Geophys. J. Int. (2009) 178, 962–975
doi: 10.1111/j.1365-246X.2009.04211.x
Normal mode coupling due to hemispherical anisotropic structure
in Earth’s inner core
J. C. E. Irving,1 A. Deuss1 and J. H. Woodhouse2
1 Institute
of Theoretical Geophysics & Bullard Laboratories, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK. E-mail: [email protected]
of Earth Sciences, University of Oxford, Parks Road, Oxford OX1 3PR, UK
2 Department
SUMMARY
We present illustrative calculations of the effect of hemispherical variation in inner core
anisotropy on Earth’s normal modes of oscillation. Body wave studies show that the anisotropy
in the inner core is not simple cylindrical anisotropy, which is often portrayed in models derived
from normal mode data, but varies with longitude. ‘Hemispherical’ or odd degree, structure
has to be studied by cross-coupling normal modes, as the self-coupling technique is sensitive
only to even degree Earth structure. A completely general definition of inner core anisotropy
would require a prohibitive number of degrees of freedom; however, we show that any existing
cylindrical anisotropy model can be confined to only one part of the inner core. Using our new
theory, we find that hemispherical anisotropy causes significant changes in the frequency and
quality factor of several inner core sensitive normal modes. The effect of hemispherical inner
core anisotropy can also be seen in synthetic seismograms. Radial, PKIKP and PKJKP modes
all respond to the presence of hemispherical variation in inner core anisotropy. If the variations
in inner core anisotropy seen in body wave data are part of a gross, large-scale pattern, then
this structure should also affect normal mode data on an observable scale.
Key words: Composition of the core; Surface waves and free oscillations; Seismic
anisotropy; Theoretical seismology.
1 I N T RO D U C T I O N
Inner core anisotropy was first suggested as a hypothesis to explain
anomalous seismic data in two back-to-back papers, the first of
which dealt with the variation of body wave traveltimes in the inner
core (Morelli et al. 1986) and the second of which discussed the
anomalous splitting functions of seven inner core sensitive normal
modes (Woodhouse et al. 1986). These two observations can both
be explained by invoking a simple model of cylindrical anisotropy
in the inner core, with the axis of anisotropy coincident with Earth’s
axis of rotation. Since these early papers, both normal mode and
body wave data have been used to probe anisotropy in the inner core
(see reviews by Song 1997; Tromp 2001).
When the anisotropy axis is aligned with Earth’s rotation axis,
cylindrical anisotropy implies that all waves travelling through the
inner core with the same angle from the rotation axis (ζ ) should
travel at the same speed. However, body wave studies have shown
that this does not appear to be the case—there is a systematic
variation in the compressional wave velocity exhibited by body
waves with the same ζ , but traversing different parts of the inner
core. Subsequent body wave studies have shown that the anisotropy
in the inner core is not the simple cylindrical anisotropy, which is
often portrayed by models derived from normal mode data, but is
dependent on longitude. Some authors have dealt with the variation
in traveltimes for rays with the same ζ by modelling the inner core
962
as having an axis of anisotropy which is tilted with respect to the
Earth’s rotation axis (Shearer & Toy 1991; Creager 1992; Su &
Dziewonski 1995; Song & Richards 1996; McSweeney et al. 1997;
Isse & Nakanishi 2002). The estimated angle between the rotation
axis and the anisotropy axis varied between 5◦ (Creager 1992) and
10.5◦ (Su & Dziewonski 1995), with the longitude of the axis at the
Earth’s surface ranging over 180◦ .
More recent studies of the inner core have suggested that traveltimes in the inner core cannot be described by cylindrical anisotropy
which is uniform across the inner core, but rather that the anisotropy
in one region of the inner core is weaker than that in the rest of the
inner core (Tanaka & Hamaguchi 1997). As the regions of weaker
and stronger anisotropy both occupy approximately half of the inner
core, they are termed ‘hemispheres’, with the two hemispheres being separated by two lines of constant longitude. The hemispherical
variation in the inner core has been observed for both velocity and
attenuation anisotropy, as well as for isotropic velocity and attenuation in the inner core (Creager 1999; Garcia & Souriau 2000; Garcia
2002; Niu & Wen 2002; Oreshin & Vinnik 2004). The ‘eastern
hemisphere’ is only weakly anisotropic (about 0.5 per cent), whilst
the ‘western hemisphere’ is more strongly anisotropic (2–4 per cent,
Creager 1999). There is considerable variation in the descriptions of
the boundaries between the two hemispheres. Moreover, the depth
to which the difference between the two hemispheres persists is
not clear. Body wave studies have suggested that the hemispherical
C 2009 The Authors
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GJI Seismology
Accepted 2009 April 9. Received 2009 April 9; in original form 2009 February 17
Normal mode coupling and hemispherical inner core anisotropy
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Journal compilation core sensitive normal modes when the anisotropy in the inner core is
independent of longitude—azimuthally symmetric (AS)—or when
it is present in one hemisphere only. We also investigate the impact
on normal modes when only a section and not a whole hemisphere
of the inner core is isotropic; this is the ‘isotropic wedge’ (IW)
scenario.
2 N O R M A L M O D E T H E O RY
Each normal mode of the Earth occurs at a specific eigenfrequency,
ω and is labelled as either spheroidal or toroidal. Spheroidal modes
produce motion both perpendicular and parallel to the surface of
the Earth and can therefore be seen in both vertical and horizontal
components of a seismogram. Only spheroidal modes are both sensitive to the inner core and observable at the surface of the Earth;
spheroidal modes are used in this study.
Spheroidal normal modes can be described using the notation n Sl
where n is the overtone number and l the angular order of the mode. l
corresponds to the number of nodal lines an eigenfunction has on the
surface of the Earth. For a radial mode, where l = 0, n corresponds
to the number of nodes in that mode’s eigenfunction as a function
of depth. The properties of a normal mode can be described by
its frequency, ω and its quality factor, Q, which is the reciprocal
of the seismic attenuation. A mode with a high Q will have a low
attenuation and will be observable long after it has been excited
by an earthquake. For an spherically symmetric, non-rotating Earth
model, each mode consists of a (2l + 1)-fold degenerate multiplet,
whose singlets all have the same frequency. In this case each mode
would contribute a single δ-function to the seismogram. Both a
finite observational time window and attenuation by an elasticity of
the Earth mean that modes are not observed as δ-functions but as
broadened peaks.
Aspects of the Earth which deviate from the spherically symmetric, non-rotating Earth model, such as Earth’s elliptical shape,
lateral heterogeneity or anisotropy within the Earth, remove the degeneracy of the normal modes so that each mode multiplet becomes
split into a set of (2l + 1) singlets with different frequencies. These
singlets are labelled by their azimuthal order, m, where m takes integer values −l ≤ m ≤ l. In addition to splitting of individual modes,
cross-coupling or resonance between different modes, also occurs,
which again changes the frequencies of the singlets.
The ‘fundamental’ branch of the normal mode spectrum consists
of those modes with an overtone number of zero. The modes which
have n > 0 are then termed ‘overtones’. The fundamental modes
are sensitive only to the upper portions of the Earth, the depth to
which they are sensitive decreases as the angular order of the mode
increases. To probe the inner core, overtone modes must be used.
