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Earth and Planetary Science Letters 262 (2007) 429 – 437
www.elsevier.com/locate/epsl
Chemical versus thermal heterogeneity in the lower mantle:
The most likely role of anelasticity
John P. Brodholt a,⁎, George Helffrich b , Jeannot Trampert c
b
a
Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, UK
Earth Sciences Department, University of Bristol, Wills Memorial Building, Queen's Road, Bristol BS8 1RJ, UK
c
Department of Earth Sciences, Utrecht University, P.O. Box 80021, TA Utrecht, The Netherlands
Received 19 February 2007; received in revised form 25 July 2007; accepted 26 July 2007
Available online 9 August 2007
Editor: R.D. van der Hilst
Abstract
A widely held belief is that anelasticity can contribute significantly to seismic wave-speed variations in the lower mantle. In
particular it has been argued that anelasticity can strongly increase the sensitivity of VS to temperature, and that anelasticity may
also increase VS relative to VP (as measured by the parameter RS/P = dlnVS/dlnVP). If true, this could significantly reduce or, in
some cases, eliminate the need to explain seismic signals in the mantle with chemical heterogeneity, and that most of the variation
in seismic velocities in the lower mantle could be attributed to temperature alone. We re-visit this view and find that a strong
anelastic effect is unlikely in the Earth's lower mantle. Given the present knowledge of the properties of perovskite under lower
mantle conditions, we find it unlikely that anelasticity can affect dlnVS/dT by more than about 30%, and it is most likely less than
20%. We also find that the probable upper bound on dlnVS/dlnVP due to thermal variations is about 1.8 in the shallow lower mantle,
and about 2.1 in the deeper lower mantle. We conclude, therefore, that anelasticity cannot be invoked to significantly reduce or
remove interpretations of lower mantle chemical heterogeneity based on large-scale seismic wave-speed variations.
© 2007 Elsevier B.V. All rights reserved.
Keywords: lower mantle; chemical heterogeneity; anelasticity; ab initio
1. Introduction
Seismic tomography presents to the Earth Science
community a three-dimensional image of seismic wavespeeds within the Earth. Typically, slow areas are
coloured in red and fast areas coloured in blue, depicting
the inference that slow areas are hot (hence red) and fast
areas are cold (hence blue). There is, however, growing
evidence that a significant part of the seismic signal
⁎ Corresponding author.
E-mail address: [email protected] (J.P. Brodholt).
0012-821X/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2007.07.054
comes from variations in chemistry as well as variations
in temperature. For instance, it has been argued that the
changes in S-wave velocity observed in the lower
mantle seismic anomaly below Africa (the African
Super-plume) occur over such short distances that it is
unlikely to be caused by temperature alone (Ni et al.,
2002). Similarly, the anti-correlation between bulksound velocity and S-wave velocity (Su and Dziewonski, 1997; Kennett et al., 1998; van der Hilst and
Karason, 1999; Masters et al., 2000; Saltzer et al., 2001)
and between density and S-wave velocity (Ishii and
Tromp, 1999) observed in some parts of the lower
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J.P. Brodholt et al. / Earth and Planetary Science Letters 262 (2007) 429–437
mantle cannot be explained by temperature variations.
Evidence for chemical heterogeneity also comes from
the observation that it is not always possible to explain
the variation in VS and VP with the same temperature
anomaly. For instance, observed values of RS/P = dlnVS/
dlnVP range from about 1.6 to greater than 2.7 (Masters
et al., 2000) whereas mineral physics estimates for
perovskite are above 1.9 at 2000 km depth (Oganov
et al., 2001). While there is good reason to suspect that
the Earth's lower mantle contains at least some chemical
heterogeneity, how much of the seismic signal is
chemical and how much is thermal is still not known.
In order to do this we must be able to account fully for
all the thermal and chemical factors that affect seismic
velocities. And, critically, this includes understanding
the effect of anelasticity.
Recent advances in high-pressure and temperature
experimental techniques, together with high-temperature
ab initio molecular dynamics (MD) or lattice dynamics
(LD) calculations, means that we now have plausible
estimates for the effect of temperature and composition on
the elastic properties of the major mantle phases. For
instance, ab initio densities and elastic constants predict
values for dlnVS/dT of about −4 × 10−5 K−1 at about
1000 km depth and −2.5 × 10−5 K−1 at about 2500 km
depth (Oganov et al., 2001; Wookey et al., 2005).
