Traveltime sensitivity kernels for PKP phases in the mantle
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Physics of the Earth and Planetary Interiors 153 (2005) 21–31
Traveltime sensitivity kernels for PKP phases in the mantle
Marie Calvet∗ , Sébastien Chevrot
Université Paul Sabatier, Laboratoire de Dynamique Terrestre et Planétaire, CNRS, UMR 5562,
Observatoire Midi-Pyrénées, 14 Avenue Edouard Belin, 31400 Toulouse, France
Received 9 November 2004; accepted 29 June 2005
Abstract
We investigate the finite-frequency effects of perturbations of compressional wave velocity on the traveltimes of PKP phases.
Owing to their long paths in the mantle, PKP phases have a large Fresnel zone at the CMB and thus sample a large volume of the
D” layer. We compute traveltime Fréchet derivatives of the three PKP phases in the mantle with respect to the ak135 reference
model, taking into account the coupling between the different branches. PKP kernels do not have a simple “banana-doughnut”
shape, because the different PKP branches interfere with each other, and because of the presence of shadow zones and caustics.
In addition, we compute the differential kernels AB–BC, BC–DF and AB–DF. The geometry of PKP differential kernels in D”
show significant qualitative differences from those predicted by simple ray tracing. This suggests that the tomographic techniques
commonly used to image lower mantle structures could be improved by using 3D Fréchet kernels.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Earth’s mantle; D” layer; PKP waves; Finite-frequency kernels; Fréchet derivatives
1. Introduction or PKP(BC)–PKP(DF) to study inner core anisotropy
(Shearer and Toy, 1991; Creager, 1992, 2000; Tanaka
PKP phases are P waves that travel through Earth’s and Hamaguchi, 1997). The PKP(DF) phase in the dis-
core. Most of what we know about inner core anisotropy tance range 148–155◦ only samples the upper 350 km
comes from the analysis of PKP(DF) which travels of the inner core. Therefore, BC–DF differential travel-
through the inner core (Poupinet et al., 1983; Morelli times have to be complemented by AB–DF differential
et al., 1986; Shearer, 1994; Su and Dziewonski, 1995; traveltimes in the distance range 150–180◦ to explore
Garcia and Souriau, 2000; Ishii and Dziewonski, 2002, the innermost part of the inner core. However, at large
2003). However, this wave also propagates through the epicentral distances, the ray paths of the PKP(DF) and
lithosphere, the whole mantle, and the very heteroge- PKP(AB) phases are very different in the lower mantle
neous region at the base of the mantle called the D” and D” heterogeneities may still contribute strongly to
layer. As a result, PKP(DF) traveltimes are not sim- the observed anomalies (Bréger et al., 1999, 2000). In
ply related to the inner core structure. To reduce the addition, PKP(AB) phases have larger sensitivity inside
contributions of mantle heterogeneities, seismologists the D” layer than PKP(DF) phases. Thus, obtaining good
often use differential traveltimes PKP(AB)–PKP(DF) models of D” structures is a first step toward reliable con-
straints on inner core anisotropy.
∗ Corresponding author. Tel.: +33 5 61 33 28 44;
The D” region is a thermal and chemical boundary
fax: +33 5 61 33 29 00. layer thought to play an important role in the dynam-
E-mail addresses: calvet@pontos.cst.cnes.fr (Marie Calvet), ics of the earth. Seismic body wave studies reveal a
sebastien.Chevrot@cnes.fr (S. Chevrot). complex velocity structure. Using diffracted P waves,
0031-9201/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.pepi.2005.06.014
22 M. Calvet, S. Chevrot / Physics of the Earth and Planetary Interiors 153 (2005) 21–31
Wysession (1996) has shown that large-scale continent- for PKP phases. Differential traveltimes are measured by
sized P-velocity structures are present in D”. Numer- cross-correlation after standardization of the waveforms,
ous studies have revealed a high level of complexity in by taking the Hilbert transform and changing the polar-
D”, including discontinuous increases in velocity at the ity of PKP(AB) (Song and Helmberger, 1993; Bréger
top of the region (e.g., Lay and Helmberger, 1983b,a; et al., 2000; Tkalčić et al., 2002). The sensitivity ker-
Houard and Nataf, 1992; Vidale and Benz, 1993; Kendall nels for PKP differential traveltimes are then obtained
and Shearer, 1994), a very-low-velocity layer just above simply by subtracting the kernels for the individual
the CMB (e.g., Garnero and Helmberger, 1996), par- PKP phases. Finally, we describe these kernels and dis-
tial melting (e.g., Lay et al., 2004), the presence of cuss the implications of the complex sensitivity of PKP
seismic anisotropy (e.g., Maupin, 1994; Kendall and Sil- phases to the lower mantle structure for studies of the D”
ver, 1996), and small-scale convection with formation layer.