Normal modes can be further categorized according to their sensitivity to different regions of the Earth. There are three main types
of modes that are both sensitive to the inner core and observable at
the surface of the Earth: radial modes, PKIKP modes and PKJKP
modes. Radial modes are those that have no longitudinal or latitudinal variations in their eigenfunctions: the displacements they
cause are dependent only on radius. Radial modes therefore have
l = m = 0 and can be written as n S0 . Radial modes with small n
often have very high quality factors; these oscillations decay much
more slowly than other normal modes. PKIKP modes are analogous
in sensitivity to PKIKP body waves—they respond to changes in
density, compressional (P-wave) and shear (S-wave) velocity in the
inner core. PKIKP modes often have high overtone number n and
low angular order l. PKJKP modes are not sensitive to inner core
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differences persist to at least 400 km (Creager 2000) or to 700 km
(Sun & Song 2008) depth. The innermost inner core (of a radius of
300–500 km) is likely to have an anisotropic texture different from
that in the rest of the inner core (Ishii & Dziewonski 2002; Cormier
& Stroujkova 2005; Cao & Romanowicz 2007), but hemispherical
variation may persist down to this region.
If the anisotropy in the inner core is hemispherical and not cylindrically symmetric, two different fields of study of the Earth will be
affected. Hemispherical structure (HS) in the inner core’s anisotropy
will rule out several different theories on how the anisotropic structure is formed. Those theories that invoke alignment of crystals by
the magnetic field (Karato 1993c; Buffett & Wenk 2001) will require
that there is some sort of degree one structure in the Earth’s magnetic field. Theories that see inner core anisotropy as ‘freezing in’ as
the inner core grows (Karato 1993b; Bergman 1998) would imply
that the conditions on the inner core boundary are not uniform, but
vary with longitude.
It is possible that variations in heat flow at the core–mantle boundary, due to thermal and chemical processes and structures there, may
affect heat flow at the inner core boundary. This variation in heat
flow may be enough to allow anisotropy to form in some parts of
the inner core and not others (Sumita 2007; Aubert et al. 2008).
The heat flux variation at the inner core boundary may be separated
by 180◦ from the variation at the core–mantle boundary (Sumita &
Olson 1999), though this work has been superseded by that published by Aubert et al. (2008). Both Aubert et al. (2008) and Douglas
(2006) suggest the heat flux variations at the inner core boundary
and the core–mantle boundary may be directly aligned.
The formation of HS in the inner core would be also difficult to
reconcile with an inner core rotating with respect to the rest of the
Earth (Song & Richards 1996). This differential rotation has been
suggested as an explanation for anomalous body wave data (Zhang
et al. 2008) but is still a matter of some debate (Laske & Masters
1999). The absence of relative rotation can be used to restrict the
current geodynamical models to those that require no such rotation,
thus it is important to get constraints on the existence of HS from
seismology.
The free oscillations of the Earth have been studied since the
1960s and are the best tool to study the largest scale structures of
the Earth. Also called the normal modes of the Earth, they can be
observed after large earthquakes, when the Earth’s oscillations can
last for weeks. They provide information about inner core structure
in addition to that which can be gathered using body waves. Normal
mode studies of the inner core have not been able to study the
variation in anisotropy with longitude in the inner core, as they have
all, thus far, used the self-coupling (SC) approximation. In Irving
et al. (2008), we have shown that full coupling (FC), where very
large groups of modes are coupled together, is important when inner
core anisotropy is considered. HS corresponds to an angular order l
= 1 term in a spherical harmonic expansion of heterogeneities and
is therefore often referred to as an odd degree structure. When FC
is used, modes can couple through odd degree structure, as they are
not constrained by the SC selection rules.
In this paper, we detail the relevant theory governing normal
mode oscillations, highlighting especially the effects of anisotropy.
We derive a method for the inclusion of hemispherical variation in
cylindrical anisotropy. This allows us to investigate the properties
of normal modes when anisotropy persists over all longitudes in the
inner core and when it is present only in part of the inner core. We
investigate whether hemispherical inner core anisotropy (HS) has a
significant effect on the frequency and quality factor of inner core
sensitive normal modes. We create synthetic seismograms for inner
963
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J. C. E. Irving, A. Deuss and J. H. Woodhouse
P-wave velocity, though they are sensitive to changes in both the
shear wave velocity and density structure of the inner core.
2.1 Mode coupling
2.2 Representing Earth’s normal modes of oscillation
The oscillations of the Earth are the oscillations of a rotating, heterogeneous, massive ellipsoid. However, the case of a spherically
symmetric, non-rotating, elastic, isotropic (SNREI) Earth is a more
straightforward proposition and the effects of rotation, ellipticity,
heterogeneity and anisotropy can then be considered as perturbations of the SNREI Earth. The theory governing the SNREI Earth
is described in Dahlen & Tromp (1998). The eigenfunctions, s, of
the SNREI Earth obey the eigenvalue problem:
H0 skm = ωk2 skm
ω2k
(1)
is the eigenfrequency of the normal mode, skm and H0
where
is the self adjoint Hamiltonian operator derived from the elastic
displacement equation of the SNREI Earth. Each normal mode can
be described by a unique combination of four indices: n, q, l and m.
k,m
where the coefficients akm can be found using
2π π re
akm =
s∗km ρ0 u sin θ dθ dφ dr,
0
0
(3)
0
where ∗ denotes a complex conjugate and r e is the radius of the
Earth.
The normal modes of the real Earth can be calculated using the
splitting matrix approach, as described in Woodhouse & Dahlen
(1978) and Woodhouse (1980). The eigenfrequencies of the real
Earth can be found by calculating the perturbations to the SNREI
eigenfrequencies caused by rotation, ellipticity, heterogeneity and
anisotropy. The perturbations of the eigenfrequencies are the eigenvalues of the splitting matrix, H, as defined in eq. 80 of Woodhouse
& Dahlen (1978). The size of the splitting matrix is a function of
the number of normal modes which are being considered, with each
normal mode contributing (2l + 1) new rows (and columns) to the
splitting matrix; each row and column of H is labelled by a unique
pair of the indices km. When the self coupling approximation is
made H takes the form of a block diagonal matrix, with each block
being (2l + 1)(2l + 1) in size. The off-diagonal blocks, which are
(2l + 1)(2l + 1) in size are empty in the self coupling approximation, but contain cross-coupling terms between the modes sk and sk when full coupling is used.
2.3 Synthetic seismograms
The solution of the eigenfunction problem of the matrix H for
the perturbations due to anisotropy, ellipticity, rotation or lateral
heterogeneity can then be used to compute the contribution of these
perturbations to the time domain synthetic seismogram:
√
u(t) = R · ei(
H0 +H)t
·S
(4)
where S is the source vector and can be obtained from the Centroid
Moment Tensor (CMT) solution for the earthquake source and R,
the receiver vector, is a function of the seismograph’s response and
orientation. The H0 term is the diagonal matrix which contains
the squared frequencies, ω2k of the SNREI Earth. This technique is
described by Woodhouse & Girnius (1982) and Deuss & Woodhouse
(2001).
2.4 Normal mode splitting due to anisotropy
Here, we are interested in the contributions of anisotropic Earth
structure, A to the splitting matrix H. This contribution of the element in the row labelled by the pair km and the column labelled
k m can be derived as follows (Mochizuki 1986; Tromp 1995).
Ek m : L : E∗km dr3 ,
(5)
Akm k m =
volume
where the volume over that the integration is carried out is the
volume of the Earth. k represents a unique combination of n, q
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Different approximations of the coupling of modes are made to simplify the calculation of the Earth’s eigenmodes and eigenfrequencies and therefore enable the creation of synthetic seismograms.
Coupling or resonance between different singlets in either the same
mode or two different modes, can be caused by rotation of the Earth,
heterogeneities in Earth structure (velocity, density or attenuation
heterogeneities), ellipticity or elastic anisotropy.
