Geophys. J. Int. (2008) 174, 919–929
doi: 10.1111/j.1365-246X.2008.03824.x
Wide-band coupling of Earth’s normal modes due to anisotropic
inner core structure
J. C. E. Irving, A. Deuss and J. Andrews
University of Cambridge, Institute of Theoretical Geophysics & Bullard Laboratories, Madingley Road, Cambridge CB3 0EZ, UK. E-mail: jcei2@cam.ac.uk
Accepted 2008 April 10. Received 2008 April 7; in original form 2007 December 18
1 I N T RO D U C T I O N
1.1 Motivation
It has long been known that Earth’s normal modes are split due
to rotation, ellipticity, 3-D heterogeneity and anisotropy (Dahlen
& Tromp 1998). A number of modes with significant sensitivity
to the core show anomalous splitting which cannot be attributed
to ellipticity, rotation or mantle heterogeneity alone. Woodhouse
et al. (1986) suggested inner core anisotropy as an explanation
for this anomalous splitting of core-sensitive modes; it also provides an explanation for the directional variation in traveltimes
of seismic body waves travelling through the inner core (Morelli
et al. 1986). The majority of models of the inner core’s anisotropy
have been produced using either body wave or normal mode data;
see Creager (2000), Tromp (2001) and Song (1997) for recent
overviews. While both normal modes and body waves sample the
same inner core, there is still significant disagreement between the
models which aim to explain either traveltime anomalies in body
waves or anomalous splitting of inner core modes, and it has been
difficult to reconcile the two data types (Durek & Romanowicz
1999).
Most normal mode studies have used isolated modes and ignored
coupling with other modes, that is, the so-called self-coupling approximation (SC). SC also forms the basis for the splitting function technique. First steps in including coupling between different
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Journal compilation modes have been taken by Resovsky & Ritzwoller (1995, 1998),
who coupled a few pairs of modes. The coupling of modes across
wide frequency bands (i.e. full-coupling) was investigated by Deuss
& Woodhouse (2001) who found that modes could couple across
wider frequency bands than had previously been thought and that
the impact of this coupling was an observable shift in both the amplitude and phase of synthetic seismograms. The calculations they
performed did not include any inner core anisotropy. Andrews et al.
(2006) showed that due to anelasticity, wide-band normal mode
coupling causes significant changes in mode attenuation and frequency when inner core shear velocity is perturbed. The impact of
full-coupling (FC) on mode characteristics when small changes are
made to inner core properties illustrates the need to consider FC for
inner core structure. The importance of the effect of FC due to inner
core anisotropy is still unclear; here we try to quantify this effect.
As several different seismic models of inner core anisotropy exist,
a FC treatment of the modes, where large numbers of modes are
coupled, could cast some insight into anisotropy of the inner core.
It is hoped that underlying similarities between the models could
highlight the aspects of the anisotropy about which the models agree.
In particular, the modes which couple together may be universal and
independent of the model used. These universal groups could allow
more rigorous inversions of seismic data, and production of better
seismic models of inner core anisotropy and also provide definitive
explanations of the anomalous coupling of some normal modes
which are still unexplained.
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Key words: Composition of the core; Surface waves and free oscillations; Seismic anisotropy.
Downloaded from http://gji.oxfordjournals.org/ at Princeton University on March 10, 2015
SUMMARY
We investigate the importance of wide-band coupling of normal modes due to inner core
anisotropy. We compare four different seismic models of inner core anisotropy, which were
obtained by others using the splitting of Earth’s normal modes. These models have been developed using a self-coupling (SC) approximation, which assumes that coupling between nearby
modes through anisotropic inner core structure is negligible. We test the SC approximation
by comparing the frequencies and quality factors of 90 inner core sensitive modes, computed
for these models using either the SC approximation or full-coupling (FC) among large groups
of modes. We find significant shifts in the quality factors and frequencies for some modes.
Groups of modes which significantly couple together are constructed for six target modes.
These groups are model dependent and in some cases contain large numbers of modes. Synthetic seismograms are calculated to show that the difference between SC and FC is observable
on the scale of seismograms and of the same order of magnitude as the difference between
synthetic and observed seismograms. Thus, future models of inner core anisotropy should take
cross-coupling between large groups of modes into account.
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J. C. E. Irving, A. Deuss, J. Andrews
1.2 Inner core anisotropy
(a) W,G&L model
α=
C−A
,
A0
β=
L−N
A0
γ =
A − 2N − F
,
A0
(1)
where A is related to P-wave velocity in the equatorial direction (and
A 0 is the value of A at the centre of the inner core), C is related to
P-wave velocity in the polar direction, L is related to S-wave velocity
in the polar direction, N to S-wave velocity in the equatorial direction
and F is dependent on off-axis properties.
