Seismic Anisotropy of the Deep Earth from a Mineral and Rock

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Mainprice D Seismic Anisotropy of the Deep Earth from a Mineral and Rock Physics Perspective. In: Gerald Schubert (editor-in-chief) Treatise on Geophysics, 2nd edition,

Vol 2. Oxford: Elsevier; 2015. p. 487-538.

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2.20 Seismic Anisotropy of the Deep Earth from a Mineral andRock Physics PerspectiveD Mainprice, Universite Montpellier II, Montpellier, France

ã 2015 Elsevier B.V. All rights reserved.

2.20.1 Introduction 4872.20.2 Mineral Physics 4922.20.2.1 Elasticity and Hooke’s Law 4922.20.2.2 Plane Waves and the Christoffel Equation 4942.20.2.3 Measurement of Elastic Constants 5012.20.2.4 Effective Elastic Constants for Crystalline Aggregates 5022.20.2.5 Seismic Properties of Polycrystalline Aggregates at High Pressure and Temperature 5042.20.2.6 Anisotropy of Minerals in the Earth’s Mantle and Core 5062.20.2.6.1 Upper mantle 5062.20.2.6.2 Transition zone 5072.20.2.6.3 Lower mantle 5082.20.2.6.4 Subduction zones 5132.20.2.6.5 Inner core 5182.20.3 Rock Physics 5212.20.3.1 Introduction 5212.20.3.2 Olivine the Most-Studied Mineral: State-of-the-Art-Temperature, Pressure, Water, and Melt 5212.20.3.3 Seismic Anisotropy and Melt 5262.20.4 Conclusions 529Acknowledgments 530References 530

2.20.1 Introduction

Seismic anisotropy is commonly defined as the direction-

dependent nature of the propagation velocities of seismic

waves. However, this definition does not cover all the seismic

manifestations of seismic anisotropy. In addition to direction-

dependent velocity, there is direction-dependent polarization

of P- and S-waves, and anisotropy can contribute to the split-

ting of normal modes. Seismic anisotropy is a characteristic

feature of the Earth, with anisotropy being present near the

surface due to aligned cracks (e.g., Crampin, 1984), in the

lower crust, upper mantle, and lower mantle due to mineral

preferred orientation (e.g., Karato, 1998; Mainprice et al.,

2000). At the bottom of the lower mantle in the D00 layer

(e.g., Kendall and Silver, 1998), and in the solid inner core

(e.g., Ishii et al., 2002a), the causes of anisotropy are still

controversial (Figure 1). In some cases, multiple physical fac-

tors could be contributing to the measured anisotropy, for

example, mineral crystal preferred orientation (CPO) and

alignment of melt inclusions at mid-ocean ridge systems

(e.g., Mainprice, 1997). In the upper mantle, the pioneering

work of Hess (1964) and Raitt et al. (1969) from Pn velocity

measurements in the shallow mantle of the ocean basins

showed azimuthal anisotropy in a shallow horizontal layer.

Long-period surface waves studies (e.g., Montagner and

Tanimoto, 1990; Nataf et al., 1984) have since confirmed

that azimuthal anisotropy and SH/SV polarization anisotropy

are global phenomena in the Earth’s upper mantle, particularly

in the top 200 km of the upper mantle. Anisotropic global

atise on Geophysics, Second Edition http://dx.doi.org/10.1016/B978-0-444-538

Treatise on Geophysics, 2nd edition

tomography, based on surface and body wave data, has

shown that anisotropy is very strong in the subcontinental

mantle and present generally in the upper mantle, but signifi-

cantly weaker at greater depths (e.g., Beghein et al., 2006;

Panning and Romanowicz, 2006). The large wavelengths

used in long-period surface wave studies mean that such

methods are insensitive to heterogeneity less than the wave-

length of about 1000 km. In an effort to address the problem

of regional variations of anisotropy, the splitting of SKS tele-

seismic shear waves that propagate vertically has been exten-

sively used. At continental stations, SKS studies show that the

azimuth of the fast polarization direction is parallel to the

trend of mountain belts (Fouch and Rondenay, 2006; Kind

et al., 1985; Silver, 1996; Silver and Chan, 1988, 1991; Vinnik

et al., 1989). From the earliest observations, it was clear that

the anisotropy in the upper mantle was caused by the CPO of

olivine crystals induced by plastic deformation related to man-

tle flow processes at the geodynamic or plate tectonic scale.

The major cause of seismic anisotropy in the upper mantle

is the CPO caused by plastic deformation. Knowledge of the

CPO and its evolution require well-characterized naturally

deformed samples, experimentally deformed samples, and

numerical simulation for more complex deformation histories

of geodynamic interest. The CPO not only causes seismic

anisotropy but also records some aspects of the deformation

history. Samples of the Earth’s mantle are readily found on the

surface in the form of ultramafic massifs and xenoliths in

basaltic or kimberlitic volcanics and as inclusions in dia-

monds. However, samples from depths greater than 220 km

02-4.00044-0 487, (2015), vol. 2, pp. 487-538

http://dx.doi.org/10.1016/B978-0-444-53802-4.00044-0

Olivine

Pyrolite

Volume fractions201.00

410

660

2000

2700

2891

5150

6371

1.05 1.10

VSH > VSV

VSH > VSV

VS

V >

VS

H

40 60 80 20 40 60 80Volume fractions Kyanite Wadeite

Orthoclase

Upper mantle

Transition zone

Lower mantle

Pressure (G

Pa)

Temp

erature (C)

Dep

th (km)

Dep

th (k

m)

P-and S-Wave anisotropy MORB Sediments

Cpx CpxCoesite

Stish-ovite

CaAlSi-phase

Ca P

vAl-phase

(CaF Type)

Al-phase(CaF Type)

Liquid iron

ε-phase HCP Iron

x= V 2SH /V 2sv

Fe-Al-M

g Persovskite

Ca P

erovskite

Stishovite

Ca P

ersovskite

Ferropericlase

New Al-phase(hexagonal)

K-Holl andite

RingwooditeWadsleyite

Fe-Al-Mg Perovskite

Fe high spin

?

? ?

?

?

? ?? ?

?

Fe low spin(radiative thermalconductor)

Post-perovskite

Vp N

E

S

W

Opx+Cpx

Garnet+Majorite

Garnet+Majorite

Garnet+Majorite

410 1400 13

660 1600 24

2000 2000 70

2700 2500 125

2891 3000 136

5150 5000 329

6371 5200 364

Inner core

Outer core

CMB

D” Layer

Figure 1 The simplified petrology and seismic anisotropy of the Earth’s mantle and core. The radially (transverse isotropic) anisotropic model of S-waveanisotropy of the mantle is taken from Panning and Romanowicz (2006). The icon at the inner core represents the fast P-wave velocities parallel tothe rotation axis of the Earth. The petrology ofmantle is taken fromOno and Oganov (2005) for pyrolite and Perrillat et al. (2006) and Ricolleau et al. (2010)for the transformed MORB and based on Irifune et al. (1994) as modified by Poli and Schmidt (2002) for the transformed argillaceous sediments.

488 Seismic Anisotropy of the Deep Earth from a Mineral and Rock Physics Perspective


are extremely rare. Upper mantle samples large enough for the

measurement of CPO have been recovered from kimberlitic

volcanics in South Africa to a depth of about 220 km estab-

lished by geobarometry (e.g., Boyd, 1973). Kimberlite mantle

xenoliths of deeper origin (>300 km) with evidence for equil-

ibrated majorite garnet, which is now preserved as pyrope

garnet with exsolved pyroxene, have been reported (Haggerty

and Sautter, 1990; Sautter et al., 1991). It has been proposed

that the Alpe Arami peridotite garnet lherzolite has been

exhumed from a minimum depth of 250 km based on clin-

oenstatite exsolution lamellae present in diopside grains

(Bozhilov et al., 1999). Samples of even deeper origin are

preserved as inclusions in diamonds. Although most dia-

monds crystallize at depths of 150–200 km, some diamonds

contain inclusions of majorite (Moore and Gurney, 1985),

enstatite and ferropericlase (Scott-Smith et al., 1984), and

CaSiO3+(Fe,Mg)SiO3+SiO2 (Harte and Harris, 1993). The

mineral associations imply transition zone (410–660 km)

and lower mantle origins for these diamond inclusions

(Kesson and Fitz Gerald, 1991). Although these samples help

to constrain mantle petrology, they are too small to provide

information about CPO. Hence, knowledge of CPO in the

transition zone, lower mantle, and inner core will be derived

from deformation experiments at high pressure and

temperature (e.g., olivine (Couvy et al., 2004), ringwoodite

(Karato et al., 1998), perovskite (Cordier et al., 2004b),

MgGeO3 postperovskite (Merkel et al., 2006a), and e-phaseiron (Merkel et al., 2005)).

It has been accepted since the PREM seismic model

(Dziewo�nski and Anderson, 1981) that the top 200 km of the

Treatise on Geophysics, 2nd edition,

Earth’s mantle is anisotropic on a global scale (Figure 1). How-

ever, there are exceptions, for example, under the Baltic shield,

the anisotropy increases below 200 km (Pedersen et al., 2006).

The seismic discontinuity at about 200 kmwas first reported by

the Danish seismologist Lehmann (1959, 1961), which now

bears her name. However, the discontinuity is not always pre-

sent at the same depth. Anderson (1979) interpreted the dis-

continuity as the petrologic change of garnet lherzolite to

eclogite. More recently, interpretations have favored an anisot-

ropy discontinuity, although, even this is controversial (see

Vinnik et al., 2005), proposed interpretations include a local

anisotropic decoupling shear zone marking the base of the

lithosphere (Leven et al., 1981), a transition from an aniso-

tropic mantle deforming by dislocation creep to isotropic man-

tle undergoing diffusion creep (Karato, 1992), simply the base

of an anisotropic layer beneath continents (e.g., Gaherty and

Jordan, 1995), or the transition from [100] to [001] direction

slip in olivine (Mainprice et al., 2005). Global tomography

studies show that the base of the anisotropic subcontinental

mantle may vary in depth from 100 to 450 km (e.g., Polet and

Anderson, 1995), but most global studies favor an anisotropy

discontinuity for S-waves at around 200–250 km, which is

stronger and deeper (300 km) beneath continents (e.g., Deuss

and Woodhouse, 2002; Panning and Romanowicz, 2006;

Ritsema et al., 2004) and weaker and shallower (200 km)

beneath the oceans. There is also evidence for weak seismic

discontinuities at 260 and 310 km that have been reported in

subduction zones by Deuss and Woodhouse (2002).

A major seismic discontinuity at 410 km is due to the

transformation of olivine to wadsleyite (e.g., Helffrich and

(2015), vol. 2, pp. 487-538

Seismic Anisotropy of the Deep Earth from a Mineral and Rock Physics Perspective 489


Wood, 1996) with a shear wave impedance contrast of 6.7%

(e.g., Shearer, 1996). The 410 km discontinuity has topogra-

phy within 5 km of the global average. The olivine to wad-

sleyite transformation will result in the lowering of anisotropy

with depth. Global tomography models (e.g., Beghein et al.,

2006; Montagner, 1994a,b; Montagner and Kennett, 1996;

Panning and Romanowicz, 2006) indicate that the strength

of anisotropy is less in the transition zone (410–660 km)

than in the upper mantle (Figure 1). A global study of the

anisotropy of transition zone by Trampert and van Heijst

(2002) has detected a weak anisotropy shear wave of about

1–2%. The surface wave overtone technique used by Trampert

and van Heijst (2002) cannot localize the anisotropy within

the 410–660 km depth range; however, the only mineral with

a strong anisotropy and significant volume fraction in the

transition zone is wadsleyite occurring between 410 and

520 km. Between 520 and 660 km, there is an increase in the

very weakly anisotropic phases, such as garnet, majorite, and

ringwoodite in the transition zone (Figure 1). Tommasi et al.

(2004) had shown that the CPO predicted by a plastic flow

model using the experimentally observed slip systems of wad-

sleyite can reproduce the weak anisotropy observed by

Trampert and van Heijst (2002). The weaker discontinuity at

520 km, with a shear wave impedance contrast of 2.9%, has

been reported by Shearer and coworkers (e.g., Flanagan and

Shearer, 1998; Shearer, 1996). The discontinuity at 520 km

depth has been attributed to the wadsleyite to ringwoodite

transformation by Shearer (1996). Deuss and Woodhouse

(2001) had reported the ‘splitting’ of the 520 km discontinuity

into two discontinuities at 500 and 560 km, they interpreted

the variations of depth of 520 km, and the presence of two

discontinuities at 500 and 560 km in certain regions can only

be explained by variations in temperature and composition

(e.g., Mg/Mg+Fe ratio), which affect the phase transition Cla-

peyron. Regional seismic studies by Vinnik and coworkers

(Vinnik and Montagner, 1996; Vinnik et al., 1997) show evi-

dence for a weakly anisotropic (1.5%) layer for S-waves at the

bottom 40 km of the transition zone (620–660 km). Some

global tomography models (e.g., Montagner, 1998; Montagner

and Kennett, 1996) also show significant transverse isotropic

anisotropy in the transition zone with VSH>VSV and

VPH>VPV. Given the low intrinsic anisotropy of most of the

minerals in the lower part of the transition zone, Karato (1998)

suggested that this anisotropy is due to petrologic layering

caused by garnet and ringwoodite rich layers of transformed

subducted oceanic crustal material. Such a transversely isotropic

medium with a vertical symmetry axis would not cause any

splitting for vertically propagating S-waves and would not pro-

duce the azimuthal anisotropy observed by Trampert and van

Heijst (2002), but would produce the difference between hori-

zontal and vertical velocities seen by global tomography.

A global study supports this suggestion, as high-velocity slabs

of former oceanic lithosphere are conspicuous structures just

above the 660 kmdiscontinuity in the circum-Pacific subduction

zones (Ritsema et al., 2004). A regional study by Wookey et al.

(2002) also finds significant shear wave splitting associated with

horizontally traveling S-waves, which is compatible with a

layered structure in the vicinity of the 660 km discontinuity.

However, recent anisotropic global tomography models do not

show significant anisotropy in this depth range (Beghein et al.,

2006; Panning and Romanowicz, 2006)


The strongest seismic discontinuity at 660 km is due to the

dissociation of ringwoodite to perovskite and ferropericlase

(Figure 1) with a shear wave impedance contrast of 9.9%

(e.g., Shearer, 1996). The 660 km discontinuity has an impor-

tant topography with local depressions of up to 60 km from

the global average in subduction zones (e.g., Flanagan and

Shearer, 1998). From 660 to 1000 km, a weak anisotropy is

observed in the top of the lower mantle with VSH<VSV and

VPH<VPV (e.g., Montagner, 1998; Montagner and Kennett,

1996). Karato (1998) attributed the anisotropy to the CPO of

perovskite and possibly ferropericlase caused by plastic defor-

mation in the convective boundary layer at the top of the lower

mantle. In this depth range, Kawakatsu and Niu (1994) had

identified a flat seismic discontinuity at 920 km with S to

P converted waves with an S-wave velocity change of 2.4% in

Tonga, Japan Sea, and Flores Sea subduction zones. They sug-

gested that some sort of phase transformation thermodynam-

ically controls this feature, or alternatively, we may suggest that

it marks the bottom of the anisotropic boundary layer pro-

posed by Montagner (1998) and Karato (1998). Reflectors in

lower mantle have been reported by Deuss and Woodhouse

(2001) at 800 km depth under North America and at 1050 and

1150 km beneath Indonesia; they only considered the 800 km

reflector to be a robust result. Karki et al. (1997c) had sug-

gested that the transformation of the highly anisotropic SiO2

polymorph stishovite to CaCl2 structure at 50�3 GPa at room

temperature may be the possible explanation of reflectivity in

the top of the lower mantle. However, according to Kingma

et al. (1995), the transformation would take place at 60 GPa at

lower mantle temperatures in the range 2000–2500 K, corre-

sponding to depth of 1200–1500 km, that is several hundred

kilometers below the 920 km discontinuity. It is highly specu-

lative to suggest that free silica is responsible for the 920 km

discontinuity as a global feature as proposed by Kawakatsu and

Niu (1994). Ringwood (1991) suggested that 10% stishovite

would be present from 350 to 660 km in subducted oceanic

crust and this would increase to about 16% at 730 km. Hence,

in the subduction zones studied by Kawakatsu and Niu (1994),

it is quite possible that significant stishovite could be present to

1200 km and may be a contributing factor to the seismic

anisotropy of the top of the lower mantle. From 1000 to

2700 km, the lower mantle is isotropic for body waves or free

oscillations (e.g., Beghein et al., 2006; Montagner and Kennett,

1996; Panning and Romanowicz, 2006). Karato et al. (1995)

had suggested by comparison with deformation experiments of

fine-grained analogue oxide perovskite that the seismically

isotropic lower mantle is undergoing deformation by super-

plasticity or diffusive creep, which traditionally has been con-

sidered to not produce a CPO; this is now being challenged by

recent experimental results (Miyazaki et al., 2013; Sundberg

and Cooper, 2008). In the bottom of the lower mantle, the D00

layer (100–300 km thick) appears to be transversely isotropic

with a vertical symmetry axis characterized by VSH>VSV

(Figure 1) (e.g., Kendall and Silver, 1996, 1998), which may

be caused by CPO of the constituent minerals, shape preferred

orientation of horizontally aligned inclusions, possibly melt

(e.g., Berryman, 2000; Williams and Garnero, 1996) or core

material. It has been suggested that the melt fraction of D00 may

be as high as 30% (Lay et al., 2004). Seismology has shown

that D00 is extremely heterogeneous as shown by globally high

fluctuations of shear (2–3%) and compressional (1%) wave

, (2015), vol. 2, pp. 487-538



velocities (e.g., Lay et al., 2004; Megnin and Romanowicz,

2000; Ritsema and van Heijst, 2001), a variation of the thick-

ness of D00 layer between 60 and 300 km (e.g., Sidorin et al.,

1999a), P- and S-wave velocity variations sometimes correlated

and sometimes anticorrelated (thermal, chemical, and melting

effects?) (e.g., Lay et al., 2004), ultralow-velocity zones at the

base of D00 with Vp 10% slower and Vs 30% slower than

surrounding material (e.g., Garnero et al., 1998), and regions

with horizontal (e.g., Kendall, 2000; Kendall and Silver, 1998)

or inclined anisotropy (e.g., Garnero et al., 2004; Maupin et al.,

2005; McNamara et al., 2003; Wookey et al., 2005a) in the

range 0.5–1.5% and isotropic regions, localized patches of

shear velocity discontinuity, that even predicted the possibility

of a globally extensive phase transformation (Nataf and

Houard,1993) and its Clapeyron slope (Sidorin et al.,

1999b). Until recently, the candidate phase for this transition

was SiO2 (Murakami et al., 2004). However, the mineralogical

picture of the D00 layer has been completely changed with the

discovery of postperovskite by Murakami et al. (2004), which

is produced by the transformation of Mg-perovskite in the

laboratory at pressures greater than 125 GPa at high tempera-

ture. Seismic modeling of the D00 layer using the new phase

diagram and elastic properties of perovskite and postperovskite

can explain many features mentioned earlier near the core–

mantle boundary (Wookey et al., 2005b). The imaging of the

layered structures within the D00 region by van der Hilst et al.

(2007) using three-dimensional inverse scattering of core-

reflected shear waves has provided a more quantitative view

of D00 heterogeneity. The layered structures imaged by van der

Hilst et al. (2007) are compatible with transverse isotropic

anisotropy reported by earlier studies (e.g., Kendall, 2000;

Kendall and Silver, 1998; Wookey et al., 2005a).

Although the outer core was discovered by British geologist

Richard Oldham in 1906 (Oldham, 1906), the inner solid core

was identified 30 years later by the Danish seismologist Inge

Lehmann in a paper published in 1936 with the short title P0.She identified P-waves that traveled through the core region

(PKP, where K stands for core) at epicentral distances of 105–

142� in contradiction to the expected travel times for a single

core model (Lehmann, 1936). She proposed a two-shell model

for the core with a uniform velocity of about 10 km s�1 with a

small velocity discontinuity between each shell and an inner

shell radius of 1400 km, close to the actually accepted value of

1221.5 km from PREM (Dziewo�nski and Anderson, 1981).

The liquid nature of the outer core was first proposed by

Jeffreys (1926) based on shear wave arrival times, and the

solid nature of inner core was first proposed by Birch (1940)

based on the compressibility of iron at high pressure. Given the

great depth (5149.5 km) and the number of layers a seismic

wave has to traverse to reach the inner core and return to the

surface, it is not surprising that the first report of anisotropy of

the inner core was inferred 50 years after the discovery of the

inner core. Poupinet et al. (1983) were the first to observe that

PKIKP (where K now stands for the outer core and I is for inner

core) P-waves travel about 2 s faster parallel to the Earth’s

rotation axis than waves traveling the equatorial plane. They

interpreted their observations in terms of a possible heteroge-

neity of the inner core. Shortly afterward, a PKIKP travel time

study by Morelli et al., 1986 and normal modes (free oscilla-

tions) by Woodhouse et al. (1986) reported new observations


and interpreted the results in terms of anisotropy. However,

the interpretation of PKIKP body wave travel times in terms of

anisotropy remained controversial, with an alternative inter-

pretation being that the inner core had a nonspherical structure

(e.g., Widmer et al., 1992). Finally, the observation of large

differential travel times for PKIKP for paths from the South

Sandwich Islands to Alaska by Creager (1992) and Song and

Helmberger (1993) and the interpretation of higher-quality

free oscillation data by Tromp (1993,1994) and Durek and

Romanowicz (1999) gave further strong support for the

homogenous transverse anisotropy interpretation. The general

consensus became that the inner core is strongly anisotropic,

with a P-wave anisotropy of about 3–4% with the fast velocity

direction parallel to the Earth’s rotation axis (see reviews by

Creager, 2000; Song, 1997; Tromp, 2001). However, many

studies have suggested variations to this simple anisotropy

model of the inner core. It has been suggested that the symme-

try axis of the anisotropy is tilted from the Earth’s rotation axis

(Creger, 1992; Shear and Toy, 1991; Su and Dziewo�nski, 1995)

by 5–10�. A significant difference in the anisotropy between

eastern and western hemispheres of the inner core has been

reported by Creger (1999) and Tanaka and Hamaguchi (1997)

with the western hemisphere having significantly stronger

anisotropy than the eastern hemisphere that is nearly isotropic.

Several recent studies concur that the outer part (100–200 km)

of the inner core is isotropic and inner part is anisotropic (e.g.,

Garcia, 2002; Garcia and Souriau, 2000, 2001; Song and

Helmberger, 1998; Song and Xu, 2002; Sun and Song, 2008).

It has also been suggested that there is a small innermost inner

core with radius of about 300 km with distinct transverse

isotropy relative to the outermost inner core by Ishii and

Dziewo�nski (2002). The innermost core has the slowest P-

wave velocity at 45� to the east–west direction, and the outer

part has a weaker anisotropy with slowest P-wave velocity

parallel to the east–west direction. Using split normal mode

constraints, Beghein and Trampert (2003) also showed that

there is a change in velocity structure with radius in the inner

core; however, their model shows that the symmetry of the P-

and S-wave changes at about 400 km radius, suggesting a rad-

ical change, such as a phase transition of iron. Much of the

complexity of the observations seems to be station- and

method-dependent (see Ishii et al., 2002a,b). In a detailed

study, Ishii et al. (2002a,b) derived a model that simulta-

neously satisfies normal mode, absolute travel time, and dif-

ferential travel time data and has allowed them to separate a

mantle signature and regional structure from global anisotropy

of the inner core. Their preferred model of homogeneous

transverse isotropy with a symmetry axis aligned with the

rotation axis contradicts many of models proposed earlier but

is similar to previous suggestions. In a study of inner core

P-wave anisotropy using both finite-frequency and ray theo-

ries, Calvet et al. (2006) found that the data can be explained

by three families of models that all exhibit anisotropy changes

at a radius between 550 and 400 km (compared to 300 km for

Ishii and Dziewo�nski (2002, 2003) and about 400 km for

Beghein and Trampert (2003)). The first model has a weak

anisotropy with a slow P-wave velocity symmetry axis parallel

to the Earth’s rotation axis. The second model has a nearly

isotropic innermost inner core. Lastly, the third model has a

strongly anisotropic innermost inner core with a fast symmetry

(2015), vol. 2, pp. 487-538



axis parallel to the Earth’s rotation axis. These models have very

different implications for the origin of the anisotropy and the

history of the Earth’s core. These divergences partly reflect the

uneven sampling of the inner core by PKP(DF) paths resulting

from the spatial distribution of earthquakes and seismographic

stations. A recent study using high-quality, 2360 handpicked

PKIKP arrival times is the largest database to be used for study-

ing the inner core to date (Lythgoe et al., 2014). Lythgoe et al.

found their data are best explained by significant hemispheri-

cal variations in anisotropy (east 0.5–1.5%; west 3.5–8.8%),

with slow Vp direction at 57–61� to the rotation axis at all

depths (Figure 2). Furthermore, there is no need for innermost

inner core to explain their data, and they suggest that observa-

tions of an innermost inner core are an artifact from averaging

over lateral anisotropy variations.

Theoretical studies of the process of generating the Earth’s

magnetic field through fluid motion of the outer core have

predicted that the electromagnetic torque would force the

inner core to rotate relative to the mantle (Glatzmaier and

Roberts, 1995, 1996; Gubbins, 1981; Steenbeck and Helmis,

1975; Szeto and Smylie, 1984). Song and Richards (1996) first

reported seismic differential travel time observations based of

three decades of data supporting the eastward relative rotation

of inner core by about 1� per year faster than the daily rotation

of the mantle and crust; this is sometimes referred to as the

superrotation of the inner core. Around the equator of the

inner core, this rotation rate corresponds to a speed of a few

tens of kilometers per year. The interpretation of the observed

travel times required using the seismic anisotropy model inner

core established by Su and Dziewo�nski (1995). Clearly, to

establish such small relative rotation rate, a detailed knowledge

of anisotropy, heterogeneity, and shape of the inner core is

required, as travel times will change with direction and time

(Song, 2000). The result was supported by some studies (e.g.,

Creager, 1997) and challenged by other studies (e.g., Souriau

et al., 1997), but all indicated a smaller rotation rate than 1� peryear. Zhang et al. (2005) reported a rotation rate of 0.3–0.5�

using techniques that avoid artifacts of poor event locations and

dt/t

Layer 1 : >750 kmLayer 2 : 550- 750 kmLayer 3 : <550 km

Western hemisphere

0 30 60 90-0.12

-0.08

-0.04

0.00

Angle from symmetry axis (°)

Isotropic

Vp low 57°–61°

Vp highest at 0°East

West40 °E

-95°W

N

Figure 2 An anisotropy model for the eastern and western hemispheres ofboundaries are at 750 and 550 km from the center of the Earth. The P-waveanisotropy from spherically symmetrical model given by Lythgoe et al. (2014ray path and the Earth’s rotation axis. Isotropic elastic behavior is equivalent


contamination by small-scale heterogeneities. Tkalcic et al.