The simplest theory used to explain normal mode coupling, SC,
makes the assumption that a mode may be treated as isolated. One
assumes that heterogeneity is a first-order quantity and that the
frequency difference between the target mode and its neighbours in
the frequency spectrum can be regarded as zeroth order. This is the
basis of SC.
When multiplets overlap in the spectrum, they clearly cannot
be treated with SC. The approximation used in this case is groupcoupling (GC). In this approximation, coupling between the modes
in each group is considered to be significant but coupling between
modes in different groups is assumed to be insignificant.
Both the self- and group-coupling approximations assume that it
is not necessary to calculate the coupling between modes which are
considered to be isolated from each other. Allowing all modes to
couple with each other is termed full coupling. If self- and groupcoupling are accurate approximations, then FC would produce the
same results, but require extra computational power to reach those
results.
The importance of FC in the mantle was investigated by Deuss
& Woodhouse (2001), who showed that there are significant differences between synthetic spectra calculated using SC, GC or FC.
Andrews et al. (2006) showed that inner core anelasticity caused
significant coupling between modes, and that FC was therefore
important; Irving et al. (2008), showed that the effects of FC are
significant when inner core anisotropy is included in the calculations. Moreover, Irving et al. (2008) shows that the groups of modes
which couple together are spread over a wide band of frequencies,
and the members of each group are dependent on the inner core
anisotropy model used. It is therefore inappropriate to use GC in
this case. In this paper, we do not use SC as selection rules prevent
modes from self-coupling through odd degree structure, so that HS
cannot be observed using SC.
The integer indices n, l and m correspond to the overtone number,
angular order and azimuthal order of the eigenfunction, respectively.
The parameter q indicates whether the mode is a spheroidal mode
or a toroidal mode. To simplify the notation used, each different
combination of n, q and l is given a unique value of k.
As the eigenfunctions the SNREI Earth are a complete set, the
eigenfunctions, u, of the real, perturbed Earth can be defined in
terms of s
akm skm
(2)
u(x, ω) =
Normal mode coupling and hemispherical inner core anisotropy
and l and m takes integer values from −l to l. Note that, for SC,
k = k, however here we are interested in the more general case
where cross-coupling between modes with different values of k is
possible. L is the elastic tensor and Ekm is the strain tensor and is
found from the eigenfunctions, ukm :
Ekm =
1
∇ukm + (∇ukm )T .
2
(6)
It is easiest to consider the tensors involved in the canonical basis,
which is explained in more detail in Appendix A. The eigenfunction
ukm (r) can then be expressed in terms of the scalar field components,
u α (r ) and the generalized spherical harmonics Y αlm (θ , φ):
ukm (r) =
α
u α (r )Ylm
(θ, φ),
(7)
α=−,0,+
Lαβγ δ (r, θ, φ) =
αβγ δ
L st
(α+β+γ +δ)
(r )Yst
2.5 Cylindrical anisotropy in the self coupling
approximation
SC allows cylindrical anisotropy to be described using just three
depth-dependent parameters (Tromp 1995). Cylindrical anisotropy
has non-zero anisotropic terms only when t = 0, that is there are no
variations of the anisotropy when the longitude is varied. This small
number of parameters makes the inversion of normal mode data
to produce an inner core anisotropy model much more straightforward and the models produced by Woodhouse et al. (1986),
Tromp (1993), Durek & Romanowicz (1999), Ishii et al. (2002)
and Beghein & Trampert (2003) are parametrized by three depthdependent variables. Here, we show how these parameters relate
αβγ δ
to L s0 . In Tromp (1995), a cylindrically symmetric, anisotropic
tensor is defined using coefficients λ1 , λ2 , λ3 , λ4 and λ5 , all of which
are functions of radius, with
√
αβγ δ
αβγ δ
αβγ δ
(12)
L 00 = 4π (g1 λ1 + g2 λ2 )
αβγ δ
L 20
(θ, φ)
=
s,t
=
αβγ δ
L st
(r )YstN (θ, φ),
(8)
s,t
where α, β, γ and δ can take the values +, 0 and −, t takes values
between − s and s and N = α + β + γ + δ. When anisotropy in the
Earth is considered, this tensor is restricted by thermodynamical and
mechanical constraints (Dahlen & Tromp 1998) to be symmetric,
so that
Lαβγ δ = Lβαγ δ = Lαβδγ = Lγ δαβ .
(9)
The contributions to the matrix A are then given by
Akm
k m =
α,β,γ ,δ,γ ,δ s,t
×
(α+β)∗
Ylm
re
αβγ δ
αβ∗
Elm
L st
o
(γ +δ )
YstN Yl m γ δ
El m gγ γ gδδ r 2 dr
d,
(10)
where d is an integral over the surface of a sphere and gδδ and
gγ γ are the elements of the canonical metric, defined in Appendix
A. This equation can then be re-cast using Wigner 3-j symbols so
that
(2l + 1)(2s + 1)(2l + 1) 2
=
(−1)m
4π
s,t
l
s
l re
×
N I r 2 dr,
−m
t
m
0
N
I
k m (11)
where N takes integer values from −4 to +4 and I takes integer
values from 1 to I N , where I 0 = 5, I ±1 = 3, I ±2 = 3, I ±3 = 1 and
I ±4 = 1. There are therefore 21 NI , these radial integrands depend
on the canonical basis equivalents of the eigenfunctions and their
derivatives, functions of l, l , s, N and components of the elastic
αβγ δ
tensor L st . The values taken by each NI are given by Mochizuki
(1986). When the anisotropic contribution to Akm k m is written as
in eq. (11), it can be easily introduced into computer code which
calculates the matrix A to find the normal modes of an anisotropic
Earth.
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Journal compilation =
(13)
4π αβγ δ
(g
λ5 ),
9 5
(14)
αβγ δ
where g h
are known numerical factors, shown in Table 1. The
αβγ δ
components of L st are zero for t = 0, as required by the cylindrical
symmetry.
It is then possible to write λi , the constituent components of the
αβγ δ
elastic tensor in the canonical basis (L s0 ), in terms of the Love
coefficients (Love 1927): A, C, L, N and F.
λ1 = 6A + C − 4L − 10N + 8F
(15)
λ2 = A + C + 6L + 5N − 2F
(16)
λ3 = −6A + C − 4L + 14N + 5F
(17)
λ4 = A + C + 3L − 7N − 2F
(18)
αβγ δ
Table 1. g h
αβγ δ
coefficients for each L s0
Table of
αβγ δ
gh
.
αβγ δ
coefficients needed to find L s0
s=0
s=2
h=4
h=1
h=2
h=3
1
15
2
15
2
15
1
15
4
21
2
− 21
8
21
4
− 21
2
− 21
0
1
− 21
0
0
1
− 21
g ±±∓0
h
0
g ±∓±0
h
− 213
12
21
√
− 2112
√
− 213
g 0000
h
1
Akm
αβγ δ
L 40
4π αβγ δ
αβγ δ
(g
λ3 + g4 λ4 )
5 3
g ±±∓∓
h
g ±∓±∓
h
g ±∓00
h
g ±0∓0
h
g ±000
h
0
1
15
1
− 15
0
1
− 15
0
√
3
21
√
g ±±00
h
√
6
21
g ±0±0
h
0
g ±±∓∓
h
− 216
g ±±±∓
h
g ±±±±
h
√
√
0
√
6
21
√
− 2124
.
s=4
h=5
8
35
2
35
2
35
4
35
4
35
√4
490
√2
490
√2
490
√4
490
√4
490
√2
490
√2
70
√2
70
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The fourth-order elastic tensor, L, can then be written in terms of
spherical harmonics with angular order s and azimuthal order t:
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J. C. E. Irving, A. Deuss and J. H. Woodhouse
λ5 = A + C − 4L − 2F
(19)
The degree two and degree four lateral variation in anisotropy is
controlled completely by the three coefficients λ3 , λ4 and λ5 . This
anisotropy has been parametrized by many authors in terms of three
coefficients, α, β and γ , whose definitions (Tromp 1993) are given
by
α = (C − A)/A0
(20)
β = (L − N )/A0
(21)
γ = (A − 2N − F)/A0 ,
(22)
where A0 is the value of A at the centre of the Earth. We can
therefore find the components of the elastic tensor in the canonical
αβγ δ
basis (L s0 ) for any self-coupling anisotropy model which is given
in terms of these three coefficients, converting them using
(23)
λ4 = α + 3β + 2γ
(24)
λ5 = α − 4β + 2γ .