of whole-mantle plumes (e.g., Bréger and Romanowicz,
1998; Montelli et al., 2004). Many 1D models use a dis- 2. The PKP branches
continuity at the top of D” to describe a variable vertical
velocity gradient. The existence of Earth’s core was inferred at the
Recent laboratory X-ray diffraction measurements begining of the twentieth century when seismolo-
performed at thermodynamical conditions of the core- gists noticed that compressional-wave amplitudes decay
mantle boundary suggest that MgSiO3 perovskite trans- rapidly beyond epicentral distances of ∼100◦ . This
forms to a new high-pressure form, increasing its density shadow zone indicates a low-velocity region. In model
by 1.0–1.2% (Murakami et al., 2004). The origin of the ak135 (Kennett et al., 1995), the P wave velocity profile
D” seismic discontinuity may be attributed to this phase shows a strong drop from 13.66 to 8.00 km/s at the CMB.
transition. This phase could also provide an explanation This major discontinuity located at 2891 km depth pro-
for seismic anisotropy below the D” discontinuity. While duces a shadow zone between approximately 100 and
there is some agreement in global tomographic models, 143◦ where, in principle, no waves can be recorded.
strong differences are observed in small-scale structures. However, diffracted P waves are observed in this dis-
Some analyses of differential traveltime residuals have tance range. For epicentral distances larger than 143◦ ,
demonstrated that the D” layer could exhibit large veloc- the liquid outer core produces two P wave arrivals: a
ity contrasts implying strong lateral velocity gradients slow phase called PKP(AB) that travels in the upper part
that are difficult to interpret with ray theory (Bréger et of the core, and a fast phase called PKP(BC) that travels
al., 1999; Luo et al., 2001; Garcia et al., 2004). Differen- in the lower part of the core (Fig. 1).
tial traveltimes of PKP phases can be used to study the At the inner core boundary (ICB), the P wave veloc-
lower part of the mantle above the Core Mantle Bound- ity increases from 10.289 to 11.043 km/s at a radius of
ary (CMB) (Sylvander and Souriau, 1996; Karason and 1217.5 km in model ak135. This high-velocity region
van der Hilst, 2001; Tkalčić et al., 2002). Some of these generates a triplication in the traveltime curves of
studies invoke strong lateral variations on scale lengths PKP(BC), producing the totally reflected phase in the
shorter than a few hundred kilometers (Bréger et al., outer liquid core PKP(CD) and the refracted phase in
1999). the inner core PKP(DF) (Fig. 2). For a source located
PKP waves have a large first Fresnel zone at the bot-
tom of the mantle because they sample regions that are far
from both the source and the receiver even though their
dominant period is short (between 1 s and 3 s on most
broadband records). The theory of 3D Fréchet kernels
for seismic wave traveltimes developed by Marquering
et al. (1999), Dahlen et al. (2000), Hung et al. (2000)
allows us to describe the effect of the finite frequency
content of PKP waves on their traveltimes. In this study,
we take a similar approach and derive the sensitivity ker-
nels of PKP traveltimes for small isotropic perturbations
of P-wave velocities with respect to the radial reference
Fig. 1. Ray geometry of PKP phases. PKP(DF) phases (yellow) travel
earth model ak135 (Kennett et al., 1995). After describ- through the inner core, and PKP(BC) (blue) and PKP(AB) (red) phases
ing the properties of the individual PKP branches, we travel through the outer core. PKP(CD) phases (green) are reflected at
compute the differential traveltime sensitivity kernels the inner core boundary (ICB).
M. Calvet, S. Chevrot / Physics of the Earth and Planetary Interiors 153 (2005) 21–31 23
geneities (Dahlen et al., 2000). A negative traveltime
anomaly (δT < 0) corresponds to an advance of the per-
turbed wave with respect to the reference wave, whereas
a positive traveltime anomaly (δT > 0) corresponds to a
delay.