(2013) found that rotation rate varies with time with mean

values of 1� per year over a 10-year period and fluctuations of

0.25–0.48 per year. An analysis reconciling the hemispherical

structure with the inner core superrotation, and avoiding the

presence of innermost inner core, has been present by Waszek

et al. (2011). The authors link the superrotation to the 3-D east–

west hemispherical boundary structure and the eastward dis-

placement of these boundaries with depth. In agreement with

other studies suggesting the melting in the eastern hemisphere

and crystallization in the west, which are responsible for the

translation of the inner core towards the east (e.g., Alboussiere

et al., 2010; Monnereau et al., 2010). The estimated super-

rotation rate based on the eastward growth rate is extremely

low at 0.1–1� per million years. If previously reportedmeasure-

ments of 0.3–0.5� per year are to be accepted, they can only

represent short-term fluctuations of the rotation rate. Shear

waves are very useful for determining the magnitude and ori-

entation of anisotropy along individual ray paths; despite sev-

eral claims to have observed shear waves (e.g., Julian et al.,

1972; Okal and Cansi, 1998), the analysis of the attenuation

(Doornbos, 1974) and frequency range (Deuss et al., 2000)

reveals that these claims are unjustified. The analysis and vali-

dation of observations by Deuss et al. (2000) provide the

first reliable observation of long-period (20–30 s) shear waves

providing new possibilities for exploring the anisotropy of

inner core. Wookey and Helffrich (2008) presented two obser-

vations of an inner core shear wave phase (PKJKP) at higher

frequencies in stacked data from the Japanese high-sensitivity

array (Hi-Net). Their model based on S-wave data is that c-axis

of the hexagonal close-packed (hcp) iron phase is aligned

normal to the Earth’s rotation axis as proposed by Steinle-

Neumann et al. (2001). As I will discuss later, it is not obvious

that this interpretation is compatible with the majority of now

available mineral physics data.

The transverse isotropy observed for P-waves traveling

through the inner core could be explained by the CPO of the

hcp form of iron (e-phase) crystals or a layered structure. The

Layer 1 : >750 kmLayer 2 : 550-750 kmLayer 3 : <550 km

Eastern hemisphere

0 30 60 90-0.12

-0.08

-0.04

0.00


IsotropicVp low above 45°

Vp high <45°

East

40 °E

-95°W

N

West

the inner core with hemispherical boundaries at 95�W and 40�E. Radialtravel time residual (dt/t) is a perturbation of a weak cylindrical). The angle from symmetry axis in this case is the angle between theto dt/t¼0, indicated by the horizontal black line.

, (2015), vol. 2, pp. 487-538



analysis of the coda of short-period inner core boundary-

reflected P-waves (PKiKP) requires only a few percent heteroge-

neity at length scales of 2 km (Vidale and Earle, 2000), which

suggests a relatively homogenous nonstructured inner core;

however, this interpretation of coda has recently been ques-

tioned by Poupinet and Kennett (2004). The mechanism

responsible for the CPO has been the subject of considerable

speculation in recent years, with the suggested mechanisms

including the alignment of crystals in the magnetic field as

they solidify from the liquid outer core (Karato, 1993), the

alignment of crystals by plastic flow under the action of

Maxwell normal stresses caused the magnetic field (Karato,

1999), faster crystal growth in the equatorial region of the

inner core (Yoshida et al., 1996), anisotropic growth driven by

strain energy (Stevenson, 1987), dendritic crystal growth aligned

with the direction of dominant heat flow (Bergman, 1997),

plastic flow in a thermally convective regime ( Jeanloz and

Wenk, 1988; Wenk et al., 1988, 2000), and plastic flow under

the action of magnetically induced Maxwell shear stresses

(Buffett and Wenk, 2001). An alternative explication that was

proposed by Singh et al. (2000) to explain the P-wave anisot-

ropy and the low shear wave velocity of about 3.6 km s�1

(Deuss et al., 2000) is the presence of a volume fraction of 3–

10% liquid iron (or FeS) in the form of oblate spherical inclu-

sions aligned in the equatorial plane in a matrix of iron crystals

with their c-axes aligned parallel to the rotation axis as originally

proposed by Stixrude and Cohen (1995). The S-wave velocity

and attenuation data are mainly from the outer part of the inner

core, and hence, it was suggested the liquid inclusions are pre-

sent in this region. Note that this is in contradiction with other

studies, which suggest that the outer core has a low anisotropy.

Several problems that are posed by all models to different

degrees are the inner core thermally convective (see Stevenson,

1987; Weber andMachetel, 1992; Yukutake, 1998), the viscosity

of the inner core, the strength of magnetic field and magnitude

Maxwell stresses necessary to cause crystal alignment, the

presence of liquids, and even the ability of the models to cor-

rectly predict the magnitude and orientation of the seismic

P-wave anisotropy. Given the range of seismic models and the

variety of physical phenomena proposed to explain these

models, better contents on the seismic data, probably using

better quality data and a wider geographic distribution of seis-

mic stations in polar regions, are urgently required.

In this chapter, I review our current knowledge of the seismic

anisotropy of the constituent minerals of the Earth’s interior and

our ability to extrapolate these properties to mantle conditions

of temperature and pressure (Figure 1). I will begin by reviewing

the fundamentals of elasticity, plane wave propagation in aniso-

tropic crystals, the measurements of elastic constants, and the

effective elastic constants of crystalline aggregates.

2.20.2 Mineral Physics

2.20.2.1 Elasticity and Hooke’s Law

Robert Hooke’s experiments demonstrated that extension of a

spring is proportional to the weight hanging from it, which was

published in de Potentia Restitutiva (or of Spring Explaining the

Power of Springing Bodies (1678)), establishing that in elastic

solids, there is a simple linear relationship between stress and


strain. The relationship is now commonly known as Hooke’s

law (Ut tensio, sic vis – which translated from Latin is ‘as is the

extension, so is the force’ – was the solution to an anagram

announced 2 years early in ‘A Description of Helioscopes and

some other Instruments 1676,’ to prevent Hooke’s rivals from

claiming to have made the discovery themselves!). In the case of

small (infinitesimal) deformations, a Maclaurin expansion of

stress as a function of strain developed to first order correctly

describes the elastic behavior of most linear elastic solids:

sij eklð Þ¼ sij 0ð Þ+ @sij@ekl

� �@ekl¼0

ekl +1

2

@sij@ekl@emn

� �@ekl¼0

@emn¼0

eklemn + L

As the elastic deformation is zero at a stress of zero, then

sij(0)¼0, and restricting our analysis to first order, then we

can define the fourth-rank elastic tensor cijkl as

cijkl ¼ @sij@ekl

� �@ekl¼0

where «kl and sij are, respectively, the stress and infinitesimal

strain tensors. Hooke’s law can now be expressed in its tradi-

tional form as

sij ¼ cijkl ekl

The coefficients of the elastic fourth-rank tensor cijkl trans-

late the linear relationship between the second-rank stress and

the infinitesimal strain tensors. The four indexes (ijkl) of the

elastic tensor have values between 1 and 3, so that there are

34¼81 coefficients. The stress tensor is symmetrical as we

assume that stresses acting on opposite faces are equal and

opposite, and hence, there are no stress couples to produce a

net rotation of the elastic material. The infinitesimal strain

tensor is also symmetrical, because we assume that pure and

simple shear quantities are so small that their squares and

products can be neglected. Due to the symmetrical symmetry

of stress and infinitesimal strain tensors, they only have six

independent values rather than nine for the asymmetrical case,

and hence, the first two (i, j) and second two (k,l) indexes of the

elastic tensor can be interchanged:

cijkl cjikl and cijkl ¼ cijlk

The permutation of the indexes caused by the symmetry of

stress and strain tensors reduces the number of independent

elastic coefficients to 62¼36 because the two pairs of indexes

(i,j) and (k,l) can only have six different values:

1� 1, 1ð Þ 2� 2, 2ð Þ 3� 3, 3ð Þ 4� 2, 3ð Þ¼ 3, 2ð Þ 5� 3, 1ð Þ¼ 1, 3ð Þ 6� 1, 2ð Þ¼ 2, 1ð Þ

It is practical to write a 6 by 6 table of 36 coefficients with

two Voigt indexes m and n (cmn) that have values between 1

and 6, whereas the representation of the cijkl tensor with

81 coefficients would be a printer’s nightmare. The relation

between the Voigt (mn) and tensor indexes (ijkl) can be

expressed most compactly by

m¼ diji + 1�dij� �

9� i� jð Þ and n¼ dklk+ 1�dklð Þ 9�k� lð Þwhere dij is the Kronecker delta (dij¼1 when i¼ j and dij¼0

when i 6¼ j).

Combining the first and second laws of thermodynamics

for stress–strain variables, we can define the variation of the

(2015), vol. 2, pp. 487-538



internal energy (dU) per unit volume of a deformed aniso-

tropic elastic body as a function of entropy (dS) and elastic

strain (deij) at an absolute temperature (T ) as

dU¼ sij deij +TdS

U and S are called state functions. From this equation, it follows

that the stress tensor at constant entropy can be defined as

sij ¼ @U

@eij

� �¼ cijklekl hence cijkl ¼ @sij

@ekl

� �and cklij ¼ @skl

@eij

� �

and finally, we can write the elastic constants in terms of

internal energy and strain as

cijkl ¼ @

@ekl

@U

@eij

� �S

¼ @2U

@eij@ekl

� �S

¼ @2U

@ekl@eij

� �S

¼ cklij

The fourth-rank elastic tensors are referred to as second-

order elastic constants in thermodynamics, because they are

defined as second-order derivatives of a state function (e.g.,

internal energy @2U for adiabatic or Helmholtz free energy @2F

for isothermal constants) with respect to strain; we obtain the

Schwarz integrability condition that allows the interchanging

of the order of partial derivatives of a function. It follows from

these mathematical and thermodynamic arguments that the

symmetry of the derivatives allows the interchange of the first

pair of indexes (ij) with second (kl):

cijkl ¼ cjikl and cijkl ¼ cijlk and now cijkl ¼ cklij

The additional symmetry of cijkl¼ cklij permutation reduces the

number of independent elastic coefficients from 36 to 21, and

tensor with two Voigt indexes is symmetrical, cmn¼ cnm.

Although we have illustrated the case of isentropic (constant

entropy, equivalent to an adiabatic process for a reversible pro-

cess such as elasticity) elastic constants that intervene in the

propagation of elastic waves whose vibration is too fast for

thermal diffusion to establish heat exchange to achieve isother-

mal conditions, these symmetry relations are also valid for

isothermal elastic constants that are used in mechanical prob-

lems. Most of the elastic constants reported in the literature are

determined by the propagation of ultrasonic elastic waves and

are adiabatic. More recently, elastic constants predicted by

atomic modeling for mantle conditions of pressure, and in

some cases temperature, are also adiabatic (see review by Karki

et al., 2001, and also see Chapter 2.08).

The elastic constants in the literature are presented in the

form of 6 by 6 tables for the triclinic symmetry with 21 inde-

pendent values; here, the independent values are shown in

bold characters in the upper diagonal of cmn with the corre-

sponding cijkl:

c11 c12 c13 c14 c15 c16c12 c22 c23 c24 c25 c26c13 c23 c33 c34 c35 c36c14 c24 c34 c44 c45 c46c15 c25 c35 c45 c55 c56c16 c26 c36 c46 c56 c66

6666666664

7777777775¼

c1111 c1122 c1133 c1123 c1113 c1112c1122 c2222 c2233 c2223 c2213 c2212c1133 c2233 c3333 c3323 c3313 c3312c1123 c2223 c3323 c2323 c2313 c2312c1113 c2213 c3313 c2313 c1313 c1312c1112 c2212 c3312 c2312 c1312 c1212

26666664

37777775

In the triclinic system, there are no special relationships

between the constants. On the other extreme is the case of

isotropic elastic symmetry that is defined by just two coeffi-

cients. Note that this is not the same as cubic symmetry, where

there are three coefficients and that a cubic crystal can be


elastically anisotropic. The isotropic elastic constants can be

expressed in the four-index system as

cijkl ¼ ldijdkl +m dikdjl + dildjk� �

where l is Lame’s coefficient and m is the shear modulus. l andm are often referred to as Lame’s constants after the French

mathematician Gabriel Lame, who first published his book

‘Lecons sur la theorie mathematique de l’elasticite des corps

solides’ in 1852. In the two-index Voigt system, the indepen-

dent values are

c11 ¼ c22 ¼ c33 ¼ l +2mc12 ¼ c23 ¼ c13 ¼ lc44 ¼ c55 ¼ c66 ¼ 1⁄2 c11� c12ð Þ¼ m

In matrix form, they are written as

c11 c12 c12 0 0 0

c12 c11 c12 0 0 0

c12 c12 c11 0 0 0

0 0 0 1=2 c11� c12ð Þ 0 0

0 0 0 0 1=2 c11� c12ð Þ 0

0 0 0 0 0 1=2 c11� c12ð Þ

66666666666664

77777777777775where the two independent values are c11 and c12. Another

symmetry that is very important in seismology is the trans-

verse isotropic medium (or hexagonal crystal symmetry). In

many geophysical applications of transverse isotropy, the

unique symmetry direction (X3) is vertical and the other

perpendicular elastic axes (X1 and X2) are horizontal and

share the same elastic properties and velocities. It is very

common in seismological papers to use the notation of

Love (1927) for the elastic constants of transverse isotropic

media where

A¼ c11 ¼ c22 ¼ c1111 ¼ c2222C¼ c33 ¼ c3333F¼ c13 ¼ c23 ¼ c1133 ¼ c2233L¼ c44 ¼ c55 ¼ c2323 ¼ c1313N¼ c66 ¼ 1⁄2 c11�c12ð Þ¼ c1212 ¼ 1⁄2 c1111�c1122ð Þand A�2N¼ c12 ¼ c21 ¼ c11�2c66 ¼ c1212 ¼ c2211 ¼ c1111�2c1212

c11 c12 c13 0 0 0

c12 c11 c12 0 0 0

c13 c13 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

0 0 0 0 0 1=2 c11�c12ð Þ

6666666666664

7777777777775¼

A A�2N F 0 0 0

A�2N A F 0 0 0

F F C 0 0 0

0 0 0 L 0 0

0 0 0 0 L 0

0 0 0 0 0 N

26666666664

37777777775

and the velocities in orthogonal directions that characterize a

transverse isotropic medium are functions of the leading diag-

onal of the elastic tensor and are given as

A¼ c11 ¼ rV2PH C¼ c33 ¼ rV2

PV L¼ c44 ¼ rV2SV N¼ c66

¼ rV2SH

where r is density, VPH and VPV are the velocities of horizon-

tally (X1 or X2) and vertically (X3) propagating P-waves, and

VSH and VSV are the velocities of horizontally and vertically

polarized S-waves propagating horizontally.

Elastic anisotropy can be characterized by taking ratios of

the individual elastic coefficients. Thomsen (1986) introduced

, (2015), vol. 2, pp. 487-538



three parameters to characterize the elastic anisotropy of any

degree, not just weak anisotropy, for transverse isotropic

medium:

e¼ c11�c33=2c33 ¼A�C=2C

g¼ c66�c44=2c44 ¼N�L=2L

and

d*¼ 1⁄2c233 2 c13 + c44ð Þ2� c33� c44ð Þ c11 + c33�2c44ð Þ� �d*¼ 1⁄2C2 2 F + Lð Þ2� C�Lð Þ A +C�2Lð Þ� �

Thomsen also proposed a weak anisotropy version of the d*parameter:

d¼ c13 + c44ð Þ2� c33�c44ð Þ2=2c33 c33�c44ð Þ� �¼ F + Lð Þ2� C�Lð Þ2=2C C�Lð Þ� �

These parameters go to zero in the case of isotropy and have

values of much less than one (i.e., 10%) in the case of weak

anisotropy. The parameter e describes the P-wave anisotropy

and can be defined in terms of the normalized difference of the

P-wave velocity in the directions parallel to the symmetry axis

(X3, vertical axis) and normal to the symmetry axis (X12,

horizontal plane). The parameter g describes the S-wave anisot-ropy and can be defined in terms of the normalized difference

of the SH wave velocity in the directions normal to the sym-

metry axis (X12, horizontal plane) and parallel to the symme-

try axis (X3, vertical axis) but also in terms SH and SV, because

SH parallel to the symmetry axis has the same velocity as SV

normal to the symmetry axis:

e¼Vp X12ð Þ�Vp X3ð Þ=Vp X3ð Þ¼VPH�VPV=VPV

g¼VSH X12ð Þ�VSH X3ð Þ=VSH X3ð Þ¼VSH X12ð Þ�VSV X12ð Þ=VSV X12ð Þ¼VSH�VSV=VSV

Thomsen (1986) found that the parameter d* controls most

of the phenomena of importance for exploration geophysics,

such as velocities inclined to the symmetry axis (vertical), some

of which are nonnegligible even when the anisotropy is weak.

The parameter d* is an awkward combination of elastic param-

eters, which is totally independent of the velocity in the direction

normal to the symmetry axis (X12 horizontal plane) and which

may be either positive or negative. Mensch and Rosolofosaon

(1997) had extended the application of Thomsen’s parameters

to anisotropic media of arbitrary symmetry and the associated

analysis in terms of the perturbation of a reference model that

can exhibit strong S-wave anisotropy.

In the domain of one- or three-dimensional radial anisotropic

seismic tomography, it has been the practice to use the parameters

f, x, and � to characterize the transverse anisotropy, where

f¼ c33=c11 ¼C=A¼VPV2=VPH

2

x¼ c66=c44 ¼N=L¼VSH2=VSV

2

�¼ c13= c11�2c44ð Þ¼ F= A�2Lð ÞFor characterizing the anisotropy of the inner core, some

authors (e.g., Song, 1997) use a variant of Thomsen’s param-

eters, e 00 ¼(c33�c11)/2c11¼(C�A)/2A (positions of c11 and

c33 reversed from Thomsen; single prime 00 has been added to

avoid confusion here with Thomsen’s parameter),

g¼(c66�c44)/2c44¼(N�L)/2L (same as Thomsen), and

s¼(c11+c33�4c44�2c13)/2c11¼(A+C�4L�2F)/2A (very


different from Thomsen’s d*); others (e.g., Woodhouse et al.,

1986) use

a¼ c33�c11ð Þ=Ao ¼ C�Að Þ=Ao

b¼ c66�c44ð Þ=Ao

g¼ c11�2c44�c13ð Þ=Ao ¼ A�2N�Fð Þ=Ao

where Ao ¼ roVpo2 is calculated using the density ro and P-wave

velocityVpo at the center of the spherically symmetrical reference

Earth model, PREM (Dziewo�nski and Anderson, 1981). With at

least four different sets of triplets of anisotropy parameters to

describe transverse isotropy in various domains of seismology,

the situations are complex for a researcher who wants to com-

pare the anisotropy from different published works. Even when

comparisons are made, for example, for the inner core (Calvet

et al., 2006), drawing conclusions may be difficult as the

parameters reflect only certain aspects of the anisotropy.

In studying the effect of symmetry of the elastic properties of

crystals, one is directly concerned with only the 11 Laue classes

and not the 32 point groups, because elasticity is a centro-

symmetric physical property. The velocity of an elastic wave

depends on its direction of propagation in an anisotropic crystal,

but not the positive or negative sense of the direction. In this

chapter, we are restricting our study to second-order elastic

constants, corresponding to small strains characteristic of elastic

deformations associated with the propagation of seismic waves.

If we wanted to consider larger finite strains or the effect of an

externally applied stress, we would need to consider third-order

elastic constants, as the approximation adopted in limiting the

components of the strain tensor to terms of the first degree in

the derivatives is no longer justified. For second-order elastic

constants, the two cubic and two hexagonal Laue classes are

not distinct (e.g., Brugger, 1965) andmay be replaced by a single

cubic and a single hexagonal class, which results in only nine

distinct symmetry classes for crystals shown in Table 1.

2.20.2.2 Plane Waves and the Christoffel Equation

There two types of elastic waves, which propagate in an isotro-

pic homogeneous elastic medium, the faster compressional (or

longitudinal) wave with displacements parallel to propagation

direction and slower shear (or transverse) waves with displace-

ments perpendicular to the propagation direction. In aniso-

tropic elastic media, there are three types, one compressional

and two shear waves with in general three different velocities.

In order to understand the displacements associated with dif-

ferent waves and their relationship to the propagation direc-

tion and elastic anisotropy, it is important to consider the

equation of propagation of a mechanical disturbance in an

elastic medium. If we ignore the effect of gravity, we can write

the equation of displacement (ui) as function of time (t) as

r@2ui@t2

� �¼ @sij

@xj

� �

where r is the density and xj is position. From Hooke’s law, we

can see that stress can be written as

sij ¼ cijkl@ul@xk

� �

and hence, elastodynamic equation that describes the inertial

forces can be rewritten with one unknown, the displacement, as

(2015), vol. 2, pp. 487-538

Table 1 Second-order elastic constants of all Laue crystalsymmetries

Cubic (3) 23,m3,432, -43m,m3m

Hexagonal (5) 6,-6, 6/m, 622, 6mmm,-62m, 6/mmm

c11 c12 c12 0 0 0c12 c11 c12 0 0 0c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44

26666664

37777775

c11 c12 c13 0 0 0c12 c11 c13 0 0 0c13 c13 c33 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 1=2 c11� c12ð Þ

266666664

377777775

Trigonal (6) 32,3m,-3mc11 c12 c13 c14 0 0c12 c11 c13 �c14 0 0c13 c13 c33 0 0 0c14 �c14 0 c44 0 00 0 0 0 c44 c140 0 0 0 c14 1=2 c11�c12ð Þ

26666664

37777775

Trigonal (7) 3,-3c11 c12 c13 c14 �c25 0c12 c11 c13 �c14 c25 0c13 c13 c33 0 0 0c14 �c14 0 c44 0 c25�c25 c25 0 0 c44 c140 0 0 c25 c14 1=2 c11�c12ð Þ

26666664

37777775

Tetragonal (6)422,4mm,-42m,4/mmm

Tetragonal (7) 4,-4,4/m

c11 c12 c13 0 0 0c12 c11 c13 0 0 0c13 c13 c33 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c66

26666664

37777775

c11 c12 c13 0 0 c16c12 c11 c13 0 0 �c16c13 c13 c33 0 0 00 0 0 c44 0 00 0 0 0 c44 0c16 �c16 0 0 0 c66

26666664

37777775

Orthorhombic (9) 222,mm2,mmm

Monoclinic (13) 2, m, 2/m

c11 c12 c13 0 0 0c12 c22 c23 0 0 0c13 c23 c33 0 0 00 0 0 c44 0 00 0 0 0 c55 00 0 0 0 0 c66

26666664

37777775

c11 c12 c13 0 c15 0c12 c22 c23 0 c25 0c13 c23 c33 0 c35 00 0 0 c44 0 c46c14 c25 c35 0 c55 00 0 0 c46 0 c66

26666664

37777775

Triclinic (21) 1,-1c11 c12 c13 c14 c15 c16c12 c22 c23 c24 c25 c26c13 c23 c33 c34 c35 c36c14 c24 c34 c44 c45 c46c15 c25 c35 c45 c55 c56c16 c26 c36 c46 c56 c66

26666664

37777775

The number in brackets is the number of independent constants.



@@2ui@t2

� �¼ cijkl

@2ul@xjxk

� �

Describing the displacement of monochromatic plane wave

by any harmonic form as a function of time, such as (e.g.,

Federov, 1968)

u¼Aexpi k:x�o tð Þwhere A is the amplitude vector, which gives the direction and

magnitude of particle motion; t time; n the propagation direc-

tion normal to the plane wave front; o the angular frequency,

which is related to frequency by f¼o/2p, and k the wave vector

that is related to the phase velocity (V) by V¼o/k and the

plane wave front normal (n) by k¼(2p/l) n, where l is the


wavelength. For plane waves, the total phase f¼(k.x�o t) is a

constant as the phase is constant along the wave front. Hence,

the equation for a surface of equal phase at any instant of time

(t) is a plane perpendicular to the propagation unit vector (n).

If now we insert the solution for the time-dependant displace-

ment into the elastodynamic equation, we find the Christoffel

equation (Christoffel, 1877) as one of his contributions to the

propagation of discontinuities as waves in elastic materials:

Cijklsjslpk ¼ rV2pi or Cijklsjslpk ¼ rpi

Cijklnjnl�rV2dik� �

pk ¼ 0 or Cijklsjsl�rgdik� �

pk ¼ 0

where V are the phase velocities, r is density, pk are polarization

unit vectors, nj are propagation unit vectors, and sj are the

slowness vectors of magnitude 1/V and the same direction as

the propagation direction (n). The polarization unit vectors pkare obtained as eigenvectors and corresponding eigenvalues of

the roots of the equation

det !Cijklnjnl�rV2dik !¼0 or det !Cijklsjsl�rdik !¼ 0

We can simplify this equation by introducing the

Christoffel (Kelvin–Christoffel or acoustic) tensor Tik¼Cijklnjnland three wave moduli M¼rV2; hence, det!Tik�Mdik!¼0.

The equation can written in full as

T11�M T12 T13

T21 T22�M T23

T31 T32 T33�M

��¼ 0

which upon expansion yields the cubic polynomial in M:

M3� ITM2 + IIT M� IIIT ¼ 0

where IT¼Tii, IIT¼½ (TiiTjj�TijTij), and ¼det!Tij! are

the first, second, and third invariants of the Christoffel tensor.

The three roots of the cubic polynomial inM are the three wave

moduli M. The eigenvectors (ej) associated with each wave

moduli can be found by solving (Tij�M dij) ej¼0. Analytic

solutions for the Christoffel tensor have been proposed in

various forms by Cerveny (1972), Every (1980), Mainprice

(1990), Mensch and Rosolofosaon (1997), and probably

others.

The Christoffel tensor is symmetrical because of the sym-

metry of the elastic constants, and hence,

Tik ¼Cijklnjnl ¼Cjiklnjnl ¼Cijlknjnl ¼Cklijnjnl ¼ Tki

The Christoffel tensor is also invariant upon the change of

sign of the propagation direction (n) as the elastic tensor is not

sensitive to the presence or absence of a center of symmetry,

being a centrosymmetric physical property. Because the elastic

strain energy (1/2 Cijkl � eij � ekl) of a stable crystal is always positiveand real (e.g., Nye, 1957), the eigenvalues of the 3�3Christoffel

tensor (being aHermitianmatrix) are three positive real values of

the wavemoduli (M) corresponding to rVp2,rVs12,rVs22 of theplane waves propagating in the direction n. The three eigenvec-

tors of the Christoffel tensor are the polarization directions (also

called vibration, particle movement, or displacement vectors) of

the threewaves; as theChristoffel tensor is symmetrical, the three

eigenvectors (and polarization) vectors are mutually perpendic-

ular. In the most general case, there are no particular angular

relationships between polarization directions (p) and the

propagation direction (n); however, typically, the P-wave

, (2015), vol. 2, pp. 487-538

4

6

8

10

12

14

0 90 180 270

Stishovite

Vp phase VSH phase VSV phase

Pha

se v

eloc

ity (k

ms−1

)

Propagation direction (�)

Vp

VSH

VSV

100 010 100001

Figure 3 The variation of velocity with direction for tetragonalstishovite as described by Weidner et al. (1982).



polarization direction is nearly parallel and the two S-wave

polarizations are nearly perpendicular to the propagation direc-

tion and they are termed quasi-P- or quasi-S-waves. If the P-wave

and two S-wave polarizations are, respectively, parallel and per-

pendicular the propagation direction, which may happen along

a symmetry direction, then thewaves are termed pure P and pure

S or pure modes. Only velocities in pure mode directions can be

directly related to single elastic constants (Neighbours and

Schacher, 1967). In general, the three waves have polarizations

that are perpendicular to one another and propagate in the same

direction with different velocities, with Vp>Vs1>Vs2.

A propagation direction for which two (or all three) of the

phase velocities are identical is called an acoustic axis, which

occurs even in crystals of triclinic symmetry. Commonly, the

acoustic axis is associated with the two S-waves having the

same velocity. The S-wave may be identified by their relative

velocity Vs1>Vs2 or by their polarization being parallel to a

symmetry direction or feature, for example, SH and SV, where

the polarization is horizontal and vertical to the third axis of

reference Cartesian frame of the elastic tensor (X3 in the termi-

nology of Nye (1957); X3 is almost always parallel to the crystal

c-axis) in mineral physics and perpendicular to the Earth’s sur-

face in seismology.