(25)
Just as the elastic tensor is free to vary over the radius of the inner
core, each of the parameters α, β and γ can be considered to be
depth-independent (as in the simplest model of Woodhouse et al.
1986) or depth-dependent.
The advantage of parametrizing inner core anisotropy using α, β
and γ is that these coefficients can be related to physical quantities.
α represents the relative speeds of inner core P-waves travelling
along and perpendicular to the Earth’s rotational axis. Similarly, β
represents the relative speeds of inner core S-waves travelling along
and perpendicular to the Earth’s rotational axis. γ describes the
speeds of waves which travel at intermediate angles to the Earth’s
rotational axis.
2.6 Hemispherical variations in inner core anisotropy
Starting from the elastic tensor described in terms of spherical
harmonics given in eq. (8) (and noting that Lαβγ δ may vary with
radius although that is not explicitly mentioned here) we attempt
to find the expansion of a field which takes the value of Lαβγ δ
(θ, φ) over longitudes φ1 → φ2 of the sphere and a value of zero
over the rest of the sphere. This expansion simulates an anisotropic
‘hemisphere’ between φ1 and φ2 and an isotropic ‘hemisphere’
between φ2 and φ1 .
Eq. (8) shows that any tensor can be written in terms of general
αβγ δ
spherical harmonics. Each coefficient, L st , of a tensor Lαβγ δ
(θ, φ), expanded in general spherical harmonics can be found by
multiplying the function by the complex conjugate of the relevant
spherical harmonic (Y ∗N
st (θ , φ)) and integrating the function over a
sphere,
2s + 1 2π π αβγ δ
αβγ δ
=
L
(θ, φ)Yst∗N (θ, φ) sin θ dθ dφ. (26)
L st
4π
0
0
For the situation we are considering, this integral is only non-zero
in the anisotropic hemisphere, between φ1 and φ2 , where the elastic
tensor has the same value as Lαβγ δ (θ , φ),
2s + 1 φ2 π αβγ δ
αβγ δ
L st
=
L
(θ, φ)Yst∗N (θ, φ) sin θ dθ dφ. (27)
4π
φ1
0
1 1
× sin θ dθ dφ.
(θ, φ) = PsNt
(28)
(cos θ) e
(cos θ) are generalized Legendre polynomials (described in Appendix A of Phinney
& Burridge 1973), the required expansion to find the spherical
αβγ δ
harmonic components of the hemispherical elastic tensor, L st ,
becomes
2s + 1 αβγ δ φ2 π N t1
αβγ δ
=
L s1 t1
Ps1 (cos θ)PsN t (cos θ )
L st
4π s ,t
φ1
0
Noting that YstN
itφ
, where PsNt
1 1
× ei(t1 −t)φ sin θ dθ dφ.
(29)
We then carry out the integral over φ, resulting in
π
2s + 1 αβγ δ
αβγ δ
(φ2 − φ1 )L s1 t
=
PsN1 t (cos θ)
L st
4π
0
s1
× PsN t (cos θ) sin θdθ
+
ei(t1 −t)φ2 − ei(t1 −t)φ1 αβγ δ π
L s1 t1
PsN1 t1 (cos θ )
i(t
−
t)
1
0
s1 ,(t1 =t)
× PsN t (cos θ) sin θ dθ
.
(30)
ss
We define the integral I N t11 t as the integral containing the generalized
Legendre functions
π
ss
PsN1 t1 (cos θ)PsN t (cos θ) sin θdθ = I N t11 t .
(31)
0
ss
The integral I N t11 t can be evaluated using Edmonds’ (1960)
(s)
eq. (4.5.2) (noting that our PsNt (cos θ) corresponds to Edmonds d Nt ),
so that PsNt (cos θ) can be defined as
π t2 t π −it2 θ
Ps cos
e
PsN t (cos θ) = i N −t
Pst2 N cos
. (32)
2
2
t =−s,s
2
ss
Using eq. (32) we can evaluate the integral in eq. (31), leaving I N t11 t
in terms of a sum over two dummy indices, t2 and t3 which are both
summed between − s and s,
π t2 t1 π t3 N π
ss
Ps1 cos
Ps
cos
i 2N −t−t1 Pst12 t cos
I N t11 t =
2
2
2
t2 ,t3
−i(t2 +t3 )π
π 1+e
(33)
× Pst3 t cos
,
2 1 − (t2 + t3 )2
noting that when t 2 + t 3 = ±1, the fraction must be replaced by its
limiting value of ∓ iπ2 . Then, substituting eq. (33) into eq. (30) the
αβγ δ
new coefficients, L st , of the elastic tensor Lαβγ δ (r , θ, φ) can be
calculated using
2s + 1 αβγ δ
αβγ δ ss =
(φ2 − φ1 )L s1 t I N tt1
L st
4π
s1
αβγ δ ei(t1 −t)φ2 − ei(t1 −t)φ1 ss (34)
L s1 t1
+
I N t11 t .
i(t1 − t)
s ,(t =t)
1
1
αβγ δ
of a
Eq. (34) shows how we can obtain the coefficients L st
hemispherical model which is anisotropic between φ1 and φ2 and
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λ3 = α − 4β − 5γ
We can then substitute the expansion in spherical harmonics of the
known elastic tensor, Lαβγ δ (θ, φ) (eq. 8) into eq. (27)
2s + 1 φ2 π αβγ δ N
αβγ δ
=
L s1 t1 Ys1 t1 (θ, φ)Yst∗N (θ, φ)
L st
4π
φ1
0 s ,t
Normal mode coupling and hemispherical inner core anisotropy
967
Table 2. Centre frequencies, mode characters and frequency ranges of normal modes coupled together.
Target mode
Mode
frequency
Mode
character
Frequency range of mantle
and crust sensitive modes
coupled with target mode
(mHz)
PKJKP
Radial
PKJKP
PKJKP
Radial
PKIKP
PKIKP
PKIKP
0.000–3.000
0.000–3.000
0.000–3.000
2.380–3.397
2.600–3.700
4.200–4.470
4.470–5.050
4.960–5.429
(mHz)
3 S2
1 S0
6 S1
6 S3
3 S0
13 S1
13 S2
13 S3
Figure 1. An expansion of a field, ψ, where ψ = 1 (0 ≤ φ ≤ π ), ψ =
0 (φ > π ), using spherical harmonics up to degree 4.
αβγ δ
3 M E T H O D S A N D D ATA
3.1 Normal mode models of inner core anisotropy
Many different seismic models of inner core anisotropy are
available in the literature, developed using either normal mode
or body wave data. Here, we use four of those models that
have been obtained using normal modes: the B&T model
(Beghein & Trampert 2003), the D&R model (Durek &
Romanowicz 1999), the Tr model (Tromp 1993) and the W,G&L
model (Woodhouse et al. 1986). These models can all be described
by the parameters α, β and γ (defined in eqs 20, 21 and 22).
