Some aspects of seismic anisotropy

Elasticity tensor and symmetries of the medium

For seismic anisotropy we start with the general relation between stress and strain according to Hooke:

σij = cijklεkl

where σij is the stress tensor, εkl the strain tensor, and cijkl is the elasticity (or stiffness) tensor. This4th order tensor has 34 = 81 elements of which only 21 are independent (cijkl = cjikl = cijlk = cklij).

The 21 coefficients for an arbitrary isotropic medium are often arranged in a 6x6 matrix, thestiffness matrix Cij

C =

c1111 c1122 c1133 c1123 c1113 c1112c2211 c2222 c2233 c2223 c2213 c2212c3311 c3322 c3333 c3323 c3313 c3312c2311 c2322 c2333 c2323 c2313 c2312c1311 c1322 c1333 c1323 c1313 c1312c1211 c1222 c1233 c1223 c1213 c1212

[Pairs (i, j) or (k, l) of the elasticity tensor convert to indices ofCij according to: (1, 1)→ 1, (2, 2)→2, (3, 3)→ 3, (2, 3)→ 4, (1, 3)→ 5, (1, 2)→ 6.]

There are many minerals in crust, mantle and core that have anisotropic properties. How many in-dependent coefficients the elasticity tensor of each of the crystals has, depends on the crystallographicstructure. A medium with hexagonal symmetry has 5 independent coefficients. Hexagonal symmetry,or, equivalently, cylindrical symmetry, is of particular interest to seismology because it is relativelysimple and it can approximate many actual situations in the Earth (e.g. alignment of olivine crystalsalong the ~a axis with randomly oriented~b and ~c axes, or laminated structure).A medium with hexagonal symmetry with a vertical symmetry axis is called transversely isotropic. Amedium with hexagonal symmetry with a horizontal symmetry axis is called azimuthally anisotropic.

Wave propagation in anisotropic media

Seismic anisotropy distorts the particle motion and, in general, no clear distinction can be made be-tween the modes of elastic waves such as P and S waves in the isotropic case. This implies seismicvelocity variations as a function of (1) propagation direction and (2) polarization direction of the seis-mic waves. This will be shown from the solutions of the wave equation in an anisotropic medium.Assume a homogeneous anisotropic medium and a plane wave solution with phase velocity c, ampli-tude and polarization vector ~a, and propagation direction ~n:

~u(~r, t) = ~af(t− ~n · ~rc

)

Our problem is now to find suitable combinations of c and ~a that satisfy the anisotropic wave equationfor a wave travelling in the ~n direction:

cijkl∂j∂kul = ρ∂ttui

Substitution of the plane wave solution ~u in the wave equation:

cijkl∂j∂k[alf(t− ~n · ~rc

)] = ρ∂tt[aif(t− ~n · ~rc

)]

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gives

cijklalnjnk1

c2= ρai

ormilal = c2ai

withmil =

1

ρcijklnjnk

The matrixM is called the Christoffel matrix. If we find the eigenvalues and eigenvectors of the matrixmil, we have the phase velocities and the propagation directions of the independent wave types. Thiswill be illustrated in the following for isotropic and hexagonal media.

Type of symmetry Number of independent Typicalelastic coefficients mineral

isotropic solid 2 volcanic glasscubic 3 garnethexagonal 5 icetrigonal I 7 ilmenitetrigonal II 6 quartztetragonal 6 stishoviteorthorhombic 9 olivinemonoclinic 13 hornblendetriclinic 21 plagioclase

Table 1: Number of independent elastic coefficients for selected symmetry systems and typical minerals orEarth’s materials

Figure 1: Examples of symmetry planes in the different symmetry systems.

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Figure 2: Compressional and shear-wave velocities in a momocrystal of olivine. Original data from Kumazawa& Anderson (1969).