The four different models that we compare are shown in Fig. 1.
Woodhouse et al. (1986) (W,G&L) created two models of inner core
anisotropy by inverting data from seven normal mode multiplets to
get estimates of α, β and γ . We use the W,G&L model (Fig. 1a)
in which α, β and γ are constant throughout the inner core, as
this is the most simple model of inner core anisotropy that exists.
This simple model was extended by Tromp (1993), who inverted
a data set of splitting functions for 18 normal modes to find depth
dependent values for α, β and γ . The Tr model (Fig. 1b) proposes a
cylindrically anisotropic structure with little anisotropic structure at
the centre of the core, because the sensitivity kernels of the modes
tend to zero towards the centre of the inner core (see Fig. 2 for some
examples of sensitivity kernels).
To improve the accuracy of seismic anisotropy models at the
centre of the inner core, where normal modes have very little sensitivity, the Durek & Romanowicz (1999) (D&R) model (Fig. 1c)
was created by adding body wave observations to the inversion of
normal mode data from four shallow and four intermediate to deep
(b) Tr model
Dimensionless
anisotropy
parameters
1200
Radius (km)
and
1000
α
β
γ
800
600
400
200
0
(c) D&R model
(d) B&T model
Radius (km)
1200
1000
800
600
400
200
0
-0.1
0.0
Amplitude
0.1
-0.1
0.0
0.1
Amplitude
Figure 1. Comparison of different inner core anisotropy models in terms of the parameters α, β and γ (see eq. 1 for definitions) for the (a) W,G&L (Woodhouse
et al. 1986), (b) Tr (Tromp 1993), (c) D&R (Durek & Romanowicz 1999) and (d) B&T (Beghein & Trampert 2003) models.
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The inner core of the Earth is known to be anisotropic: it exhibits
different properties in different directions. Most inner core models,
including the ones discussed below, assume cylindrical anisotropy,
that is, anisotropy which is independent of longitude. There are a
variety of different ideas about the dynamic origin of the anisotropy,
these have been discussed in detail by several authors (see reviews by
Tromp 2001; Song 1997). The crystal structure of the inner core, responsible for the anisotropic behaviour, is also currently unknown.
Although ab initio calculations suggest that pure iron is stable in
the hexagonal close packed structure under inner core conditions
(Stixrude 1995; Vočadlo et al. 2003), the presence of lighter elements in the core may alter the energistic landscape sufficiently that
another crystalline structure, such as body-centred cubic, is more
stable (Oganov et al. 2005; Vočadlo 2007). Different seismic models
suggest different crystal structures so that further knowledge concerning seismic anisotropy in the inner core would provide information about its crystal structure. Until the seismologically anisotropic
structure of the Earth’s inner core is known in more detail the cause
of the anisotropy and its implications will remain an enigma.
Many different seismic models of inner core anisotropy are available in the literature, developed using either normal mode or body
wave data. We are interested in models obtained using normal modes
and we investigate the effect of coupling due to inner core anisotropy
on four models: those produced by Woodhouse et al. (1986), Tromp
(1993), Durek & Romanowicz (1999) and Beghein & Trampert
(2003). All of these seismic models can be described using the same
parameters: α, β and γ (as in Tromp 1993). 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. These
three parameters can be defined in terms of the Love coefficients
(Love 1927) of the inner core, so that
Normal mode coupling in the inner core
2 T H E O RY
2.1 Normal modes
The Earth’s free oscillations, or normal modes, can be observed after large earthquakes. Any arbitrary motion of the Earth can be
written as a superposition of these normal modes. There are
(a) Radial mode 2 S0
(d) PKIKP mode
11 S1
two types of normal mode, described using the notation n S l for
spheroidal modes and n T l for toroidal modes, where n is the overtone number and l the angular order of the mode. The modes oscillate
at discrete frequencies. Toroidal modes involve shear motion of the
Earth and do not exist in the fluid outer core. Spheroidal modes
are sensitive to both the outer and inner core and will be used in
this study. For a 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. The finite observational time window and attenuation by the anelasticity of the Earth
mean that modes are not observed as δ-functions but as broadened
peaks. The properties of a normal mode singlet 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.