3.2 Data
In this study, example data are shown for two earthquakes: the 1994
June 9 Bolivia earthquake (060994A) and 2004 December 26 Suma
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Journal compilation Note: The frequency quoted for each mode is that found using PREM
(Dziewonski & Anderson 1981) with SC, with no inner core or mantle
structure and no ellipticity or rotation corrections. As only a 1-D velocity
model is used, all of the singlets in each mode oscillate at exactly the same
frequency. The method of Deuss (2008) was used to determine the
character of each mode.
tra earthquake (122604A). The time, location and CMT mechanisms
are taken from the global CMT catalogue (www.globalcmt.org).
The data from the earlier of these events were used by Durek &
Romanowicz (1999) and also were part of the splitting function
data set used by Beghein & Trampert (2003) in their inversion for
an inner core anisotropy model. We use vertical component seismograms, with a 10 s sample interval, which have been Fourier
transformed to provide the frequency domain spectra for a specified
time window after each event.
3.3 Calculations
The frequencies and quality factors of all of the 97 inner core
modes sensitive between 0 and 7.0 mHz were calculated for mantle
and inner core structure with the modes permitted to fully couple.
Frequency bands of mantle and crust sensitive normal modes that
couple with the mode of interest (the target mode) were included in
the calculations performed in Sections 4.2, 5 and 6. The frequency
bands of modes coupled, together with the frequency of each target
mode, are shown in Table 2.
Whilst the preliminary calculations in Section 4.1 do not include
the effects of coupling through ellipticity, rotation and mantle structure, these effects are included in all further calculations. In each
calculation inner core anisotropy was also included. Four different
inner core anisotropy models were used and for each mode the calculations of modal frequency and quality factor were carried out for
(1) azimuthal symmetry (AS) in inner core anisotropy and (2) HS
in inner core anisotropy. When HS was used, the anisotropic region
was between 180◦ W and 0◦ W and the isotropic region was between
0◦ W and 180◦ E. When an isotropic west (IW), in the inner core was
used, the IW occupied region 40–180◦ E and the anisotropic region
was between 180◦ W and 40◦ E. HS and IW are expanded to degree
4, which is the maximum degree to which the models of cylindrical
inner core anisotropy are expanded. When HS or IW are imposed
on the inner core, the global average inner core anisotropy is left
unchanged. Thus, any difference between the modal frequencies
quality factors and synthetic seismograms for AS, IW and HS are
caused only by the presence of HS.
Shear wave model S20RTS (Ritsema et al. 1999) was used to
describe lateral variations in mantle velocity and density. The shear
Downloaded from http://gji.oxfordjournals.org/ at Princeton University on March 10, 2015
isotropic elsewhere, using the coefficients, L st , of a cylindrically
symmetric anisotropy model described only in terms of the parameters α, β and γ (defined in eqs 20–22). An example of a divided
inner core can be seen in Fig. 1. The scalar field, ψ, represented
takes a value of 1 between the longitudes 0 ≤ φ ≤ π and zero
elsewhere. A spherical harmonic expansion up to degree s = 4 is
used to obtain this image.
The four normal mode models of inner core anisotropy, which we
use in this study have all been made using SC, and contain angular
degree two and four cylindrical anisotropic structure only, so that
possible values for s1 in eq. (34) are s1 = 2 or s1 = 4 and t 1 = 0.
Whilst they do not describe HS in the inner core, it is expected that
they will represent the global average of the anisotropic structure
with some degree of accuracy. If the inner core was split into an
anisotropic and an isotropic hemisphere using the procedure aforementioned, the global average anisotropy would be halved. To split
the inner core into two hemispheres whilst maintaining the global
average, it is necessary to double the inner core anisotropy model
before using eq. (34) to separate the inner core into an isotropic
half and an anisotropic half. The resulting anisotropy model will
then give the same amount of coupling due to degree 2 and 4 structure as the original model and also give the coupling that would
be produced if the anisotropy present in the inner core is confined
to one hemisphere. Likewise, if the anisotropy is to be constrained
to some region φa → φb , the model parameters α, β and γ should be
before using eq. (34) to find the new model of
multiplied by φb2π
−φa
inner core anisotropy.
1.106
1.631
1.980
2.822
3.270
4.499
4.845
5.194
968
J. C. E. Irving, A. Deuss and J. H. Woodhouse
2000
(a)
B&T AS
B&T HS
(b)
D&R AS
D&R HS
(c)
Tr AS
Tr HS
(d)
W,G&L AS
W,G&L HS
1600
Q
1200
800
400
0
2000
1600
Q
1200
800
0
0
1
2
3
4
5
6
Frequency (mHz)
7
0
1
2
3
4
5
6
7
Frequency (mHz)
Figure 2. Frequencies and quality factors (Q) for inner core sensitive modes when the inner core model is azimuthally symmetric (AS, filled shapes) or
contains hemispherical structure (HS, outlined shapes) for the (a) B&T model, (b) D&R model, (c) Tr model and (d) W,G&L model. Only inner core sensitive
modes have been coupled together, and no mantle structure, rotation or ellipticity have been included in these calculations. Those modes with Q greater than
200 and frequency less than 6 mHz are displayed here.
wave velocity perturbations were scaled to obtain compressional
velocity, v p and density, ρ, with scaling of the form δv p /v p =
0.5δvs /vs (Li et al. 1991) and δρ/ρ = 0.3δvs /vs (Karato 1993a).
The Preliminary Reference Earth Model (PREM, Dziewonski &
Anderson 1981) was used to describe the Earth’s 1-D velocity
and density structure. Synthetic seismograms were calculated using
eq. (4); each synthetic seismogram contains the phase and amplitude
information of the Fourier transformed, cosine tapered synthetic
seismogram for vertical motion.
4 H E M I S P H E R I C S T RU C T U R E
IN THE INNER CORE
4.1 Inner core sensitive modes
The singlet frequencies and quality factors, Q, of 97 inner core
sensitive modes have been calculated using full coupling for AS and
HS inner core anisotropy distributions and are shown in Fig. 2. No
mantle structure or coupling due to ellipticity or rotation has been
included in these FC calculations. Changes in modal frequency and
Q between AS and HS are seen using each model. The effects of
HS are greatest when the B&T model (Fig. 2a) is used, whilst the
W,G&L model (Fig. 2d) shows the smallest changes when HS is
introduced. Calculations using the Tr model (Fig. 2c) show that one
singlet of 13 S2 , is especially strongly affected by the inclusion of
HS. The frequency of the singlet changes from 4.8478 to 4.8489
mHz, and its Q nearly doubles, going from 864 to 1640. Mode
13 S2 exhibits atypical behaviour, with its splitting function being
difficult to determine (Durek & Romanowicz 1999). The change
in this mode’s properties when HS is present in the inner core
may explain some of the complications in observations of 13 S2 . We
investigate this mode in more detail in Section 4.2.
4.2 Response of individual modes
When mantle structure, ellipticity and rotation are included in the
FC calculations a more accurate description of the properties of
each normal mode are found. In each of the calculations that follows, all of these effects are included and the ‘target’ inner core
sensitive mode is permitted to couple with mantle and crust sensitive modes close in frequency, to account for coupling due to 3-D
mantle structure.
All five of the radial modes investigated in this study show
changes when the anisotropy in the inner core is only permitted
to exist in the western hemisphere. Fig. 3 shows how the frequency
and Q of mode 1 S0 vary when HS (solid circle in Fig. 3) is applied to
inner core anisotropy, instead of allowing the whole inner core to be
anisotropic (AS, open square in Fig. 3). The changes are non-linear
and depend on the coupling interactions caused by the anisotropy
model. If 1 S0 was only permitted to self-couple there would be no
change to the frequency or Q of the mode; SC rules do not permit
radial modes to respond to 3-D structure (as discussed in Irving
et al. 2008).