Using the Born approximation, we can relate wave-
form perturbations δu to model perturbations δmi /mi ,
and write the traveltime anomalies to first order by the
volume integral
δmi
δT = Kmi dV (2)
V mi
i
where Kmi is the 3D Fréchet derivative of T with respect
Fig. 2. PKP traveltimes as a function of epicentral distance in model to the perturbation of the parameter mi .
ak135 for a source located at the surface. In an isotropic medium, P wave traveltimes depend
on a combination of three different parameters: the den-
sity ρ, P-wave velocity α and S-wave velocity β. In the
at the surface, the PKP(DF) is observed from approx- following, we only consider the sensitivity kernels for
imately 120 to 180◦ , PKP(BC) from 148 to 155◦ , and perturbations δα of the P-wave velocity. The traveltime
PKP(AB) from 150 to 180◦ . anomalies are thus simply given by
If the ray parameter decreases as a function of epi-
central distance, the branch is called prograde. When δα
δT = Kα dV (3)
the epicentral distance decreases as the ray parameter V α
decreases, the traveltime curve is called retrograde. The Dahlen et al. (2000), using a far-field approximation of
transition from prograde to retrograde and back to pro- the Green’s tensor, express the 3D kernels as a summa-
grade generates a triplication in the traveltime curve. tion over all the possible paths between the source and
Caustics are found at the endpoints of the triplication. the receiver involving a single scattering event. Follow-
The PKP(AB) phase is retrograde whereas the PKP(BC) ing Dahlen et al. (2000), we write the Fréchet kernel Kα
phase is prograde and point B is a caustic. Moreover, as
PKP(AB) is the Hilbert transform of PKP(DF) (Choy
1 1 R Π1 Π2
and Richards, 1975). Kα = − ΩαP→P
Depending on the epicentral distance, several PKP 2π αX αR R1 R2 Π
AB,BC,CD,DF
phases can be recorded (Figs. 1 and 2). In addition, PKP +∞
branches may be present or absent in a given depth and 0 ω3 |ṁ(ω)|2 sin[ω(T1 + T2 − T )
epicentral distance range (Fig. 1) which makes the three- − (M1 + M2 − M)π/2] dω
× +∞ (4)
dimensional sampling of mantle and core by these waves ω2 |ṁ(ω)|2 dω
0
particularly complicated.
where ΩαP→P is the isotropic P → P scattering ampli-
3. Sensitivity kernels for PKP waves in the tude for a perturbation δα/α, αX the P wave velocity at
mantle the scatterer in the reference medium, αR the P wave
velocity at the receiver in the reference medium, R the
3.1. Definitions geometrical spreading for the reference ray, R1 the geo-
metrical spreading for the ray between the source and the
The finite frequency traveltime anomaly δT measured scatterer and R2 is the geometrical spreading for the ray
by cross-correlation between the observed and reference between the receiver and the scatterer, Π the reflexion-
waveforms is given by transmission coefficients product for the reference ray,
+∞ 0 Π1 the reflexion-transmission coefficients product for
u̇ (t) δu(t) dt the ray between the source and the scatterer, Π2 the
δT = −∞+∞ 0 (1)
−∞ ü (t) u (t) dt
0 reflexion-transmission coefficients product for the ray
between the receiver and the scatterer, T the reference
where u0 is the wavefield in the reference 1D model ray traveltime, T1 the traveltime between the source and
and δu the change in displacement produced by hetero- the scatterer, and T2 the traveltime between the scatterer
24 M. Calvet, S. Chevrot / Physics of the Earth and Planetary Interiors 153 (2005) 21–31
Fig. 3. Epicentral distance as a function of ray parameter and depth. The positions of the points A–D vary with depth.
and the receiver. The Maslov index of the reference ray is each particular point of the mantle. Because the ∆(p)
denoted M whereas M1 and M2 are the Maslov indices curves vary strongly as a function of depth (Fig. 3), the
for the source-to-scatterer and the receiver-to-scatterer ray parameter and epicentral distance corresponding to
paths. The fourier transform of the source time func- the limits of each branch have to be determined accu-
tion is ṁ(ω). From the reciprocity theorem, we can write rately. In particular, both ray parameter and distance
the geometrical spreading factors as (Aki and Richards, corresponding to the B caustic vary as a function of
1980) depth.