Only some modes are sensitive to the inner core, others have
sensitivity only to the upper regions of the Earth’s interior. PKIKP
modes are those which are sensitive to the P-wave velocity in the
inner core (as PKIKP body waves are). PKIKP modes normally have
high overtone number n and low angular order l. Radial modes are
those which have an angular order of zero, that is, n S 0 . For a radial
mode the overtone number, n, corresponds to the number of nodes
of the standing wave along the radius of the Earth. Examples of three
radial modes ( 2 S 0 , 4 S 0 , 6 S 0 ) and three PKIKP modes ( 11 S 1 , 13 S 1 ,
15 S 1 ) are shown in Fig. 2. This figure shows the sensitivity of the
different modes to variations in P-wave velocity, S-wave velocity
and density. It can be seen that some modes are more sensitive
(b) Radial mode 4 S0
(e) PKIKP mode
13 S1
(c) Radial mode 6 S0
(f) PKIKP mode
15 S1
Figure 2. Sensitivity kernels calculated using PREM for three radial modes: (a) 2 S 0 , (b) 4 S 0 , (c) 6 S 0 ; and three PKIKP modes (d) 11 S 1 , (e) 13 S 1 , (f) 15 S 1 .
Solid line shows P-wave sensitivity, dashed line S-wave sensitivity, and dot–dashed line density sensitivity. The black bars at the edge of each kernel show the
depths to which a P wave (left-hand side) and S wave (right-hand side) with sensitivity to the Earth’s structure equivalent to the normal mode would travel.
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earthquakes (including the 1994 Bolivia event). Data concerning 20
different modes were used in the inversion. The use of body wave
data, in addition to normal mode data, resulted in the placement of
strong anisotropic structure in the centre of the inner core of the
D&R model.
The Beghein & Trampert (2003) (B&T) model (Fig. 1d) was
developed using the ‘Neighbourhood Algorithm’ (Sambridge 1999),
a specific type of Monte Carlo technique which was used to explore
possible inner core anisotropic models using normal mode splitting
functions. The sensitivity kernels for normal modes tend towards
zero at the centre of the Earth (Fig. 2) but, using the Neighbourhood
Algorithm, Beghein & Trampert (2003) suggested an anisotropic
structure towards the centre of the inner core, without using any
body wave data.
The amplitudes of α, β and γ are larger in the D&R and B&T
models than in the W,G&L and Tr models. Whilst the models are
in some agreement at the inner core boundary, further towards the
centre of the core large discrepancies occur. The differences between
the models are greatest there, with α, β and γ varying in both
magnitude and sign. These discrepancies can occur because the
sensitivity of normal modes to this region is very limited.
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J. C. E. Irving, A. Deuss, J. Andrews
2000
(a)
W,G&L SC
W,G&L FC
(b)
Tr SC
Tr FC
(c)
D&R SC
D&R FC
(d)
B&T SC
B&T FC
1600
Q
1200
800
400
0
2000
1600
Q
1200
400
0
0
1
2
3
4
5
Frequency (mHz)
6
7 0
1
2
3
4
5
Frequency (mHz)
6
7
Figure 3. Eigenfrequencies and Q for inner core sensitive modes using the self-coupling (SC) approximation and full-coupling (FC) for the inner core anisotropy
models of (a) W,G&L (Woodhouse et al. 1986), (b) Tr (Tromp 1993), (c) D&R (Durek & Romanowicz 1999) and (d) B&T (Beghein & Trampert 2003). Only
those modes with a frequency of less than 6 mHz and Q greater than 200 are shown. Mantle structure, ellipticity and rotation have not been included in these
calculations—all variations in frequency and Q are due to inner core anisotropy.
to P-wave velocity in the uppermost inner core (e.g. 2 S 0 , 11 S 1 )
whereas other modes (e.g. 4 S 0 , 6 S 0 ) have a greater sensitivity to
the intermediate region of the inner core. The symmetry exhibited
by normal modes requires than no modes have any sensitivity to
velocity or density perturbations at the very centre of the inner core.
Aspherical structure, rotation and ellipticity of the Earth all
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 between different modes also
occurs, which again changes the frequencies of the singlets. Rotation couples spheroidal modes with the same angular order and also
pairs n S l and n T l±1 where n and n can take any value. Ellipticity
couples the same pairs of modes as rotation and also pairs of modes
n S l – n S l±2 and n T l – n T l±2 .
If the frequencies of two multiplets are close, the modes will also
be coupled as a result of the 3-D heterogeneous structure of the Earth.