The scale of the frequency and quality factor changes vary
strongly between the four models—there is a Q change of 6 per cent
for the W,G&L model (Fig. 3d), but the change in Q for the Tr model
(Fig. 3c) is only 0.1 per cent. The quality factor of 1 S0 decreases
for all of the different models. This is because 1 S0 is coupling,
through the odd degree structure, with other modes which have a
much lower quality factor and therefore decay away more quickly.
The frequency changes caused by the HS also vary in size, from
0.01 μHz (D&R model, Fig. 3b) to 0.35 μHz (W,G&L model). The
inclusion of HS in the B&T (Fig. 3a) and W,G&L models causes the
frequency of 1 S0 to increase, whereas HS in the D&R and Tr models
causes the frequency to decrease. The frequency and Q of 1 S0 have
been measured by several authors: Masters & Gilbert (1983) found
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400
Normal mode coupling and hemispherical inner core anisotropy
PKJKP mode 6S1
Radial mode 1S0
1500 (a) B&T
969
700
(b) D&R
(a) B&T
(b) D&R
(c) Tr
(d) W,G&L
1450
Q
Q
650
600
550
1400
500
1500
(c) Tr
700
(d) W,G&L
1450
Q
Q
650
600
550
1400
500
Figure 3. Frequency and quality factor of radial mode 1 S0 when the inner core model is azimuthally symmetric (AS - open square) or contains
hemispherical structure (HS - solid circle) for the (a) B&T model, (b) D&R
model, (c) Tr model and (d) W,G&L model. Mantle heterogeneities, rotation
and ellipticity and coupling with mantle sensitive modes have been included
in these calculations.
ω = 1.63151 mHz ± 0.05 μHz; He & Tromp (1996) found ω =
1.63164 mHz ± 0.01 μHz. The error bounds on measurements of
the frequency of this mode are much smaller than the frequency difference shown when HS is imposed on the W,G&L model; and are
of the same order as the differences caused by HS when the other
three models are used. The changes in frequency caused by HS
are thus sufficiently large that the effects of HS should be observable in real data. Although the self-coupling approximation would
prevent radial modes exhibiting sensitivity to 3-D structure, radial
modes are affected by degree one structure in the inner core and
full-coupling should therefore be applied to include these effects.
The variations in frequency and Q of the three singlets for PKJKP
mode 6 S1 are shown in Fig. 4. The variations are again model dependent, as was seen for radial modes. The changes in Q for the
B&T model (Fig. 4a) are much greater than those for the D&R, Tr
and W,G&L (Figs 4b–d) models, which show only very small effects when HS is included in the inner core. The coupling caused by
the imposition of hemispherical anisotropic structure on the B&T
model permits 6 S1 to couple strongly with other inner core modes,
whilst the imposition of HS on the other three models does not
cause such strong coupling. Observations of the quality factor of
6 S1 are limited; those which have been made vary from 293 ± 29
(Resovsky & Ritzwoller 1998) to 564 ± 182 (quoted on the Reference Earth Model web-page http://mahi.ucsd.edu/Gabi/rem.html)
whilst the value predicted by PREM (Dziewonski & Anderson 1981)
is 650.
Mode 6 S1 is sensitive only to shear wave velocity in the inner
core, and not to inner core compressional wave velocity. As the
observation of PKJKP body waves is very difficult (Deuss et al.
2000), no HS in S-waves anisotropy has ever been detected using
body waves. As can be seen in Fig. 4, PKJKP modes are sensitive
to HS and will be the best way to determine whether HS is also
present in S-wave anisotropy.
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Journal compilation 1.980
1.985
1.975
Frequency (mHz)
HS
1.980
1.985
Frequency (mHz)
AS
HS
Figure 4. Frequency and quality factor of PKJKP mode 6 S1 when the
inner core model is azimuthally symmetric (AS - open square) or contains
hemispherical structure (HS - solid circle) for the (a) B&T model, (b) D&R
model, (c) Tr model and (d) W,G&L model. Mantle heterogeneities, rotation
and ellipticity and coupling with mantle sensitive modes have been included
in these calculations.
Finally, we investigate 13 S3 (Fig. 5), a PKIKP mode, which is
sensitive to both v p and vs structure in the inner core. When hemispherical anisotropic structure is included in the inner core, the
frequency and attenuation of the singlets that make up this mode
PKIKP mode 13S3
950 (a) B&T
(b) D&R
900
Q
AS
1.975
Frequency (mHz)
850
800
750
700
950
(c) Tr
(d) W,G&L
900
Q
Frequency (mHz)
1.63100 1.63125 1.63150
850
800
750
700
5.18
5.19
5.20
Frequency (mHz)
AS
5.21 5.18
5.19
5.20
5.21
Frequency (mHz)
HS
Figure 5. Frequency and quality factor of PKIKP mode 13 S3 when the
inner core model is azimuthally symmetric (AS - open square) or contains
hemispherical structure (HS - solid circle) for the (a) B&T model, (b) D&R
model, (c) Tr model and (d) W, G&L model. Mantle heterogeneities, rotation
and ellipticity and coupling with mantle sensitive modes have been included
in these calculations.
Downloaded from http://gji.oxfordjournals.org/ at Princeton University on March 10, 2015
1.63100 1.63125 1.63150
5 THE EFFECT OF HEMISPHERES
ON SPECTRA
As we have shown in the previous section, the inclusion of HS
can have a dramatic effect on the frequency and Q of a mode. It
is therefore expected that the shape and position of a mode in a
frequency domain seismogram may likewise be affected. Here, we
show examples of seismograms for three such modes, which are
representative of the general influence of HS in the inner core.
Radial mode 3 S0 exhibits frequency changes of up to 0.14 μHz for
the W,G&L anisotropy model, which can also be seen in the spectra
(Fig. 6). The changes caused in the frequency of 3 S0 are in all cases
greater than the error quoted in the most recent measurement of
its frequency (He & Tromp 1996, measured the frequency to be
3.27259 mHz ± 0.03 μHz). Differences in frequency between HS
and AS seismograms, of the magnitude seen in Fig. 6, are therefore
sufficiently large to be robustly observed.
6 S3 is a PKJKP mode; the differences between synthetic seismograms for AS and HS are therefore due to the presence of HS
in S-wave anisotropy. Fig. 7 shows that HS causes changes in the
Amplitude
0
0
2.80
2.81
2.82
Frequency (mHz)
Data
AS
2.83
2.84
HS
Figure 7. Data (solid line) and synthetic seismograms for PKJKP mode
6 S3 when the inner core is azimuthally symmetric (AS, dotted line) or
contains hemispherical anisotropic structure (HS, dashed line) data for event
122604A. The inclusion of HS in the B&T model causes changes in the
frequency and attenuation of the peak formed by the singlets that comprise
this mode.
synthetic seismogram for the B&T model, which are much larger
than the difference between the data and the AS seismogram. The
influence of HS on this mode is again large enough to be observed
in real data.
