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the standard approach to estimating the anelastic
effect, noting that the assumptions used are rarely
stated explicitly. The parameters
S
ijkl
,
S
ijkl
(
anelasticity over this frequency range is very
difficult, the following approach is used.
Let us suppose that anelasticity follows a simi-
larity rule. This means that a reference timescale
τ
r
can be appropriately defined as a function
of
T
,
d
,
ω
)and
X
ijkl
(
), introduced in Section 3.3.2, are generally
expressed as
S
ijkl
(
T
,
d
,
τ
,
C
),
S
ijkl
(
φ
ω
,
T
,
d
,
φ
,
C
), and
X
ijkl
(
,
C
) when formulating the effects on
seismic wave velocities of temperature
T
, grain
size
d
, melt fraction
τ
,
T
,
d
,
φ
,and
C
, and that the relaxation spec-
trum can be given by a single-valued function
of a normalized timescale
φ
, and the concentration of
water
C
(hydrogen-related defects). For simplicity,
I consider anelastic relaxation of the shear mod-
ulus
N
in an isotropic system. I introduce scalar
parameters
N
e
,
N
1
,
Q
N
-
1
,and
X
N
defined by
φ
τ
n
=
τ/τ
r
(
T
,
d
,
φ
,
C
)as
X
N
(
τ
,
T
,
d
,
φ
,
C
)
=
X
N
(
τ
n
). Then, Equation (3.26)
becomes
Q
−
1
S
π
∂
ln
V
S
1
2
∂
ln
N
e
∂
ln
τ
r
ln
x
=
ln
x
+
(3.27)
∂
∂
∂
ln
x
N
e
T
,
d
,
,
C
=
(4
S
xyxy
)
φ
(1
−
φ
)
/
and
C
, and where
Q
−
1
S
=
Q
−
N
for
x
=
T
,
d
,
φ
=
(4
S
xyxy
)
N
1
(
ω
,
T
,
d
,
φ
,
C
)
=
(1
−
φ
)
/
S
xyxy
=
π
.
(3.25)
S
xyxy
/
2, because the effect of vis-
cosity on
S
ijkl
is negligible at the seismic fre-
quency range. Therefore, using the similarity
rule, the anelastic effect can be calculated from
our knowledge of
Q
-
1
S
at the frequency of the
seismic wave, which can be measured seismo-
logically. Therefore, Equation (3.27) is practically
useful, and for this reason has already been used in
previous studies. For example, Karato (1993) used
Equation (3.27) with
x
N
1
X
N
/
Q
−
N
(
S
xyxy
/
S
xyxy
ω
,
T
,
d
,
φ
,
C
)
=
X
N
(
τ
,
T
,
d
,
φ
,
C
)
=
4
X
xyxy
/
(1
−
φ
)
At the seismic frequency range, attenuation is
small (
Q
N
-
1
1). Hence, the shear wave velocity
V
S
and attenuation
Q
-
1
S
are approximately given
by
V
S
=
=
Q
−
N
(e.g., McCarthy
et al
., 2011). The following relationship is
obtained by calculating the partial derivative
of the first equation of (3.7) with respect to one of
the four factors
T
,
d
,
)
1
/
2
and
Q
−
1
S
(
N
1
/ρ
=
T
and
∂
ln
τ
r
/∂
ln
T
=
−
RT
,where
H
and
R
represent the activation
enthalpy and gas constant, respectively.
The application of the similarity rule with
τ
r
(
T
,
d
)
H
/
φ
,and
C
, keeping the other
three factors fixed
f
M
(
T
,
d
) to anelastic relaxation caused
by diffusionally accommodated grain boundary
sliding has been confirmed both theoretically and
experimentally (Raj, 1975; Gribb & Cooper, 1998;
Jackson
et al
., 2002; Morris & Jackson, 2009a,b;
McCarthy
et al
., 2011). However, the confirma-
tion is limited to the temperature and grain size
effects, and also to the relatively low normal-
ized frequencies (
f
=
1
/
τ
=
1
/ω
∂
ln
V
S
1
2
∂
ln
N
e
∂
N
1
2
∂
X
N
∂
=
−
ln
x
d
ln
τ
,
∂
ln
x
ln
x
τ
=
0
(3.26)
where
x
=
T
,
d
,
φ
,and
C
, and where the effect of
x
on
is neglected. The first term on the RHS
of (3.26) represents the effect of the factor
x
on
elasticity. When
x
ρ
T
, this term represents the
anharmonic effect, and when
x
=
10
5
). Behavior at higher
normalized frequencies, and the effects of melt
and water, are still poorly understood. For a melt
(
x
/
f
M
<
, it represents
the poroelastic effect (Table 3.1). Quantitative
assessment of the poroelastic effect is one of the
major subjects in the previous sections of this
chapter. The second term on the RHS of (3.26)
represents the anelastic effect. It demonstrates
that if one is to assess the anelastic effect, it is
necessary to know the change in the relaxation
spectrum
X
N
(
=
φ
), experiments using the partially molten
rock analogue (borneol
=
φ
+
melt) showed that the
similarity rule with
) ap-
plies (McCarthy & Takei, 2011), whereas experi-
ments with partially molten mantle rock showed
a significant change of the attenuation spectrum
with a peak unique to the melt phase (Jackson
et al
., 2004). The reason for this large difference
τ
r
(
T
,
d
,
φ
)
=
1
/
f
M
(
T
,
d
,
φ
,
C
) over
all timescales from 0 to the seismic period
ω
−
1
. Because experimental measurement of
τ
) by the factor
x
(
=
T
,
d
,
φ
