,
K
the
wave
number
,
t
the
travel
time,
x
the
distance
and
c
=
wj
K
the
phase
velocity.
If
spatial
attenuation
occurs
,
then
K
is
complex
.
The
imaginary
part
of
""
·
K *
is
called
the
spatial
attenuation
coefficient
.
The
elastic
moduli
,
M,
are
now
also
complex:
M
=
M
+
iM
*
The
specific
quality
factor,
a
convenient
dimen-
sionless
measure
of
dissipation,
is
Q-
1
=M*/M
This
is
related
to
the
energy
dissipated
per
cycle.
Since
the
phase
velocity
c
=
~
=
jMjp
k
it
follows
that
k *
M
*
Q
_,
=
2-
=
-
for
Q
»
1
k
M
In
general,
all
the
elastic
moduli
are
complex,
and
each
wave
type
has
its
own
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
started
to
be
used
in
seismology,
it
was
noted
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
SEISMIC-WAVE
ATTENUATION
247
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
seism.ic
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
tim.e
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
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
dash
pot
,
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
248
DISSIPATION
-~
I E
0
+
0
u
II
]:
u
]:
I
0
Coo
Co
o-1
m
WT
Q
- I
and
phase
velocity
as
a function
of
frequency
the
temperature
dependence
of
moduli
or
seis-
mic
velocities.
The
Q
- I
of
such
a
system
is
_
1
k
2
wr
Q
(w)
=
k;
1
+
(wr)
2
where
k
2
and
k
1
are,
respectively,
the
spring
constants
(or
moduli)
of
the
series
spring
and
the
parallel
spring
and
r
is
the
relaxation
time.
Clearly,
Q
-
l,
the
dimensionless
attenuation,
is
a
maximum
at
wr
=
1,
and
- 1
2
_ ,
wr
Q
(w)
=
Qm
ax
1
+
(wr)
2
(1)
The
resulting
absorption
peak
is
shown
in
Figure
19.1.111is
can
be
considered
a
plot
of
atten-
uation
and
velocity
vs.
either
frequency,
or
tem-
perature,
since
r
is
a
function
of
temperature,
an
exponential
function
for
thermally
activated
processes.
The
phase
velocity
is
approximately
given
by
c(w)
=Co
(
1
+
Q~,~
x
1
~(~:)
2
)
(2)
where
c
0
is
the
zero-frequency
ve
locity.
The
high-
frequency
or
elastic
velocity
is
(3)
Far
away
from
the
absorption
peak,
the
velocity
can
be
written
(
1
k,
2)
c(w)
~C
o
1
+
--
Q -
2 k2
for
wr
«
1
=
C
00
(1
-
k
~
Q -
2
)
for
w r
»
1
(2k,
+
k2
)k
2
and
the
Q
effect
is
only
second
order.
In
these
lim-
its,
velocity
is
nearly
independent
of
frequency,
but
Q
is
not;
Q
and
c
cannot
both
be
inde-
pendent
of
frequency.
Velocity
depends
on
the
attenuation
.
When
Q
is
constant,
or
nearly
so,
the
fractional
change
in
phase
velocity
becomes
a first-order
effect.
For
activated
processes,
r
=
r
0
exp
E
*/R
T
(4)
where
E*
is
an
activat
ion
energy.
This
is
where
the
temperature
dependence
of
seismic
velocities
comes
in,
in
anelastic
processes.
Velocity
is
not
a
simple
linear
function
of
temperature.
For
activated
processes,
then,
Q- ' (
w)
=
2Q~
1
1
0
x
{
w
ro
exp
E
*/
RT
}/
{1
+
(wr
0
)
2
x
exp2E
*/
RT}
(5)
The
relaxation
peak
can
be
defined
either
by
changing
w
or
changing
T.
At
high
temperatures,
or
low
frequencies
,
(6)
This
is
contrary
to
the
general
intuition
that
attenuation
must
increase
with
temperature.