What is the difference between an elastic isotopic medium

and an anisotropic medium for wave propagation? For an

isotropic medium, the propagation direction is parallel to the

Vp polarization and the Vs polarizations normal to propagation

direction; all are associatedwith the same S velocity asVs1¼Vs2

and in general Vp>Vs. Even in isotropic medium, the S-wave

polarizations are normal to P-wave polarization. What is differ-

ent for anisotropic medium is that propagation direction is no

longer parallel to theVp polarization by an angle of few degrees,

and the two S-wave polarizations normal to P-wave polariza-

tion are now associated different velocities (Vs1>Vs2). The

angle between propagation direction and the Vp polarization,

as well as the difference between the S-wave velocities, depends

on themagnitude of the elastic anisotropy.While discussing the

difference between isotropic and anisotropic media, it is perti-

nent to mention the case of the ratio between Vp and Vs, which

is a parameter frequentlymeasured in seismology, especially for

regions where fluids may be present. It is a well-known math-

ematical fact that for an isotropic medium, Vp/Vs ratio can be

related to Poisson’s ratio (n) by nonlinear relationship

v¼ 1

2

Vp=Vsð Þ2�2

Vp=Vsð Þ2�1

" #

The mechanical Poisson’s ratio is defined by the negative

radial strain over the longitudinal strain v¼� er/el. Clearly,

Poisson’s ratio does not have any physical connection with

Vp/Vs, which are the velocity of a P-wave with extensional

and contraction strains and that of an S-wave with shear strains

even in the isotropic case; the analogy with anisotropic case is

even less convincing as there are Vs1, Vs2, and Vp where the

polarization strains are inclined to the propagation direction;

and the mechanical situation is very far from Poisson’s ratio.

Hence, I strongly recommend seismologists and rock and min-

eral physicists to report the measurable Vp/Vs1 and Vp/Vs2

ratios of anisotropic media rather than Poisson’s ratio derived

from a calculation of dubious physical significance.


The velocity at which energy propagates in a homogenous

anisotropic elastic medium is defined as the average power

flow density divided by average total energy density (e.g.,

Auld, 1990) and can be calculated from the phase velocity

using the following relationship given by Federov (1968)

Vie ¼Cijklpjplnk=rV

The phase and energy velocities are related by a vector

equation Vie �ni¼Vi. It is apparent from this relationship that

Ve is not in general parallel to propagation direction (n) and

has a magnitude equal or greater to the phase velocity (V¼o/k). The propagation of waves in real materials occurs as

packets of waves typically having a finite band of frequencies.

The propagation velocity of wave packet is called the group

velocity, and this is defined for plane waves of given finite-

frequency range as

Vig ¼ @w=@kið Þ

The group velocity is in general different to the phase veloc-

ity except along certain symmetry directions. In lossless aniso-

tropic elastic media, the group and energy velocities are

identical (e.g., Auld, 1990); hence, it is not necessary to evaluate

the differential angular frequency versus wave vector to obtain

the group velocity as Vg¼Ve. The group velocity has direct

measurable physical meaning that is not apparent for the

energy or phase velocities.

Various types of plot have been used to illustrate the varia-

tion of velocity with direction in crystals. Velocities measured

by the Brillouin spectroscopy are displayed using graphs of

velocity as a function of propagation directions used in the

experiments. The phase velocities Vp, VSH, and VSV of stishovite

are shown in Figure 3, using the elastic constants from

Weidner et al. (1982); although this type of plot may be useful

for displaying the experimental results, it does not convey the

(2015), vol. 2, pp. 487-538



symmetry of the crystal. In crystal acoustics, the phase velocity

and slowness surfaces have traditionally illustrated the anisot-

ropy of elastic wave velocity in crystals as a function of the

propagation direction (n) and plots of the wave front (ray or

group) surface given by tracing the extremity of the energy

velocity vector defined earlier. The normal to the slowness

surface has the special property of being parallel to the energy

velocity vector. The normal to the wave front surface has the

special property of being parallel to the propagation (n) and

wave vector (k). We can illustrate these polar reciprocal prop-

erties using the elastic constants of the hcp e-phase of iron,

which is considered to be the major constituent of the inner

core, determined by Mao et al. (1998) at high pressure

(Figure 4). Notice that the twofold symmetry along the

a 2110� �

axis of hexagonal e-phase is respected by the slowness

and wave front surfaces of the SHwaves. The wave front surface

can be regarded as a recording after one second of the propa-

gation from a spherical point source at the center of the

0.1 s km −1

Slowness surface

Ve, enevector

n

SH-wave surfaces of e-phase iron

(2110)

(011

(0001)

Figure 4 The polar reciprocal relation between the slowness and the waveconstants determined by Mao et al. (1998). The normal to the slowness surfacthe propagation direction (parallel to the wave vector). Note the twofold sym

P

SH

SV

5 km s−1

Min. P =10.28 Max. P =12.16 Min. SH =5.31 Max. SH =8.39 Min. SV =7.66 Max. SV = 7.66

0.1 s km−1

Min. P = 0.08 Min. SH =0.12 Min. SV = 0.13

P

SH

SV

SlownesPhase velocity surface

(100)

(010)

(001)(100)

(0

(0

Stishovite surfac

Figure 5 The three surfaces used to characterize acoustic properties, the pstishovite. Note fourfold symmetry of the surfaces and the cusps on the SH wWeidner et al. (1982) at ambient conditions.


diagram. The wave front is a surface that separates the dis-

turbed from the undisturbed regions. Anisotropic media have

velocities that vary with direction and hence phase velocity and

slowness surfaces with concave and convex undulations in

three dimensions. The undulations are not sharp as velocities

and slowness change slowly with orientation. In contrast, the

wave surface can have sharp changes in direction, called cusps

or folded wave surfaces in crystal physics (e.g., Musgrave,

2003) and triplications or caustics in seismology (e.g.,

Vavrycuk, 2003), particularly for S-waves, which correspond

in orientation to undulations in the phase velocity and slow-

ness surfaces. The high-pressure form of SiO2 called stishovite

illustrates the various facets of the phase velocity, slowness,

and wave front surfaces in a highly anisotropic mineral

(Figure 5). Velocity clearly varies strongly with propagation

direction; in the case of stishovite, the variation for SH is very

important, whereas SV is constant in the (001) plane. Stisho-

vite has tetragonal symmetry, and hence, the c-axis has fourfold

5 km s−1

Wavefront surface

rgy (group)n, progation

direction(wave vector)

Ve

at a pressure of 211 GPa

0)

(2110)

(0110)

(0001)

front surfaces in hexagonal e-phase iron at 211 GPa using the elastice is the energy vector (brown), and the normal to the wave front surface ismetry on the surfaces.

Max. P =0.10 Max. SH =0.19 Max. SV = 0.13

5 km s−1

Min. P =10.28 Max. P =12.16 Min. SH =5.31 Max. SH =9.40 Min. SV = 7.66 Max. SV = 7.66

P

SH

SV

s surface Wavefront surface

10)

01)(100)

(010)

(001)

es in the (001) plane

hase velocity, and slowness and wave front surfaces for tetragonalave front surface. The elastic constants of stishovite were measured by

, (2015), vol. 2, pp. 487-538



symmetry that can clearly be identified in the various surfaces

in (001) plane. There are orientations where the SH and SV

surfaces intersect, and hence, there is no shear wave splitting

(S-wave birefringence) as both S-waves have the same velocity.

The phase velocity and slowness surfaces have smooth changes

in orientation corresponding to gradual changes in velocity. In

contrast, along the a[100] and b[010] directions, the SH wave

Wavefront cusps on SH-waves in Stishovite in the (001) plane

(100)

(010)

(001)(100)

A

B�

CB

C�

A�

Slowness Wavefront

S

Figure 6 Cusps on the wave front surface of tetragonal stishovite andits relation to the slowness surface in the [100] direction. Thepropagation directions of the wave front are marked by arrows every 10�.See the text for detailed discussion.

e-phase iron (hexagonal) surfac

5 km s−1

5 km s−1

0.1 s km−1

0.1 s km−1

SlownesPhase velocity surface

P SHSV

P

SH

SV

P

SH

SV

10°20°30°40°

50°60°

(000

1) P

lane

(0001) (000

(2110) (211

(0001) (000

(011(0110)

(2110)

(211

0) P

lane

Polarization Energy vector

Polarization Energy vector

Figure 7 Velocity surfaces of e-phase of iron in the second-order prism andisotropic structure. Note the perfectly isotropic (circular) velocity surfaces invelocity surfaces, and for the basal plane, the polarizations for P are normal tS, they are normal and vertical for SV and tangential and horizontal for SH asnormal to the slowness and wave front surfaces in the basal plane.


front has sharp variations in orientation called cusps. The

cusps on the SH wave front are shown in more detail in

Figure 6, where the cusps on the wave front are clearly related

to minima of the slowness (or maxima on the phase velocity)

surface. The propagation of SH in the a[100] direction is

instructive; if one considers seismometer at the point S, then

seismometer will record first the arrival of wave front AA0, thenBB0, and finally CC0. The parabolic curved nature of the cusp

AA0 is also at the origin of the word caustic to describe this

phenomenon by analogy with the convergent rays in optics,

whereas the word triplication evokes the arrival of the three

wave fronts. Although we are dealing with homogeneous

anisotropic medium, a single crystal of stishovite, the seis-

mometer will record three arrivals for SH, plus of course SV

and P, giving a total of five arrivals for a single mechanical

disturbance. Media with tetragonal elastic symmetry are not

very common in seismology, whereas media with hexagonal

(or transverse isotropic) symmetry are very common and have

been postulated, for example, for the D00 layer above the core–mantle boundary (CMB) (e.g., Kendall, 2000). The three sur-

faces of the hexagonal e-phase of iron are shown in Figure 7 in

the a 2110� �

and c(0001) planes, which are, respectively, the

perpendicular and parallel to the elastic symmetry axis (c-axis

or X3) of transverse isotropic elastic symmetry. I will consider

es in the (100) and (001) planes

5 km s−1

5 km s−1

s surface Wavefront surface

PSHSV

P

SH

SV

1) (0001)

(0110)

0) (2110)

1) (0001)

(0110)0)

Propagation vector

Propagation vector

basal plane illustrating the anisotropy in a hexagonal or transversethe base plane. The polarizations are marked on the phaseo the surface and parallel to the propagation direction, whereas forin an isotropic medium. The energy vector and propagation direction are

(2015), vol. 2, pp. 487-538



first the wave properties in the a 2110� �

plane. In this plane,

the maximum P-wave velocity is 45� from the c-axis and min-

imum parallel to the a- and c-axes. The maximum SH velocity

is parallel to the c-axis and the minimum parallel to b-axis,

whereas the SV velocity has a maximum parallel to the a- and

c-axes and a minimum 45� to the c-axis. The polarization

direction of P-waves is not perpendicular to the phase velocity

surface in general and not parallel to the propagation direction,

except along the symmetry directions m 0110� �

and c[0001]

axes. The SH polarizations are all normal to the c-axis, and

hence, they appear as points in the a 2110� �

plane. The SV

polarizations are inclined to the a-axis, and hence, they appear

as lines of variable length depending on their orientation. The

SH and SV velocity surfaces intersect parallel to the c-axis and at

60� from the c-axis, where they have the same velocity. The

minima in the SV slowness surface along the a- and c-axes

correspond to the cusps seen on the wave front surface. The

wave properties in the (0001) plane are completely different as

both the P- and S-waves display a single velocity, hence the

name transverse isotropy as the velocities do not vary with

direction in this plane perpendicular (transverse) to the unique

elastic symmetry axis (c-axis or X3). The isotropic nature of this

plane is also shown by the polarizations of the P-waves, which

are normal to the phase velocity surface and parallel to the

propagation direction, as in the isotropic case. Similarly for the

S-waves, the polarization directions for SV are parallel and

those of SH are normal to the symmetry axis, and both are

perpendicular to the propagation direction.

6.67 6.29

11.36

6.67 6.29

11.36

6.80 6.80

11.07

6.675.22

11.03

6.00

5.29

11.84

6.48

5.72

11.65

5.22 6.80

11.03

6.67 6.29

11.36

6.8

6.6

6.

11.36

5.53

5.89

11.54

5.89 5.35

11.81

5.22

6.80 11.03

6.00

5.29

11.84

5.35

__

ε-Phase iron (hexag

(2110)

(0001)

6.80

Figure 8 A 3-D illustration of the propagation direction (black), P-wave polairon at 211 GPa. The sphere is marked with grid at 10� intervals. The SH wavpattern. The P-wave polarization is not in general parallel, and hence, the S-w

illustrated for the 1213h i

direction. However, along symmetry directions, su

polarizations are perpendicular to the propagation.


The illustrations used so far are only two-dimensional sec-

tions of the anisotropic wave properties. Ideally, we would like

to see the three-dimensional form of the velocity surfaces and

polarizations. A three-dimensional plot of the P, SH, and SV

velocities in Figure 8 shows the geometric relation of the

polarizations to the crystallographic axes but is too compli-

cated to see the variation in velocities; only a few directions

have been plotted for clarity. A more practical representation

that is directly related to spherical plot in Figure 8 is the pole

figure plot of contoured and shaded velocities with polariza-

tions shown in Figure 9. The circular nature of the velocity and

polarization around the sixfold symmetry axis c[0001] is

immediately apparent. The maximum shear wave splitting

and SV velocity are in the basal (0001) plane. From the plot,

we can see that e-phase of iron is very anisotropic at the

experimental conditions of Mao et al. (1998) with a P-wave

anisotropy of 7.1%. The shear wave anisotropy has a maxi-

mum of 26.3%, because in this transverse isotropic structure,

SH has a minimum and SV has a maximum velocity in the

basal plane. At first sight, the pole figure plot of polarizations

appears complex. To illustrate the representation of S-wave

polarizations on a pole figure, I have drawn a single propaga-

tion direction in Figure 10. I have chosen stishovite as my

example mineral because it is very anisotropic, and hence, the

angles between the polarizations are clearly not parallel (qP) or

perpendicular (qS1, qS2) to the propagation direction.

It is important to note that for seismic and laboratory

ultrasonic applications, one should use adiabatic elastic

6.67

6.29

11.36

5.22

6.80 11.03

6.40

5.57

11.72

6.29

11.36

5.89

5.35

11.81

5.22

0 11.03

7

29

5.22

6.80 11.03

6.00

5.29

11.84

5.89

11.81

6.40

5.57

11.72

SH

SV

_ _

_

(1213)P

onal)

(0110)

Propagationdirection

rization (red), SH wave (green), and SV wave (blue) in hexagonal e-phasee polarizations are organized around the [001] direction in a hexagonalave polarizations are not perpendicular to the propagation, as

ch as 21 10h i

, the P-wave polarization is parallel, and hence, the S-wave

, (2015), vol. 2, pp. 487-538

ε-Phase iron (hexagonal)

Upperhemisphere

11.84

11.03

Vp Contours (km s−1)

Max. velocity = 11.84 Anisotropy = 7.1%

Max.velocity = 6.80Anisotropy = 26.3%

Min. velocity = 11.03

Min. anisotropy = 0.00

Min.velocity = 5.22 Min.velocity = 5.30

11.12

11.28

11.44

11.60

26.29

0.00

AVs Contours (%)

Max. anisotropy = 26.29

3.0

9.0

15.0

21.0

6.80

5.22

VSH Polarization planes

6.80

5.22

VSH Contours (km s−1)

Max.velocity = 6.80Anisotropy = 24.8%

5.60 5.80 6.00 6.20

6.80

5.30

VSV Contours (km s−1)

5.60 5.80 6.00 6.20

6.80

5.30

VSV Polarization planes

(0110)

(2110)

P velocity SH velocity SV velocity

dVs anisotropy SH polarization SV polarization

Figure 9 The pole figure of plot for hexagonal e-phase iron where the circular symmetry around the [0001] axis is clearly visible for velocities, dVsanisotropy, and S-wave polarizations.

8.04

6.69

45°

45°

12.54

8.04

6.69 12.54

X3 = (001)

Upper hemisphere

Lower hemisphere

qP qS2

qS1

Stishovite

X1 = (100)

X2 = (010)

qS1 polarization

Projection plane

qS1

qS1

UH

LH

Figure 10 The pole figure of plot for strongly anisotropic stishovite illustrating the projection of the qS1 polarization onto the equatorial plane ofprojection. On the left-hand side, the upper hemisphere projection down on to the equatorial plane is shown. On the right-hand side, the lowerhemisphere projection up towards the equatorial plane is shown. The velocities and the polarizations for qP, qS1, and qS2 have the same velocities (inkilometer per second) and orientations, respectively, for the propagation direction in the positive (upper hemisphere) or negative (lower hemisphere)sense due to the centrosymmetric property of elasticity. Note the change from an upper hemisphere to a lower hemisphere projection for acentrosymmetric property is achieved by a 180� rotation in azimuth around X3.


Treatise on Geophysics, 2nd edition, (2015), vol. 2, pp. 487-538




constants, as the timescale of elastic deformation is relatively

short compared with timescale of thermal diffusion. Hence,

thermal diffusion is too slow to obtain isothermal conditions.

We can determine the correction necessary to obtain adiabatic

constants from isothermal ones by considering the elastic

strains caused by stresses and temperature changes:

deij ¼ @eij@skl

� �T

dskl+

@eij@T

� �s@T

In this equation given by Nye (1957), the first term in

brackets on the right-hand side is the elastic compliance tensor

at constant temperature (T ), and the second term is the ther-

mal expansion tensor at constant entropy (s). A second equa-

tion is required to define change in entropy due to changes in

stress and temperature:

dS¼ @S

@skl

� �T

dskl +@S

@T

� �s@T

In this equation, the first term in brackets is the piezocaloric

effect and the second term is heat capacity. If we assume that

entropy is constant (dS¼0) in the second equation and sub-

stitute the result in the first equation, we can eliminate dT and

we obtain

deij ¼ @eij@skl

� �T

dskl� @eij@T

� �s

@S

@skl

� �T

dskl=@T

@S

� �e

Dividing the resulting equation by dskl and using the

relation

@eij@T

� �s¼ @S

@skl

� �T

gives the final result

@eij@skl

� �S

� @eij@skl

� �T

¼� @S

@skl

� �s

@ekl@T

� �s

@T

@S

� �s

¼ @eij@T

� �s

@ekl@T

� �s

@T

@S

� �s

In tensor notation, this is the equation for the difference

between adiabatic and isothermal compliance tensors:

SSijkl�STijkl ¼�aijakl T=Ceð Þ¼�aijakl T=rCPð ÞSSijkl ¼ STijkl�aijakl T=rCPð Þ

where the tensor superscript S stands for adiabatic and T for

isothermal, aij is thermal expansion tensor, T is absolute tem-

perature in Kelvin, Ce is heat capacity per unit volume at

constant strain, CP is specific heat capacity at constant

pressure, and r is density. In a similar way, we can derive the

result for stiffness tensors as

CSijkl ¼CT

ijkl + lijlkl T=rCVð Þl¼ aijCTijkl

where CV is specific heat capacity at constant volume. The

adiabatic stiffness tensor will have greater values than isother-

mal stiffness tensor in general. Stiffness tensors determined by

ab initio modeling in static conditions (T¼0 K) are a special

case as there is no difference between adiabatic and isothermal

tensors. At ambient temperature (T¼300 K), the difference

between adiabatic and isothermal tensors is typically less

than 1%, whereas at 3500 K and 48 GPa, it increases to 7%


for Mg-perovskite (Zhang et al., 2013) and is 13% at 6000 K

and 360 GPa for hcp iron in the inner core (Steinle-Neumann

et al., 2001). For the discussion of the effect of pressure on

these tensors, see Chapter 2.08.

2.20.2.3 Measurement of Elastic Constants

Elastic properties can be measured by a various methods,

including mechanical stress–strain, ultrasonic, Brillouin spec-

troscopy, nonhydrostatic radial x-ray diffraction (RXD), x-ray

and neutron inelastic scattering, and shock measurements.

In addition to physical measurements, atomic-scale first-

principles methods can predict elastic properties of crystals

(see review by Karki et al., 2001; also see Chapter 2.08). The

classical mechanical stress–strain measurements of elastic con-

stants are no longer used due to the large errors, and most

compilations of single-crystal elastic constants (e.g., Anderson

and Isaak, 1995; Bass, 1995; Isaak, 2001) are mainly based on

ultrasonic measurements, the traditional technique at ambient

conditions for large specimens. The measurement of the elastic

constants for minerals from the deep Earth using classical

techniques requires large (�1 cm3) gem quality crystals. How-

ever, many minerals of the deep Earth are not stable, or meta-

stable, at ambient conditions and no large gem quality crystals

are available. Furthermore, the main applications of elastic

constants are for the interpretation of seismological data at

the high-pressure and high-temperature conditions of the

Earth’s mantle and core. Although ultrasonic techniques are

still widely used, new methods are constantly being developed

and refined for measurements at higher pressures and temper-

atures ( Jacobsen et al., 2005; Li et al., 2004; Liebermann and

Li, 1998).

Ultrasonic measurements require a mechanical contact

between the transducers that produce and detect the ultrasonic

signal and the sample. For experiments at high pressure and

temperature, the contact is generally made via the high-

pressure pistons and a high-temperature ceramic buffer rod

or directly with the diamond anvils. Various corrections are

necessary to take into account the ray paths through the pis-

tons and buffer rods and transducer-bond phase shift effects.

The technique most commonly used is based on the ultrasonic

interferometry method introduced by Jackson and Niesler

(1982) to obtain accurate pressure derivatives of single-crystal

MgO to 3 GPa in a piston cylinder apparatus. The use of

ultrasonic interferometry in conjunction with synchrotron

x-radiation in multianvil devices permits a more accurate mea-

surement of elastic constants at simultaneous high pressure

and temperature (Li et al., 2004), as the sample length can be

directly measured by x-radiography in situ, thereby reducing

uncertainties in velocity measurements. A new gigahertz ultra-

sonic interferometer has been recently developed for the dia-

mond anvil cell (DAC; Jacobsen et al., 2005). The gigahertz

frequency reduces the wavelength in minerals to a few

micrometers, which allows the determination of velocity and

the elastic constants for samples of a few tens of micrometers

in thickness. Another ultrasonic technique called resonant

ultrasound spectroscopy has been used to study the elastic

constants of minerals with high precision to very high temper-

atures at ambient pressure (e.g., Isaak, 1992; Isaak et al., 2005).

, (2015), vol. 2, pp. 487-538



Brillouin scattering spectroscopy has become an established

tool for measuring elastic constants since the introduction of

laser sources and multipass Fabry–Perot interferometers (e.g.,

Vacher and Boyer, 1972). Unlike the ultrasonic techniques that

usually measure the velocity in relatively few crystallographic

directions, and where possible in pure mode directions, using

Brillouin scattering, the velocities can be measured along many

propagation directions and the numerous velocities are inverted

to obtain a least-squares determination of the elastic constants

(Weidner and Carleton, 1977), and hence, it is very suitable for

the measurement of low-symmetry crystals. Brillouin scattering

has several advantages formeasurements of transparentminerals

at high pressure and temperature in a DAC, it only requires a

small sample, no physical contact with the sample is required,

and the Brillouin peaks increase with temperature (Sinogeikin

et al., 2005). The elastic constants ofMgOhave been determined

to high pressure (55 GPa, Zha et al., 2000) and high temperature

(1500 K, Sinogeikin et al., 2005) using Brillouin scattering.

A related technique using laser-induced phonon spectroscopy

in the form of impulsive scattering in a DAC has been used

to measure elastic constants of mantle minerals to 20 GPa (e.g.,

Abramson et al., 1977; Chai et al., 1977a,b).

Several x-ray and neutron diffraction and scattering tech-

niques (e.g., Fiquet et al., 2004) have recently been developed

to explore elastic behavior at extreme pressures (>100 GPa) in

diamond anvils. Experiments using radial RXD of polycrystal-

line samples under nonhydrostatic stress have been used to

estimate the single-crystal elastic constants by using the mea-

sured lattice strains and CPO combined with model polycrys-

talline stress distribution (e.g., Reuss uniform stress field) (e.g.,

Mao et al., 1998, 2008a,b,c; Merkel et al., 2005, 2006b). The

uncertainties in the absolute value of the elastic constants may

be on the order of 10–20%, and model-dependent stress or

strain distributions may be limited by the presence of elastic

and plastic strain in some cases; however, this technique pro-

vides valuable information at extreme pressures. Inelastic x-ray

scattering (IXS) has provided volume-averaged P-wave veloci-

ties of hcp iron as a function of pressures to 112 GPa and

temperatures up to 1100 K (Antonangeli et al., 2004, 2010,

2012; Fiquet et al., 2001). Using knowledge of the CPO com-

bined with different scattering geometries, Antonangeli et al.

(2004) had determined the P-wave velocities in two directions

and C11 elastic constant of hcp iron. The nuclear-resonant IXS

technique provides a direct probe of the phonon density of

states. By the integration of the measured phonon density of

states, the elastic and thermodynamic parameters are obtained,

and when combined with a thermal equation of state (EOS),

the P- and S-wave velocities of hcp iron have been determined

to 73 GPa and 1700 K in a laser-heated DAC by Lin et al.

(2005). However, it has only recently been realized that RXD

and IXS elastic constant or velocity determinations can be

compromised by nonhydrostatic stresses in DACs and the

elastic lattice strain equation is violated by the presence of

plastic strain caused by differential stresses above the plastic

yield stress of the crystals in some cases (Antonangeli et al.,

2006; Mao et al., 2008a,b,c; Merkel et al., 2006b, 2009).

2.20.2.4 Effective Elastic Constants for CrystallineAggregates

The calculation of the physical properties from microstructural

information (crystal orientation, volume fraction, grain shape,


etc.) is important for upper mantle rocks because it gives

insight into the role of microstructure in determining the

bulk properties and it is also important for synthetic aggregates

experimentally deformed at simulated conditions of the Earth’s

interior. A calculation can be made for the in situ state at high

temperature and pressure of the deep Earth for samples where

the microstructure has been changed by subsequent chemical

alteration (e.g., the transformation from olivine to serpentine)

or mechanically induced changes (e.g., fractures created by

decompression). The in situ temperatures and pressures can

be simulated using the appropriate single-crystal derivatives.

Additional features not necessarily preserved in the recovered

microstructure, such as the presence of fluids (e.g., magma),

can be modeled (e.g., Blackman and Kendall, 1997; Mainprice,

1997; Williams and Garnero, 1996). Finally, the effect of phase

change on the physical properties can also be modeled using

these methods (e.g., Mainprice et al., 1990). Modeling is essen-

tial for anisotropic properties as experimental measurements in

many directions necessary to fully characterize anisotropy are

not currently feasible for the majority of the temperature and

pressure conditions found in the deep Earth.

In the following, we will only discuss the elastic properties

needed for seismic velocities, but the methods apply to all

tensorial properties where the bulk property is governed by the

volume fraction of the constituent minerals. Many properties of

geophysical interest are of this type, for example, thermal con-

ductivity, thermal expansion, elasticity, and seismic velocities.

However, these methods do not apply to properties determined

by the connectivity of a phase, such as the electrical conductivity

of rocks with conductive films on the grain boundaries (e.g.,

carbon). We will assume the sample may be microscopically

heterogeneous due to grain size, shape, orientation, or phase

distribution but will be considered macroscopically uniform.