Coupling rules described by Luh (1974) show that heterogeneous
structure of degree L (when L is even) will allow a normal mode n S l
to couple with modes n S L+l , n S L+l−2 , . . . , n S l , . . . , n S |L−l| . This
means that when a model contains only even structure (as is the case
for the four inner core anisotropy models discussed here), coupling
modes will always have an even difference in angular order. This
coupling due to heterogeneity has been considered to be negligible
by many authors. Deuss & Woodhouse (2001), on the other hand,
have shown that coupling through heterogeneous mantle structure is
important for normal modes sensitive to the mantle. Consequently,
when the effect of inner core anisotropy is considered, mode coupling may also be significant. In this context we define significant
changes to be those changes which are of the same magnitude as
the differences between the different models using the SC approxi-
mation, or as large as the differences between the model predictions
and data. The purpose of this paper is to investigate the effect of
normal mode coupling for seismic models of inner core anisotropy.
2.2 Coupling approximations
The SC approximation assumes that frequency differences between
modes are much greater than the magnitude of possible coupling
due to heterogeneous structure, so that each mode is effectively independent and can be isolated from the other modes. SC has been
used by most authors in normal mode studies and forms the basis
for the use of splitting functions. Group-coupling (GC) approximations allow very small groups (or pairs) of modes to couple. In this
approximation, coupling between modes in the groups is considered
to be significant but coupling between modes in different groups is
assumed to be insignificant. Resovsky & Ritzwoller (1995, 1998)
used this technique to find small groups and make observations of
generalised splitting functions. FC schemes (Deuss & Woodhouse
2001) allow all modes to couple together. If the assumptions of SC
(or GC) were valid it would be expected that the SC, GC and FC
schemes would give the same results, although GC and FC schemes
would require greater computational power to obtain those results.
3 METHODOLOGY
The Preliminary Earth Reference Model (PREM) (Dziewonski &
Anderson 1981) is used to calculate degenerate normal modes for a
spherically symmetric and non-rotating Earth. Perturbation theory is
then used to add mantle and inner core structure, rotation and ellipticity (Dahlen & Tromp 1998). We identified 90 inner core sensitive
modes at frequencies under 10 mHz and performed our calculations
comparing SC, GC and FC for all modes in this group. Whilst inves
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800
Normal mode coupling in the inner core
(a) Radial Mode 2S0
1300
(i) Self-Coupling
(b) Radial Mode 4S0
1000
(i) Self-Coupling
1000
(i) Self-Coupling
Q
Tr
700
1000
800
600
900
500
800
400
(ii) Full-Coupling
1000
B&T
D&R
900
800
1100
1300
(c) Radial Mode 6S0
900
1200
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W,G&L
PREM
700
600
(ii) Full-Coupling
1000 (ii) Full-Coupling
900
1200
900
800
Q
700
1000
800
600
900
500
800
400
2.5075
700
600
2.5100
2.5125
Frequency (mHz)
4.105
4.110
Frequency (mHz)
5.740
5.745
Frequency (mHz)
Figure 4. Eigenfrequencies and Q for the radial modes (a) 2 S 0 , (b) 4 S 0 and (c) 6 S 0 using (i) self-coupling and (ii) full-coupling. Mantle structure, ellipticity
and rotation have not been included in these calculations—all variations in frequency and Q are due to inner core anisotropy. For comparison, degenerate values
are also shown for the isotropic Preliminary Reference Earth Model (PREM). For radial modes using the SC approximation, all five symbols lie on top of each
other.
800
750
(a) PKIKP Mode 11S1
(b) PKIKP Mode 13S1
(c) PKIKP Mode 15S1
(i) Self-Coupling
(i) Self-Coupling
(i) Self-Coupling
B&T
D&R
650
Tr
Q
700
600
W,G&L
550
PREM
500
450
800
750
(ii) Full-Coupling
(ii) Full-Coupling
(ii) Full-Coupling
700
Q
650
600
550
500
450
3.68
3.69
3.70
Frequency (mHz)
4.49
4.50
Frequency (mHz)
4.51
5.29
5.30
5.31
Frequency (mHz)
Figure 5. Eigenfrequencies and Q for PKIKP modes (a) 11 S 1 , (b) 13 S 1 and (c) 15 S 1 using (i) self-coupling and (ii) full-coupling, similar to Fig. 4. Mantle
structure, ellipticity and rotation have not been included in these calculations—all variations in frequency and Q are due to inner core anisotropy. For comparison,
degenerate values are also shown for the isotropic Preliminary Reference Earth Model (PREM).
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J. C. E. Irving, A. Deuss, J. Andrews
4 R E S U LT S
Table 1. Modes used in the creation of each inner core
anisotropy model.