However,
if
r
differs
greatly
from
seismic
peri-
ods,
it
is
possible
that
we
may
be
on
the
low-
temperature
or
high-frequency
portion
of
the
absorption
peak,
and
Q -
1
(w)
=
2Q
;;:,~x/
(wr
0
expE
*/
RT)
,
wr
»
1
(7)
In
that
case
Q
does
decrease
with
an
increase
in
T,
and
in
that
regime
Qincreases
with
frequency.
This
appears
to
be
the
case
for
short-period
waves
in
the
mantle.
It
is
also
generally
observed
that
low-Qand
low-velocity
regions
of
the
upper
man-
tle
are
in
tectonically
active
and
high
heat-flow
areas.
Thus,
seismic
frequencies
appear
to
be
near
the
high-frequency,
low-temperature
side
of
the
absorption
peak
in
the
Earth's
upper
mantle.
This
may
not
be
true
in
the
lower
mantle;
the
absorption
band
shifts
with
frequency,
as
in
the
ab
so
rpti
o n
ban
d
mo
del
f o r
mantle
a
tte
nua
tio
n.
SEISM
I
C-WAVE
ATTENUATION
249
The
characteristic
relaxation
time
also
]:
changes
with
pressure,
r
= r
0
exp(E
*
+
PV
*)/ RT
(8)
where
V",
the
activation
volume,
controls
the
effect
of
pressure
on
rand
Q,
and
seismic
velocity.
Most
mechanisms
of
attenuation
at
seismic
frequencies
and
mantle
temperatures
can
be
described
as
activated
relaxation
effects.
Increas-
ing
temperature
drives
the
absorption
peak
to
higher
frequencies
(characteristic
frequencies
increase
with
temperature).
Increasing
pressure
drives
the
peak
to
lower
frequencies.
Absorption
in
a
medium
with
a
single
charac-
teristic
frequency
gives
rise
to
a
bell-shaped
Debye
peak
centered
at
a
frequency
wr
=
1,
as
shown
in
Figure
19.1.
The
specific
dissipation
function
and
phase
velocity
satisfY
the
differential
equation
for
the
standard
linear
solid
and
can
be
written
c
2
(w)
=
c ~
(1
+
w
2
r
2
c
;,;
c ~
)
/
[(1
+
w2
r 2)2
+
2
w2
r
2
Q
;;;
~xf
/
2
The
high-frequency
(c)
and
low-frequency
velocities
are
related
by
c;,-
c ~
_
1
---
2Q
C
C
-
ma
x
0
""
so
that
the
total
dispersion
depends
on
the
mag
-
nitude
of
the
peak
dissipation.
For
a Q
of
200,
a
typical
value
for
the
upper
mantle,
the
total
velocity
dispersion
is
2
%.
Solids
in
general,
and
mantle
silicates
in
particular,
are
not
characterized
by
a
single
relaxation
time
and
a
single
Debye
peak.
A
dis-
tribution
of
relaxation
times
broadens
the
peak
(
~
)
=
10
5
s ,
w tS
1
~1&03IIIIIWI
Absorption
band
with
a distribution
of
relaxation
1
t<.otnaimon
and
Anderson,
1977).
and
gives
rise
to
an
absorption
band
(Figure
19.2)
.
Q
can
be
weakly
dependent
on
frequency
in
such
a
band
.
Seismic
Q
values
are
nearly
con-
stant
with
frequency
over
much
of
the
seismic
band.
A
nearly
constant
Q
can
be
explained
by
involving
a
spectrum
of
relaxation
times
and
a
superposition
of
elementary
relaxation
peaks,
g1vmg
and
For
r
1
«
w -
1
«
r
2
the
value
of
Q
- I
is
con
-
stant
and
equal
to
Q;;;~
The
total
dispersion
in
this
case
is
c""
-Co
(
_
1
)
--
c-
0
-
=
Qm
axf
n
ln(r
1
/ r
2
)
Which
depends
on
the
ration
r
J/
r
2
,
which
is
the
width
of
the
absorption
band
.
The
spread
in
r
can
be
due
to
a
distribution
of
r
0
or
of
E
* .