13 S2 , a PKIKP mode, is identified by several authors (Durek &
Romanowicz 1999; Deuss 2008) as an unusual mode. The changes
in frequency and quality factor for this mode when HS is included
are shown in Fig. 8, and synthetic seismograms together with data
are shown in Fig. 9. 13 S2 exhibits differences between HS and AS
for all four models. The most striking changes in frequency and Q
are for the Tr model. The quality factor for the highest frequency
singlet is doubled and the frequency increases by 1.2 μHz, when
HS is introduced to the Tr model (Fig. 8c). This drastic difference
PKIKP mode 13S2
1800
(a) B&T
(b) D&R
(c) Tr
(d) W,G&L
1600
1400
1200
1000
Phase
Event 122604A, station PAB, 40-120 hours, W,G&L model
800
1800
0
1600
Amplitude
Q
1400
1200
1000
800
4.835 4.840 4.845 4.850 4.855 4.835 4.840 4.845 4.850 4.855
0
Frequency (mHz)
3.265
3.270
3.275
Frequency (mHz)
Data
AS
3.280
HS
Figure 6. Data (solid line) and synthetic seismograms for radial mode 3 S0
when the inner core is azimuthally symmetric (AS, dotted line) or contains
hemispherical anisotropic structure (HS, dashed line) for event 122604A for
the W,G&L model.
AS
Frequency (mHz)
HS
Figure 8. Frequency and quality factor of PKIKP mode 13 S2 when the
inner core model is azimuthally symmetric (AS, open square) or contains
hemispherical structure (HS, solid circle) for the (a) B&T model, (b) D&R
model, (c) Tr model and (d) W,G&L model. Mantle heterogeneities, rotation
and ellipticity and coupling with mantle sensitive modes have been included
in these calculations.
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change by different amounts depending on the inner core anisotropy
model used. Both the B&T (Fig. 5a) and the D&R (Fig. 5b) models exhibit large drops in quality factor when HS is included:
14 per cent and 22 percent, respectively. The changes in frequency
for these two models (0.6 μHz for the B&T model and 0.8 μHz for
the D&R model) are also larger than for the other two models. The
Tr model (Fig. 5c) exhibits the smallest changes, with the singlet
frequency changing by up to 0.2 μHz and changes in Q of up to
1 per cent. The W,G&L model (Fig. 5d) is the only one for which HS
causes a substantive increase in the Q of any of this mode’s singlets,
with increases in Q of up to 3 per cent and in frequency of up to
0.3 μHz.
These three example modes described here: one radial mode, one
PKJKP mode and one PKIKP mode typify the changes in frequency
and quality factor caused by the inclusion of HS in inner core
anisotropy. The model which shows the largest shifts is different for
each mode and the direction in which the frequency and Q change
is also dependent on the inner core anisotropy model used. The
variation with inner core anisotropy model of the effect of HS will
permit discrimination between different HS models.
Phase
J. C. E. Irving, A. Deuss and J. H. Woodhouse
Q
970
(a) B&T
(b) D&R
(c) Tr
(d) W,G&L
971
0
Amplitude
Phase
Normal mode coupling and hemispherical inner core anisotropy
0
0
4.830
4.835
4.840
4.845
4.850
4.855
4.860
4.830
Frequency (mHz)
Data
4.835
4.840
4.845
4.850
4.855
4.860
Frequency (mHz)
Azimuthal symmetry
Hemispherical structure
Figure 9. Data (solid line) and synthetic seismograms for PKIKP mode 13 S2 when the inner core is azimuthally symmetric (AS, dotted line) or contains
hemispherical anisotropic structure (HS, dashed line) for event 060994A. The inner core anisotropy models used are the: (a) B&T model, (b) D&R model,
(c) Tr model and (d) W,G&L model.
corresponds to a substantive difference between the HS and AS
seismograms (Fig. 9c). The changes in frequency and Q for the
D&R model (Fig. 8b) cause the split peak of 13 S2 that is seen
when AS is used to become a single peak when HS is present
(Fig. 9b). Changes in both the phase and amplitude of the synthetic
seismograms for the B&T model are produced when HS is included
(Fig. 9a). Also, the frequency of the modal peak as a whole increases
when HS is used, because all of the singlets increase their frequency,
by up to 1 μHz, (Fig. 8a) when AS is replaced by HS. The changes
induced in the synthetic seismogram by HS using the W,G&L model
(Fig. 9d) are much smaller than when the other three models are
used. This smaller difference is consistent with the smaller changes
in frequency (up to 0.2 μHz) and Q seen than when other models
are used (Fig. 8d).
We find that the response of 13 S2 to the inclusion of HS in the inner core is strong and highly dependent on the inner core anisotropy
model used. Whilst this mode is likely to be very useful in constraining inner core properties, it is clear that all of the possible
causes of splitting of 13 S2 —changes in vs (as in Deuss 2008), the
presence of an isotropic layer at the top of the inner core and azimuthal variation in anisotropic inner core structure—will need to
be taken into account simultaneously so that models of the inner
core can be completely reconciled with data for this mode.
6 A N I S O T RO P I C W E D G E
IN THE INNER CORE?
In Sections 4 and 5, we have investigated the effect of HS on modes
sensitive to the inner core, with the western hemisphere (180◦ W
to 0◦ E) exhibiting anisotropy, and the eastern hemisphere (0◦ E
to 180◦ E) being isotropic. Previous studies by other authors have
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Journal compilation Table 3. Definitions of the location of the edges of the two inner core
hemispheres.
Paper
Tanaka & Hamaguchi (1997)
Creager (1999)
Garcia & Souriau (2000)
Garcia (2002)
Niu & Wen (2002)
Oreshin & Vinnik (2004)
Longitude of
first boundary
Longitude of
second boundary
43◦ E
40◦ E
60◦ E
60◦ E
40◦ E
50◦ E
177◦ E
160◦ E
160◦ E
180◦ W
180◦ W
120◦ W
suggested that the boundaries of the isotropic region are around
40◦ E and 180◦ E (see Table 3). They suggest that the inner core is
not divided into true hemispheres, but into an anisotropic major
sector, with an IW below the Indian Ocean, Asia and the Pacific
Ocean. These two different ideas are depicted in Fig. 10, where
the inner core, as seen from the north pole, is divided by two
lines of constant longitude into an isotropic and an anisotropic
region.
As there is some discrepancy between the proposed locations of
the boundaries between the isotropic and anisotropic hemispheres
(Table 3), it would be very helpful if normal modes were sufficiently
sensitive to detect the difference between an inner core that contains
an IW, and the one which exhibits truly hemispherical inner core
anisotropy structure. To investigate this possibility, we have also
calculated modal frequencies, quality factors and synthetic seismograms for inner core sensitive modes when the inner core contains
an IW occupying the region between 40◦ E and 180◦ W.
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Amplitude
Phase
0
972
J. C. E. Irving, A. Deuss and J. H. Woodhouse
AS, so that IW does not simply represent a midpoint between the
frequencies and quality factors. In the B&T model (Fig. 11a) the
IW results are very close to the HS results, but both of the lower
frequency singlets have a lower quality factor.
The introduction of an IW into the inner core has different effects on different modes when the different models are used. The
frequencies and quality factors of the modes cannot be described by
a simple interpolation between their values using HS and AS, but
rather they must be calculated for each model.
6.2 The effect of an isotropic wedge on spectra
Figure 10. A cartoon explaining the different ideas of regional variation
in inner core anisotropy: (a) HS and (b) an IW. The coloured region is
anisotropic and the white region is isotropic. The hemispherical structure
has boundaries at 0◦ E and 180◦ W; the IW has boundaries at 40◦ E and
180◦ W. The cartoon represents the view of the inner core from a latitude
of 90◦ .
(a) B&T
(b) D&R
(c) Tr
(d) W,G&L
Q
750
700
650
600
800
Q
750
700
650
600
4.49
4.50
4.51
4.49
Frequency (mHz)
AS
4.50
4.51
Frequency (mHz)
IW
HS
Figure 11. Frequency and quality factor of PKIKP mode 13 S1 when the
inner core model is azimuthally symmetric (AS, open square), contains
hemispherical structure (HS, solid circle) or an isotropic wedge (IW, grey
triangle) for the (a) B&T model, (b) D&R model, (c) Tr model and
(d) W,G&L model. Mantle heterogeneities, rotation and ellipticity and coupling with mantle sensitive modes have been included in these calculations.