B&T
D&R
Tr
W,G&L
2S3
3S2
2S3
3S2
3S2
5S2
3S2
6S3
5S3
5S3
6S3
9S3
7S4
8S1
8S5
11 S 4
7S5
8S5
9S3
11 S 5
8S1
9S2
11 S 4
13 S 2
8S5
9S4
11 S 5
13 S 3
9S3
11 S 4
13 S 2
11 S 4
11 S 5
13 S 3
11 S 5
13 S 1
14 S 4
13 S 1
13 S 2
15 S 3
13 S 3
13 S 3
16 S 6
15 S 7
15 S 3
18 S 4
16 S 5
16 S 5
20 S 5
17 S 1
18 S 2
21 S 6
18 S 3
18 S 3
23 S 5
18 S 4
18 S 4
25 S 2
21 S 6
22 S 1
27 S 2
21 S 7
23 S 4
21 S 8
23 S 5
23 S 4
23 S 5
27 S 1
Note: Anisotropy models are B&T: Beghein & Trampert
(2003); D&R: Durek & Romanowicz (1999); Tr: Tromp
(1993) and W,G&L: Woodhouse et al. (1986).
4.1 The differences between full-coupling and
self-coupling
We first investigate the importance of FC due to inner core
anisotropy, comparing it to SC. We calculate the eigenfrequencies
and quality factors for each of the four anisotropy models (see Fig. 1)
using different coupling schemes—(i) SC approximation for the individual inner core sensitive modes and (ii) FC between all 90 inner
core sensitive modes. The effects of mantle heterogeneity, rotation
and ellipticity were excluded so that it would be possible to isolate
those modes which were coupling due to the anisotropy model used.
Fig. 3 shows the eigenfrequencies and quality factors (Q) for
inner core sensitive modes using SC and FC for each model. The
multiplets of many modes are split more strongly when FC is used
than under the SC approximation. There are large changes in Q for
many modes, and there are also variations in frequency (although
these are not easily visible at the scale used in Fig. 3). The frequency
and Q differences between FC and the SC approximation depend
on the model used; the changes are not uniform between the four
models.
We identified six modes which showed a marked difference between SC and FC, including three radial ( 2 S 0 , 4 S 0 and 6 S 0 ) and
three PKIKP modes ( 11 S 1 , 13 S 1 and 15 S 1 ). It should be noted that
these are not the only modes which change their properties, rather,
they have been chosen to illustrate the differences between SC and
FC. The sensitivity kernels of these six modes are shown in Fig. 2.
All of these six modes have sensitivity to the structure of the mantle and the inner core. None of the modes are sensitive to the very
centre of the inner core which is where the models disagree most
strongly.
The eigenfrequencies and quality factors (Q) for each of these
modes, together with those calculated using the 1-D isotropic PREM
are shown in Figs 4 and 5. The shifts in Q for FC are significant—up
to a 35 per cent change in attenuation. The shifts of the frequencies of
the modes are smaller, up to about 0.5 per cent. We also find that the
effect of coupling on inner core sensitive modes is model dependent.
The W, G&L model produces the largest shifts for modes 2 S 0 and
6 S 0 (Figs 4a and c), the Tr model for modes 11 S 1 and 15 S 1 (Figs 5a
and c) and the B&T model for modes 4 S 0 and 13 S 1 (Figs 4b and
5c). The changes in the frequencies and attenuations of the radial
modes are especially noteworthy. Radial modes are not sensitive to
3-D structure when SC is used (Fig. 4i), but FC allows coupling
with other modes and allows the radial modes to respond to the
anisotropic structure in the inner core (Fig. 4ii).
4.2 Model-building modes
The four anisotropy models used here have all been developed by
inverting data extracted from seismic records. The modes inverted
to form each model are shown in Table 1. Of the six modes which
have been investigated in detail here, both the B&T and the D&R
models used only mode 13 S 1 . Neither the W,G&L model, nor the Tr
model use any of these six modes.
The four modes 13 S 3 , 11 S 4 , 11 S 5 and 3 S 2 were used in the construction of all four anisotropy models. Of these four modes, three
( 13 S 3 , 11 S 4 and 11 S 5 ) show differences between SC and FC (frequency changes of 0.0005 mHz and changes in Q of up to 42). More
significantly 3 S 2 (Fig. 6a) shows frequency and Q factor changes as
large as those seen for the previous three PKIKP modes discussed
in this paper. Singlets in the mode 3 S 2 have frequency changes of
up to 0.003 mHz and Q factor changes of up to 78 (27 per cent).