Isotropic medium

First we solve the system for an isotropic medium

mil =1

ρcijklnjnk

withcijkl = λδijδkl + µ(δikδjl + δilδjk)

Using ~n = (n1, n2, n3)T with n21 + n22 + n23 = 1:

mil =1

ρ[λδijδkl + µ(δikδjl + δilδjk)]njnk

we findmil =

1

ρ[λninl + µ(ninl + δil(n1n1 + n2n2 + n3n3))]

mil =1

ρ[λninl + µ(ninl + δil)]

Thus

M =1

ρ

(λ+ µ)n21 + µ (λ+ µ)n1n2 (λ+ µ)n1n3(λ+ µ)n1n2 (λ+ µ)n22 + µ (λ+ µ)n2n3(λ+ µ)n1n3 (λ+ µ)n2n3 (λ+ µ)n23 + µ

The eigenvalues are obtained from det(M − ΛI) = 0, giving 1

ρ3(µ − Λ)2(λ + 2µ − Λ) = 0 where

Λ denotes the eigenvalue. We find the eigenvalues Λ1 = Λ2 = µρ and Λ3 = λ+2µ

ρ . Checking theexpression of the Christoffel matrix milal = c2ρai we find that the eigenvalues are related to thephase velocities: c1 =

√µρ , c2 =

√µρ , and c3 =

√λ+2µρ .

The eigenvectors ~a corresponding to the first two eigenvalues Λ1 and Λ2 are found to satisfy n1a1 +n2a2 + n3a3 = 0. This means that the two orthogonal eigenvectors are perpendicular to the prop-agation direction ~n. The eigenvector corresponding to Λ3 satisfies ~a = ~n. Thus we have found

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3 independent plane wave solutions of the isotropic wave equation: one P-wave and two S-waves.The phase velocities are obtained from the eigenvalues of the Christoffel matrix, the polarizations aredetermined by the eigenvectors.

Note that the stiffness matrix Cij for the isotropic case is:

C =

λ+ 2µ λ λ 0 0 0λ λ+ 2µ λ 0 0 0λ λ λ+ 2µ 0 0 00 0 0 µ 0 00 0 0 0 µ 00 0 0 0 0 µ

Hexagonal symmetry

For hexagonal symmetry with a vertical (x3) symmetry axis (transverse isotropy) the stiffness matrixis:

C =

A A− 2N F 0 0 0A− 2N A F 0 0 0F F C 0 0 00 0 0 L 0 00 0 0 0 L 00 0 0 0 0 N

with the coefficients A,C, F, L, and N according to Love (1927).If we choose

~n =

100

we obtain

mil =1

ρcijklnjnk =

1

ρci11l

or

M =

m11 m12 m13

m21 m22 m23

m31 m32 m33

=1

ρ

c1111 c1112 c1113c2111 c2112 c2113c3111 c3112 c3113

=1

ρ

C11 C16 C15

C61 C66 C65

C51 C56 C55

or

M =1

ρ

A 0 00 N 00 0 L

We immediately recognize the eigenvalues:

λ1 = Aρ

λ2 = Nρ

λ3 = Lρ

and find the corresponding phase velocities:

c1 =√

Aρ ‘P ′

c2 =√

Nρ ‘S′

c3 =√

Lρ ‘S′

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with the polarization vecors:

~a1 =

100

~a2 =

010

~a3 =

001

This means that for propagation along the x1 axis (in a medium with hexagonal symmetry with avertical symmetry axis such as a layered medium), we have 3 waves with different velocity and po-larization, one of P-type with phase velocity αH =

√Aρ , and two of S-type. If an S-wave enters the

medium along the x1 axis with components of polarization along both the x2 and x3 axes, the twocomponents along these axes will propagate at different speeds. This is called shear-wave splitting.

Figure 3: (Top) Three possible seismic plane-waves propagating in the plane perpendicular to the symmetryaxis (x3 here). (Bottom) P and S waves propagating in the direction of the symmetry axis.

Now we choose

~n =

001

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for vertically propagating waves. We obtain

mil =1

ρcijklnjnk =

1

ρci33l =

1

ρ

L 0 00 L 00 0 C

and find waves with

λ1 = Lρ c1 =

√Lρ ‘S′

λ2 = Lρ c2 =

√Lρ ‘S′

λ3 = Cρ c3 =

√Cρ ‘P ′

with polarization vectors

~a1 =

100

~a2 =

010

~a3 =

001

The first two solutions are of S-type (polarization ⊥ propagation). Note that there is no shear wavesplitting for propagation in the x3 direction. However, we see that the P-wave in the x1 directiontravels with a different speed than the one in the x3 direction.For a stack of isotropic layers, one can show thatA > C andN > L, so that αH > αV and βH > βV .For intermediate directions the polarization vector ~a is not necessarily parallel (or perpendicular) tothe propagation direction ~n.