The complete structural details of the sample are in general

never known, but a ‘statistically uniform’ sample contains

many regions, which are compositionally and structurally

similar, each fairly representative of the entire sample. The

local stress and strain fields at every point r in a linear elastic

polycrystal are completely determined by Hooke’s law as

follows:

sij rð Þ¼Cijkl rð Þ ekl rð Þwhere sij(r) is the stress tensor, Cijkl(r) is the elastic stiffness

tensor, and ekl(r) is the strain tensor at point r. The evaluation

of the effective constants of a polycrystal would be the sum-

mation of all components as a function of position, if we know

the spatial functions of stress and strain. The average stress hsiand strain hei of a statistically uniform sample are linked by an

effective macroscopic modulus C* that obeys Hooke’s law of

linear elasticity:

C*¼ sh i eh i�1

where eh i¼ 1V

ðe rð Þdr and < s>¼ 1

V

ðs rð Þdr where V is the

volume and the notation hi denotes an ensemble average.

The stress s(r) and strain e(r) distribution in a real polycrystal

varies discontinuously at the surface of grains. By replacing the

real polycrystal with a ‘statistically uniform’ sample, we are

assuming that s(r) and strain e(r) are varying slowly and

continuously with position r.

A number of methods are available for determining the

effective macroscopic effective modulus of an aggregate.

(2015), vol. 2, pp. 487-538



We will briefly present these methods, which try to take into

account an increasing amount of microstructural information,

which of course results in increasing theoretical complexity but

yields estimates, which are closer to experimental values. The

methods can be classified by using the concept of the order of

the statistical probability functions used to quantitatively

describe the microstructure (Kr€oner, 1978). A zero-order

bound is given when one has no statistical information of

the microstructure of the polycrystal and, for example, we do

not know the orientation of the component crystals, and in

this case, we have to use the single-crystal properties. The

maximum and minimum of the single-crystal property are the

zero-order bounds. The simplest and best-known averaging

techniques for obtaining estimates of the effective elastic

constants of polycrystals are the Voigt (1928) and Reuss

(1929) averages. These averages only use the volume fraction

of each phase, the orientation, and the elastic constants of the

single crystals or grains. In terms of statistical probability func-

tions, these are first-order bounds as only the first-order correla-

tion function is used, which is the volume fraction. Note that no

information about the shape or position of neighboring grains is

used. The Voigt average is found by simply assuming that the

strain field is everywhere constant (i.e., e(r) is independent of r).The strain at every position is set equal to the macroscopic strain

of the sample. C* is then estimated by a volume average of local

stiffnesses C(gi) with orientation gi and volume fraction Vi:

C*CVoigt ¼X

iViC gið Þ

h iReuss average is found by assuming that the stress field is

everywhere constant. The stress at every position is set equal to

the macroscopic stress of the sample. C* or S* is then estimated

by the volume average of local compliances S(gi):

C*CReuss ¼X

iVi S gið Þ

h i�1

S* SReuss ¼X

iVi S gið Þ

h iCVoigt 6¼CReuss and CVoigt 6¼ SReuss

� ��1

These two estimates are not equal for anisotropic solids

with the Voigt being an upper bound and the Reuss a lower

bound. A physical estimate of the moduli should lie between

the Voigt and the Reuss average bounds as the stress and

strain distributions are expected to be somewhere between

uniform strain (the Voigt bound) and uniform stress (the

Reuss bound). Hill (1952) observed that arithmetic mean

(and the geometric mean) of the Voigt and Reuss bounds,

sometimes called the Hill or Voigt–Reuss–Hill (VRH) average,

is often close to experimental values. The VRH average has no

theoretical justification. As it is much easier to calculate the

arithmetic mean of the Voigt and Reuss elastic tensors, all

authors have tended to apply the Hill average as an arithmetic

mean. In Earth sciences, the Voigt, Reuss, and Hill averages

have been widely used for averages of oriented polyphase

rocks (e.g., Crosson and Lin, 1971). Although the Voigt and

Reuss bounds are often far apart for anisotropic materials,

they still provide the limits within which the experimental

data should be found.

Several authors have searched for a geometric mean of

oriented polycrystals using the exponent of the average of the

natural logarithm of the eigenvalues of the stiffness matrix

(Matthies and Humbert, 1993). Their choice of this averaging

procedure was guided by the fact that the ensemble average


elastic stiffness hCi should equal the inverse of the ensemble

average elastic compliances hSi�1, which is not true, for exam-

ple, of the Voigt and Reuss estimates. A method of determining

the geometric mean for arbitrary orientation distributions has

been developed (Matthies and Humbert, 1993). The method

derives from the fact that a stable elastic solid must have an

elastic strain energy that is positive. It follows from this that the

eigenvalues of the elastic matrix must all be positive. Compar-

ison between the Voigt, Reuss, and Hill and the self-consistent

estimates shows that the geometric mean provides estimates

very close to the self-consistent method but at considerably

reduced computational complexity (Matthies and Humbert,

1993). The condition that the macroscopic polycrystal elastic

stiffness hCi must equal the inverse of the aggregate elastic

compliance hSi�1 would appear to be a powerful physical

constraint on the averaging method (Matthies and Humbert,

1993). However, the arithmetic (Hill) and geometric means

are very similar (Mainprice and Humbert, 1994), which tends

to suggest that they are just mean estimates with no additional

physical significance.

The second set of methods uses additional information

on the microstructure to take into account the mechanical

interaction between the elastic elements of the microstruc-

ture. Mechanical interaction will be very important for rocks

containing components of very different elastic moduli, such

as solids, liquids, gases, and voids. The most important

approach in this area is the ‘self-consistent’ (SC) method

(e.g., Hill, 1965). The SC method was introduced for mate-

rials with a high concentration of inclusions where the inter-

action between inclusions is significant. In the SC method, an

initial estimate of the anisotropic homogeneous background

medium of the polycrystal is calculated using the traditional

volume-averaging method (e.g., Voigt). All the elastic ele-

ments (e.g., grains and voids) are inserted into the back-

ground medium using Eshelby’s (1957) solution for a single

ellipsoidal inclusion in an infinite matrix. The elastic moduli

of the ensemble, inclusion, and background medium are used

as the ‘new’ background medium for the next inclusion. The

procedure is repeated for all inclusions and repeated in an

iterative manner for the polycrystal until a convergent solu-

tion is found. The interaction is notionally taken into account

by the evolution of the background medium that contains

information about the inclusions, albeit in a homogenous

form. As the inclusion can have an ellipsoidal shape, an

additional microstructural parameter is taken into account

by this type of model. A hybrid self-consistent method that

combines elements of the geometric mean for the calculation

of the elastic properties of multiphase rock samples has been

introduced by Matthies (2010, 2012).

Several people (e.g., Bruner, 1976) have remarked that the

SC progressively overestimates the interaction with increasing

concentration. They proposed an alternative differential effec-

tive medium (DEM) method in which the inclusion concen-

tration is increased in small steps with a reevaluation of the

elastic constants of the aggregate at each increment. This

scheme allows the potential energy of the medium to vary

slowly with inclusion concentration (Bruner, 1976). Since

the addition of inclusions to the background material is

made in very small increments, one can consider the concen-

tration step to be very dilute with respect to the current effective

medium. It follows that the effective interaction between

inclusions can be considered negligible and we can use the

, (2015), vol. 2, pp. 487-538



inclusion theory of Eshelby (1957) to take into account the

interaction. In contrast, the SC uses Eshelby’s theory plus an

iterative evaluation of the background medium to take into

account the interaction. Mainprice (1997) had compared the

results of SC and DEM for anisotropic oceanic crustal and

mantle rocks containing melt inclusions and found the results

to be very similar for melt fractions of less than 30%. At higher

melt fractions, the SC exhibits a threshold value around 60%

melt, whereas the DEM varies smoothly up a 100% melt. The

presence of a threshold in the SC calculations is due to the

specific way that the interaction is taken into account. The

estimates of both methods are likely to give relatively poor

results at high fractions of a phase with strong elastic contrast

with the other constituents as other phenomena, such as

mechanical localization related to the percolation threshold,

are likely to occur.

The third set of methods uses higher-order statistical corre-

lation functions to take into account the first- or higher-order

neighbor relations of the various microstructural elements. The

factors that need to be statistically described are the elastic

constants (determined by composition), orientation, and rela-

tive position of an element. If the element is considered to be

small relative to grain size, then grain shape and the heteroge-

neity can be accounted for the relative position correlation

function. Nearest neighbors can be taken into account using

two-point correlation function, which is also called an auto-

correlation function by some authors. If we use the ‘statistically

uniform’ sample introduced earlier, we are effectively assuming

that all the correlation functions used to describe the micro-

structure up to order infinity are statistically isotropic; this is

clearly a very strong assumption. In the special case where all

the correlation functions up to order infinity are defined,

Kr€oner (1978) had shown that the upper and lower bounds

converge for the self-consistent method so that Csc¼(Ssc)�1.

The statistical continuum approach is the most complete

description and has been extensively used for model calcula-

tions (e.g., Beran et al., 1996; Mason and Adams, 1999). Until

recently, it has been considered too involved for practical

application. With the advent of automated determination of

crystal orientation and positional mapping using electron

backscattered diffraction (EBSD) in the scanning electron

microscope (Adams et al., 1993), digital microstructural

maps are now available for the determination of statistical

correlation functions. This approach provides the best possible

estimate of the elastic properties but at the expense of consid-

erably increased computational complexity.

The fact that there is a wide separation in the Voigt and Reuss

bounds for anisotropic materials is caused by the fact that the

microstructure is not fully described by such methods. How-

ever, despite the fact that these methods do not take into

account such basic information as the position or the shape of

the grains, several studies have shown that the Voigt average or

the Hill average is within 5–10% of experimental values for

low-porosity rocks free of fluids. For example, Barruol and Kern

(1996) showed for several anisotropic lower crust and upper

mantle rocks from the Ivrea zone in Italy that the Voigt average

is within 5% of the experimentally measured velocity. Finally,

from a practical point of view, a free and open-source MTEX

toolbox is now available (https://code.google.com/p/mtex/)

for effective medium calculations, such as Voigt, Reuss, and


Hill averages for second-, third-, and fourth-rank tensors, and,

plotting the results, a wide variety of graphic formats. A detailed

description of the application of MTEX to elastic and seismic

properties can be found in Mainprice et al. (2011).

2.20.2.5 Seismic Properties of Polycrystalline Aggregatesat High Pressure and Temperature

The orientation of crystals in a polycrystal can be measured by

volume diffraction techniques (e.g., x-ray or neutron diffrac-

tion) or individual orientation measurements (e.g., U-stage

and optical microscope, electron channeling, or EBSD). In

addition, numerical simulations of polycrystalline plasticity

also produce populations of crystal orientations at mantle

conditions (e.g., Tommasi et al., 2004). An orientation, often

given the letter g, of a grain or crystal in sample coordinates can

be described by the rotation matrix between crystal and sample

coordinates. In practice, it is convenient to describe the rota-

tion by a triplet of Euler angles, for example, g¼(’1 f ’2)

used by Bunge (1982). One should be aware that there are

many different definitions of Euler angles that are used in the

physical sciences. The orientation distribution function (ODF)

f(g) is defined as the volume fraction of orientations with an

orientation in the interval between g and g+dg in a space

containing all possible orientations given by

DV=V ¼ðf gð Þdg

where DV/V is the volume fraction of crystals with orientation

g, f(g) is the texture function, and dg¼1/8p2 sin f d’1dfd’2

is the volume of the region of integration in orientation space.

To calculate the seismic properties of a polycrystal, one

must evaluate the elastic properties of the aggregate. In the

case of an aggregate with a crystallographic fabric, the anisot-

ropy of the elastic properties of the single crystal must be taken

into account. For each orientation g, the single-crystal proper-

ties have to be rotated into the specimen coordinate frame

using the orientation or rotation matrix gij:

Cijkl gð Þ¼ gip:gjq:gkr :glt Cpqrt goð Þ

where Cijkl(g) is the elastic property in sample coordinates,

gij¼g(’1 f ’2) is the measured orientation in sample co-

ordinates, and Cpqrt(go) is the elastic property in crystal

coordinates.

The elastic properties of the polycrystal may be calculated

by integration over all possible orientations of the ODF. Bunge

(1982) had shown that integration is given as

Cijkl

m¼

ðCijklm gð Þf gð Þdg

where hCijklim is the elastic properties of the aggregate of min-

eral m. Alternatively, it may be determined by simple summa-

tion of individual orientation measurements:

Cijkl

m¼

XCijklm gð Þ:v gð Þ

where v(g) is the volume fraction of the grain in orientation g.

For example, the Voigt average of the rock formmineral phases

of volume fraction v(m) is given as

(2015), vol. 2, pp. 487-538

https://code.google.com/p/mtex/

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0 200 400 600 800 1000 1200 1400

Cub

ic a

niso

trop

y fa

ctor

(A)

Temperature (°C)

MgO at 8 GPa

A = ((2C44+ C12)/C11) − 1

dCij /dPdT = 0

Figure 11 An illustration of the importance of the cross pressure–temperature derivatives for cubic MgO at 8 GPa pressure. Withincreasing temperature at constant pressure, the anisotropy measuredby the cubic anisotropy factor (A) increases by a factor of 2. Data fromChen et al. (1998).



Cijkl

Voigt¼

Xv mð Þ Cijkl

m

The final step is the calculation of the three seismic phase

velocities by the solution of the Christoffel equation, details of

which are given earlier.

To calculate the elastic constants at pressures and

temperatures, the single-crystal elastic constants are given at

the pressure and temperature of their measurement by using

the following relationship:

Cij PTð Þ¼Cij PoToð Þ + dCij=dP� �

DP +1=2 d2Cij=dP2

� �DP2

+ dCij=dT� �

DT + d2Cij=dPdT� �

DPDT

where Cij(PT ) are the elastic constants at pressure P and tem-

perature T, Cij(PoTo) are the elastic constants at a reference

pressure Po (e.g., 0.1 MPa) and temperature To (e.g., 25 �C),dCij/dP is the first-order pressure derivative, dCij/dT is the first-

order temperature derivative, DP¼P�Po, and DT¼T�To. The

equation is a Maclaurin expansion of the elastic tensor as a

function of pressure and temperature, which is a special case of

a Taylor expansion as the series is developed about the elastic

constants at the reference condition Cij(PoTo). The series only

represent the variation of the Cij in their intervals of pressure

and temperature of convergence, in other words the pressure

and temperature range of the experiments or atomic modeling

calculations used to determine the derivatives. Note that this

equation is not a polynomial and care has to be taken when

using the results of data fitted to polynomials, as, for example,

the second-order derivatives fitted to a polynomial should be

multiplied by two for use in the equation mentioned earlier,

for example, second-order pressure derivatives for MgO given

by Sinogeikin et al., (2001). Also note that this equation is not

a Eulerian finite strain EOS (e.g., Davies, 1974; see succeeding

text), and data fit to such an equation will not have derivatives

compatible with the equation mentioned earlier, for example,

the pressure derivatives of brucite determined by Jiang et al.

(2006). The second-order pressure derivatives d2Cij/dP2 are

available for an increasing number of mantle minerals (e.g.,

olivine, orthopyroxene, garnet, and MgO), and the first-order

temperature derivatives seem to adequately describe the tem-

perature dependence of most minerals, although the second-

order derivatives are also available in a few cases (e.g., garnet,

fayalite, forsterite, and rutile; see Isaak, 2001 for references).

Experimental measurements of the cross pressure–temperature

derivatives d2Cij/dPdT (i.e., the temperature derivative of the

Cij/dP at constant temperature) are still very rare. For example,

despite the fact that MgO (periclase) is a well-studied reference

material for high-pressure studies, the complete set of single-

crystal cross derivatives were measured for the first time by

Chen et al. (1998) to 8 GPa and 1600 K. The effect of the

cross derivatives on the Vp and dVs anisotropy of MgO is

dramatic, and the anisotropy is increased by a factor of 2

when cross derivatives are used (Figure 11). Note that when

a phase transition occurs, then the specific changes in elastic

constants at pressures near the phase transition will have to be

taken into account, for example, the SiO2 polymorphs

(Carpenter, 2006; Cordier et al., 2004a; Karki et al., 1997c).

The seismic velocities also depend on the density of the min-

erals at pressure and temperature, which can be calculated

using an appropriate EOS (Knittle, 1995). The Murnaghan

EOS derived from finite strain is sufficiently accurate at


moderate compressions (Knittle, 1995) of the upper mantle

and leads to the following expression for density as a function

of pressure:

r Pð Þ¼ ro 1 + K 0=Kð Þ: P�Poð Þð Þ1=K 0

where K is the bulk modulus, K0 ¼dK/dP is the pressure deriv-

ative of K, and ro is the density at reference pressure Po and

temperature To. For temperature, the density varies as

r Tð Þ¼ ro 1�ðav Tð ÞdT

� � ro 1�aav T�Toð Þ½

where av(T )¼1/V(@V/@T ) is the volume thermal expansion

coefficient as a function of temperature and aav is an average

value of thermal expansion that is constant over the tempera-

ture range (Fei, 1995). According to Watt (1988), an error of

less than 0.4% on the P and S velocity results from using aav to1100 K for MgO. For temperatures and pressures of the mantle,

the density is described for this chapter by

r P, Tð Þ¼ ro 1 + K 0=Kð Þ: P�Poð Þð Þ1=K 01�aav T�Toð Þ½

n oAn alternative approach for the extrapolation elastic con-

stants to very high pressures is Eulerian finite strain theory

(e.g., Davies, 1974). The theory is based on a Maclaurin expan-

sion of the free energy in terms of Eulerian finite volumetric

strain. For example, Karki et al. (2001) reformulated Davies’s

equations for the elastic constants in terms of finite volumetric

strain (f ) as

Cijkl fð Þ¼ 1+2fð Þ7=2 Coijkl + b1f + 1⁄2 b2f + � � �� PDijkl

where

f ¼ 1⁄2 Vo=Vð Þ2=3�1h i

b1 ¼ 3Ko dCoijkl=dP� ��5Coijkl

b2 ¼ 9K2o d2Coijkl=dP

2� �

+3 Ko=dPð Þ b1 + 7Coijkl

� ��16b1�49Coijkl

Dijkl ¼�dijdkl�dikdjl�dildjk

, (2015), vol. 2, pp. 487-538



This formulation in terms of finite strain can be very useful

for finding the elastic constants a given pressure from series of

ab initio calculations at different pressures, for example, room

pressure. With the advent of practical computational methods

for applying first-principles (ab initio) methods to calculation

of elastic constants of minerals at extreme pressures reduces the

Eulerian finite strain theory to a descriptive tool, if tensors

are available at the appropriate pressure (and temperature)

conditions. The simple Maclaurin series expansions given

earlier for pressure and temperature are a compact way of

describing the variation of the elastic tenors in experimenta-

tions at high pressure and temperature. Extrapolation of the

simple Maclaurin series expansions outside the range of exper-

imental (or computational) data is not recommended, as the

formulation is descriptive. The Eulerian finite volumetric strain

formulation has the merit of a physical basis, and hence,

extrapolation beyond the experimental data range may be

undertaken with caution. In practice, applications of the

Eulerian finite volumetric strain formulation have been mainly

limited to high-symmetry crystals (e.g., Li et al., 2006b, cubic

Ca-perovskite, trigonal brucite Jiang et al., 2006) but can

be applied to lower symmetries (e.g., monoclinic chlorite

Mookherjee and Mainprice, 2014). A thermodynamically cor-

rect formulation of the problem intermediate values in terms

of pressure and temperature has been proposed by Stixrude

and Lithgow-Bertelloni (2005a,b).

2.20.2.6 Anisotropy of Minerals in the Earth’s Mantleand Core

To understand the anisotropic seismic behavior of polyphase

rocks in the Earth’s mantle, it is instructive to first consider the

properties of the component single crystals. In this section, I

will emphasis the anisotropy of individual minerals rather

than the magnitude of velocity. The percentage anisotropy

(A) is defined here as A¼200 (Vmaximum�Vminimum)/

(Vmaximum+Vminimum), where the maximum and minimum

are found by exploring a hemisphere of all possible propaga-

tion directions. Note that for P-wave velocities, the anisotropy

is defined by the maximum and minimum velocities in two

different propagation directions, for example, the maximum

A is given by the maximum and minimum Vp in a hemisphere

or for Vp in two specific directions such as the vertical and

horizontal can be used. For S-waves in an anisotropic medium,

there are two orthogonally polarized S-waves with different

velocities for each propagation direction; hence, A can be

defined for each direction. The consideration of the single-

crystal properties is particularly important for the transition

zone (410–660 km) and lower mantle (below 660 km) as the

deformation mechanisms and resulting preferred orientation

of these minerals under the extreme conditions of temperature

and pressure are very poorly documented by experimental

investigations. In choosing the anisotropic single-crystal prop-

erties, where possible, I have included the most recent experi-

mental determinations. A major trend in recent years is the use

of computational modeling to determine the elastic constants

at very high pressures and more recently at high temperatures.

The theoretical modeling gives a first estimate of the pressure

and temperature derivatives in a range not currently accessible

to direct measurement (see review by Karki et al., 2001; also see


Chapter 2.08). Although there is an increasing amount of

single-crystal data available to high temperature or high pres-

sure, no data are available for simultaneous high temperature

and pressure of the Earth’s lower mantle or inner core (see

Figure 1 for the pressure and temperatures).

2.20.2.6.1 Upper mantleThe upper mantle (down to 410 km) is composed of three

anisotropic and volumetrically important phases: olivine,

enstatite (orthopyroxene), and diopside (clinopyroxene). The

other volumetrically important phase is garnet, which is nearly

isotropic and hence not of great significance to our discussion

of anisotropy.

Olivine – A certain number of accurate determinations of

the elastic constants of olivine are now available, which all

agree that the anisotropy of Vp is 25% and maximum anis-

otropy of Vs is 18% at ambient conditions for a mantle com-

position of about Fo90. The first-order temperature derivatives

have been determined between 295 and 1500 K for forsterite

(Isaak et al., 1989a) and olivine (Isaak, 1992). The first- and

second-order pressure derivatives for olivine were first deter-

mined to 3 GPa by Webb (1989). However, a determination to

17 GPa by Zaug et al. (1993) and Abramson et al. (1997) has

shown that the second-order derivative is only necessary for

elastic stiffness modulus C55. The first-order derivatives are in

good agreement between these two studies. The second-order

derivative for C55 has proved to be controversial. Zaug et al.

(1993) were the first to measure nonlinear variations of C55

with pressure, but other studies have not reproduced this

behavior (e.g., for olivine Chen et al., 1996; Zha et al., 1998)

or forsterite Zha et al., 1996). The anisotropy of the olivine

single crystal increases slightly with temperature (+2%) using

the data of Isaak (1992) and reduces slightly with increasing

pressure using the data of Abramson et al. (1997).

Orthopyroxene – The elastic properties of orthopyroxene

(enstatite or bronzite) with a magnesium number (Mg/

Mg+Fe) near the typical upper mantle value of 0.9 have also

been extensively studied. The Vp anisotropy varies between

15.1% (En80 bronzite; Frisillo and Barsch, 1972) and 12.0%

(En100 enstatite; Weidner et al., 1978) and the maximum Vs

anisotropy between 15.1% (En80 bronzite; Webb and Jackson,

1993) and 11.0% (En100 enstatite; Weidner et al., 1978).

Some of the variation in the elastic constants and anisotropy

may be related to composition and structure in the orthopyr-

oxenes (Duffy and Vaughan, 1988). Near-end-member

enstatite containing Ca shows only small changes in the bulk

and shear moduli by 5% and 3% from the pure Mg end-

member (Perrillat et al., 2007). The first-order temperature

derivatives have been determined over a limited range between

298 and 623 K (Frisillo and Barsch, 1972). An extended range

of temperature derivatives has been given by Jackson et al.

(2007) up to 1073 K at ambient pressure. The first- and

second-order pressure derivatives for Enstatite have been deter-

mined up to 12.5 GPa by Chai et al. (1977b). This study

confirms an earlier one of Webb and Jackson to 3 GPa that

showed that the first- and second-order pressure derivatives are

needed to describe the elastic constants at mantle pressures.

The anisotropy of Vp and Vs does not vary significantly with

pressure using the data of Chai et al. (1977a) to 12.5 GPa. The

anisotropy of Vp and Vs does increase by about 3% when

(2015), vol. 2, pp. 487-538



extrapolating to 1000�C using the first-order temperature

derivatives of Frisillo and Barsch (1972).

Clinopyroxene – The elastic constants of clinopyroxene

(diopside) of mantle composition have only been experimen-

tally measured at ambient conditions (Collins and Brown,

1998; Levien et al., 1979); both studies show that Vp anisot-

ropy is 29% and Vs anisotropy is between 20% and 24%. There

are no measured single-crystal pressure derivatives. In one of

the first calculations of the elastic constants of a complex

silicate at high pressure, Matsui and Busing (1984) predicted

the first-order pressure derivatives of diopside from 0 to 5 GPa.

The calculated elastic constants at ambient conditions are in

good agreement with the experimental values, and the pre-

dicted anisotropy for Vp and Vs of 35.4% and 21.0%,

respectively, is also in reasonable agreement. The predicted

bulk modulus of 105 GPa is close to the experimental value

of 108 GPa given by Levien et al. (1979). The pressure deriva-

tive of the bulk modulus 6.2 is slightly lower than the value of

7.8�0.6 given by Bass et al. (1981). Using the elastic constants

of Matsui and Busing (1984), the Vp anisotropy decreases from

35.4% to 27.7% and Vs increases from 21.0% to 25.5% with

increasing pressure from ambient to 5 GPa. A recent ab inito

study of the effect of pressure on the elastic constants of diop-

side and jadeite from 0 to 20 GPa shows a slight decrease in Vp

anisotropy from 23.4% to 17.9% and a slight increase in Vs

anisotropy from 20.1% to 25.2% (Walker, 2012). Isaak and

9.35

7.87

8.20 8.40 8.60 8.80

Diopside

9.65

7.59 Max. velocity = 9.65Anisotropy = 23.8%

Max. velocity = 8.81Anisotropy = 13.7 %





7.8

8.2

8.6

9.0

Max. aniso

Max. aniso

Max. aniso

Olivine

8.81

7.68

8.00

8.20

8.40

Enstatite

(100)

(010)(001)

(100)

(010)(001)

Vp (km s−1)

Figure 12 Single-crystal anisotropic seismic properties of upper mantle mi(monoclinic) at about 220 km (7.1 GPa, 1250 �C).


Ohno (2003) and Isaak et al. (2005) had measured the tem-

perature derivatives of chrome diopside to 1300 K at room

pressure that are notably smaller than other mantle minerals.

The single-crystal seismic properties of olivine, enstatite, and

diopside at 220 km depth are illustrated in Figure 12. Garnet is

nearly isotropic with Vp anisotropy of 0.6% and Vs of 1.3%.

The determination of the elastic constants of diopside with

upper mantle composition (Collins and Brown, 1998) at

ambient pressure agrees closely with values for chrome diop-

side, except for a 3% decrease in the shear modulus (C44).

2.20.2.6.2 Transition zoneOver the last 20 years, a major effort has been made to exper-

imentally determine the phase petrology of the transition zone

and lower mantle. Whereas single crystals of upper mantle

phases are readily available, single crystals of transition zone

and lower mantle for elastic constant determination have to

be grown at high pressure and high temperature. The petrology

of the transition zone is dominated by garnet, majorite, wad-

sleyite, ringwoodite, calcium-rich perovskite, clinopyroxene,

and possibly stishovite.