Mode 13 S 2 (Fig. 6b) was used in the creation of three of the four
anisotropy models (D&R, Tr and W,G&L). 13 S 2 is known to be
a poorly understood mode, about which there is some ambiguity
concerning the splitting function (Durek & Romanowicz 1999). It
was not used in the construction of the B&T model for this reason. It
shows frequency changes of up to 0.0011 mHz for individual singlets
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tigating the differences between FC and SC (Sections 4.1 and 4.2),
and finding groups of modes which couple together (Section 4.3),
mantle structure, rotation and ellipticity were not included in the
computations so that the only differences between SC and FC would
be those due to inner core anisotropy. When comparing the seismic
models using SC and FC with real data (Section 4.4), mantle structure, rotation and ellipticity were used in addition to inner core
anisotropy to achieve realistic seismograms.
Data from the 1994 June 9 Bolivia earthquake have been used for
comparison with the synthetic data, which were calculated using the
Centroid Moment Tensor solution for this event (Dziewonski et al.
1995). The 1994 Bolivia event provided some of the best quality
normal mode data to be recorded thus far. As a result, data collected
from this earthquake has been used in many different normal modes
studies (He & Tromp 1996; Resovsky & Ritzwoller 1998; Deuss
& Woodhouse 2001), as well as in the inversions for inner core
anisotropy performed by Durek & Romanowicz (1999) and Beghein
& Trampert (2003).
Model S20RTS, developed by Ritsema et al. (1999), was used to
provide mantle heterogeneity in shear wave velocity, v s . The shear
wave velocity perturbations were then scaled to obtain compressional velocity, v p and density, ρ, with scaling of the form δv p /v p =
α(r )δvs /vs , δρ/ρ = β(r )δvs /vs . We use α = 0.5 (Li et al. 1991) and
β = 0.3 (Karato 1993).
Normal mode coupling in the inner core
4.3 Group-coupling
To discover the modes that are the most significant contributors to
changes in frequency and quality factors in the FC computation,
groups of modes were sought for each of the modes 2 S 0 , 4 S 0 , 6 S 0 ,
11 S 1 , 13 S 1 and 15 S 1 . These groups contain all modes which couple
with the lead mode and reproduce the FC modal frequencies to
450
(a) PKIKP Mode 3S2
(i) Self-Coupling
400
925
within 0.01 per cent and the corresponding Q values to within 0.04
per cent. The coupling groups were found by adding and removing
individual modes from the group until the group was able to replicate
the FC frequency and Q of the lead mode to within the required
accuracy. It was noted that some modes, whilst not coupling directly
with the lead mode, coupled strongly with other modes in the group,
so that their omission would have significantly decreased the ability
of the coupling group to replicate the FC properties. These indirect
interactions should be taken into account in any inversion scheme
which is based on group coupling.
Fig. 7 displays the groups for the four different inner core
anisotropy models as a function of frequency. It can be seen that
the groups are model dependent and that they are of drastically
different sizes: those for the Tr model are the smallest and those
for the B&T model are the largest. This result suggests that the
more complex models also exhibit more complex coupling in larger
groups.
There are some modes which couple in every model, the common modes of the group; these common modes are also displayed
in Fig. 7. The groups of common modes are much smaller than
the groups needed to account for all of the coupling. The common
groups already show significant wide-band coupling over ranges up
to 5 mHz. The coupling needed to completely account for all effects
occurs over even wider frequency bands—for the W,G&L, D&R and
B&T models the bands for some modes are over 8 mHz in width.
Similar coupling over broad frequency bands was previously found
for mantle structure by Deuss & Woodhouse (2001) and for inner
core anelasticity by Andrews et al. (2006).
Modes which couple within these groups always differ by an even
angular order, as is required by the symmetries of the system and
the corresponding coupling rules. The modes which couple together
(b) PKIKP Mode 13S2
(i) Self-Coupling
900
(c) PKIKP Mode 9S3
(i) Self-Coupling
B&T
900
Q
800
350
D&R
Tr
875
W,G&L
700
300
850
250
450
825
925
(ii) Full-Coupling
400
PREM
(ii) Full-Coupling
600
900
(ii) Full-Coupling
900
Q
800
350
875
300
850
700
250
1.10
1.11
Frequency (mHz)
825
1.12 4.840
4.845
Frequency (mHz)
600
4.850
3.55
3.56
Frequency (mHz)
Figure 6. Eigenfrequencies and Q for PKIKP modes (a) 3 S 2 , (b) 13 S 2 and (c) 9 S 3 using (i) self-coupling and (ii) full-coupling, similar to Figs 4 and 5.
Mantle structure, ellipticity and rotation have not been included in these calculations—all variations in frequency and Q are due to inner core anisotropy. For
comparison, degenerate values are also shown for the isotropic Preliminary Reference Earth Model (PREM).