Compositional derivatives have also been derived from
ab initio calculations, with Stackhouse et al. (2006)
predicting a value for dlnVS/dlnXFe of about −0.22 at
2500 km depth. Taken at face value, these results suggest
that the −3% S wave-speed anomaly seen in the African
super-plume could be caused either by a thermal anomaly
of about 1000° or an enrichment in the absolute
concentration in iron of about 10% (i.e. from 10 to
20%). As mentioned above, thermal anomalies of 1000°
over relatively short distances as seen in the African
super-plume anomaly (b 100 km) are generally felt to be
too large, and a chemical anomaly is perhaps the more
likely explanation. This analysis, however, assumes a
purely elastic effect of temperature on seismic wavespeeds and ignores any anelastic effects. The potential
contribution of anelasticity to seismic wave-speeds was
described by Karato (1993) for the upper mantle, where he
argued that in an upper mantle based on a dominantly
olivine rheology, an anelastic contribution may be
significant. He also argued that it would be true in the
lower mantle as well. This result has been widely
accepted, and strongly influences the interpretation and
modelling of seismic data (Goes et al., 1999; Bijwaard
and Spakman, 1999; Karato and Karki, 2001; Forte and
Mitrovica, 2001; Schmid et al., 2002). In an extension of
the work to lower mantle minerals, Karato and Karki
(2001) concluded that when anharmonicity and anelasticity are fully accounted for, dlnρ/dlnVS at 2300 km may
be reduced from about 0.75 down to values of as low as
0.3 to 0.15 (see Plate 2 in Karato and Karki, 2001). Since
anelasticity does not affect density, the low dlnρ/dlnVS
value implies that the sensitivity of VS to temperature (as
given by dlnVS/dT) is increased by as much as 2.5 to 5
times above the elastic contribution. Changing dlnVS/dT
this much means that VS anomalies of 3% could be caused
by temperature anomalies of as low as 200° rather than the
1000° needed if only the purely elastic effect is accounted
for. A 200° anomaly over a distance of 100 km or so is
perhaps not so extreme, and so strong anelasticity would
reduce, or even eliminate, the need to explain the African
Super-plume anomalies with chemical heterogeneity.
In addition to predicting that anelasticity has a strong
effect on the temperature dependence of shear-wave
speeds, Karato and Karki (2001) also concluded that
anelasticity affects dlnVS/dT significantly more than
dlnVP/dT. This would be manifested in a strongly
increased RS/P. In fact they concluded that even values
of RS/P as large as 2.7 would require nothing but a
thermal origin (see Plate 3, in Karato and Karki, 2001).
In other words, RS/P cannot be used as a discriminator
for chemical heterogeneity in the Earth's mantle at all.
It is clear from experiments that anelasticity and
visco-elasticity can have a very significant effect on
seismic properties, particularly when temperatures
approach the solidus or when the grain size is very
small (Jackson et al., 2005), but the conclusion that
anelasticity has a significant effect in the Earth's lower
mantle is not so clear. Trampert et al. (2001), for
instance, found that anelasticity can affect dlnVS/dT and
dlnVP/dT by no more that 30%. Similarly Cammarano
et al. (2003) concluded that the effect of anelasticity
decreases strongly with depth and accounts for no more
than 20% to the temperature derivatives in the lower
mantle. These results are also echoed by Stacey and
Davis (2004). We have, therefore, decided to re-visit the
issue. In particular we test a wide range of material
parameters to see by how much they can affect seismic
velocities. We also construct probability density functions to test the combined contribution of the different
parameters, and thereby evaluate how likely it is for
anelasticity to contribute to a significant fraction of the
temperature derivatives of seismic velocities.
2. Background
Attenuation of seismic wave speeds in Earth materials
depends on frequency. Combining observations and
physical models of possible mechanisms leads to an
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J.P. Brodholt et al. / Earth and Planetary Science Letters 262 (2007) 429–437
Table 1
Activation energies for perovskite and periclase
Reference
Substance
H
(kJ/mol)
Note
Poirier (1995)
BaTiO3
KNbO3
KTaO3
NaNbO3
CaTiO3
CaTiO3
CaTiO3
CaTiO3
469
415
292
192
274
444
500–750
550–580
SrTiO3
SrTiO3
LaAlO3
520
720
84
Ca1−
96–103
Creep
″
″
″
″
″
″
Forced
oscillations
″
Micro-creep
Motion of twin
walls — O diffusion?