6.1 Frequency changes for an isotropic wedge
The frequency changes exhibited when an IW is introduced into
the inner core are again model and mode dependent. Fig. 11
shows the changes in frequency and Q for PKIKP mode 13 S1 when
the inner core exhibits azimuthal symmetry (AS, open square), contains hemispherical structure (HS, solid circle) or an isotropic wedge
(IW, grey triangle). There is a difference between AS and HS for
each of the four different anisotropy models. When there is an IW in
the inner core, the frequencies and quality factors can resemble either those of AS, as occurs with the W,G&L model (Fig. 11d), or be
very close to those of HS, as happens with the Tr model (Fig. 11c).
When the D&R model (Fig. 11b) is used, the lower frequency singlets move closer in frequency and separate in quality factor. The
difference between IW and AS is greater than that between HS and
7 D I S C U S S I O N A N D C O N C LU S I O N S
We have derived new theory which allows us to calculate the effect
of hemispherical anisotropic structure on normal modes sensitive
to the inner core. The theory can be used, in conjunction with
existing models of azimuthally symmetric, cylindrical anisotropy
in the inner core, to simulate the effect of an inner core which
contains one anisotropic hemisphere and one isotropic hemisphere.
The inclusion of HS in models of inner core anisotropy changes
the frequencies, quality factors and spectra of whole Earth oscillations. Not all inner core sensitive modes are affected by the introduction of HS into the inner core, but those that are affected include
radial, PKIKP and PKJKP modes. Only one estimation of shear
wave anisotropy using PKJKP body waves has been made so far
(Wookey & Helffrich 2008). PKJKP modes are the only tool that
can currently be used to observe variation in shear wave anisotropy
in the inner core. The difference between HS and AS is smaller for
PKJKP modes than for radial modes, but non-negligible. Normal
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PKIKP mode 13S1
800
3 S2 is a PKJKP mode which is strongly affected by inner core
anisotropy. It was the dominant mode in the inversion for inner core
anisotropy by Woodhouse et al. (1986) to create the W,G&L model,
and has also been used in the inversions which created the D&R, Tr
and B&T models of azimuthally symmetric inner core anisotropy.
In Irving et al. (2008) we showed that the frequency and quality
factor of this mode are strongly affected by full-coupling.
The effect of an IW on the 3 S2 peaks in a synthetic seismogram
is shown in Fig. 12. For the B&T model (Fig. 12a) there are large
differences between AS, IW and HS. The differences between the
amplitudes of these three synthetic seismograms are of the same
magnitude as the difference between synthetic seismograms and
data. The main differences between HS, AS and IW for the Tr
model (Fig. 12c) are in the phase of the seismogram; the phase
of HS and IW are very close, but diverge from that of AS between
1.106 and 1.109 mHz, the frequency range where two of the singlets
are located. Conversely, AS and IW are very similar when the D&R
model (Fig. 12b) is used for mode 3 S2 , but the phase of the HS
seismogram is different from the phase for AS and IW at lower
frequencies. The amplitude at higher frequencies for HS is also
slightly lower then for AS and IW. The HS, IW and AS synthetic
seismograms when the W,G&L model (Fig. 12d) is used are virtually
indistinguishable, as would be expected because the differences in
the frequency and Q of mode 3 S2 caused by the presence of HS or
an IW are very small.
The spectra show that the presence of an IW is not equivalent
to simply a midpoint between AS and HS. The modes which can
couple when there is an IW in the inner core are model dependent.
The imposition of longitudinally varying structure, either in the
form of HS or IW, sometimes fits the data better than AS does.
(a) B&T
(b) D&R
(c) Tr
(d) W,G&L
973
0
Amplitude
Phase
Normal mode coupling and hemispherical inner core anisotropy
0
0
1.090
1.095
1.100
1.105
1.110
1.115
1.120
1.090
1.095
1.100
Frequency (mHz)
Data
1.105
1.110
1.115
1.120
Frequency (mHz)
AS
HS
IW
Figure 12. Data (solid line) and synthetic seismograms for mode 3 S2 when the inner core is azimuthally symmetric (AS, dotted line), contains hemispherical
anisotropic structure (HS, dashed line) or contains an isotropic wedge (IW, dot-dashed line) for event 122604A. The inner core anisotropy models used are
(a) B&T model, (b) D&R model, (c) Tr model and (d) W,G&L model.
modes will provide the only tool with which to probe hemispherical
structure in shear wave anisotropy for the foreseeable future.
The changes induced by the imposition of hemispherical structure
into the inner core are highly model dependent. HS in the D&R
model causes the biggest changes in radial modes, whilst HS in the
B&T and Tr models cause the biggest changes for PKJKP modes.
The presence of HS causes an observable effect on synthetic spectra.
For several different modes the magnitude of this effect is of the
order of the differences between synthetic and observed spectra.
For some radial, PKIKP and PKJKP modes, the presence of an IW
in the inner core produces changes in frequency and Q of the modes,
and observable changes in synthetic seismograms. The presence of
an IW is not equivalent to simply a midpoint between AS and HS.
The coupling between modes when there is an IW in the inner core
may be similar to that present when there is AS or HS in the inner
core or different to either of these models.
As normal modes will be able to provide information about hemispherical structure in the inner core, it will become possible to differentiate between different mechanisms of inner core anisotropy. To
create hemispherical structure in inner core anisotropy, the mechanism would need to be affected by some type of degree one structure.
Aubert et al. (2008) have suggested that a degree one pattern in the
heat flow at the inner core boundary could be caused by coupling
of the structure in the lower mantle with the inner core through
heat and mass flux anomalies; this may account for the seismically
observable hemispherical pattern in the inner core. The presence of
different chemical compositions at the inner core boundary seems
unlikely, given the rapid convection in the outer core. Likewise, no
large degree one pattern in the magnetic field has been observed.
Mechanisms which require these properties (Singh et al. 2000;
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Journal compilation Karato 1993b; Buffett & Wenk 2001) would therefore be ruled out
if degree one structure in the inner core could be confirmed using
normal modes.
AC K N OW L E D G M E N T S
We would like to thank Caroline Beghein and an anonymous reviewer for their constructive reviews. JCEI was supported by NERC
Studentship NER/S/A/2005/13491 and by a Research Grant from
Trinity College, Cambridge. The research leading to these results
has received funding from the European Research Council under the European Community’s Seventh Framework Programme
(FP7/2007-2013)/ERC grant agreement number 204995.
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APPENDIX A: THE CANONICAL BASIS
This basis is explained in Phinney & Burridge (1973) and summarized here. The canonical basis (ê− , ê0 and ê+ ) can be related to the
spherical basis (θ̂, φ̂ and r̂) by
1
ê− = √ (θ̂ − i φ̂)
2
(A1)
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Normal mode coupling and hemispherical inner core anisotropy
ê0 = r̂
(A2)
−1
ê+ = √ (θ̂ + i φ̂).
2
(A3)
The canonical basis vectors are orthonormal:
ê∗α · êβ = δαβ ,
(A4)
975
where α and β can take the values of +, 0 and −. The metric used for tensor contraction in this basis set is given by
[G]αβ = gαβ = êα · êβ , so that
⎤
⎡
0
0
−1
⎥
⎢
(A5)
G=⎣ 0
1
0 ⎦.
−1
0
0
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