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and Q changes of up to 33 (4 per cent). Likewise, mode 9 S 3 (Fig. 6c)
was used in the inversions to produce three of the four anisotropy
models (B&T, Tr and W,G&L). It was not used in the creation of
the D&R model as it showed poor convergence in the inversion.
Singlets of 9 S 3 display frequency changes of up to 0.0013 mHz and
Q changes of up to 96 (14 per cent).
Beghein & Trampert (2003) comment that the B&T model prediction for the 3 S 2 mode fits the data used in the inversion well.
Durek & Romanowicz (1999) published the residual variances of
each mode after subtracting the predictions made by their anisotropy
model. For the D&R model, predictions for modes 3 S 2 and 13 S 2 fit
the data used in the inversion better than the average, and 13 S 1 fit
the data used less well than the average. Tromp (1993) fits 3 S 2 and
13 S 2 poorly, 11 S 4 , 11 S 5 and 13 S 3 reasonably well and 9 S 3 very well.
Information about the degree to which the W,G&L model fits the
data from which it was created is unavailable. The modes for which
the SC approximation work poorly are in some cases fitted well by
the SC derived anisotropy model.
The investigation of the precise variation of each inner core mode
is beyond the scope of this paper, but all of the models have been
produced using at least two modes which display large differences
between FC and SC, and at least three modes which are also changed
by the SC approximation.
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J. C. E. Irving, A. Deuss, J. Andrews
model used. As the coupling groups are different for every seismic
anisotropy model, inversions which use a GC approximation will
clearly produce models which are dependent on the coupling groups
selected.
in the B&T and Tr models have angular order differences of up to
six and eight respectively, whereas the D&R and W,G&L models
have several groups where the greatest angular order (l) difference
between coupling modes is only two.
Although the W,G&L model only includes degree 0, 2 and 4
structure, the angular order differences between the main mode in
each group and the other modes in the same group are larger than
would be expected by the coupling rules (Luh 1974). Consider the
mode 2 S 0 . When the W,G&L model is used, this mode couples
with fourteen other modes, including 7 S 2 and 4 S 6 . Whilst coupling
between 2 S 0 and 4 S 6 would be forbidden by the coupling rules,
4 S 6 is allowed to couple to 7 S 2 . To successfully approximate the FC
behaviour of 2 S 0 , the mode 4 S 6 , normally considered to be forbidden
from coupling with 2 S 0 , must be added to the group to successfully
replicate the interaction between 2 S 0 and 7 S 2 , which does couple to
4 S 6.
These results demonstrate that the inner core modes which couple
together in a group differ depending on the inner core anisotropy
4.4 Normal mode spectra - the consequences
of full-coupling
Synthetic seismograms and corresponding normal mode spectra
were calculated for the 1994 June 9 Bolivia event at various stations, using the SC approximation, GC for the common modes
and FC to illustrate the differences between both the inner core
anisotropy models and also the different coupling approximations.
Here, we have also included mantle structure, ellipticity and rotation.
The synthetic seismograms, as shown in Figs 8 and 9, are displayed
along with vertical motion (VHZ) observed data seismograms. Both
synthetic and observed seismograms are cosine tapered and padded
with zeroes before Fourier transformation to the frequency domain.
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6S0
15S1
B&T model
13S1
4S0
11S1
2S0
6S0
15S1
D&R model
13S1
4S0
11S1
2S0
6S0
15S1
Tr model
13S1
4S0
11S1
2S0
6S0
15S1
W,G&L model
13S1
4S0
11S1
2S0
6S0
15S1
Common to all models
13S1
4S0
11S1
2S0
0
1
2
3
4
5
6
7
8
9
10
Frequency (mHz)
Figure 7. The groups of modes which couple with modes 2 S 0 , 4 S 0 , 6 S 0 , 11 S 1 , 13 S 1 and 15 S 1 . The small dots each represent a mode which couples with the
main mode, whilst the large diamonds mark the main mode on each line. The length of the line indicates the width of the group across which coupling occurs.
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(c)Mode 13S1, Station KIP: 5-100 hours.
Phase
Phase
(a)Mode 13S1, Station KIP: 5-100 hours.