Motion of twin
walls — O diffusion?
Fe–Mg
inter-diffusion
Si self-diffusion bulk
Si self-diffusion
grain-boundary
0 self-diffusion
Li et al. (1996)
Webb et al.
(1999)
Harrison and
Redfern (2002)
Harrison et al.
(2003)
Holzapfel
et al. (2005)
Yamazaki et al.
(2000)
(Mg,Fe)
SiO3
MgSiO3
MgSiO3
414
Dobson (2003)
MgSiO3
154
xSrxTiO3
336
311
absorption band model (Minster and Anderson, 1981;
Anderson and Given, 1982) which describes this
frequency dependence. We assume, as is generally the
case, that the lower mantle falls within the absorption
band, and that the dependence of Q on frequency is
small. Following Karato (1993) and references therein,
the dependence of wave-speed on temperature is given by
FðaÞ 1
V ðT Þ ¼ V0 ðT Þ 1 Q ðT Þ
ð1Þ
pa
which consists of an elastic (and anharmonic) part, V0(T),
and an
FðaÞ ¼
paanelastic correction that depends on
−1
pa
,
the
reciprocal
attenuation
factor
Q
cot
(T),
and
2
2
on α, the frequency exponent in the frequency dependent
attenuation, Q. The temperature dependence of Q−1 can
be written as
Q1 ðT Þ ¼ Axa e
aH ⁎
RT
ð2Þ
with 0 b α b 1, H⁎ an activation energy, R the gas
constant, and A is a material constant. The normalised
(logarithmic) temperature derivative at constant pressure
for either VP or VS is given as
1
AlnV
AlnV0
V0 ðT Þ
Q ðx; T Þ H ⁎
FðaÞ
¼
AT P
p
RT 2
AT P V ðT Þ
ð3Þ
where the subscript P is pressure (Karato, 1993). Since
lateral temperature variations in the lower mantle are
431
likely to be large, Karato and Karki (2001) recognised
that the differential form could not be compared directly
with seismological variations in VS and VP and integrated Eq. (3) over a finite temperature difference 〈T 〉 + δT to
〈T 〉 − δT, where 〈T 〉 is the average mantle temperature at
a particular depth, and ±δT is the variation about that
average. During the integration, the V0 /V term was
assumed to be 1.0 and was dropped from the integration.
In most cases this simplification is valid. However, with
large temperature variations or low values of Q, as is
shown below, the simplification tends to exaggerate the
effect of anelasticity.
An alternative but exact approach which is used here is
to simply take the normalised finite difference of Eq. (1).
In other words we consider the expression
dlnV
V ðhT i þ dT Þ V ðhT i dT Þ
ð4Þ
¼
dT P
2dT V ðhT iÞ
where V(T) is given in Eq. (1). This is in fact a more
natural approach since observed variations in velocities
from tomography are also presented as normalised finite
differences. By substituting Eq. (1) into Eq. (4) we can get:
ð Þ
dlnV
dT
¼
P
½
1
dlnV0
F2
2dT
1 F2 dT
ð
V0 ðhT i þ dT Þ
aH ⁎
dT
exp
R hT i2 þ hT idT
V0 ðhT iÞ
V0 ðhTi dT Þ
aH ⁎
dT
exp
R hT i2 hT idT
V0 ðhT iÞ
!
ÞÞ
ð5Þ
F2 ¼
FðaÞQ1 ðhT iÞ
ap
dlnV0 V0 ðhTi þ dTÞ V0 ðhTi dTÞ
¼
2dT V0 ðhTiÞ
dT
ð6Þ
ð7Þ
In order to use Eq. (4) (or (3)) we need to have
mineral physics values for the activation energy of the
anelastic process H⁎, the frequency dependence of Q,
α, and the temperature dependence of the elastic part
of the wave velocities (P and S) at the average
temperature of interest, 〈T 〉. We also need parameters
for the state of the mantle. These are the average mantle
temperature 〈T 〉, the deviation from the average, δT, and
mantle values of the P and S quality factors at the
average temperature, Q(〈T 〉). By using seismic observations, the dependence on the unknown material
parameter, A, is removed. Eq. (4) only strictly holds
for the frequency corresponding to Q(〈T 〉), but by
testing various values of Q(〈T 〉) as is done below, we
implicitly cover a range of frequencies.
