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has yet to be clarified. Therefore, it is worth
emphasizing that although the similarity rule is
usually used to estimate the anelastic effect, use
of the similarity rule does not necessarily have
any theoretical or experimental basis. This was
also pointed out by Cooper (2002) for the temper-
ature effect. For various relaxation mechanisms,
it is necessary to investigate whether a similarity
rule applies, and if so, a detailed functional form
of the reference timescale
found a systematic change of pore geometry from
a nearly equilibrium geometry at about 90 km
depth (
R
SP
∼
0.1) to a highly disequilib-
rium geometry at shallower depths (
R
SP
>
1-1.2,
α
∼
α
0.1). Their results are preliminary, because the
anelastic effects of melt and temperature used in
the analysis of the data are still subject to large un-
certainties. However, this study has shown that
the constraints imposed by the seismological data
on the melt migration process in the Earth are im-
portant and complementary to those derived from
dynamical and geochemical approaches.
2,
τ
r
must be developed,
over a wide frequency range from purely elastic
to seismic frequencies.
3.7.2 Seismological detection of small
amounts of melt
3.7 Applications
3.7.1 The seismological observability of pore
geometry by R
SP
The structure-sensitive character of the poroelas-
tic effect is summarized in Figure 3.9. In seis-
mological data, unlike experimental data, both
the melt fraction and the pore geometry are un-
known. It used to be the case that
There has been increasing recognition of the geo-
logical importance of partially molten rocks con-
taining very small amounts of melt (
1%). Recent
geochemical studies have shown that the melt
phase can be removed from the residual matrix at
a very small melt fraction (e.g., McKenzie, 2000).
Hirschman (2009) discussed the widespread
existence of small amounts of melt in the oceanic
mantle. Here, following McCarthy and Takei
(2011), I discuss a seismological detectability of
such small amounts of melt. As an example, I con-
sider a reduction of the shear wave velocity caused
by just 0.25%melt. Based on the result of the con-
tiguity model,
<
ln
V
S
alone was available from the seismological obser-
vations and hence that neither the melt fraction
nor the pore geometry could be estimated inde-
pendently. However, due to recent progress in
seismology, it is now possible to obtain highly
resolved seismic tomographic images for both
V
P
and
V
S
structures. When both
ln
V
P
or
ln
V
S
due to the poroelastic effect
ln
V
P
and
ln
V
S
caused by the poroelastic effect are ob-
tained, the equivalent aspect ratio (
is estimated as
0.0025 (Figure 3.8).
The anelastic effect is calculated on the basis
of the recent experimental data in McCarthy
and Takei (2011) for a partially molten rock
analogue. For simplicity, a similarity rule with
τ
r
(
T
,
d
,
−
0.9% at
φ
=
α
)andthe
melt fraction (
φ
) of that region can be estimated
independently:
α
can be estimated by applying
R
SP
=
ln
V
S
/
ln
V
P
to Figure 3.9a, and
φ
can
be estimated from
ln
V
S
by reading the value of
φ
)
=
1
/
f
M
(
T
,
d
,
φ
)
∝
η
(
T
,
d
,
φ
)
experimen-
10
3
/
f
M
<
from Figure 3.9b.
In other words,
R
SP
can be used as a seismological
indicator of pore geometry. An important result
of the theoretical models is that the value of
R
SP
, corresponding to the texturally equilibrated
partially molten rock, is shown to be consider-
ably smaller than the value corresponding to the
cracks and dikes filled withmelt (
ln
V
S
/φ
corresponding to this
α
tally obtained at
f
is assumed to apply
10
6
). Then, from the
second term on the RHS of (3.27), the anelastic
effect can be estimated as
to the seismic waves (
f
/
f
M
>
Q
-
1
S
π
-
1
,
demonstrating that the singular behavior of
viscosity causes the singular behavior of anelas-
ticity. From
Q
-
1
S
=
ln
V
S
=
ln
η
80 (PREM; Dziewonski &
Anderson, 1981) and from the reduction of
viscosity by a factor of 40 (melt
2; Figure 3.9a).
Nakajima
et al
. (2005) applied this method to the
low-velocity regions of the upper mantle beneath
the northeastern Japan subduction zone, and they
>
chemical
effects, Section 3.5.2) or a factor of 5 (melt effect),
I obtain
+
80
-
1
-
1
ln(40
-
1
)
ln
V
S
=
π
∼−
1.5% or