RX−R αR = RR−X αX (5)
3.2.2. The AB caustic
where RR−X is the geometrical spreading from the The PKP(AB) phase goes through the AB caustic,
receiver R to the scattering point X and RX−R is for strongly altering the kernels. At the cusps in the trip-
the path from the scattering point X to the receiver R. lications of traveltimes, d∆/dp tends to zero, a large
number of rays with different ray parameters collapse
3.2. Computation of the Fréchet kernels into one, and the amplitude becomes infinite. The caus-
tic also shifts the harmonic phase by π/2 (Choy and
3.2.1. Several PKP phases contributions Richards, 1975; Choi and Hron, 1981). The Maslov
To compute the Fréchet kernel Kα for the PKP phases, index M accounts for the number of π/2 phase shifts
we have to evaluate the traveltimes T, T1 and T2 , the due to passage of the wave through caustics (Choi and
Maslov indices M, M1 and M2 , and the geometrical Hron, 1981). The Maslov indices for both the PKP(DF)
spreading R, R1 and R2 for every scattering position in and PKP(BC) reference wave are M = 0. In contrast, the
the mantle. The method we use is derived from Buland Maslov index of a PKP(AB) is M = 1 because the AB
and Chapman (1983). We tabulate the delay or intercept caustic in the liquid outer core causes a phase shift of
time τ as a function of ray parameter p in the spher- π/2.
ically symmetric model ak135 (Kennett et al., 1995). The PKP(DF), PKP(BC) and PKP(AB) contribu-
For a source at a given depth, the τ branches are mono- tions to their own sensitivity kernels are characterized
tonic and single valued functions of the ray parameter. by M1 + M2 − M = 0 and the oscillatory term in the
Computing kernels for PKP is difficult because different numerator of 4 is sin[ω (T1 + T2 − T )]. On the other
branches occur for the same epicentral distance. Thus, hand, the PKP(AB) contributions to the PKP(DF) and
to evaluate the complete traveltime sensitivity kernel of PKP(BC) kernels are both characterized by M1 + M2 −
a given PKP phase, we have to consider the possible M = 1 and the oscillatory term in the numerator of
contributions of the other PKP phases. For this purpose, 4 becomes − cos[ω(T1 + T2 − T )]. Similarly, for the
we need to determine all the PKP phases that can reach PKP(BC) or the PKP(DF) contributions to the PKP(AB)
M. Calvet, S. Chevrot / Physics of the Earth and Planetary Interiors 153 (2005) 21–31 25
Fig. 4. Cross-section in the mantle of traveltime sensitivity kernels for a PKP(DF) with a dominant period of 2.25 s. The kernels are calculated for
epicentral distances 140◦ (a), 145.5◦ (b), 150◦ (c) , 160◦ (d) and 170◦ (e).
kernel we have M1 + M2 − M = −1 and + cos[ω(T1 + core. Thus, a scattered phase coming from the outer core
T2 − T )]. When the wave shape of the wavelet is the nth is strongly delayed compared to the reference PKP(DF)
derivative of a Gaussian, the power spectrum is given by and does not contribute to the sensitivity kernel. At
∼145◦ epicentral distance, the Fresnel zone width is
ω2n τc2 −ω2 τc2 /8π2
|ṁ(ω)|2 = e (6) about 1400 km at the CMB for a PKP(DF) wave with
4π a dominant period of 2.25 s a typical value for these
where τc is the characteristic period of the wave and waves observed on broadband records. At 140◦ epicen-
is related√to the dominant (or “visual”) period Td by tral distance, the contributions of the outer core phases
τc /Td = 2n (Favier and Chevrot, 2003). begin to modify the PKP(DF) kernel, especially in the
lower mantle. When the epicentral distance increases,
3.3. Description of the sensitivity kernels there is no contribution from the outer core phases
because PKP(DF) waves are much faster than the scat-
The sensitivity kernels for PKP phases are similar in tered PKP(AB) or PKP(BC) waves.