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3. Parameters
The material parameter having the greatest uncertainty is the activation enthalpy H⁎ of the anelastic
process, with higher activation enthalpies resulting in
higher anelasticity. Activation energies of a wide range
of perovskites for different processes are given in
Table 1. The activation energies of creep in analogue
perovskites span a range from about 200 to 750 kJ/mol
(see Table 1). Activation energies measured at seismic
frequencies for viscoelastic dissipation in CaTiO3 and
SrTiO3 perovskites using forced oscillations give
activation energies of around 500 kJ/mol (Webb et al.,
1999), consistent with the activation energy of the slow
diffusing Ti ion. Activation energies of Si, O and Mg
diffusion in MgSiO3 are on the order of 150 to 400 kJ/
Fig. 1. dlnVS/dT plotted against enthalpy for three different values of QS (〈T 〉) at a depth of about 2000 km. The average temperature,〈T 〉, is taken as
2500 K, and α = 0.3. The purely elastic (and anharmonic) results are at H⁎ = 0 kJ/mol. The solid lines are from Eq. (4) whereas the dashed lines use the
formulation of Karato and Karki (2001). At low values of H⁎ the two methods agree, but at high values of H⁎, the method of Karato and Karki (2001)
overestimates the effect of anelasticity. The top plot uses 〈T 〉 = ±500 K, while the bottom plot shows the result for a more extreme variation of 〈T 〉 =
±1000 K. At δT = ±500 K, anelasticity can provide as much as 50% of the total temperature derivative, but only at very high values of H⁎ and low
values of QS (〈T 〉). (Figures in colour in on-line version. Top solid and dashed lines are Q b T N = 400, middle lines are Q b T N = 300 and bottom lines
are Q b T N = 200.)
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J.P. Brodholt et al. / Earth and Planetary Science Letters 262 (2007) 429–437
433
Fig. 2. This plot shows calculated values of dlnVS/dlnVP using similar parameters as in Fig. 1. However, in this case the solid lines are the results for δT=±500 K
and the dashed lines are for δT =±1000 K. Only for very large values of δTand low values of QS (〈T 〉) can anelasticity affect dlnVS/dlnVP significantly. (Figures
in colour in on-line version. Top solid and dashed lines are Q b T N = 200, middle lines are Q b T N = 300 and bottom lines are Q b T N = 400.)
mol (Yamazaki et al., 2000; Dobson, 2003; Holzapfel
et al., 2005). Activation volumes have not been
measured in MgSiO3 perovskite yet, but they could
add as much as 400 kJ/mol to the activation energy at the
bottom of the mantle. We have, therefore, decided to
consider a wide range of plausible activation enthalpies.
In addition, we have used the same activation enthalpy
for both P and S wave anelasticity, based on the
assumption that anelasticity primarily affects the shear
modulus.
Although we have decided to test activation
enthalpies as high as 750 kJ/mol, it is worth noting
that Matas and Bukowinski (2007) have shown that H⁎
is unlikely to be as high as this. Since Q depends on H⁎
via Eq. (2), seismological models of Q can be used to
constrain the variation of H⁎ with depth. By taking
reasonable values of Q at the top of the lower mantle,
Matas and Bukowinski (2007) showed that very high
values of H⁎ lead to Q models for the mantle that are
inconsistent with those observed. It is likely, therefore,
that activation enthalpies may be no higher than 400 kJ/
mol throughout the lower mantle.
Many studies show that α b 0.4 with typical values of
0.1–0.4 (Romanowicz and Durek, 2000). As with H⁎,
we have considered a wide range of plausible values.
Values of dlnV0/dT for both P and S waves were taken
from the ab initio molecular dynamics results of Oganov
et al. (2001). These are very similar to those calculated
using ab initio lattice dynamics (Wentzcovitch et al.,
2004) and VIB molecular dynamics (Marton et al.,
2001). However, the exact values used are not important
for the purposes of this paper, which is to evaluate the
anelastic effect.