Majorite – The pure Mg end-member majorite of the

majorite–pyrope garnet solid solution has tetragonal symme-

try and is weakly anisotropic with 1.8% for Vp and 9.1% for Vs

(Pacalo and Weidner, 1997). A study of the majorite–pyrope

system by Heinemann et al. (1997) shows that tetragonal form

26.05

1.02

6.0 9.0 12.0 15.0 18.0 21.0

26.05

1.02

16.01

0.28 tropy = 16.01

tropy = 17.59

tropy = 26.05




4.0 6.0 8.0 10.0 12.0 14.0

16.01

0.28

17.59

0.50

4.0

8.0

12.0

17.59

0.50

dVs (%) Vs1 polarization

nerals olivine (orthorhombic), enstatite (orthorhombic), and diopside

, (2015), vol. 2, pp. 487-538



of majorite is restricted to a composition of less 20% pyrope

and hence is unlikely to exist in the Earth’s transition zone.

Majorite with cubic symmetry is nearly isotropic with Vp

anisotropy of 0.5% and Vs of 1.1%. Pressure derivatives for

majorite and majorite–pyrope have been determined by

Sinogeikin and Bass (2002a) and temperature derivatives by

the same authors (2002b). Cubic majorite has very similar

properties to pyrope garnet (Chai et al., 1997a) as might be

expected. The elastic properties of sodium-rich majorite have

been studied by Reichmann et al. (2002).

Wadsleyite – The elastic constants of Mg2SiO4 wadsleyite

were first determined by Sawamoto et al. (1984), and this early

determination was confirmed by Zha et al. (1997) with a Vp

anisotropy of 16% and Vs of 17%. The (Mg,Fe)2SiO4 wad-

sleyite has slightly lower velocities and higher anisotropies

(Sinogeikin et al., 1998). The first-order pressure derivatives

determined from the data of Zha et al. (1997) to 14 GPa show

that the anisotropy of Mg2SiO4 wadsleyite decreases slightly

with increasing pressure. At pressures corresponding to the

410 km seismic discontinuity (ca. 13.8 GPa), the Vp anisotropy

would be 11.0% and Vs 12.5%.

Ringwoodite – The elastic constants of Mg2SiO4 ringwoo-

dite were first measured by Weidner et al. (1984) and

(Mg,Fe)2SiO4 ringwoodite by Sinogeikin et al. (1998) at ambi-

ent conditions with Vp anisotropy of 3.6 and 4.7% and Vs of

7.9 and 10.3%, respectively. Kiefer et al. (1997) had calculated

the elastic constants of Mg2SiO4 ringwoodite to 30 GPa. Their

constants at ambient conditions give a Vp anisotropy of 2.3%

and Vs of 4.8% very similar to the experimental results of

Weidner et al. (1984). There is a significant variation (5–0%)

of the anisotropy of ringwoodite with pressure 15 GPa (ca.

500 km depth), the Vp anisotropy is 0.4%, and Vs is 0.8%;

hence, ringwoodite is nearly perfectly isotropic at transition

zone pressures. Single-crystal temperature derivatives have

10.15

9.82

9.90

9.95

10.00

10.05

10.10

Ringwoodite

11.36

10.26

10.4

10.6

10.8

11.0

Wadsleyite

(100)

(010)(001)

Vp (km s−1)


Min. velocity = 10.26 Max. anisotr


Min. velocity = 9.82 Max. anisot

Figure 13 Single-crystal anisotropic seismic properties of transition zone mand ringwoodite (cubic) at about 550 km (19.1 GPa, 1520 �C).


been measured for ringwoodite ( Jackson et al., 2000;

Sinogeikin et al., 2001), but none are available for wadsleyite.

Olivine transforms to wadsleyite at about 410 km, and wad-

sleyite transforms to ringwoodite at about 500 km; both trans-

formations result in a decrease in anisotropy with depth. The

gradual transformation of clinopyroxene to majorite between

400 and 475 km would also result in a decrease in anisotropy

with depth. The seismic anisotropy of wadsleyite and ringwoo-

dite is illustrated in Figure 13. An ab initiomolecular dynamics

study by Li et al. (2006a) has shown that ringwoodite is nearly

isotropic at transition zone conditions with P- and S-wave

anisotropies close to 1%; extrapolated experimental values to

a depth of 550 km (19.1 GPa, 1520 �C) suggest that the Vp

anisotropy and Vs anisotropy are 3.3% and 8.2%.

2.20.2.6.3 Lower mantleThe lower mantle is essentially composed of perovskite,

ferropericlase, and possibly minor amount of SiO2 in the

form of stishovite in the top part of the lower mantle (e.g.,

Ringwood, 1991). Ferropericlase is the correct name for

(Mg,Fe)O with small percentage of iron, less than 50% in Mg

site; previously, this mineral was incorrectly called magnesiow-

ustite, which should have more than 50% Fe. It is commonly

assumed that there is about 20% Fe in ferropericlase in the

lower mantle. MgSiO3 may be in the form of perovskite or

possibly ilmenite. The ilmenite-structured MgSiO3 is most

likely to occur at the bottom of the transition zone and the

top of the lower mantle. In addition, perovskite transforms to

postperovskite in the D00 layer, although extract distribution

with depth (or pressure) of the phases will depend on the local

temperature and their iron content.

Perovskite (MgSiO3 and CaSiO3) – The first determination

of the elastic constants of pure MgSiO3 perovskite at ambient

conditions was given by Yeganeh-Haeri et al. (1989). However,

8.22

0.00

1.0

3.0

5.0

7.0

8.22

0.00

13.63

0.15


4.0

6.0

8.0

10.0

13.63

0.15


opy = 13.63

Min. anisotropy = 0.00ropy = 8.22

inerals wadsleyite (orthorhombic) at about 450 km (15.2 GPa, 1450 �C)

(2015), vol. 2, pp. 487-538



this determination has been replaced by a more accurate study

of a better quality crystal (Yeganeh-Haeri, 1994). The 1994

study gives Vp anisotropy of 13.7% and Vs of 33.0%. The

[010] direction has the maximum dVs anisotropy. A new mea-

surement of the elastic constants of MgSiO3 perovskite at

ambient conditions was made by Sinogeikin et al. (2004)

and gives Vp anisotropy of 7.6% and dVs of 15.4%, which

has a very similar velocity distribution to the determination

of Yeganeh-Haeri (1994), but the anisotropy is reduced by a

factor of 2. Karki et al. (1997a) calculated the elastic constants

of MgSiO3 perovskite 140 GPa at 0 K. The calculated constants

are in close agreement with the experimental measurements of

Yeganeh-Haeri (1994) and Sinogeikin et al. (2004). Karki et al.

(1997a) found that significant variations in anisotropy

occurred with increasing pressure, first decreasing to 6% at

20 GPa for Vp and to 8% at 40 GPa for Vs and then increasing

to 12% and 16%, respectively, at 140 GPa. At the 660 km

seismic discontinuity (ca. 23 GPa), the Vp and Vs anisotropies

would be 6.5% and 12.5%, respectively. Recent progress in

finite temperature first-principles methods for elastic constants

has allowed their calculation at lower mantle pressures and

temperatures. Oganov et al. (2001) calculated the elastic con-

stants of Mg-perovskite at two pressures and three tempera-

tures for the lower mantle (Figure 14). More recently,

Wentzcovitch et al. (2004) had calculated the elastic constants

over the complete range of lower mantle conditions and pro-

duced pressure and temperature derivatives. The results for

pure Mg-perovskite from Oganov et al. and Wentzcovitch

et al. agree quite closely for P-wave anisotropy, but Oganov’s

elastic constants give a higher S-wave anisotropy. At ambient

conditions, the results from all studies are very similar with the

Vp anisotropy that is 13.7% and the Vs anisotropy that is

33.0%.When extrapolated along a geotherm using the pressure

and temperature derivatives of Wentzcovitch et al. (2004), the

P- and S-wave anisotropies about the same at 8% at 1000 km

MgO

13.34

12.38

12.50 12.60 12.70 12.80 12.90 13.00 13.10 13.20

Mg-Perovskite

16.07

14.02

14.4

14.8

15.2

15.6

(100)

(010)(001)

Vp (km s−1)

Min. velocity = 12.38Max. velocity = 13.34Anisotropy = 7.5%

Max. an

Min. velocity = 14.02Max. velocity = 16.07Anisotropy = 13.7%

Max. an

Figure 14 Single-crystal anisotropic seismic properties of lower mantle min2000 km (88 GPa, 3227 �C) using the elastic constants determined at high P


depth and again similar anisotropies of 13% at 2500 km depth.

The plasticity of Mg-perovskite has been studied using ab initio

dislocation modeling to determine the critical resolved shear

stresses at lower mantle pressures and viscoplastic self-

consistent (VPSC) modeling (Mainprice et al., 2008a). Using

the VPSC-predicted CPO and ab initio elastic constants of

Oganov et al. (2001), the predicted anisotropy at lower mantle

pressures and temperatures is weak (>3% for P-waves and>2%

for S-waves) and decreases with increasing temperature and

pressure.

The other perovskite structure present in the lower mantle is

CaSiO3 perovskite; recent ab initiomolecular dynamics study by

Li et al. (2006b) has shown that this mineral is nearly isotropic

at lower mantle conditions with P- and S-wave anisotropies

close to 1%. Recent extension of experimental measurements

to in situ lower mantle conditions and more refined ab initio

modeling at finite temperature are starting to challenge previous

results. The measurement of sound velocities at lower mantle

conditions by Murakami et al. (2012) suggests that the lower

mantle is composed of 93%Mg-perovskite. However, a state-of-

the-art ab initio modeling of Mg-perovskite elastic properties

(Zhang et al., 2013) with the current view of a pyrolite mantle

composition for lower mantle provides an alterative view. The

work of Zhang et al. (2013) also allows a better understanding

of the P- and S-wave anisotropies along geotherms at 1500,

2500, and 3500 K. In Figure 15, you can see that for P-wave

anisotropy, there is little effect of temperature and the anisot-

ropy increases with depth. For S-wave, there is strong increase of

anisotropy with temperature and depth. Mg-perovskite may be

playing more important role in the seismic anisotropy of the D00

layer than we have considered so far.

MgSiO3 ilmenite – Experimental measurements byWeidner

and Ito (1985) have shown that MgSiO3 ilmenite of trigonal

symmetry is very anisotropic at ambient conditions with Vp

anisotropy of 21.1% and Vs of 36.4%. Pressure derivatives to

16.25

0.29

4.0 6.0 8.0 10.0 12.0 14.0

16.25

0.29

27.54

0.04

6.0

12.0

18.0

24.0

27.54

0.04


Min. anisotropy = 0.29isotropy = 16.25

Min. anisotropy = 0.04isotropy = 27.54

erals Mg-perovskite (orthorhombic) and MgO (periclase – cubic) at about–T by Oganov et al. (2001) and Karki et al (2000), respectively.

, (2015), vol. 2, pp. 487-538

P- and S-wave anisotropy (%)D

epth

(km

)

6 8 10 12 14 16 18 20500

1000

1500

2000

2500

3000

Vp(%) 1500KVs(%) 1500kVp(%) 2500kVs(%) 2500kVp(%) 3500kVs(%) 3500k

VsVp

3500 K2500 K1500 K

MgSiO3 - perovskite

Figure 15 P- and S-wave seismic anisotropy for MgSiO3 perovskitewith depth in the lower mantle predicted by the ab initio calculations ofZhang et al. (2013).



30 GPa have been obtained by first-principles calculation

(Da Silva et al., 1999) and the anisotropy decreases with

increasing pressure to 9.9% for P-waves and 24.8% for

S-waves at 30 GPa.

Ferropericlase – The other major phase is ferropericlase

(Mg,Fe)O, for which the elastic constants have been determined.

The elastic constants of the pure end-member periclase MgO of

theMgO–FeO solid solution series have beenmeasured to 3 GPa

by Jackson and Niesler (1982). Isaak et al. (1989a) had mea-

sured the temperature derivatives for MgO to 1800 K. Both these

studies indicate a Vp anisotropy of 11.0% and Vs of 21.5% at

ambient conditions. Karki et al. (1997a,b,c) calculated the elastic

constants of MgO to 150 GPa at 0 K. The thermoelasticity of

MgO at lower mantle temperatures and pressures has been

studied by Isaak et al. (1990) and Karki et al. (1999, 2000)

(Figure 16), and more recently Sinogeikin et al. (2004, 2005);

there is good agreement between these studies for the elastic

constants and pressure and temperature derivatives. However,

the theoretical studies do not agree with experimentally mea-

sured cross pressure–temperature derivatives of Chen et al.

(1998). At the present time, only the theoretical studies permit

the exploration of the seismic properties ofMgOat lowermantle

conditions. They find considerable changes in anisotropy pre-

served at high temperature with increasing pressure, along a

typical mantle geotherm; MgO is isotropic near the 670 km

discontinuity, but the anisotropy of P- and S-waves increases

rapidly with depth reaching 17% for Vp and 36% for Vs at the

D00 layer. The anisotropy of MgO increases linearly from 11.0%

and 21.5% for Vp and Vs, respectively, at ambient conditions to

20% and 42%, respectively, at 1800 K according to the data of

Isaak et al. (1989a,b). The effect of temperature on anisotropy is

more important at low pressure than at lower mantle pressures,

where the effect of pressure dominates according to the results of

Karki et al. (1999). Furthermore, not only the magnitude of the

anisotropy of MgO but also the orientation of the anisotropy


changes with increasing pressure according to the calculations of

Karki et al. (1999), for example, the fastest Vp is parallel to [111]

at ambient pressure and becomes parallel to [100] at 150 GPa

pressure and fastest S-wave propagating in the [110] direction

has a polarization parallel to [001] at low pressure that changes

to [1–10] at high pressure.

Ferropericlase – Magnesiowustite solid solution series has

been studied at ambient conditions by Jacobsen et al. (2002);

for ferropericlase with 24% Fe, the P-wave anisotropy is 10.5%

and maximum S-wave is 23.7%, slightly higher than pure MgO

at the same conditions. Data are required at lower mantle

pressures to evaluate if the presence of iron has a significant

effect on anisotropy of ferropericlase of mantle composition.

The effect of anomalous compressibility of ferropericlase

through the iron spin crossover has been studied experimen-

tally at room temperature and high pressure; in these

conditions, the high to low spin produces a sharp reduction

in C11 and C12 elastic moduli (e.g., Marquardt et al., 2009).

Theoretical studies using ab initio methods (e.g., Wentzcovitch

et al., 2009) at lower mantle conditions show that the change

in elasticity is smaller and occurs in a broad depth range of

1200–2000 km for 2000 K geotherm. The elastic anomaly is

increasingly attenuated with higher temperatures. It has been

suggested that iron spin crossover would be seismically trans-

parent (Antonangeli et al., 2011). It should be noted that iron

spin crossover has also been proposed for perovskite, this

crossover has much weaker effect on elastic properties as

shown by experimental studies (e.g., Lundin et al., 2008),

and ab initio modeling of the elastic properties (Caracas et al.,

2010) suggests this will be difficult to detect seismically.

SiO2 polymorphs – The free SiO2 in the transition zone and

the top of the lower mantle (to a depth of 1180 km or 47 GPa)

will be in the form of stishovite. The original experimental

determination of the single-crystal elastic constants of stisho-

vite by Weidner et al. (1982) and the more recent calculated

constants of Karki et al. (1997b) both indicate Vp and Vs

anisotropies at ambient conditions of 26.7–23.0% and 35.8–

34.4%, respectively, making this a highly anisotropic phase.

The calculations of Karki et al. (1997a,b,c) show that the

anisotropy increases dramatically as the phase transition to

CaCl2-structured SiO2 is approached at 47 GPa. The Vp anisot-

ropy increases from 23.0% to 28.9% and Vs from 34.4% to

161.0% with increasing pressure from ambient to 47 GPa. The

maximum Vp is parallel to [001] and the minimum parallel to

[100]. Themaximum dVs is parallel to [110] and the minimum

parallel to [001]. I discuss the properties of stishovite in the

section on subduction zones.

Postperovskite – Finally, this new phase is present in the D00

layer. Discovered and published in May 2004 by Murakami

et al. (2004), the elastic constants at 0 K were rapidly estab-

lished at low (0 GPa) and high (120 GPa) pressure by static

atomistic calculations (Iitaka et al., 2004; Oganov and Ono,

2004; Tsuchiya et al., 2004). From these first results, we can see

that Mg-postperovskite is very different to Mg-perovskite as

there are substantial changes in the distribution of the velocity

anisotropy with increasing pressure (Figure 16). At zero

pressure, the anisotropy is very high, 28% and 47% for

P- and S-waves, respectively. The maximum for Vp is parallel

to [100] with small submaxima parallel to [001] and mini-

mum near [111]. The shear wave splitting (dVs) has maxima

(2015), vol. 2, pp. 487-538

12.35

9.32

10.0

10.5

11.0

11.5

47.28

0.66

10.0

20.0

30.0

40.0

47.28

0.66

15.38

13.23

13.60

14.00

14.40

14.80

22.74

0.74

4.0

8.0

12.0

16.0

20.0

22.74

0.74

14.54

12.69

13.00

13.40

13.80

24.41

0.64

6.0

12.0

18.0

24.41

0.64

0 GPa 0 K

120 GPa 0 K

136 GPa 4000 K

(100)

(010)(001)

Vp (km s−1) dVs (%) Vs1 Polarization

Mg-Post-perovskite

Min. anisotropy = 0.64Max. velocity = 14.54Anisotropy = 13.6%

Min. velocity = 12.69 Max. anisotropy = 24.41

Min. anisotropy = 0.66Max. velocity = 12.35 Min. velocity = 9.32 Max. anisotropy = 47.28

Min. anisotropy = 0.74Max. velocity = 15.38Anisotropy = 15.0 %


Figure 16 Single-crystal anisotropic seismic properties of the D00 layer mineral Mg-postperovskite (orthorhombic). Increasing the pressurefrom 0 to 120 GPa at a temperature of 0 K decreases the anisotropy and also changes the distribution the maximum velocities and S-wave polarizations.Elastic constants calculated by Tsuchiya et al. (2004). Increasing the pressure to 136 GPa and temperature to 4000 K, there are relatively minorchanges in the anisotropy. Elastic constants calculated by Stackhouse et al. (2005).



parallel to h101i and h110i. At 120 GPa, the anisotropy has

reduced to 15% and 22% for P- and S-waves, respectively. The

distribution of velocities has changed, with the P maximum

still parallel to [100] and submaximum parallel to [001],

which is now almost the same velocity as parallel to [100];

the minimum is now parallel to [010]. The S-wave splitting

maxima have also changed and are now parallel to h111i.Apparently, the compression of the postperovskite structure

has caused important elastic changes as in MgO. An ab initio

molecular dynamics study by Stackhouse et al. (2005) at high

temperature revealed that the velocity distribution and anisot-

ropy were little effected by increasing the temperature from

0 to 4000 K when at a pressure of 136 GPa (Figure 16).

Wentzcovitch et al. (2006) produced a more extensive set of

high-pressure elastic constants and pressure and temperature

derivatives, similar P distributions, and slightly different shear

wave splitting pattern with maximum along the [001] axis that

is not present in the Stackhouse et al. velocity surfaces or in

high pressure 0 K results.

In conclusion, for the mantle, we can say that the general

trend favors an anisotropy decrease with increasing pressure and

increase with increasing temperature; olivine is a good example

of this behavior for minerals in upper mantle and transition

zone. The changes are limited to a few percent in most cases.

The primary causes of the anisotropy changes are minor crystal

structural rearrangements rather than velocity changes due to


density change caused by compressibility with pressure or ther-

mal expansion with temperature. The effect of temperature is

almost perfectly linear in many cases; some minor nonlinear

effects are seen in diopside, MgO, and SiO2 polymorphs. There

may be some perturbation in the seismic anisotropy due to the

iron spin crossover around 1200–2000 km. Nonlinear effects

with increasing pressure on the elastic constants cause the anisot-

ropy of wadsleyite and ringwoodite to first decrease. In the case

of the lower mantle minerals Mg-perovskite and MgO, there is a

steady increase in the anisotropy in increasing depth; this is a

verymarked effect forMgO. Stishovite also showsmajor changes

in anisotropy in the pressure range close to the transformation

to the CaCl2 structure. The single-crystal temperature derivatives

of wadsleyite, ilmenite MgSiO3, and stishovite are currently

unknown, whichmakes quantitative seismic anisotropic model-

ing of the transition zone and upper part of the lower mantle

speculative. To illustrate the variation of anisotropy as a function

of mantle conditions of temperature and pressure, I have

calculated the seismic properties along a mantle geotherm

(Figure 17). The mantle geotherm is based on the PREM

model for the pressure scale. The temperature scale is based on

the continental geotherm of Mercier (1980) from the surface to

130 km and Ito and Katsura (1989) for the transition zone and

Brown and Shankland (1981) for the lower mantle. The upper

mantle minerals olivine (Vp, Vs) and enstatite (Vp) show a slight

increase of anisotropy in the first 100 km due to the effect of

, (2015), vol. 2, pp. 487-538

10 15 20 25 30 35

0

100

200

300

400

Single crystal anisotropy : Upper mantle

Anisotropy (%)

Dep

th (k

m)

Vs Cpx

Vp Cpx

Vp

Vs Opx

Vp Opx

0 10 20 30 40 50 60

0

400

800

1200

1600

2000

2400

2800

Single crystal anisotropy : Transition zone and lower mantle

Anisotropy (%)

Dep

th (k

m)

VpVs

670 km

Vs

Vs

Vp CaCl2

1170 kmStishovite->CaCl2

DiopsideVp520 km

Stishovite

Vp Vs

Colum bite

CaCl2->Columbite2230 km

410 km

VpVs

Vp Vs

Vp Vs

MgO

VsVp

Vp Vs

VpVs

Mg-Perovskite

MgO

Wadsleyite

Ringwoodite

Olivine

Olivine

Vs

Lowspin

Highspin

Highspin

Lowspin

Figure 17 Variation of single-crystal seismic anisotropy with depth. The phases with important volume factions (see Figure 1) (olivine, wadsleyite,ringwoodite, Mg-perovskite, and MgO) are highlighted by a thicker line. P-wave anisotropy is the full line and the S-wave is the dashed line.See text for details.



temperature. With increasing depth, the trend is for decreasing

anisotropy except for Vs of enstatite and diopside. In the transi-

tion zone and lower mantle, the situation is more complex due

to the presence of phase transitions. In the transition zone,

diopside may be present to about 500 km with an increasing

Vs and decreasing Vp anisotropy with depth. Wadsleyite is less

anisotropic than olivine at 410 km, but significantlymore aniso-

tropic than ringwoodite found below 520 km. Although the

lower mantle is known to be seismically isotropic, the


constituent minerals are anisotropic. MgO shows important

increase in anisotropy with depth (10–30%) at 670 km (it is

isotropic) and 2800 km (it is very anisotropic), possibly being

candidate mineral to explain anisotropy of the D00 layer.

Mg-perovskite is strongly anisotropic (ca. 10%) throughout the

lower mantle, but now, it is strongest in the lowermost mantle

(Zhang et al., 2013). The SiO2 polymorphs are all strongly

anisotropic, particularly for S-waves. If free silica is present in

the transition zone or lowermantle, due perhaps to the presence

(2015), vol. 2, pp. 487-538



of subducted basalt (e.g., Ringwood, 1991), then even a small

volume fraction of the SiO2 polymorphs could influence the

seismic anisotropy of the mantle. However, to do so, the SiO2

polymorphswould have to be oriented, due to either dislocation

glide (plastic flow), oriented grain growth, or anisometric crystal

shape (viscous flow) (e.g., Mainprice and Nicolas, 1989). Given

that the SiO2 polymorphs are likely to be less than 10% by

volume (Ringwood, 1991) and hence would not be the load-

bearing framework of the rock, it ismore likely that the inequant

shape of SiO2 polymorphs would control their orientation dur-

ing viscous flow.

2.20.2.6.4 Subduction zonesSubduction zones are mainly vertical structures in the Deep

Earth, with a major section in the upper mantle and transition

zone. In certain regions, the subducted plates have nearly

horizontal sections just below 670 km seismic discontinuity.

Slabs continue to descend until they reach the D00 layer andCMB. During the decent of a plate, the mineralogy changes

almost continuously with the temperature and pressure

changes. The subducted plate is populated with many very

specific minerals because the temperature of the plate is gener-

ally lower that the surrounding mantle, whereas the depth

controls the pressure. The subducting plate is a major vector

for chemical exchange with deep Earth, not only hydrated

minerals providing hydrogen but also minerals derived from

sediments and oceanic basalts rich in potassium, sodium,

aluminum, and silicon. Subduction zones are also the most

seismically active regions of the planet.

Here, we will focus on the seismic anisotropy of a few key

subduction zone minerals. The upper part of the subduction

zone is dominated by the mantle wedge, where dehydration of

the plate and fluid-triggered melting are very active processes,

which are directly associated with earthquake activity and

volcanism, respectively. One of the reactions occurring within

the mantle wedge is hydration of minerals by fluids released

from the descending slab, for example, the transformation

olivine to the high-pressure serpentine mineral antigorite.

Bezacier et al. (2010) had reported experimental measure-

ments of antigorite single-crystal elasticity at ambient condi-

tions. Antigorite is monoclinic layered-structured mineral that

has fast Vp in the (001) plane and low Vp normal it. Vs1 is fast

in the (001) plane with a polarization parallel to the (001)

plane, whereas Vs2 is slow in the (001) plane with a polariza-

tion normal to the (001) plane. The Vp and Vs anisotropies are

high at 47% and 75%, respectively, with the highest S-wave

splitting in the (001) plane (Figure 18). One key seismic

observation in the mantle wedge because of the presence of

fluids is the Vp/Vs ratio; for anisotropic crystals, there are two

ratios, Vp/Vs1 and Vp/Vs2, where Vp/Vs1 corresponds to ratio

with the first S-wave arrival and is probably themost frequently

measured. Vp/Vs1 ranges from 2.3 to 1.1 with direction, being

highest normal and lowest parallel to the (001) plane. Vp/Vs2

range is 3.8–1.4, the magnitude being highest parallel and

lowest normal to the (001) plane, that is, opposite in distribu-

tion to Vp/Vs1; hence, measuring Vp/Vs2, Vp/Vs1 could provide

a complementary set of data. All the velocity-related properties

just described are typical of layered-structured minerals in

widest sense or phyllosilicates in the present case. Information

on the pressure dependence of antigorite has been provided by


an ab initio study by Mookherjee and Capitani (2011) up to

13.8 GPa, and more recently, an experimental study by

Bezacier et al. (2013) measured a subset of the moduli to

9 GPa. Bezacier et al. (2013) indicated that there is decrease

in P- and S-wave anisotropies from 47% to 75%, respectively,

at ambient conditions to 22% and 63% at 7 GPa. Mookherjee

and Capitani (2011) found that Vp anisotropy is 57% at zero

pressure and that it decreases to 27% at 7 GPa; however, Vs

anisotropy is 54% at zero pressure and increases to 63% at

7 GPa, so that both studies agree at 7 GPa, but not ambient

conditions. Another potential mantle wedge mineral that has

stable higher-temperature conditions than antigorite is talc;

this industrially very important mineral has been studied

using ab initio methods by Mainprice et al. (2008b). At zero

pressure, talc is very anisotropic with P- and S-wave anisotrop-

ies of 80% and 85%, respectively; with increasing pressure,

both P- and S-wave anisotropies decrease. At pressures where

talc is known be stable in the Earth (up to 5 GPa), the Vp and

Vs anisotropies are reduced to about 40% for both velocities,

which is still a very high value. The third important layered-

structured mineral is chlorite; Mookherjee and Mainprice

(2014) have studied the elastic properties to a pressure of

18.4 GPa. Chlorite is stable to higher temperatures and pres-

sures than antigorite and talc. Chlorite has velocity character-

istics very similar to antigorite, with P- and S-wave anisotropies

of 30% and 52%, respectively, at zero pressure, which decrease

with pressure for P-waves to 25% but increase with pressure for

S-waves to 72%. Antigorite can often be seen associated with

chlorite as intergrowths with the same crystal orientation (e.g.,

Morales et al., 2013); there are also topotactic relationships

between antigorite and olivine during the phase transforma-

tion (Boudier et al., 2010; Morales et al., 2013). Both antigorite

and chlorite contain 13 wt.% H2O, and hence, they are more

important for the water cycle than talc, which has only

4.8 wt.% H2O. All these layered-structured minerals form

CPO with strong point maximum of the (001) normal to the

foliation and girdles of both (100) and (010) in the foliation

plane, which will result in very strong transverse isotropic

velocity distribution.