Phase and group velocity

The fact that the phase velocity varies with the direction of propagation leads to a further complicationwhich does not exist in isotropic media. The direction of propagation of the energy of a wave packetmay be different from the direction ~n of propagation of the phase. The group-velocity vector ~v doesnot coincide with the phase-velocity vector ~c except for particular directions. Consider a plane wavepropagating in an anisotropic medium and a screen with a large slot parallel to the plane wave as shownin the figure. If Huygens principle is used to propagate the plane wave from the elementary pointsources in the slot, we obtain non-circular wavefronts due to directional dependent phase velocities,and see that constructive interference may take place in an obliques direction with regard to the normal~n of the incident plane wave. For weak anisotropy the departure of the group-velcity vector from thenormal ~n remains quite small.

Weak anisotropy

In general, if a P or S wave is incident on an anisotropic structure, three mutually orthogonal wavesare excited which travel at different speeds. However, when the anisotropy is weak, the nature ofanisotropy can be described as perturbations to the isotropic situation. We then speak of quasi P- andS-waves: qP and qS. For weak anisotropy one finds (Backus, 1965; Crampin, 1977):

ρV 2P = A+Bc cos 2θ +Bs sin 2θ + Cc cos 4θ + Cs sin 4θ

ρV 2SH = D + Ec cos 4θ + Es sin 4θ

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Figure 4: Surface wave anisotropy.

ρV 2SV = F +Gc cos 2θ +Gs sin 2θ

where VP is the velocity of the (quasi) P wave, VSH the velocity of the horizontally polarized (quasi)shear wave, VSV the velocity of the vertically polarized (quasi) shear wave, θ the azimuth, andA,D,F,Bc,s, Cc,s, Ec,s, Gc,s are constants. For a transversely isotropic material we have Bc,s =Cc,s = Ec,s = Gc,s = 0.

Similar expressions may be obtained for surface waves (Smith and Dahlen, 1973; Montagner andNataf, 1986):

c(ω, θ) = c0(ω) + c1(ω) cos 2θ + c2(ω) sin 2θ + c3(ω) cos 4θ + c4(ω) sin 4θ

Since Love waves mainly involve the SH component of movement, the azimuthal anisotropy of Lovewaves is dominated by the 4θ-terms. On the other hand, Rayleigh waves mainly involve the SVcomponent, and therefore the 2θ-terms dominate the azimuthal dependence of Rayleigh waves.

Seismogical observations of anisotropy

As can be inferred from the above, there are two ways of detecting anisotropy:(1) by detecting velocity variations as a function of direction,(2) by detecting differences in velocity as a function of polarization.A very brief overview of observations is listed below:

1. Azimuthal anisotropy of compressional waves. The primary data in this case are travel timeanomalies observed by controlled-source experiments using refracted (Pn) waves. The fastdirection in oceanic lithosphere coincides with the (paleo)spreading direction.

2. Azimuthal anisotropy of surface waves. Rayleigh waves have significant azimuthal anisotropy.The direction of the fast Rayleigh waves (with a period of 200 s) does not always agree withthe direction of plate motion, but shorter period Rayleigh waves seem to have the fast directionnearly parallel to the paleospreading direction.

3. Polarization anisotropy of surface waves or the so called ‘Love-Rayleigh discrepancy’. In somestudies phase velocity measurements of both Rayleigh and Love waves for a region cannot beexplained by a smoothly varying isotropic model. Most of the oceanic upper mantle showsVSH > VSV anisotropy, and transverse isotropy is usually asumed to describe the velocity

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structure. However, more detailed studies show that the polarization anisotropy depends on thedepth and age of the oceanic lithosphere.

4. Shear wave splitting. The most decisive observation of anisotropy is shear-wave splitting, and alarge number of publications has recently appeared on splitting of SKS phases.