shape to the traveltime sensitivity kernels for P waves in PKP(BC) is recorded in a narrow epicentral distance
the mantle calculated by Hung et al. (2000). However, range. In this range, PKP(AB) is slower than PKP(BC),
this “normal” shape is perturbed by the contributions of and its contribution is only significant for scatterers at
the other PKP phases. epicentral distances close to the caustic B. However, a
Fig. 4 shows the PKP(DF) traveltime sensitivity ker- scattered PKP(DF) wave can contribute even for longer
nels for epicentral distances varying from 140 to 170◦ . paths because it travels faster than PKP(BC). Neverthe-
The contributions of PKP(AB) and PKP(BC) are signif- less, this contribution oscillates quickly and the con-
icant only at epicentral distances of ∼145◦ near the B tribution of intermediate- and long-wavelength hetero-
caustic. PKP(AB) and PKP(BC) travelling in the outer geneities in the vicinity of the PKP(DF) path is negligi-
core are slower than PKP(DF) travelling in the inner ble.
26 M. Calvet, S. Chevrot / Physics of the Earth and Planetary Interiors 153 (2005) 21–31
Fig. 5. Cross-section in the mantle of traveltime sensitivity kernels for a PKP(BC) wave with a dominant period of 2.25 s for epicentral distances
145.5◦ (a) and 150◦ (b).
Fig. 5 shows the PKP(BC) traveltime sensitivity ker- 4. Differential kernels
nels for two epicentral distances in the incident plane.
For an epicentral distance near B, both PKP(AB) and AB–DF and BC–DF differential traveltimes are often
PKP(DF) contribute to the PKP(BC) kernel. used for studying the inner core anisotropy to reduce the
Fig. 6 shows the PKP(AB) sensitivity kernels in the contributions of the heterogeneities of the mantle near
incident plane. PKP(AB) is slower than PKP(DF) or the source and the receiver. In addition, AB–BC differen-
PKP(BC). Therefore, away from the path of PKP(AB), tial traveltimes can be used to study D” and lower mantle
scattered PKP(BC) and PKP(AB) can contribute. For structures. Therefore, it is important to compute the cor-
epicentral distances near B, the different PKP branches responding differential traveltime kernels and examine
interfere strongly. The sensitivity is distribued over a their properties.
region of about 1000 km at the CMB. Since PKP(AB)
is retrograde, it has a grazing incidence when epicentral 4.1. Principle
distance increases. For scatterers at shorter epicentral
distances, only PKP(DF) can contribute to the PKP(AB) The Fréchet kernel for a differential traveltime
kernel (see Fig. 1), but the inner core phase is too measurement δ(TAB ) = δTB − δTA , is the difference
fast compared to PKP(AB) to contribute significantly. between the two kernels KA and KB , if the Maslov
Note that the shape of the kernel at the base of the indices of the phases A and B are identical. For waves that
mantle is strongly affected by the proximity of the B have different Maslov indices, an initial equalization of
caustic located in the fluid outer-core at all epicentral the waveforms is necessary. For example, to measure the
distances. differential traveltime between PKP(AB) and PKP(BC),
Because we only take into account geometric rays we must take the Hilbert transform of PKP(AB) and
that do not penetrate into the shadow zone, PKP(AB) change the polarity before crosscorelating the two phases
has zero sensitivity to velocity perturbations beyond (Song and Helmberger, 1993; Bréger et al., 2000; Tkalčić
the critical point. Incorporating diffracted waves at et al., 2002). This precaution taken, we can write the dif-
the CMB in the computation of PKP kernels is an ferential traveltime sensitivity kernels of two phases A
important goal that should be pursued in the future. and B as
This will require more sophisticated numerical meth-
KA−B = KA − KB (7)
ods such as full wave theory (Richards, 1973), a com-
bination of Born and Langer approximations (Emery In the following, we calculate the differential sensitiv-
et al., 1999), or even a spectral-element method ity kernels AB–DF, AB–BC and BC–DF for different
(Tromp et al., 2005). epicentral distances.
M. Calvet, S. Chevrot / Physics of the Earth and Planetary Interiors 153 (2005) 21–31 27
Fig. 6. Cross-section in the mantle of traveltime sensitivity kernels for a PKP(AB) wave with a dominant period of 2.25 s. The kernels are calculated
for the epicentral distances 145.5◦ (a), 150◦ (b), 160◦ (c) and 170◦ (d).