At greater depths and higher pressures, there is transition to

the 10 A phase (13 wt.% H2O) followed with increasing depth

by a series of dense hydrogen magnesium silicate (DHMS) or

alphabet phases. The DHMSs have only to be observed in

laboratory experiments, but hypothetically could exist along a

cold slab geotherm (e.g., Mainprice and Ildefonse, 2009), and

mainly present at depths of the mantle transition zone.

Sanchez-Valle et al. (2006, 2008) had studied the elasticity of

the phase A (12 wt.% H2O) in the laboratory at pressures up to

12.4 GPa. The velocity distribution is very different to the

layered-structured minerals described earlier with fast Vp nor-

mal to (0001) and slow Vp in the (0001); Vs1 is slow in the

(0001) plane, the polarization parallel to the (0001) plane,

and the maximum Vs1 at 45� to the (0001) plane; and Vs2 also

is slow normal to the (0001) and but fast normal to the (0001)

plane. The P- and S-wave anisotropies are quite modest at 9%

and 18%, respectively, at 9 GPa in the stability field of this

mineral. Pressure has only a very modest effect on the anisot-

ropy of phase A. The phase D is the DHMS that occurs at the

greatest pressures and depths; being stable below 670 km

discontinuity, it is the only mineral to potentially recycle

, (2015), vol. 2, pp. 487-538

upper hemisphere

8.92

5.52


6.5

7.0

7.5

8.0

75.20

0.10

AVs Contours (%)

20.0

30.0

40.0

50.0

60.0

75.20

0.10

2.30

1.13

1.40

1.60

1.80

2.00

3.79

1.39

1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50

b c*

a

Antigorite single crystal

Vp (km s−1) dVs (%) Vs1 polarization

Vp/Vs1 Vp/Vs2

5.15

2.65

3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75

4.43

2.32

2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20

Vs1(km s−1) Vs2(km s−1)

Min. anisotropy = 0.10Max. velocity = 8.92Anisotropy = 47.1 %




Max. ratio = 2.30Anisotropy = 68.1 %

Min. ratio = 1.13 Max. ratio = 3.79Anisotropy = 92.4 %

Min. ratio = 1.39



Figure 18 The seismic properties of a single crystal of antigorite at room pressure (Bezacier et al., 2010). Vp and Vs1 are both fast in the basal plane(normal to c*). Vs1 polarization is parallel to the basal plane. Vs2 is fast in directions inclined to the basal plane. Vp/Vs1 has the maximumnormal to the basal plane, whereas Vp/Vs2 is highest parallel to the basal plane. Antigorite is monoclinic, so the plane normal to b and b* is a mirrorplane (black vertical line containing a and c*). Note that there is some imperfect threefold symmetry.



hydrogen into the lower mantle. It is a candidate mineral for

horizontal or stagnant slabs. The elasticity of phase D has been

measured experimentally at ambient conditions (Rosa et al.,

2012) for Mg- and Al–Fe-bearing crystals. Using ab initio

methods, the elasticity of the phase D has been studied by

Mainprice et al. (2007) to 84 GPa and as function of hydrogen

bond symmetrization by Tsuchiya and Tsuchiya (2008) to

60 GPa. The phase D has a velocity distribution that has

more in common with a layered-structured mineral than the

phase A. The P-wave velocity is fast in (0001) plane and slow

normal to the (0001) plane. Vs1 is fast in the basal plane and

polarized parallel to the (0001) plane and Vs2 is the opposite.

At ambient conditions, the P- and S-wave anisotropies are 18%

and 19%, and the agreement between experimental and theor-

etical values is very good. At pressure of 24 GPa in the stability

field of phase D, the anisotropies of P- and S-waves are 9%


and 18%, respectively, showing a decrease for the P-wave

anisotropy with increasing pressure. The spin transition of

Fe3+ in Al-bearing phase D occurs at around 40 GPa (Chang

et al., 2013) and causes a decrease in bulk modulus from 253

to 147 GPa. The decrease in bulk modulus corresponds to bulk

sound velocity drop of 2 km s�1, which could explain the

exceptionally strong small-scale heterogeneity detected by

direct P-waves in the mid-lower mantle beneath the circum-

Pacific subduction zones (e.g., Kaneshima and Helffrich,

2010), a region with horizontal slabs below 670 km discontin-

uity. A recent study using first-principles methods has pre-

dicted a new DHMS form from breakdown of the phase D to

give MgSiO4H2 plus stishovite (Tsuchiya, 2013) at pressure of

about 40 GPa; at the present time, there is no experimental

confirmation of this new phase. Finally, comparison with

brucite Mg(OH)2 is interesting as it is a model system for

(2015), vol. 2, pp. 487-538



understanding DHMS under hydrostatic compression and an

important structural unit of many layer silicates, such as chlo-

rite, antigorite, and talc. The single-crystal elastic constants of

brucite have been measured to a pressure of 15 GPa by Jiang

et al. (2006). The seismic anisotropy of brucite is exceptionally

high at ambient conditions with P-wave and S-wave anisotrop-

ies of 57% and 46%, respectively. Both forms of anisotropy

decrease with increasing pressure being 12% and 24% at

15 GPa for P-wave and S-wave anisotropies, respectively. The

stronger decrease of P-wave anisotropy compared to S-wave

is probably related to the very important linear compressibility

along the c-axis. The apparent symmetry of the seismic anisot-

ropy of brucite changes with pressure (Figure 19). Brucite

is stable to 80 GPa and modest temperatures, so it could

be an anisotropic component of cold subducted slabs.

A mineral called the phase X, which occurs as hydrous and

anhydrous forms and is stable at transition zone pressures and

temperatures, could contribute to mineralogy of slabs. The

phase X is the reaction product of the breakdown of potassium

rich amphibole (K-richterite) (Konzett and Fei, 2000). The

elastic properties of the phase X have been predicted by

ab initio methods to a pressure of 30 GPa (Mookherjee and

Steinle-Neumann, 2009a). The anhydrous phase X has hexag-

onal symmetry (space group P63cm); hence, it has the equiva-

lent elastic symmetry to a transverse isotropic sample.

At transition zone pressure of 22 GPa, anhydrous phase X has

P- and S-wave anisotropies of 13.1% and 12.2%, respectively.

In addition to these hydrous phases, some of normally

anhydrous minerals in the subducted slab can become

hydrated to a limited degree; the important ones in terms of

volume fraction are olivine, wadsleyite, and ringwoodite for

the pyrolite mantle (Figure 1). For the upper mantle, olivine is

the most important mineral being between 50% and 70% by

Upper hemisphere

8.17

4.56

5.5

6.0

6.5

7.0

7.5

(0110)

(2110)P velocity (km s−1)

9.31

8.25

8.40

8.60

8.80

9.00

9.20

P = 0 GPa

P = 14.6 GPa





Figure 19 Single-crystal anisotropic seismic properties of brucite (trigonal)velocity distribution of P-waves, dVs anisotropy, and S1 polarization orientatielastic properties at low pressure and the increasingly trigonal nature (threef


volume of a typical mantle peridotite. Single-crystal elastic

constants for forsterite and olivine with 0.9 and 0.8 wt.%

water have been measured by Jacobsen et al. (2008, 2009) at

ambient conditions; 0.9 wt.% water is considered the maxi-

mum water storage capacity for olivine (Smyth et al., 2006).

The combined effect of 3 mol% Fe and 0.8 wt.% water causes a

reduction of the bulk and shear moduli of 2.9% and 4.5%,

respectively. Greater modulus reductions are expected for more

iron-rich than Fo90 olivine mantle compositions. New mea-

surements for forsterite with 0.9 wt.% water to 14 GPa (Mao

et al., 2010) reveal some unexpected behavior with pressure.

Below�3.5 GPa, the Vp and Vs of hydrous forsterite are slower

than anhydrous crystal by 0.6% and 0.4%, respectively,

whereas above this pressure range, the hydrous crystal has

faster velocities by 1.1% and 1.9%, respectively. At room

pressure, the P- and S-wave anisotropies are, respectively,

24.7% and 18.0% for anhydrous forsterite and 23.9% and

17.8% for hydrous forsterite, indicating a decrease of 0.8%

and 0.2% upon hydration. At 14 GPa, the P- and S-wave

anisotropies are, respectively, 21.2% and 13.5% for anhydrous

forsterite and 15.4% and 12.1% for hydrous forsterite, indicat-

ing a decrease of 6.0% and 1.4% upon hydration. Hence, the

main effect of hydration on olivine elasticity is a change of

velocity and reduction of P-wave anisotropy and almost no

effect on S-wave anisotropy.

In the transition zone (410–670 km depth), the high-

pressure polymorphs of olivine, wadsleyite, and ringwoodite,

in their hydrous forms, have the potential to store more

water as hydroxyl than the oceans on the Earth’s surface (e.g.,

Jacobsen, 2006). Wadsleyite and ringwoodite can be synthe-

sized with up to 3.1 and 2.8 wt.% H2O, respectively, which

is nearly three times that of olivine. The maximum water

solubility in wadsleyite at transition zone conditions

45.99

0.00

5.0

15.0

25.0

35.0

45.99

0.00

Brucite


23.63

0.00

3.0 6.0 9.0 12.0 15.0 18.0

23.63

0.00



at 0 and 14.6 GPa (data from Jiang et al., 2006). Note the change inons, which reflect the nearly hexagonal symmetry of brucite’sold c-axis at the center of the pole figure) with increasing pressure.

, (2015), vol. 2, pp. 487-538



(�15 GPa, 1400 �C) is however only about 0.9 wt.%

(Demouchy et al., 2005) as temperature has an important

affect on solubility. In ringwoodite, water solubility decreases

with increasing temperature. In subducting slabs, hydrous

wadsleyite and ringwoodite can represent volume fractions of

about 50–60%, resulting in the shallower region of the mantle

transition zone having a larger water storage potential than the

deeper region (Ohtani et al., 2004).

In upper part of the transition zone (410–520 km depth),

anhydrous wadsleyite is replaced by hydrous wadsleyite in the

hydrated slab. The elastic constants of hydrous wadsleyite have

been determined at ambient conditions as a function of water

content by Mao et al. (2008a,b). Anhydrous wadsleyite

(Zha et al., 1997) has P- and S-wave anisotropies of 15.4%

and 16.8%, respectively, whereas hydrous wadsleyite with

1.66 wt.% H2O has a P-wave and S-wave anisotropies of

16.3% and 16.5%, respectively. Hence, hydration increases

Vp anisotropy by 0.9% and decreases Vs anisotropy by 0.3%.

Anhydrous wadsleyite has Vp/Vs1 and Vp/Vs2 anisotropy

values of 22.0% and 19.1%, whereas hydrous wadsleyite has

Vp/Vs1 and Vp/Vs2 of 20.9% and 20.2%, respectively. For

Vp/Vs1 and Vp/Vs2 anisotropies, the hydration of wadsleyite

decreases Vp/Vs1 by 0.1% and increases Vp/Vs2 by 1.1%.

Clearly, it is not possible to distinguish between anhydrous

wadsleyite and hydrous wadsleyite on the basis of anisotropy,

although there is a linear decrease of velocity with water con-

tent at room pressure (Mao et al., 2008a). A first-principles

study of wadsleyite as function of water content (Tsuchiya

and Tsuchiya, 2009) successfully reproduces the experimental

elastic results for anhydrous wadsleyite (Zha et al., 1997) and

hydrous wadsleyite (Mao et al., 2008a,b) using a Mg vacancy

model for the protonation of wadsleyite. Tsuchiya and

Tsuchiya (2009) also studied the effect of pressure on hydrous

wadsleyite. The pressure derivatives (dCij/dP) for hydrous wad-

sleyite are similar to anhydrous derivatives of Zha et al. (1997).

At a pressure of 20 GPa, the anhydrous wadsleyite has P- and

S-wave anisotropy of 10.1% and 11.9%, whereas hydrous

wadsleyite (3.3 wt.% H2O) has P- and S-wave anisotropy

0

2

4

6

8

10

12

0 5 10Press

HydoRw

Vp

Vs

Rw

HydoRw

0

20

40

60

80

0 5 10 15Pressure (GPa)

Brucite Vs

phase A Vs

phase A Vp

Talc Vp

Talc Vs

Brucite Vp

Chlorite Vs

Chlorite Vp

Antigorite Vs

Antigorite Vp

3-5 GPa

Tran

sitio

n zo

ne

Up

per

man

tle

Ani

sotr

opy

of V

p a

nd V

s (%

)

Figure 20 The evolution of seismic anisotropy of hydrous or hydrated minelower mantle. The pressure range 3–5 GPa is typical of the mantle wedge. In thtalc, antigorite, and chlorite contrasts with that of the DHMS phase A. Antigowith pressure. Increasing pressure reduces anisotropy with pressure for anhsmall effect on the Vs anisotropy for DHMS phase D, but increasing pressure


of 11.3% and 12.8%. At the same pressure, the anhydrous

wadsleyite has Vp/Vs1 and Vp/Vs2 anisotropies of 13.4% and

9.6%, respectively, whereas hydrous wadsleyite has Vp/Vs1 and

Vp/Vs2 anisotropies of 14.6% and 11.9%. Hence, hydration of

P- and S-wave anisotropies increases by 1.2% and 0.9%,

respectively, and for Vp/Vs1 and Vp/Vs2 anisotropies increases

by 2.2% and 1.6%, respectively. So once again hydration has

more effect on velocities than the anisotropy.

In the lower part of the transition zone (520–670 km

depth), anhydrous ringwoodite will be replaced by its hydrated

counterpart hydrous ringwoodite in the hydrated slab. The

elastic constants of hydrous ringwoodite have been measured

as a function of pressure to 9 GPa by Jacobsen and Smyth

(2006) and to 24 GPa by Wenk et al. (2006). More recently,

Mao et al. (2011, 2012) had determined the elastic constants of

hydrous ringwoodite to 16 GPa and 673 K. The presence of

1.1 wt.% water lowers the elastic moduli by 5–9%, but does

not affect the pressure derivatives. The reduction caused by

1.1 wt.% water is significantly enhanced by high temperature

when at high pressure. Hydrous ringwoodite at 16.3 GPa and

673 K has a very low anisotropy, 1.3% for P-waves and 3.0%

S-waves. The anisotropy of Vp/Vs1 and Vp/Vs2 is slightly stron-

ger at 3.1% and 3.9%, respectively. The pressure sensitivity of

the seismic anisotropy of hydrous or hydrated minerals at

upper mantle and transition zone pressures is illustrated in

Figure 20.

Apart from hydrated phases associated with subduction,

there are the phases derived from MORB and sediments

(Figure 1). Compared to the pyrolite mantle composition,

MORB is chemically enriched in incompatible elements (sili-

con, aluminum, calcium, and sodium) and depleted in com-

patible elements like magnesium. At deep mantle temperatures

and pressures, MORB transforms to mineral assemblages with

high SiO2 and aluminum content. Mineral assemblages change

with pressure and depth at the top of the lower mantle in

the pressure range 23–50 GPa (depth range 670–1300 km);

the assemblage is composed of the new aluminum phase

(NAL), aluminum-rich calcium ferrite structure (Al-CF),

0

5

10

15

20

25

0 10 20 30 40 50

phase D Vs

phase D Vp

Tran

sitio

n zo

ne

Low

er m

antle

Up

per

man

tle

Pressure (GPa) 15 20 25

ure (GPa)

Vs

VpRw

Rw

HydoRw

Tran

sitio

n zo

ne

Low

er m

antle

Up

per

man

tle

rals in the pressure range of the upper mantle, transition zone, ande upper mantle pressure range, the strong pressure sensitivity of brucite,rite and chlorite are exceptions as the anisotropy for Vs increasesydrous and hydrated ringwoodite in the transition zone. Pressure has adecreases Vp anisotropy. See text for discussion and references.

(2015), vol. 2, pp. 487-538



Mg-perovskite, Ca-perovskite, and stishovite that are stable;

and of these minerals, NAL, Al-CF, and stishovite are specific

to MORB chemistry. Above a pressure of 45–50 GPa (below a

depth of 1200 km), NAL (hexagonal, space group P63/m) is no

longer stable due to the increased in the solubility of alumin-

um with pressure in the Mg-perovskite (Perrillat et al., 2006;

Ricolleau et al., 2010; Sanloup et al., 2013). Stishovite with

rutile structure transforms to the CaCl2 structure of SiO2 at

55 GPa (depth of about 1300 km) (Andrault et al., 1998),

whereas Al-CF (orthorhombic, space group Pbnm) is stable to

at least 130 GPa (Hirose et al., 2005; Ono et al., 2005a,b). The

aluminum-rich calcium titanate structure (Al-CT) (orthorhom-

bic, space group Cmcm) mineral is only observed by Ono et al.

(2005a,b) at CMB pressure of 143 GPa. Sediments are the

other chemical source in subduction zones that can produce

minerals like K-hollandite (Irifune et al., 1994).

The elasticity of the relatively low-pressure NAL phase has

been studied by Mookherjee et al. (2012) using ab initio

methods to a pressure of 160 GPa. According to Ricolleau

et al. (2010), NAL is only present in the MORB mineral assem-

blage in the pressure range from 22 to 50 GPa. At 27 GPa, the

P- and S-wave anisotropies are 11.3% and 16.0%, respectively.

As the NAL phase is hexagonal, the anisotropy has the same

elastic symmetry as a transverse isotropic sample. The Al-CF

phase has been studied by Tsuchiya (2011), Mookherjee

(2011a), and Mookherjee et al. (2012) all using ab initio

methods. The Al-CF phase has stability over a wide range of

pressure in the lower mantle from 22 to 130 GPa (Hirose et al.,

2005; Ono et al., 2005a,b; Perrillat et al., 2006; Ricolleau et al.,

2010; Sanloup et al., 2013). The orthorhombic symmetry of

Al-CF phase results in a more complex velocity distribution

than the NAL phase. The elasticity of MgAl2O4 CF phase has

been calculated by Tsuchiya (2011) and Mookherjee (2011a),

the results for the P- and S-wave anisotropies as function of

pressure are shown in Figure 21, both P- and S-wave anisotrop-

ies decrease with increasing pressure, and the results of these

two studies agree very closely. At low-pressure end of the

5

10

15

20

25

30

35

0 40 80 120 160

MgAl2O4 CF phase

Vp(%) M Vs(%) M Vp(%) T Vs(%) T

P-

and

S-w

ave

anis

otro

py

(%)

Pressure (GPa)

Vp

VsStability fieldin pressure ofCF phase

(a) (

TZ LM

Figure 21 The effect of pressure on the seismic anisotropy of anhydrous p(T¼Tsuchiya, 2011; M¼Mookherjee, 2011a) shows a typical decrease of antalc and brucite. Hollandite I (Mookherjee and Steinle-Neumann, 2009b) is atand chlorite. In hollandite II, the Vp and Vs anisotropies decrease slightly wit


stability field, P-wave anisotropy is 15.2% and S-wave is

25.0%, and at the high-pressure end of the stability field,

6.5% and 15.3%, respectively. The P- and S-wave anisotropies

of the Al-rich calcium titanate structure (Al-CT) at 140 GPa are

11.6% and 28.1%, respectively, near the CMB. So Al-CT has

quite high S-wave anisotropy even at extreme pressures.

High-pressure studies on sediments of average continental

crust composition between 9 and 24 GPa found that hollan-

dite (now called hollandite I or lingunite) represents �30% by

volume of the high-pressure assemblage (Irifune et al., 1994),

indicating the importance of the hollandite I phase for sub-

ducted sediments. Under ambient conditions, hollandite I has

tetragonal symmetry I4/m. Using high-pressure in situ RXD

techniques up to 32 GPa (Ferroir et al., 2006; Sueda et al.,

2004) showed that hollandite I undergoes a phase transition

from the tetragonal to a monoclinic hollandite II phase with

I2/m symmetry at 20 GPa at room temperature. Phase

equilibrium studies on the system KAlSi3O8–NaAlSi3O8 by

Liu (2006) show that hollandite I is stable up to 18 GPa and

2200 �C and hollandite II is stable up to 25 GPa and 2000 �C.Hollandite I and hollandite II can be present in the transition

zone and the upper part of the lower mantle and perhaps to

greater depths. An elasticity study using ab initio methods by

Mookherjee and Steinle-Neumann (2009b) reports the P-wave

anisotropy for hollandite I as about 26% at zero pressure,

which decreases at the transition pressure of about 30 GPa to

22%. The P-wave anisotropy of hollandite II at the transition

pressure is 20%, and this slowly decreases to 12% at 100 GPa.

The S-wave anisotropy of hollandite I is 42% at zero pressure,

but rises sharply to 70% at the 30 GPa. Hollandite II at the

transition pressure has Vs anisotropy of 30%, which slowly

decreases with increasing pressure to 22% at 100 GPa

(Figure 21). Hollandite I and hollandite II have very

high S-wave anisotropies compared to other minerals in the

transition zone and lower mantle. Mussi et al. (2010) had

shown that hollandite I deforms plastically at transition zone

temperatures and pressures and predicted CPO that would

0

10

20

30

40

50

60

70

0 20 40 60 80 100

Hollandite I and II

Vp (%) Hollandite IVs (%) Hollandite IVp (%) Hollandite IIVs (%) Hollandite II

P-

ans

S-w

ave

anis

otro

py

(%)

Pressure (GPa)b)

Vs

Vp

Vp

Vs

Hollandite I Hollandite II

TZ LM

hases, CF phase and hollandite I and hollandite II. The CF phaseisotropy of Vp and Vs with increasing pressure seen in minerals likeypical with increasing Vs anisotropy with pressure, like antigoriteh pressure.

, (2015), vol. 2, pp. 487-538



result in seismic anisotropy very similar to stishovite (Cordier

et al., 2004a), as both minerals have the same rutile structure.

Aggregates of hollandite I are predicted to have P- and S-wave

anisotropies of 12.8% and 15.1%, compared to stishovite

aggregate with 8.9% and 7.1% at the same pressure of

17 GPa, which corresponds to a depth of about 500 km.

Subducted MORB will produce SiO2 phase such as coesite,

stishovite, and the poststishovite high-pressure phases. Stisho-

vite is stable from 9 to 24 GPa in the experiments of Irifune

et al. (1994) and can be stable until 55 GPa (Andrault et al.,

1998). In experiments on sediments by Irifune et al. (1994),

stishovite has a volume fraction between 20% and 40%,

whereas in transform MORB, stishovite has volume fractions

of 15–20% (Ricolleau et al., 2010). Using ab initio methods

Karki et al. (1997a,b,c) has given the elastic constants of SiO2

polymorphs, and recently, full set experimental values to

12 GPa have been measured by Jiang et al. (2009), and these

are in close agreement with the predictions of Karki et al.

(1997a,b,c). The transformation pressure and temperature of

stishovite to CaCl2-type SiO2 phase have been given by ab initio

methods by Tsuchiya et al. (2004). From the anisotropy point

of view, stishovite starts to become exceptional at about 670 km

discontinuity when the S-wave anisotropy is greater than 50%

(Figure 22). From 670 km discontinuity to the eventual trans-

formation to CaCl2-type SiO2 at about 1200 km depth, the

S-wave anisotropy sharply increases to an exceptional value of

about 150% near the transition. In parallel with the evolution

of the S-wave anisotropy, the Vp/Vs2 ratio also reaches unprec-

edented value of 10 (Figure B). Stishovite deforms plastically at

high temperature and pressure (Cordier et al., 2004a,b) and is

expected to formCPOas predicted by polycrystallinemodeling,

and a strong anisotropy is expected in real mantle samples.

Kawakatsu and Niu (1994) and Vinnik et al. (2001) had

observed seismic reflectivity in the lower mantle, which may

be explained by the SiO2 phase transformation at around

1100–1200 km.

0

20

40

60

80

100

120

140

160

0 200 400 600 800 1000 1200

Stishovite at high pressure : Anisotropy

P-

and

S-w

ave

anis

otro

py

(%)

Depth (km)

AVs AVp/Vs1

AVp

AVp/Vs2

Tran

sitio

n zo

ne

Low

er m

antle

(a) (

Figure 22 The effect of depth on the seismic anisotropy of anhydrous stishKarki et al. (1997a,b,c). The rutile-structured stishovite reaches exceptionallyfrom of SiO2 at about 1100 km depth. In contrast, Vp anisotropy increases was the ratio of Vp/Vs2 and anisotropy of Vp/Vs2, both reach exceptional value


A wide range of temperature and pressure associated with

several chemical sources represented by the pyrolite mantle,

MORD, and sediments create a complex pattern of minerals

with depth in subduction zones. The subduction zones are

compared to the extensive volumes involved with convective

mantle and are quite narrow and restricted volumes. The

thermomechanical constraints of a subduction zone are a

number of minerals that are very anisotropic: antigorite, talc,

chlorite, brucite, hollandite I, Al-CF phase, and stishovite. In

almost all minerals, the P-wave anisotropy decreases with

increasing pressure. The very anisotropic minerals fall into

two groups: (a) a group where the S-wave anisotropy decreases

with pressure that includes talc, brucite (Figure 20), and Al-CF

phase (Figure 21) and (b) a group where the S-wave anisot-

ropy increases with pressure that includes antigorite, chlorite,

hollandite I, and stishovite (Figures 20, 21, and 22,

respectively).

2.20.2.6.5 Inner coreUnlike the mantle, the Earth’s inner core is composed primar-

ily of iron, with about 5 wt.% nickel and very small amounts of

other siderophile elements such as chromium, manganese,

phosphorus, and cobalt and some light elements such as are

oxygen, sulfur, and silicon. The stable structure of iron at

ambient conditions is body-centered cubic (bcc); when the

pressure is increased above 15 GPa, iron transforms to an hcp

structure called the e-phase; and at the high pressure and

temperature, iron most likely remains hcp; however, there

have been experimental observations of a double hexagonal

close-packed structure (dhcp) (Saxena et al., 1996) and a dis-

torted hcp structure with orthorhombic symmetry (Andrault

et al., 1997). Atomic modeling at high pressure and high

temperature suggests that the hcp structure is still stable at

temperatures above 3500 K in pure iron (Vocadlo et al.,

2003a), although the energy differences between the hcp and

the bcc structures are very small and the authors speculate that

0

2

4

6

8

10

12

0 200 400 600 800 1000 1200

Stishovite at high pressure : Vp/Vs ratio

Depth (km)

Vp/Vs2

Vp/Vs1

Max

imum

Vp

/Vs1

and

Vp

/Vs2

b)

Tran

sitio

n zo

ne

Low

er m

antle

ovite SiO2 using the ab initio predicted elastic constants ofhigh Vs anisotropy (150%) near phase transition to the CaCl2 structuredith depth to about 30%. The parameters associated with Vs2, suchs near the phase transition.