5. Inner core anisotropy. The main data used to study the inner core are short period PKIKP wavesand long period normal mode splitting functions. Both data show cylindrical anisotropy (i.e.hexagonal) with the symmetry axis aligned with the Earth’s rotation axis.

Figure 5: Azimuthal dependence of Pn-wave velocities in the upper most mantle near Hawaii (after Morris etal, 1969).

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Figure 6: (a) Azimuthal variations of a maximum Rayleigh wave phase velocity at a period of 200 seconds(Tanimoto & Anderson, 1984). (b) Flow lines at 260 km depth in the kinematic model of Hager & )’Connell(1979).

Figure 7: Love and Rayleigh wave measurements for the Iberia peninsula. Dashed lines are isotropic modelpredictions, which give too low velocities for the Love wave measurements (from Maupin & Cara, 1992).

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Figure 8: (Sketch showing S-wave splitting due to the transmission through an anisotropic medium. An initialS wave split into two waves S1 and S2 when entering the anisotropic medium. Ray tgheory predicts that the twomutually perpendicular polarization vectors can rotate slowly around the ray in a smoothly varying medium.

Figure 9: Example of SKS split phases. The transverse component of SKS should not appear if the Earth wererepresentable by a isotropic symmetrically spherical model, and disappares when correcting for anisotropy.

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a Equatorial path

0−10 10 20 30 40−20

Travel time (s) w.r.t. PKPbc arrival

b Polar path

Station LVC, event data 22/02/2004

Station LVC, event data 22/02/2004

= 22.7 o

o

o

oEpicentral distance = 148.5

Epicentral distance = 150.1

ζ

= 57ζ

PKIKP

PKIKP PKPbc

PKPbc

Figure 10: Example of two PKIKP observations (a) for an equatorial path where PKIKP arrives on the predictedtime and (b) for a polar path, where PKIKP arrives early because of inner core anisotropy.

0

30

60

90

ζ (

de

g)

-2 0 2δt (sec)

data0.6% ani

0

30

60

90

-2 0 2δt (sec)

data1.6% ani

0

30

60

90

-4 -2 0 2 4δt (sec)

data3.8% ani

0

30

60

90

-6 -4 -2 0 2 4 6δt (sec)

data3.3% ani

b PKPbc−PKIKPa PKiKP−PKIKP c PKPab−PKIKP

0−57km 150−350km 220−900km57−100km

Figure 11: PKIKP data shown as a function of angle ζ with the Earth’s rotation axis.

a Observation

Mode 16S5

b Inner core anisotropy

Durek & Romanowicz (1999)

c Mantle structure only

S20RTS + CRUST5.1

-16 -8 0 8 16µHz

Figure 12: Normal mode splitting function observations of inner core anisotropy. (a) Zonal anisotropy is seenin the data and in (b) Synthetic prediction for inner core anisotropy. Only mantle structure does not match theobserved anisotropy.

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Causes of anisotropy

There are two causes of anisotropy:

1. Preferred orientation of crystallographic axes of elastically anisotropic minerals such as olivineor orthopyroxene. This type of anisotropy is generally referred to as lattice preferred orientation(LPO). There are two mechanisms to achieve LPO.

(a) Dislocation glide. Crystallographic axes of each grain rotate because of the constraints ofdeformation imposed by the surrounding grains. The nature of LPO due to this mecha-nism depends on the geometrical constraints on the deformation and is related to the flowgeometry.

(b) Grain boundary migration. When there is a large variation in the free energies of thegrains and when the grain boundaties are mobile, grains with low free energy will grow atthe expense of high free energy grains. Stress controlled seismic anisotropy may thus beformed.

2. Anisotropic shape distribution of isotropic materials, such as laminated structure or fluid filledcracks. The shape of melt pockets is related to the stress distribution. It is expected that insuboceanic lithosphere, where the flow is presumably simple shear in a horizontal plane, themelt-filled cracks will be inclined at 45 degrees with respect to the horizontal plane.

Figure 13: Two mechanisms to get lattice preferred orientation (LPO), (a) dislocation glide and (b) grainboundary migration.

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