4.2. Descriptions of the differential sensitivity tive and negative sensitivity regions. However, smaller
kernels structures may contribute. A surprising property of the
kernel is that the sensitivity to small P-wave perturba-
All the kernels are computed for a wavelet shape tions seems to be smaller close to the CMB than in the
which is the second derivative of a Gaussian and a dom- middle-mantle. That probably indicates that the contri-
inant period of 2.25 s. Fig. 7 represents the three PKP bution of waves diffracted at the CMB (not included in
differential kernels for an epicentral distance of 150◦ . our calculations) is significant even at distance as small
The BC–DF differential kernel plotted on top shows a as 150◦ .
region with no sensitivity between two regions of sen- The AB–DF differential traveltime sensitivity kernel
sitivity with opposite signs. The PKP(DF) contribution for an epicentral distance of 150◦ is plotted on the mid-
to the PKP(BC) kernels is negligible because its small- dle of Fig. 7. At epicentral distances smaller than 120◦ ,
scale oscillations average to zero. The width of the region there is a shadow zone for PKP(AB). The “normal” shape
where the sensitivity is significant is ∼900 km. In the of the traveltime sensitivity kernel for a P wave with
middle of the mantle, we observe positive and negative negative amplitudes in the first Fresnel zone, positive
sensitivities in alternation. Therefore, an heterogeneity amplitudes in the second Fresnel zone, and no sensitivity
in P wave velocity with a lateral scale larger than 5◦ on the geometric ray path is recognizable, but this shape
will produce almost no effect on differential traveltimes, is perturbed by the contribution of scattered PKP(BC).
because of destructive interference between the posi- This contribution is significant, in contrast to the con-28 M. Calvet, S. Chevrot / Physics of the Earth and Planetary Interiors 153 (2005) 21–31 Fig. 7. PKP(BC)–PKP(DF) (top), PKP(AB)–PKP(DF) (middle) and PKP(AB)-PKP(BC) (bottom) differential traveltime sensitivity kernels for an epicentral distance of 150◦ . On the left: cross-section in the great circle plane. On the right: horizontal slice at 50 km above the CMB. tribution of PKP(DF). The PKP(BC) signature drops in such a complicated way that indeed 3D Fréchet ker- sharply at a distance of ∼145◦ , which corresponds to the nels show strong differences with geometrical ray theory. position of the C caustic at this particular depth. Sensitiv- This could have important consequences for the tomog- ity at the base of the mantle is distributed over 1400 km raphy of the lower mantle.
M. Calvet, S. Chevrot / Physics of the Earth and Planetary Interiors 153 (2005) 21–31 29 Fig. 8. PKP(AB)–PKP(DF) differential traveltime sensitivity kernels for an epicentral distance of 160◦ (top) and 170◦ (bottom). On the left: cross-section in the great circle plane. On the right: horizontal slice at 50 km above the CMB. The AB–BC differential traveltime sensitivity ker- sponding to the B caustic, but this strongly neg- nel for an epicentral distance of 150◦ is plotted in the ative sensitivity region shrinks at larger epicentral bottom of Fig. 7. The shape of the kernel is similar to distances. the AB–DF differential kernels, but the PKP(BC) and PKP(AB) ray paths are closer to each other. This kernel 5. Conclusions and perspectives shows a sensitivity distributed over the whole mantle. At 50 km above the CMB, sensitivity is distributed over Traveltime sensitivity kernels for PKP phases in the 1900 km. mantle do not show the simple “banana-doughnut” shape Fig. 8 shows the AB–DF differential travel- at all epicentral distances, because the different PKP time kernels for an epicentral distance of 160 and branches can interfere with each other and because of 170◦ . The incidence angle of the PKP(AB) at the the presence of shadow zones and caustics. Even though CMB increases with epicentral distance and the PKP phases are short-period waves, their first Fresnel PKP(AB) contribution spreads over 10–20◦ . There zone in the lower mantle is quite large. The shape of is a strong interference with the PKP(BC) contri- the Fréchet kernels show many complications that are bution for scatterers at epicentral distances corre- not accounted by standard ray theory. In addition, sensi-
30 M. Calvet, S. Chevrot / Physics of the Earth and Planetary Interiors 153 (2005) 21–31
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