(2015), vol. 2, pp. 487-538



the bcc structure may be stabilized by the presence of light

elements. A suggestion is echoed by the seismic study of

Beghein and Trampert (2003). To make a quantitative aniso-

tropic seismic model to compare with observations, one needs

either velocity measurements or the elastic constants of single-

crystal hcp iron at the conditions of the inner core. The mea-

surement or first-principles calculation of the elastic constants

of iron is a major challenge for mineral physics. The conditions

of the inner core are extreme with pressures from 325 to

360 GPa and temperatures from 5300 to 5500 K. To date,

experimental measurements have been using DACs to achieve

the high pressures on polycrystalline hcp iron. IXS has been

used to measure Vp at room temperature and high pressure up

to 112 GPa and temperatures up to 1100 K (Antonangeli et al.,

2004, 2010, 2012; Fiquet et al., 2001), and the anisotropy of

Vp has been characterized in two directions up to 112 GPa

(Antonangeli et al., 2004). Vp has been determined at simul-

taneous high pressure and temperature up to 300 GPa and up

to 1200 K from x-ray Debye–Waller temperature factors

(Dubrovinsky et al., 2001) and up to 73 GPa and 1700 K

using IXS by Lin et al. (2005). x-Ray radial diffraction (XRD)

has been used to measure the elastic constants of polycrystal-

line iron with simultaneous measurement of the CPO at room

temperature and pressures up to 211 GPa (Singh et al., 1998;

Mao et al, 1998, with corrections 1999; Merkel et al., 2005),

which is still well below inner core pressures. However, as

mentioned before, the high nonhydrostatic stress present in

DACs has exceeded the plastic yield stress in some experiments

and produced plastic strain, which violates the elastic stress

analysis used to determine the elastic parameters (Antonangeli

et al., 2006; Mao et al., 2008a,b,c; Merkel et al., 2006a,b,

2009). The experimental Vp data used to describe the anisot-

ropy of textured polycrystalline samples are shown in Fig-

ure 23. The texture or CPO of the iron is induced by the

compression in DAC with an axial symmetry. The original

experiments of H.K. Mao et al. (1998) show the characteristic

bell-shaped Vp curve with peak velocity near 45� from the

compression direction, which is also the symmetry axis

(Figure 23(a)). Note also that even the IXS data points of

Antonangeli et al. (2004) have similar symmetry to the data

Experimental dat

0 10 20 30 40 50 60 70 80 907.07.58.08.59.09.5

10.010.511.0

11.512.0


Vp

(km

s−1)

H.K. Mao et al. 1998 RXD 210 GPa

Antonangeli et al. 2004 IXS 112 GPa

W.L. Mao et al. 2008 RXD 52 GPa

W.L. Mao et al. 2008 IXS+EOS 52 GPa

(a) (

Figure 23 The seismic velocities at ambient temperature in single-crystal pexperimentally determined elastic constants of Mao et al. (1998, with correctAntonangeli et al. (2004). RXD, radial x-ray diffraction; IXS, x-ray scattering;


of Mao et al. (1998). Recently, a new combined method was

introduced by Mao et al. (2008a,b,c) to avoid problems in

DAC, which uses IXS and the EOS. In the 2008 paper, they

reproduced the old method with bell-shaped curve and intro-

duced the new method, which has the maximum Vp nearer the

symmetry axis (Figure 23(b)). More recent work using IXS uses

gas-loaded DAC to avoid or reduce nonhydrostatic stresses

(Antonangeli et al., 2010, 2012), but has not studied

anisotropy.

In order simulate the in situ conditions, the static (0 K)

elastic constants have been calculated at inner core pressures

(Stixrude and Cohen, 1995). The calculated elastic constants

predict maximum P-wave velocity parallel to the c-axis, and the

difference in velocity between the c- and a-axes is quite small

(Figure 23(a)). The anisotropy of the calculated elastic con-

stants being quite low required that the CPO is very strong, and

it was even suggested that inner core could be a single crystal of

hcp iron to be compatible with the seismic observations and

that c-axis is aligned with the Earth’s rotation axis. The first

attempt to introduce temperature into first-principles methods

for iron by Laio et al. (2000) produced estimates of the isotro-

pic bulk and shear modulus at inner core conditions (325 GPa

and 5400 K) and single-crystal elastic constants at conditions

comparable with the experimental study of Mao et al. (1998)

(210 GPa and 300 K). The next study to simulate inner core

temperatures (4000–6000 K) and pressures by Steinle-

Neumann et al. (2001) produced two unexpected results:

firstly the increase of the unit cell axial c/a ratios by a large

amount (10%) with increasing temperature and secondly the

migration of the P-wave maximum velocity to the basal plane

and normal to the c-axis at high temperature. These new high-

temperature results required a radical change in the seismic

anisotropy model with one-third of c-axes being aligned nor-

mal to the Earth’s rotation axis (Steinle-Neumann et al., 2001)

giving an excellent agreement with travel time differences.

However, more recent calculations (Gannarelli et al., 2003,

2005; Sha and Cohen, 2006) have failed to reproduce the

large change in c/a axial ratios with temperature, which casts

some doubt on the elastic constants of Steinle-Neumann et al.

at high temperature. It should be said that high-temperature

a : Vp curves

0 10 20 30 40 50 60 70 80 90


7.37.47.57.67.77.87.98.08.1

8.28.3

Vp

(km

s−1)

W.L. Mao et al. 2008 IXS+EOS 52 GPa

W.L. Mao et al. 2008 RXD 52 GPa

b)

ure hexagonal close-packed e-phase iron calculated from theion 1999) and Mao et al. (2008a,b,c) or measured velocity in the case ofEOS, equation of state (see text for discussion).

, (2015), vol. 2, pp. 487-538



first-principles calculations represent frontier science in this

area. What is perhaps even more troublesome is that there is

very poor agreement, or perhaps one should say total disagree-

ment, between experimental results and first principles for

P- and S-wave velocity distribution in single-crystal hcp iron

at low temperature and high pressure where methods are con-

sidered to be well established. The experimental techniques

can be criticized as mentioned before. However, differences

can be seen in the magnitude of the anisotropy and the posi-

tion of the minimum velocity, minimum at 50� from the c-axis

(Steinle-Neumann et al., 2001; Stixrude and Cohen, 1995) or

at 90� (Laio et al., 2000; Vocadlo et al., 2003b). In recent years,

a new consistency has appeared in the ab initio models for the

inner core. In Figure 24, we have selected some original mile-

stones like the Mao et al. (1998) experimental curve with bell

shape, which today is considered an experimental artifact. The

first OK ab initiomodel of hcp iron at core pressures by Stixrude

and Cohen (1995) and an early finite temperature ab initio

model by Steinle-Neumann et al. (2001) show to illustrate

the degree of disagreement between these pioneering studies.

All the other studies are recent from 2010 to 2013 all at inner

temperatures and pressures, all for hcp iron, a part from one for

bcc iron from Mattesini et al. (2010). Using the weak cylindri-

cal seismic model for hemispherical inner core anisotropy of

Lythgoe et al. (2014), we can compare with travel time resid-

uals (dt/t) predicted by ab initio elasticity models of iron. From

Figure 24, we can see that seismic anisotropy between the

hemispheres is very different, and single model will not explain

both hemispheres. Several models are in agreement with stron-

ger anisotropy of the western hemisphere (hcp Stixrude and

Cohen, 1995; hcp and bcc from Mattesini et al., 2010), corre-

sponding to Vp anisotropies between 4.1% and 6.3%. Two

models are also in agreement with the weaker seismic anisot-

ropy observations for eastern hemisphere, notably (hcp,

Stixrude and Cohen, 1995; hcp, Sha and Cohen, 2010)

10.0

10.5

11.0

11.5

12.0

12.5

13.0

Ab initio models : Vp Curves

0 10 20 30 40 50 60 70 80 90


Vp

(km

s−1)

SN2001

M1998 RXD

SC1995

M2010

M2010 bcc

M2013M2013 FeNi

SC2010

4.1%

20.5%

7.2%

8.3%7.5%

6.3%

5.7%

2.4%

((a)

Figure 24 (a) The seismic velocities at 0 K or high temperature in single-crelastic constants at inner core pressures, with experimental Vp of Mao et al.(b) The predicted P-wave anisotropy for the inter core using the ab initio elast(dark gray region with black dashed line outlines) and western (light gray) hefor each ab initiomodel and the experimental data of Mao et al. (1998, 1999). Airon, except where otherwise specified. Bcc stands for pure body-centered cuSN2001¼Steinle-Neumann et al. (2001) at 6000 K, SC2010¼Sha and Cohenhas the same authors and temperature with bcc structure, M2013¼Martoreland temperature with a composition of 87.5% Fe 12.5% Ni. See text for disc


corresponding to Vp anisotropies between 2.4% and 4.1%.

All the ab initio models are present as transverse isotropic

symmetry with the symmetry axis parallels to the Earth’s rota-

tion axis. In the hcp iron, which has hexagonal symmetry, this

requires that c-axis of the single crystal to be parallel to the

rotation axis. In the case of bcc iron, the direction of maximum

Vp is the [111] direction, and all the crystals in the model

aggregate are rotated about [111] to produce transverse isotro-

pic symmetry. So for hcp iron, the ab initio curves represent the

maximum possible single-crystal anisotropy, and in the

western hemisphere, there are seismic observations with

anisotropy stronger than perfectly aligned c-axis of the hcp

iron single crystals or bcc iron [111] axis is perfectly aligned.

The bcc model of Mattesini et al. (2010) produces the strongest

Vp anisotropy at 6.3%, but it is less strong than the seismic

western hemisphere measurements of Lythgoe et al. (2014) at

the layer three of their model (Figure 2), which corresponds to

less than 550 km from the center of the Earth. In addition, it is

interesting to note that recent experimental studies at 340 GPa

and 4700 K favor the hcp rather than bcc as the stable structure

of iron in the inner core (Tateno et al., 2012). It is not clear

what mechanism would produce such perfect alignment; it

seems unlikely to be caused by plasticity as this results in non-

perfect statistical alignment, unless there is more anisotropic

iron crystal structure than presently known in the inner core or

the extreme conditions near the center of the Earth. The pro-

cesses related to melting and crystallization has been recently

invoked by several studies of hemispherical nature of the inner

core (Alboussiere et al., 2010; Aubert et al., 2011; Deguen and

Cardin, 2009; Gubbins et al., 2011; Monnereau et al., 2010). It

is possible that other compositions of iron-related phases

might have greater anisotropy than pure iron and currently

known alloys, for example, orthorhombic cementite (Fe3C),

which has Vp anisotropy of 10.3% at inner core pressures

(Mookherjee, 2011b).

0 10 20 30 40 50 60 70 80 90


Ab initio models : Vp Anisotropy (δt/t)

−0.15

−0.10

−0.05

0.05

0.10

Vp

Ani

sotr

opy

(δt/

t)

0.00SC2010 2.4%

SN2001 20.5% M2013M2013 FeNi

8.3%7.5%

M2010M2010 bcc

6.3%5.7%

SC1995

4.1%

Isotropic

b)

Eastern

Western

ystal hexagonal e-phase iron calculated from the ab initio determined(1998, 1999) at 210 GPa at room temperature for reference.ic constants, with the seismic results of Lythgoe et al. (2014) for easternmispheres. Numbers with percent sign are the Vp-wave anisotropiesll models are for pure crystal hexagonal close-packed hexagonal e-phasebic a-phase iron. SC1995¼Stixrude and Cohen (1995) at 0 K,(2010) at 6000 K, M2010¼Mattesini et al. (2010) at 6000 K, M2010 bccl et al. (2013) at 5500 K, and M2013 FeNi has the same authorsussion.

(2015), vol. 2, pp. 487-538



From this brief survey of recent results in this field, it is clear

there is still much to do to unravel the meaning of seismic

anisotropy of the inner core and physics of iron at high pres-

sure and temperature in particular. Although the stability of

hcp iron at inner core conditions has been questioned from

time to time on experimental or theoretical grounds, which the

inner core may not be pure iron (e.g., Poirier, 1994), the major

problem at the present time is to get agreement between theory

and experiment at the same physical conditions. Interpretation

of the mechanisms responsible for inner core seismic anisot-

ropy is out of the question without a reliable estimate of elastic

constants of pure iron and its alloys; nowhere in the Earth is

Francis Birch’s ‘high-pressure language’ (positive proof¼vague

suggestion, Birch, 1952) more appropriate.

2.20.3 Rock Physics

2.20.3.1 Introduction

In this section, I will illustrate the contribution of CPO to

seismic anisotropy in the deep Earth with cases of olivine and

the role of melt. The CPO in rocks of upper mantle origin is

now well established (e.g., Mainprice et al., 2000; Mercier,

1985; Nicolas and Christensen, 1987) as direct samples are

readily available from the first 50 km or so, and xenoliths

provide further sampling down to depths of about 220 km.

Ben and Mainprice (1998) created a database of olivine CPO

patterns from a variety of the upper mantle geodynamic envi-

ronments (ophiolites, subduction zones, and kimberlites) with

a range of microstructures. However, for the deeper mantle

(e.g., Wenk et al., 2004) and inner core (e.g., Merkel et al.,

2005), we had to rely traditionally on high-pressure and high-

temperature experiments to characterize the CPO at extreme

conditions. In recent years, the introduction of various types of

polycrystalline plasticity models to stimulate CPO develop-

ment for complex strain paths has allowed a high degree of

forward modeling using either slip systems determined from

studying experimentally deformed samples using transmission

electron microscopy (e.g., wadsleyite – Thurel et al., 2003;

ringwoodite – Karato et al., 1998), RXD peak broadening

analysis for electron radiation sensitive minerals (e.g.,

Mg-perovskite – Cordier et al., 2004b), or predicted systems

from atomic-scale modeling of dislocations (e.g., olivine,

Durinck et al., 2005a,b and ringwoodite, Carrez et al., 2006).

The polycrystalline plasticity modeling has allowed forward

modeling of upper mantle (e.g., Blackman et al., 1996;

Chastel et al., 1993; Tommasi, 1998), transition zone (e.g.,

Tommasi et al., 2004), lower mantle (e.g., Mainprice et al.,

2008a,b; Wenk et al., 2006), D00 layer (e.g., Merkel et al.,

2006a), and the inner core (e.g., Jeanloz and Wenk, 1988;

Wenk et al., 2000).

2.20.3.2 Olivine the Most-Studied Mineral: State-of-the-Art-Temperature, Pressure, Water, and Melt

Until the papers by Jung and Karato (2001), Katayama et al.

(2004), and Katayama and Karato (2006) were published,

the perception of olivine-dominated flow in the upper mantle

was quite simple with [100] {0kl} slip being universally

accepted as the mechanism responsible for plastic flow and


the related seismic anisotropy (e.g., Mainprice et al., 2000).

The experimental deformation of olivine in hydrous condi-

tions at 2 GPa pressure and high temperature by Karato and

coworkers produced a new type of olivine CPO developed at

low stress with [001] parallel to the shear direction and (100)

in the shear plane, which they called C type, which is associ-

ated with high water content. They introduced a new olivine

CPO classification that illustrated the role of stress and water

content as the controlling factors for the development of five

CPO types (A, B, C, D, and E) (Figure 25). The five CPO

types are assumed to represent the dominant slip system

activity on A� [100](010),B� [001](010),C� [001](100),

D� [100]{0kl}, and E� [100](001). I have taken the Ben

and Mainprice (1998) olivine CPO database with 110 sam-

ples and estimated the percentages for each CPO type and

added an additional class called AG type (or axial b-[010]

girdle by Tommasi et al., 2000), which is quite common in

naturally deformed samples, particularly in samples that have

been associated with melt–rock interaction (e.g., Higgie and

Tommasi, 2012). The CPO types in percentage of the data-

base are A type (49.5%), D type (23.8%), AG type (10.1%), E

type (7.3%), B type (7.3%), and C type (1.8%). It is clear that

CPO associated with [100] direction slip (A, AG, D, and E

types) represents 90.8% of the database and therefore only

9.2% is associated with [001] direction slip (B and C types).

Natural examples of all CPO types taken from the database

are shown in Figure 26, with the corresponding seismic prop-

erties in Figure 27. There are only one unambiguous C-type

sample and another with transitional CPO between B and

C types. The database contains samples from paleo-mid-ocean

ridges (e.g., Oman ophiolite), the circum-Pacific subduction

zones (e.g., Philippines, New Caledonia, Canada), and sub-

continental mantle (e.g., kimberlite xenoliths from South

Africa). There have been some recent reports of the new olivine

B-type CPO (e.g., Mizukami et al., 2004) associated with high

water content, and other B and C types from ultrahigh-pressure

(UHP) rocks (e.g., Xu et al., 2006) have relatively low water

contents. It is instructive to look at the solubility of water in

olivine to understand the potential importance of the C-type

CPO. In Figure 28, the experimentally determined solubility of

water in nominally anhydrous upper mantle silicates (olivine,

cpw, opx, and garnet) in the presence of free water is shown

over the upper mantle pressure range. The values given in the

review by Bolfan-Casanova (2005) are in H2O ppm wt. using

the calibration of Bell et al. (2003), so the values of Karato et al.

in H/106 Si using the infrared calibration of Paterson (1982)

have to be multiplied by 0.22 to obtain H2O ppm wt. If free

water is available, then olivine can incorporate, especially

below 70 km depth, many times the concentration necessary

for C-type CPO to develop according to the results of Karato

and coworkers.

Why is it that the C-type CPO is relatively rare? It is certain

that deforming olivine moving slowly towards the surface will

lose its water due to the rapid diffusion of hydrogen. For

example, even xenoliths transported to the surface in a matter

of hours lose a significant fraction of their initial concentration

(Demouchy et al., 2005). Hence, it very plausible that in the

shallow mantle (less than about 70 km depth), the C type will

not develop because the solubility of water is too low in olivine

at equilibrium conditions and that ‘wet’ olivine upwelling

, (2015), vol. 2, pp. 487-538

600

500

400

300

200

100

00 500 1000 1500 2500

Str

ess

(MP

a) C/E

B

BB

A

A

A

A

A

A

A

C/E

B

AADry Wet

D-type

B-type

C-typeE-type B�

B�

B�

B�

B�

A-type

Water content (ppm H/Si)

Mantle stress levels2.5GPa

3.1GPa

3.1GPa

3.6GPa

Effect ofpressure B

Effect ofwater content

B

B

BA

C

DE

Figure 26 In the left-hand panel the experimental data of Ohuchi et al. (2012) on the effect of water and stress for the development of olivine CPO. Inthe right-hand panel the experimental data of Jung et al. (2008) on the effect of pressure and stress on the development of dry olivine CPO. The CPOfields proposed by Jung and Karato (2001) and Karato et al. (2008) as a function of stress and water content are marked with black solid or dashed lines.In the right-handed panel the symbols with no letter next to them come from Karato et al. (2008) and symbols with a letter come from Ohuchi et al.(2012). CPO types C/E and B’ are transitional CPO defined by Ohuchi et al. (2012). Open symbol represents dry samples or no water detected. In the left-hand panel the pressure is indicated in GPa next to each symbol. See text for discussion.

500

400

300

200

100

200 400 600 800 1000 1200 1400

B-type[001](010)

C-type[001](001)

E-type[100](001)

A-type [100](010)

D-type [100]{0kl}

Water content (ppm H/Si)

Str

ess

(MP

a)

00

[100] [010] [001]

[100] [010] [001]

[100) [010] [001][100] [010] [001]

[100] [010] [001]

A

D

[100] [010] [001]AG-type [100](010)

(10.1%)

(49.5%)

(23.8%)

(7.3%)

(7.3%)

(73.3%)

(1.8%)

Water content (ppm wt)

2001000 300

Figure 25 The classification of olivine CPO originally proposed by Jung and Karato (2001) and Karato et al. (2008) as a function of stress and watercontent. The water content scale in ppm H/Si is that originally used by Jung and Karato. The water content scale in ppm wt is more recentcalibration used by Bolfan-Casanova (2005). The numbers in brackets are the percentage of samples with the fabric types found in the databaseof Ben Ismail and Mainprice (1998). On the pole figure, red indicates [100] parallel to lineation and blue indicates [001] parallel to lineation in samplesfrom the olivine CPO database.



from greater depths and moving towards the surface by slow

geodynamic processes will lose their excess water by hydrogen

diffusion. In addition, any ‘wet’ olivine coming into contact

with basalt melt will tend to ‘dehydrate’ as the solubility of

water in the melt phase is hundreds to thousands of times

greater than olivine (e.g., Hirth and Kohlstedt, 1996). The

melting will occur in upwelling ‘wet’ peridotites at a well-

defined depth when the solidus is exceeded and the volume

fraction of melt produced is controlled by the amount of water


present. Karato and Jung (1998) estimated that melting is

initiated at about 160 km in the normal mid-ocean ridge

source regions and at greater depth of 250 km in back-arc-

type MORB with the production of 0.25–1.00% melt, respec-

tively. Given the small melt fraction, the water content of

olivine is unlikely to be greatly reduced. As more significant

melting will of course occur when the ‘dry’ solidus is exceeded

with 0.3%melt per kilometer melting at about 70 km, then 3%

or more percent melt is quickly produced and the water

(2015), vol. 2, pp. 487-538

6.42Contours (x uni.)

0.00 Max. density = 6.42 Min. density = 0.00

1.0

2.0

3.0

4.0

5.0

8.55 Contours (x uni.)


1.02.03.04.05.06.07.0



0.51.01.52.02.53.03.54.0

[100] [010] [001]



1.0

2.0

3.0

4.0

5.0



1.0 2.0 3.0 4.0 5.0



0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5



0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0



0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5



1.0

1.5

2.0



1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0



0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0



1.0 1.5 2.0 2.5 3.0 3.5

E-Type

DC334A



2.0 4.0 6.0 8.0 10.0 12.0



1.0 2.0 3.0 4.0 5.0 6.0



1.0

2.0

3.0

4.0

5.0

D-Type

CR24

C-Type

Optidsp42

B-Type

NO8B

AG-Type

90OF22

A-Type

90OA61CX

Z



1.0 1.5 2.0 2.5



1.0

1.5

2.0

2.5



1.0 1.5 2.0 2.5 3.0

Figure 27 Natural examples of olivine CPO types from the Ben Ismail and Mainprice (1998) database, except the B-type sample NO8B fromK. Michibayashi (personal communication, 2006). X marks the lineation; the horizontal line is the foliation plane. Contours given in times uniform.




8.71

7.57 Shading - inverse log

Max. velocity = 8.71 Min. velocity = 7.57 Anisotropy = 14.0%

10.01


Max. anisotropy = 10.01 Min. anisotropy = 0.37

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

10.01


8.55

7.72


7.84 7.92 8.00 8.08 8.16 8.24 8.32

6.92

0.37


2.0

3.0

4.0

5.0

6.92

0.37

8.76

7.77


7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60

8.74

0.48


2.0 3.0 4.0 5.0 6.0 7.0

8.74

0.48

9.03

7.64



8.00 8.20 8.40 8.60

11.26

0.17

AVs Contours (%)


4.0

6.0

8.0

11.26

0.17

8.47

7.66

Max. velocity = 8.47 Min. velocity = 7.66

7.76 7.84 7.92 8.00 8.08 8.16 8.24

7.17

0.29


2.0 3.0 4.0 5.0 6.0

7.17

0.29

8.30

7.91


8.00 8.05 8.10 8.15 8.20

3.71

0.04


1.0 1.5 2.0 2.5 3.0

3.71

0.04

E-Type

DC334A

D-Type

CR24

C-Type

Optidsp42

B-Type

NO8B

AG-Type

90OF22

A-Type

90OA61C

X

Vp (km s−1) dVs (%) Vs1 polarizationZ

7.80

8.00

8.20

Anisotropy = 10.0%

Figure 28 Anisotropic seismic properties of olivine at 1000˚C and 3 GPa with CPO of the samples in figure 27. X marks the lineation, the horizontal lineis the foliation plane.




Olivine

Diopside

Enstatite

Pyrop

e

70

120

180

240

300

360

420

0

0

2

4

6

8

10

12

14

0

Pressure (G

Pa)

Dep

th (k

m)

1000 2000 3000 4000

Water content (ppm wt)

Uppermantle

410 km

C-type CPO

Figure 29 The variation in the water content of upper mantle silicates inthe presence of free water from Bolfan-Casanova (2005). The watercontent of the experimentally produced C-type CPO by Jung and Karato isindicated by the light blue region.



content of olivine will be greatly reduced. So the action of

reduced solubility and selective partitioning of water into the

melt phase are likely to make the first 70 km of oceanic mantle

olivine very dry. The other region where water is certainly

present is in subduction zones, where relatively water-rich

sediments, oceanic crust, and partly hydrated oceanic litho-

sphere will add to the hydrous budget of the descending slab.

Partial melting is confined in the mantle wedge to the hottest

regions, where the temperatures approach the undisturbed

asthenospheric conditions at 45–70 km depth, and 5–25 wt.

%melt will be produced by a lherzolitic source (Ulmer, 2001).

Hence, even in the mantle wedge, the presence of large volume

factions of melt is likely to reduce the water content of olivine

to low levels, possibly below the threshold of the C-type CPO,

for depths shallower than 70 km.

Is water the only reason controlling the development of the

C-type fabric or [001] direction slip in general? The develop-

ment of slip in the [001] direction on (010) and (100) planes

at high stresses in olivine has been known experimentally since

transmission electron microscopy study of Phakey et al.

(1972). More recently, Couvy et al. (2004) had produced B-

and C-type CPO in forsterite at very high pressure (11 GPa)

using nominally dry starting materials. However, as pointed

out by Karato (2006), the postmortem infrared spectropho-

tometry of the deformed samples revealed the presence of

water, presumably due to the dehydration of sample assembly

at high pressure. However, concentration of water in the for-

sterite increases linearly with time at high pressure, whereas the

CPO is acquired at the beginning of the experiment in the

stress-relaxation tests conducted by Couvy et al. (2004), so it

is by no means certain that significant water was present at the

beginning of the experiment. Mainprice et al. (2005) suggested

it was pressure that was the controlling variable, partly inspired

by recent atomic modeling of dislocations in forsterite

(Durinck et al., 2005b), that shows the energy barrier for

[100] direction slip increases with hydrostatic pressure, where-

as for [001], it is constant, which could explain the transition

from [100] to [001] with pressure. More recent experiments by

Jung et al. (2009) on dry olivine at pressures of 2.1–3.6 GPa

report a transition from A-type CPO to B-type CPO with

increasing pressure at about 3.1 GPa. Ohuchi et al. (2011)

also working on dry olivine samples found the transition pres-

sure from A type to B type is 7.1 GPa. Axial compression

experiments using two single crystals of dry San Carlos olivine

in series, one favorably oriented for a-direction slip and other

for c-slip, find that faster c-slip occurs above 8 GPa (Raterron

et al., 2007, 2009). New experiments of Ohuchi et al. (2012)

on the deformation of olivine samples in the presence of

water show much more complex distribution of CPO types

(Figure 29) than reported by Karato et al. (2008). In Figure 29,

A types are found with water contents over 500 ppm H/Si, B

types are found at lower stress and very high water contents,

and no definitive C types are reported by Ohuchi et al. (2012).

It appears that both pressure and water content play a role in

controlling CPO types, but the experimental scatter is now

considerable (Figure 29) and exact relation to water content

and pressure needs further calibration.

If we are to accept the experimental results of Karato et al.

(2008), what are the seismic consequences of the recent dis-

covery of C-type CPO in experiments in hydrous conditions?


The classic view of mantle flow dominated by with [100] {0kl}

slip is not challenged by this new discovery as most of

the upper mantle will be dry and at low stress. The CPO

associated with [100] slip, which is 90.8% of the Ben Ismaıl

and Mainprice (1998) database, produces seismic azimuthal

anisotropy for horizontal flow with maximum Vp, polarization

of the fastest S-wave parallel to the flow direction, and

VSH>Vsv. The seismic properties of all the CPO types are

shown in Figure 27. It remains to apply the C-, B-, and possibly

E-type fabrics to hydrated section of the upper mantle.