We also need estimates for the average temperature,
〈T 〉, and values for the P and S quality factors at that
temperature, Q(〈T 〉). One option open to us is to use a
particular geotherm and attenuation profile as done by
Matas and Bukowinski (2007). However, as with the
material parameters, we feel it is better to explore a full
range of possible values. We have, however, kept the
ratio between the P and S quality factors the same as is
given in PREM. So, for instance, QP (〈T 〉) = QS (〈T 〉)/
0.41 at 1000 km.
4. Results
Fig. 1 is a plot of dlnVS /dT against activation
enthalpy for three different average values of QS, at
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Fig. 3. Probability density functions (left axis) and normalised cumulative probabilities (right axis) for dlnVS/dT. In order to show the effect of anelasticity,
dlnVS/dT has been normalised to the elastic (and anharmonic) values. The probability densities were constructed by picking randomly between α =0.1 to 0.4,
H⁎ =300 to 750 kJ/mol, QS(〈T 〉) = 250 to 350 and 〈T 〉 = 2500 to 3500 K. Results for two values for δT are also shown for comparison. At the lower value of
δT = ±500 K, the most probable value for dlnVS/dT is 1.17 (a 17% increase). The distribution peak is quite narrow indicating that extreme values of all the
parameters are needed to produce a significant affect. The cumulative probabilities show that in 90% of the case, dlnVS/dT is changed by less that 30%, and
in 99.6% of the cases, by less than 50%. The larger values of δT = ±1000 K results in a larger anelastic affect, but still the probability of increasing the
magnitude of dlnVS/dT by more than 50% is only 5%. (Figures in colour in on-line version. Top line and dark shaded histogram are dT= +/- 500 K.)
about 2000 km depth. The purely elastic contribution
is at H⁎ = 0 kJ/mol. The average temperature is taken
to be 2500 K. Although this temperature may be
slightly low for the mantle, in fact higher temperatures
decrease the anelastic effect so this lowish temperature
actually accentuates the effect of anelasticity. Two
values for δT are used; ± 500 K and ± 1000 K.
Although it is unlikely for temperatures to range from
1500 K to 3500 K at 2000 km depth, the larger δT is
to show how extreme values are needed to produce
very large anelasticity. For temperatures ranging from
2000 to 3000 K (δT = ± 500), however, the anelastic
contribution to dlnVS/dT can reach 60% or so of the
elastic part only at high activation enthaplies
(H⁎ = 700 kJ/mol) and low values of Q (QS = 200). A
QS of 200 is quite low and in the recent estimate of
average QS of Lawrence and Wysession (2006) from
body waves, QS between 1700 and 2400 km ranges
from 307 to 383. A similar range for QS (280–400) is
found from the analysis of normal mode data
(Resovsky et al., 2005). And indeed PREM also has
a Q of 312 in the lower mantle (Dziewonski and
Anderson, 1981). For these higher values of Q, the
anelastic contribution to dlnVS/dT can contribute no
more than about 40% of the elastic part. Although this
is an important contribution to the total dlnVS/dT, it is
certainly far less profound than the factor of least
250% suggested by Karato and Karki (2001).
Moreover, as shown by Matas and Bukowinski
(2007) and discussed above, a H⁎ of 700 kJ/mol is
not consistent with the observed Q of the lower
mantle. A more reasonable H⁎ of 400 kJ/mol leads to
an anelastic contribution of less than 20% of the
elastic part.
It is worth noting that while the results in Fig. 1 have
been calculated between 〈T 〉 +δT and 〈T 〉 −δT, the majority
of the anelastic effect in Fig. 1 is actually due to the 〈T 〉 +δT
part. This can be seen by considering Eq. (5). In other
words, anelasticity will only affect warm areas of the lower
mantle.
Also shown in the Fig. 1 is dlnVS/dT using the
formulation of Karato and Karki (2001). At low values of
H⁎ and moderate mantle temperatures, there is little
difference between our results using Eq. (4) and theirs,
but as H⁎ increases, the formulation of Karato and Karki
(2001) starts to greatly exaggerate the effect of anelasticity.
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J.P. Brodholt et al. / Earth and Planetary Science Letters 262 (2007) 429–437
This is particularly apparent at high δT and can lead to some
extreme and erroneous anelastic contributions to the
temperature derivatives. This was also noted by Matas
and Bukowinski (2007).
Fig. 2 shows dlnVS/dlnVP for a similar range of
parameters as Fig. 1, but at two depths. Even for a very
low QS and very high δT, dlnVS/dlnVP is only increased
by about 10% at 2000 km and 20% at 1000 km. If we
consider the more likely range of parameters (δT =
± 500 K and Q between 300 and 400), anelasticity
changes dlnVS/dlnVP by less than 10% from the purely
elastic value. Observed values of dlnVS/dlnVP greater
than 1.8 at 1000 km and 2.1 at 2000 km cannot,
therefore, be caused by temperature alone and must have
another cause, most likely chemical.
In order to consider a large range of parameters
simultaneously, it is instructive to construct probability
density functions for dlnVS/dT and dlnVS/dlnVP. The
probability density functions have been made by
drawing randomly from within a range of values for
all the parameters discussed in the previous section. The
range of values within which we have randomly drawn
are b T N = 2500 to 3500, α = 0.1 to 0.4, H⁎ = 300 to
750 kJ/mol, and QS = 250 to 350. We used one million
435
randomly selected combinations of parameters, with
each parameter being drawn from a flat probability (i.e.
every value for a parameter has an equal probability of
being picked). The results for dlnVS and dlnVS/dlnVP are
shown in Figs. 3 and 4 for two values of δT. In order to
show the effect of anelasticity, we have normalised them
to the fully anharmonic elastic values. Also shown in the
figures are the normalised cumulative probabilities.
These allow one to read off the probability of picking
above (or below) a certain value.
For a variation in temperature of ± 500 K, we find
that the peak in the normalised probability density
function of dlnVS /dT is at 1.17. In other words, the most
probable effect of anelasticity is to increase dlnVS /dT by
about 17%. The distribution is also quite sharp which
indicates that extreme values for all the choice of
parameters are needed to significantly increase dlnVS/
dT. For instance, the cumulative distribution shows that
in 90% of the cases, dlnVS /dT is increased over the
elastic value by less than a factor of 1.3. And in 99.6%
of the cases, dlnVS /dT is increased by less than a factor
of 1.5. Increases on the order of 3 or 4 as suggested by
Karato and Karki (2001) are impossible for this range of
parameters. Increasing the variation in temperature
Fig. 4. Probability density functions (left axis) and normalised cumulative probabilities (right axis) for dlnVS/dlnVP. The range of parameters from
which the samples were chose is the same as in Fig. 3. The effect of anelasticity is even less than for dlnVS/dT and even for extreme temperature
variations (δT = ±1000 K), there is only a 4% probability of increasing dlnVS/dlnVP by more than 10%, and the most probable value is significantly
less.
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to ± 1000 K does lengthen the tail on the probability
density and increases the chances of changing dlnVS/dT,
however, even with this very high variation in temperature, the probability of increasing dlnVS/dT by more
than a factor of 1.5 is less than 5%.
The results in Fig. 3 are also in agreement with the
results of Matas and Bukowinski (2007) who showed
that while the anelastic part of dlnVS/dT can be as large
as the elastic part, it is only possible for extreme and
unlikely material and mantle parameters. For more
reasonable values, they predict that the anelastic
contribution to dlnVS/dT is only about 20%.
Fig. 4 shows the probability distribution and
cumulative probability for dlnVS/dlnVP. In this case,
for a temperature variation of ± 500 K, the peak in
the probability density is at about 1.04, with the tail
extending only out to about 1.2 at best. Again, increasing the temperature variation to ± 1000 K increases the tail, but only out to about 1.16. In other
words, dlnVS/dlnVP is relatively insensitive to anelasticity and, even at the higher δT, there is only about a
4% probability of anelasticity increasing dlnVS/dlnVP
by more than 10%.
5. Conclusions
Our conclusions are relatively straightforward. By
considering a reasonable range of material parameters,
together with estimates of mantle parameters, we show
that:
1) Anelasticity is very unlikely to affect RS/P by more than
about 10% from the purely elastic (and anharmonic)
value and therefore can be ignored when using RS/P as
a measure of chemical heterogeneity.
2) Anelasticity can affect dlnVS/dT but it is unlikely to
affect it by more than about 30%, and most probably
by only 17%.
Therefore, while lower mantle anelasticity cannot be
totally ignored, it only adds a relatively small proportion
to the logarithmic temperature derivatives of seismic
wave-speeds.
Acknowledgement
We thank the reviewers for their helpful comments.
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