Katayama and Karato (2006) proposed the mantle wedge in

subduction zones is a region where the new CPO types are

likely to occur, with the B-type CPO occurring in the low-

temperature (high stress and wet) subduction where an old

plate is subducted (e.g., NE Japan). The B-type fabric would

result anisotropy parallel normal to plate motion (i.e., parallel

to the trench). In the back-arc, the A-type (or E-type reported

by Michibayashi et al., 2006 in the back-arc region of NE

Japan) fabric is likely dominant because water content is sig-

nificantly reduced in this region due to the generation of island

arc magma. Changes in the dominant type of olivine fabric can

result in complex seismic anisotropy, in which the fast shear

wave polarization direction is parallel to the trench in the

forearc but is normal to it in the back-arc (e.g., Kneller et al.,

2005). In higher-temperature subduction zones (e.g., NW

America, Cascades), the C-type CPO will develop in the mantle

wedge (low stress and wet) giving rise to anisotropy parallel to

plate motion (i.e., normal to the trench). Although these

models are attractive, trench-parallel flow was first described

by Russo and Silver (1994) for flow beneath the Nazca Plate,

where there is a considerable path length of anisotropic mantle

to generate the observed differential arrival time of the S-waves.

On the other hand, a well-exposed peridotite body analogue of

arc-parallel flow of in south central Alaska reveals horizontal

stretching lineations, and olivine [100] slip directions are sub-

parallel to the Talkeetna arc for over 200 km, clearly indicating

that mantle flow was parallel to the arc axis (Mehl et al., 2003).

, (2015), vol. 2, pp. 487-538



The measured CPO of olivine shows that the E-type fabric is

dominant along the Talkeetna arc; in this case, foliation is

parallel to the Moho suggesting arc-parallel shear with a hori-

zontal flow plane. Tommasi et al. (2006) also reported the

E-type fabric with trench-parallel tectonic context in a highly

depleted peridotite massif from the Canadian Cordillera in

dunites associated with high degrees of melting and hence

probably dry, whereas harzburgites have an A-type fabric. In

limiting the path length to the region of no melting, in coldest

part of the mantle wedge, above the plate, you are also con-

straining the vertical thickness to about 45–70 km depth, if

one accepts the arguments for melting (Ulmer, 2001). For NE

Japan, the volcanic front is about 70 km above the top of the

slab defined by the hypocenter distribution of intermediate-

depth earthquakes (Nakajima and Hasegawa, 2005, their

Figure 1), and typical S-wave delay times are 0.17 s (maximum

0.33 s, minimum 0.07 s). The delay times from local slab

sources are close to the minimum of 0.07 s for trench-parallel

fast S-wave polarizations to the east of the volcanic front (i.e.,

above the coldest part of the mantle wedge) and are over 0.20 s

for the trench normal values (i.e., the back-arc side). Are these

CPOs capable of producing a recordable seismic anisotropy

over such a short path length? The vertical S-wave anisotropy

can be estimated from the delay time given by Nakajima and

Hasegawa (2005) (e.g., 0.17 s), the vertical path length (e.g.,

70 km), and the average S-wave velocity (e.g., 4.46 km s�1) to

give a 1.1% S-wave anisotropy, which is less than maximum

S-wave anisotropy of 1.7% given for a B-type CPO in a vertical

direction given by Katayama and Karato (2006) for horizontal

flow, so B-type CPO is compatible with seismic delay time,

even if we allow some complexity in the flow pattern. The case

for C type in the high-temperature subduction zones is more

difficult to test, as the S-wave polarization pattern will be the

same for C and A types (Katayama and Karato, 2006). The clear

seismic observations of fast S-wave polarizations parallel to

plate motion (trench normal) given for the Cascadia (Currie

et al., 2004) and Tonga (Fischer et al., 1998) subduction zones,

which would be compatible with A or C types. In general, some

care has to be taken to separate below slab, slab, and above slab

anisotropy components to test mantle wedge anisotropy. The

interpretations above follow the logic of Karato et al. (2008),

but as mentioned earlier, the role of C-type CPO as indicator of

wet olivine is seriously questioned by the experiments of

Ohuchi et al. (2012). Most of the reported B-type and C-type

CPOs are associated with garnet peridotite bodies and eclogites

fromUHPmetamorphic terranes, for example, the Sulu terrane

in China (Xu et al., 2006), Alpe Arami and Cima di Gagnone in

the central Alps (Dobrzhinetskaya et al., 1996; Frese et al.,

2003; M€ockel, 1969; Skemer et al., 2006), Val Malenco peri-

dotite in northern Italy ( Jung, 2009), and the Western Gneiss

Region of the Norwegian Caledonides (Katayama et al., 2005;

Wang et al., 2013).

2.20.3.3 Seismic Anisotropy and Melt

The understanding of the complex interplay between plate sep-

aration, mantle convection, adiabatic decompression melting,

and associated volcanism at mid-ocean ridges in the upper

mantle (e.g., Solomon and Toomey, 1992) and the presence of

melt in the deep mantle in the D00 layer (e.g., Williams and


Garnero, 1996) and inner core (Singh et al., 2000) are chal-

lenges for seismology and mineral physics. For the upper

mantle, two contrasting approaches have been used to study

mid-ocean ridges: on one hand marine geophysical (mainly

seismic) studies of active ridges and on the other hand geologic

field studies of ophiolites, which represent ‘fossil’ mid-ocean

ridges. These contrasting methods have yielded very different

views about the dimensions of the mid-ocean ridge or axial

magma chambers. The seismic studies have given us three-

dimensional information about seismic velocity and attenua-

tion in the axial region. The critical question is, how can this

data be interpreted in terms of geologic structure and processes?

To do so, we need data on the seismic properties at seismic

frequencies of melt containing rocks, such as harzburgites, at

the appropriate temperature and pressure conditions. Until

recently, laboratory data for filling these conditions were limited

for direct laboratory measurements to isotropic aggregates (e.g.,

Jackson et al., 2002), but deformation of initially isotropic

aggregates with a controlled melt fraction in shear (e.g.,

Holtzman et al., 2003; Zimmerman et al., 1999) allows simul-

taneous development of the CPO and anisotropic melt distribu-

tion. To obtain information concerning anisotropic rocks, one

can use various modeling techniques to estimate the seismic

properties of idealized rocks (e.g., Mainprice, 1997; Jousselin

andMainprice, 1998; Taylor and Singh, 2002) or experimentally

deformed samples in shear (e.g., Holtzman et al., 2003). This

approach has been used in the past for isotropic background

media with random orientation distributions of liquid-filled

inclusions (e.g., Mavko, 1980; Schmeling, 1985a,b; Takei,

2002). However, their direct application to mid-ocean ridge

rocks is compromised by two factors. Firstly, field observations

on rock samples from ophiolites show that the harzburgites

found in the mid-ocean ridges have strong CPOs (e.g., Boudier

and Nicolas, 1995), which results in the strong elastic anisot-

ropy of the background medium. For the case of the D00 layerand the inner core, the nature of the background media is not

well defined and an isotropic medium has been assumed (Singh

et al., 2000; Williams and Garnero, 1996). Secondly, field obser-

vations show that melt films tend to be segregated in the folia-

tion or in veins, so that the melt-filled inclusions should be

modeled with a shape preferred orientation.

The rock matrix containing melt inclusions is modeled

using effective medium theory, to represent the overall elastic

behavior of the body. The microstructure of the background

medium is represented by the elastic constants of the crystal-

line rock, including the CPO of the minerals and their volume

factions. Quantitative estimates of how rock properties vary

with composition and CPO can be divided into two classes.

There are those that take into account only the volume frac-

tions with simple homogenous strain or stress field and upper

and lower bounds for anisotropic materials such as Voigt–

Reuss bounds, which give unacceptably wide bounds when

the elastic contrast between the phases is very strong, such

as a solid and a liquid. The other class takes into account

some simple aspects of the microstructure, such as inclusion

shape and orientation. There are two methods for the

implementation of the inclusions in effective medium theory

to cover a wide range of concentrations; both methods are

based on the analytic solution for the elastic distortion due to

the insertion of a single inclusion into an infinite elastic

(2015), vol. 2, pp. 487-538



medium given by Eshelby (1957). The uniform elastic strain

tensor inside the inclusion (eij) is given by

eij ¼ 1⁄2 Gikjl +Gjkil

� �Cklmn e*mn

where Gikjl is the tensor Green’s function associated with dis-

placement due to a unit force applied in a given direction, Cklmn

are the components of the backgroundmedium elastic stiffness

tensor, and e*mn is the eigenstrain or stress-free strain tensor

due to the imaginary removal of the inclusion from the con-

straining matrix. The symmetrical tensor Green’s function Gikjl

is given by Mura (1987) as

Gikjl ¼ 1

4p

ðp0

sinydyð2p0

K�1ij xð Þxkxl

�df

with Kip (x)¼Cijpl xj xl, the Christoffel stiffness tensor for

direction (x), and x1¼ sin ycos f/a1,x2¼ sin y sin f/a2 and

x3¼cos y/a3.The angles y and f are the spherical coordinates that define

the vector x with respect to the principal axes of the ellipsoidal

inclusion. The semiaxes of the ellipsoid are given by a1, a2, and

a3. The integration to obtain the tensor Green’s function must

be done by numerical methods, as no analytic solutions exist

for a general triclinic elastic background medium. Greater

numerical efficiency, particularly for inclusions with large

axial ratios, is achieved by taking the Fourier transform of

Gikjl and using the symmetry of the triaxial ellipsoid to reduce

the amount of integration (e.g., Barnett, 1972). The self-

consistent (SC) method introduced by Hill (1965) uses the

solution for a single inclusion and approximates the interac-

tion of many inclusions by replacing the background medium

with effective medium.

In the formulation of SC scheme by Willis (1977), a ratio of

the strain inside the inclusion to the strain in the host medium

can be identified as Ai:

Ai ¼ I +G Ci�Cscsð Þ½ �1

eSCS ¼Xi¼n

i¼1

ViAi sSCS ¼Xi¼n

i¼1

ViCiAi

CSCS ¼ sSCS

eSCS �1

where I is the symmetrical fourth-rank unit tensor, Iijkl¼1/2

(dikdjl+dil djk),dik is the Kronecker delta, Vi is the volume

fraction, and Ci are the elastic moduli of the ith inclusion.

The elastic constants of the SC scheme (Cscs) occur on both

sides of the equation because of the stain ratio factor (A), so

that solution has to be found by iteration. This method

is the most widely used in Earth sciences, being relatively

simple to compute and well established (e.g., Kendall and

Silver, 1996, 1998). Certain consider that when the SC is

used for two phases, for example, a melt added to a solid

crystalline background matrix, the melt inclusions are isolated

(not connected) below 40% fluid content, and the solid and

fluid phases can only be considered to be mutually fully inter-

connected (biconnected) between 40% and 60%. For our

application to magma bodies, one would expect such intercon-

nection at much lower melt fractions. The second method is

DEM. This models a two-phase composite by incrementally

adding inclusions of melt phase to a crystalline background


phase and then recalculating the new effective background

material at each increment. McLaughlin (1977) derived the

tensorial equations for DEM as follows:

dCDEM

dV¼ 1

1�Vð Þ Ci�CDEM� �

Ai

Here, again the term Ai is the strain concentration factor

coming from Eshelby’s formulation of the inclusion problem.

To evaluate the elastic moduli (CDEM) at a given volume

fraction V, one needs to specify the starting value of CDEM

and which component is the inclusion. Unlike the SC, the

DEM is limited to two components A and B. Either A or

B can be considered to be the included phase. The initial

value of CDEM is clearly defined at 100% of phase A or B.

The incremental approach allows the calculations at any com-

position irrespective of starting concentrations of original

phases. This method is also implemented numerically and

addresses the drawback of the SC in that either phase can be

fully interconnected at any concentration. Taylor and Singh

(2002) attempted to take advantage of both of these methods

and minimize their shortcomings by using a combined effec-

tive medium method, a combination of the SC and DEM

theory. Specifically, they used the formulation originally pro-

posed by Hornby et al. (1994) for shales; an initial melt-

crystalline composite is calculated using the SC with melt

fraction in the range 40–60% where they claim that each

phase (melt and solid) is connected and then uses the DEM

method to incrementally calculate the desired final composi-

tion that may be at any concentration with a biconnected

microstructure.

To illustrate the effect on oriented melt inclusions, I will use

the data from the study of a harzburgite sample (90OF22)

collected from the Moho transition zone of the Oman ophio-

lite (Mainprice, 1997). The CPO and petrology of the sample

have been described by Boudier and Nicolas (1995), and the

CPO of the olivine (AG Type) is given in Figure 26. The

mapping area records a zone of intense melt circulation

below a fast-spreading paleo-mid-ocean ridge at a level

between the asthenospheric mantle and the oceanic crust.

I use the DEM effective medium method combined with

Gassmann’s (1951) poroelastic theory to ensure connectivity

of the melt system at low frequency relevant to seismology; see

Mainprice (1997) for further details and references. The harz-

burgite (90OF22) has a composition of 71% olivine and 29%

opx. The composition combined with CPO of the constituent

minerals and elastic constants extrapolated to simulate condi-

tions of 1200 �C and 200 MPa where the basalt magma would

be liquid predicts the following P-wave velocities in principal

structural direction: X¼7.82, Y¼7.69, and Z¼7.38 km s�1

(X¼ lineation, Z¼normal to foliation, and Y is perpendicular

to X and Z). The crystalline rock with nomelt has essentially an

orthorhombic seismic anisotropy. Firstly, I have added basalt

spherical basalt inclusions; the velocities for P- and S-waves

decrease and attenuation increases (Figure 30) with increasing

melt faction and the rock becomes less anisotropic, but pre-

serves its orthorhombic symmetry. When ‘pancake’-shaped

basalt inclusions with X :Y :Z¼50:50:1 are added, to simulate

the distribution of melt in the foliation plane observed by

Boudier and Nicolas (1995), certain aspects of the original

orthorhombic symmetry of the rock are preserved, such as

, (2015), vol. 2, pp. 487-538

P-waves S-wavesMelt inclusion shape X:Y:Z =50:50:1

P-wavesMelt inclusion shape X:Y:Z =1:1:1

4

5

6

7

8

0.0001

0.001

0.01

0 10 20 30 40 50

Vp

(km

s−1)

Q−1

Q−1

Q−1

Q−1

Melt (%)

Q−1

Vp(X)

Vp(Y)

Vp(Z)

2.00

2.50

3.00

3.50

4.00

4.50

0.0001

0.001

0.01

0.1

0 10 20 30 40 50

Vs

(km

s−1)

Melt (%)

Vs1(X) = Vs1(Y)

Vs2(X) = Vs1(Z)

Vs2(Y) = Vs2(Z)

3

4

5

6

7

8

0.0001

0.001

0.01

0.1

0 10 20 30 40 50

Vp

(km

s−1)

Melt (%)

Vp(X)

Vp(Y)

Vp(Y)

(X) and Q−1 (Y)

(Z)

0

1

2

3

4

5

10−6

10−5

10−6

10−5

0.0001

0.001

0.01

0.1

1

0 10 20 30 40 50

Vs

(km

s−1)

Melt (%)

Vs1(X) = Vs1(Y) = VSH

VSV

s1(X) = Q−1 s1(Y)

= Q−1SH

svQ−1

Q−1

Q−1

Q−1

Q−1

Figure 30 The effect of increasing basalt melt fraction on the seismic velocities and attenuation (Q�1) of a harzburgite. See text for details.



the difference between Vp in the X direction and Vp in the Y

direction. However, many velocities and attenuations change

illustrating the domination of the transverse isotropic symme-

try with Z direction symmetry axis associated with the

‘pancake’-shaped basalt inclusions. Vp in the Y direction

decreases rapidly with increasing melt fraction, causing the

seismic anisotropy of P-wave velocities between Z and X or Y

to increase. The contrast in behavior for P-wave attenuation is

also very strong with attenuation (Q�1) increasing for Z and

decreasing for X and Y directions. For the S-waves, the effects

are even more dramatic and more like a transverse isotropic

behavior. The S-waves propagating in X direction with a Y

polarization, VsX(Y), and Y direction with polarization X,

VsY(X) (see Figure 31 for directions), are fast velocities because

they are propagating the XY (foliation) plane with polarizations

in XY plane; we can call these VSH waves for a horizontal folia-

tion. In contrast, all the other S-waves have either their


propagation or polarization (or both) direction in the Z direc-

tion and have the same lower velocity; these we can call VSV.

Similarly for the S-wave attenuation,VSH are less attenuated than

VSV. From this study,we can see that a fewpercent of alignedmelt

inclusions with high axial ratio can change the symmetry and

increase the anisotropy of crystalline aggregate (see Figure 31 for

summary), completely replacing the anisotropy associated with

crystalline background medium in the case of S-waves. Taylor

and Singh (2002) came to the same conclusion that S-wave

anisotropy is an important diagnostic tool for the study of

magma chambers and regions of partial melting.

One of the most ambitious scientific programs in recent

years was the mantle electromagnetic and tomography (MELT)

experiment that was designed to investigate the forces that

drive flow in the mantle beneath a mid-ocean ridge (MELT

Seismic Team, 1998). Two end-member models often pro-

posed can be classified into two groups; the flow is a passive

(2015), vol. 2, pp. 487-538

Y

X

Z

Vp x - high

Vp z - low

Vp y - intermediate

Vs z(x) - low

Vs

x(z)

-low

V s y ( x ) - h i g h

Vs y(z)- low

Vs x(y) - high

Vs z(y) -

lowVelocities

VSH

Vsh

VSV

VSV

VSVVSV

Y

X

Z

Vp x - low

Vp z - high

Vp y - intermediate

Vs z(x) - high

Vsx

(z)-

high

V s y ( x ) - l o w

Vs y(z)- high

Vsx(y) - low

Vs z(y) - high

Attenuation 1/Q

Figure 31 A graphical illustration of the ‘pancake’-shaped melt inclusions (red) in the foliation (XY) plane (where X is the lineation) and the relationbetween velocity and attenuation (Q�1). The melt is distributed in the foliation plane. The velocities have an initial orthorhombic symmetry with VpX>Y>Z. The directions of high velocity are associated with low attenuation. P-waves normal to the foliation have the lowest velocity. S-waves withpolarizations in the foliation (XY) plane have the highest velocities (VSH). Diagram inspired a figure in the thesis of Barroul (1993).



response to diverging plate motions, or buoyancy forces sup-

plied a plate-independent component variation of density

caused by pressure release partial melting of the ascending

peridotite. The primary objective in this study was to constrain

the seismic structure and geometry of mantle flow and its

relationship to melt generation by using teleseismic body

waves and surface waves recorded by the MELT seismic array

beneath the superfast-spreading southern East Pacific Rise

(EPR). The observed seismic signal was expected to be the

product of elastic anisotropy caused by the alignment of oliv-

ine crystals due tomantle flow and the presence of alignedmelt

channels or pockets of unknown structure at depth (e.g.,

Blackman and Kendall, 1997; Blackman et al., 1996; Kendall

et al., 2004; Mainprice, 1997). Observations revealed that on

the Pacific Plate (western) side, the EPR had lower seismic

velocities (Forsyth et al., 1998; Toomey et al., 1998) and

greater shear wave splitting (Wolfe and Solomon, 1998). The

shear wave splitting showed that the fast shear polarization was

consistently parallel to the spreading direction and at no time

parallel to the ridge axis with no null splitting being recorded

near the ridge axis. In addition, the delay time between S-wave

arrivals on the Pacific Plate was twice that of the Nazca Plate.

P delays decreased within 100 km of the ridge axis (Toomey

et al., 1998), and Rayleigh surface waves indicated a decrease in

azimuthal anisotropy near the ridge axis. Any model of the

EPR must take into account that the average spreading rate

at 17�S on the ridge is 73 mm year�1 and the ridge migrates

32 mm year�1 to the west. Anisotropic modeling of the P and

S data within 500 km of the ridge axis by Toomey et al. (2002)

and Hammond and Toomey (2003) showed that a best fitting

finite strain hexagonal symmetry 2-D flow model had an

asymmetrical distribution of higher melt fraction and temper-

ature, dipping to the west under the Pacific Plate and lower

melt fraction and temperature with an essentially horizontal

structure under the Nazca Plate. Hammond and Toomey

(2003) introduced low melt fractions (>2%), in relaxed (con-

nected) cuspate melt pockets (Hammond and Humphreys,

2000) to match the observed velocities. Blackman and

Kendall (2002) used a 3-D texture flow model to predicted

pattern of upper mantle flow beneath the EPR oceanic


spreading center with asymmetrical asthenospheric flow pat-

tern. Blackman and Kendall (2002) explored a series of models

for the EPR and found that asymmetrical thermal structure

proposed by Toomey et al. (2002) produced the model in

closest agreement with seismic observations. The 3-D model

shows that shear wave splitting will be lowest at about 50 km

to the west of the EPR on the Pacific Plate with similar low

value at 400 km to the east on the Nazca Plate and not the ridge

axis because of the underlying asymmetrical mantle structure.

The MELT experiment has showed that melt flow beneath a

fast-spreading ridge is more complicated than originally pre-

dicted, with a deep asymmetrical structure present to 200 km

depth. The influence of the near surface configuration (e.g.,

ridge migration) was also important in controlling the

asthenospheric return flow towards the Pacific superswell in

the west (Hammond and Toomey, 2003). The influence of

melt geometry appears to be small in the case of the EPR, as

the essential anisotropic seismic structure is captured by

models that do not have melt geometries with strong shape

preferred orientation. The situation may be different for the

oceanic crust at fast oceanic spreading centers. The seismic

anisotropy in regions of important melt production, such as

Iceland (Bjarnason et al., 2002), does not show the influence

of melt geometry on anisotropy, but rather the influence of

large-scale mantle flow. In other contexts, such as the rifting,

for example, the Red Sea (Vauchez et al., 2000) and East

African Rift (Kendall et al., 2004), the melt geometry does

seem to have an important influence of seismic anisotropy.

2.20.4 Conclusions

In this chapter, I have reviewed some aspects of seismology

that have a bearing of the geodynamics of the deep interior of

the Earth. In particular, I have emphasized the importance of

seismic anisotropy and the variation of anisotropy on a global

and regional basis. In the one-dimensional PREM model

(Dziewo�nski and Anderson, 1981), anisotropy was confined

to the first 250 km of the upper mantle. Subsequently, other

, (2015), vol. 2, pp. 487-538



studies of the mantle found various additional forms of global

anisotropy associated with the transition zone, the 670 km

boundary layer, and the D00 layer (e.g., Montagner, 1994a,b;

Montagner and Kennett, 1996; Montagner, 1998). The most

recent global studies using more complete data sets and new

methods of analysis emphasize the exceptionally strong nature

of the upper mantle anisotropy, the anisotropy of the D00 layer,and no significant deviation from the original isotropic PREM

for the rest of the mantle (e.g., Beghein et al., 2006; Panning

and Romanowicz, 2006). To explain velocity variations that

are observed in the mantle studies using probabilistic tomog-

raphy places the emphasis on chemical heterogeneity and

lateral temperature variations in the mantle (Deschamps and

Trampert, 2003; Trampert et al., 2004). Trampert and van der

Hilst (2005) argued that spatial variations in bulk major ele-

ment composition dominate buoyancy in the lowermost man-

tle, but even at shallower depths, its contribution to buoyancy

is comparable to thermal effects. The case of the D00 layer isperhaps even more challenging, as it is clearly a region with not

only strong temperature and compositional gradients (e.g., Lay

et al., 2004) but also a regionally varying seismic anisotropy

(e.g., Maupin et al., 2005; van der Hilst et al., 2007; Wookey

et al., 2005a) with an overall global signature (e.g., Beghein

et al., 2006; Montagner and Kennett, 1996; Panning and

Romanowicz, 2006). The inner core has well-known travel

time variations that can be modeled to fit various single or

double concentric layered anisotropy scenarios. Some studies

(Beghein and Trampert, 2003; Ishii and Dziewo�nski, 2002)

tend to favor a difference in anisotropy between the outermost

inner core and innermost inner core; however, they disagree in

the magnitude and symmetry of the anisotropy. Calvet et al.

(2006) suggested that the data set is too poor to distinguish

between several of the current models, whereas new high-

quality data (Lythgoe et al., 2014) clearly favor a strongly

anisotropic western hemisphere and weakly anisotropic east-

ern one. Lythgoe et al. (2014) also questioned if the outermost

and innermost inner core model is an artifact of not taking into

account the hemispherical structure of the inner core. In the

mantle and the inner core, there are often differences between

studies at the global and regional scales and differences

between 1-D and 3-D global models. The seismic sampling

over different radial and lateral length scales using surface

and body waves of variable frequency has made reference

models very important in the reporting and understanding of

complex data sets. Kennett (2006) had shown, for example, it

is difficult to achieve comparable P- and S-wave definition for

the whole mantle. Mineral physics can play important role as a

representation based on elastic moduli rather than wave speeds

that would provide a better comparator for interpretation in

terms of composition, temperature, and anisotropy.

In addressing the basics of elasticity, wave propagation in

anisotropic crystals, and the nature of the anisotropy polycrys-

talline aggregates with CPO, I hope I have provided some of

keys necessary for the interpretation of seismic anisotropy.

CPO produced by plastic deformation is the link between

deformation history and the seismic anisotropy of the Earth’s

deep interior. We have seen earlier that many regions of the

mantle (e.g., lower mantle) do not have a pronounced seismic

anisotropy. However, from mineral physics, we have seen that

in the upper mantle, olivine has a strong elastic anisotropy; in


the transition zone, wadsleyite is quite anisotropic; in the

lower mantle, Mg-perovskite and MgO have increasing anisot-

ropy with depth; in the D00 layer, postperovskite is very

anisotropic; and the inner core, hcp iron is moderately aniso-

tropic. In addition, if we add minerals from the hydrated

mantle in subduction regions, such as the antigorite, chlorite,

talc, and brucite, they can be very anisotropic at low pressure

and S-wave anisotropy and can be very high for antigorite and

chlorite even at high pressure. Subduction-related nonhydrous

minerals, such as Al-CF phase, hollandite I, and stishovite, can

have exceptionally high anisotropy in the transition zone and

lower mantle. Potentially, the mineralogy suggests that seismic

anisotropy could be present if these minerals have a CPO.

Aligned melt inclusions and compositional layers can also

produce anisotropy. To understand why there are regions in

the deep Earth that have no seismic anisotropy is clearly a

challenge for mineral physics, seismology, and geodynamics.

Acknowledgments

I thank Guilhem Barruol, Francois Boudier, Patrick Cordier,

Brian Kennett, Bob Liebermann, Katsuyoshi Michibayashi,

Sebastien Merkel, Adolphe Nicolas, Andrea Tommasi, and the

late Paul Silver for their helpful discussions. I thank Don Issak

for providing a copy of his 2001 publication on Elastic Properties

of Minerals and Planetary Objects, which is an excellent compli-

ment to this work. I also thank Steve Jacobsen for providing

preprints of his work on hydrous minerals. I thank Mainak

Mookherjee for providing data in digital format from his publi-

cations. I thank James Wookey for his constructive review of the

first version of this chapter and useful suggestions.

Finally, I thank volume editors G. David Price and Lars

Stixrude for providing the occasion to write this chapter and

both them and series editor Gerald Schubert for their patience

during long gestation of this manuscript.

I dedicate this chapter to the memory of Paul G. Silver,

extraordinary seismologist, genial colleague, and great friend.

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Seismic Anisotropy of the Deep Earth from a Mineral and Rock