0
0
Amplitude
Data
W,G&L, full coupling
D&R, full coupling
Amplitude
Data
W,G&L, self coupling
D&R, self coupling
0
4.48
4.49
Frequency (mHz)
4.50
0
4.48
4.51
4.49
Frequency (mHz)
4.50
0
0
Amplitude
Amplitude
Data
Tr, full coupling
B&T, full coupling
4.49
Frequency (mHz)
4.50
4.51
0
4.48
4.49
Frequency (mHz)
4.50
4.51
Figure 9. Observed and synthetic seismograms for mode 13 S 1 at Station KIP, time window 5–100 hr for the 1994 June 9 Bolivia event. (a) SC for the W,G&L
and D&R models and observed data, (b) SC for the Tr and B&T models and observed data, (c) FC for the the W,G&L and D&R models and observed data and
(d) FC for the Tr and B&T models and observed data. Mantle structure, rotation and ellipticity have also been included in addition to the anisotropy model.
Fig. 8 shows synthetic and observed spectra for the radial mode
using both SC and FC. It can be seen from Fig. 8a that there is a
significant difference between the seismograms produced using SC
and FC, in agreement with results for mantle structure by Deuss &
Woodhouse (2001). For station KIP, using the W,G&L model, the
differences between 2 S 0 mode when SC or FC is used are about
twice that between SC and the data. Fig. 4(a) shows a 0.1 per cent
frequency difference for this mode, which may seem small but is
clearly significant in the synthetic spectrum of Fig. 8(a). The differences between FC and SC are significant and observable on the
scale of seismograms from real events.
We also compared synthetic spectra using GC with only the common modes. Fig. 8(b) shows that GC with the common modes is
unable to completely replicate the effect of FC on 2 S 0 , although GC
with common modes produces seismograms which are much closer
to FC than SC. Fig. 4(a) shows that the fully coupled D&R model
produces a 2 S 0 mode which is quite close to the SC result, but even
this small shift is still observable in Fig. 8(b).
We have computed synthetic seismograms for all four inner core
anisotropy models for mode 13 S 1 using SC (Figs 9a and b) and
FC (Figs 9c and d); the synthetic seimograms are shown together
with data. The SC and FC synthetic seismograms produced from
the four seismic anisotropy models are noticeably different from
each other and the data. Using SC, the amplitudes of the 13 S 1 mode
synthetic seismograms are similar for all of the models; this is because all models produce three singlets with Q values of around
740. This high Q results in large amplitude peaks even when a long
time window, such as the window used here, is considered. There
are substantial variations in the frequencies of the three singlets be2S0
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Journal compilation tween the models, however all of the singlets fall within the window
shown.
When FC is used (Figs 9c and d), the total amount of energy in
this mode and therefore the overall amplitude of the mode is smaller
than under SC, as the quality factor is lower for 13 S 1 when FC is
used instead of SC (as can be seen in Fig. 5b). There is therefore
less energy in the mode 13 S 1 in FC than in SC over the long time
window used to observe normal modes. When the phase of mode
13 S 1 is considered, all four models show discrepancies between the
SC approximation and FC. None of the models reproduce the phase
observed in the data of the 13 S 1 mode accurately.
Using FC, the D&R model gives a peak at about 4.500 mHz because there is a singlet of the 13 S 1 mode at this frequency. This peak
is at a higher frequency in the other models: 4.505 mHz in the Tr
model, 4.507 mHz in the W,G&L model; it has been shifted out of
the observation window to a frequency of 4.510 mHz in the B&T
model. The widely varying location of this one singlet peak gives
an indication of the size of the differences between the currently
existing inner core anisotropy models and shows that these differences are readily observable.
5 C O N C LU S I O N S
We have shown that inner core anisotropy causes significant coupling between large numbers of inner core sensitive normal modes,
and that the differences between SC and FC can be large. If the SC
approximation was accurate enough, there would be no change in
singlet frequency and quality factor when inner core modes are allowed to couple. As there is a significant change in the quality factor
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Data
Tr, self coupling
B&T, self coupling
0
4.48
4.51
(d)Mode 13S1, Station KIP: 5-100 hours.
Phase
(b)Mode 13S1, Station KIP: 5-100 hours.
Phase
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J. C. E. Irving, A. Deuss, J. Andrews
pared to observed seismograms from the 1994 June 9 Bolivia event.
Using two modes sensitive to inner core structure, 2 S 0 and 13 S 1 ,
the differences between the seismograms produced using the SC
approximation, FC and the data from the Bolivia event are considerable.
It is clear that none of the models fit particularly well under the FC scheme. This is not surprising, since they have
been created by inversion or forward modelling using the SC
approximation. To better understand the causes of inner core
anisotropy, the anisotropy must first be accurately described. Further work is clearly needed to determine better models of inner core
anisotropy.
Phase
(a)Mode 2S0, Station KIP: 5-100 hours.
0
Amplitude
Data
W,G&L full coupling
W,G&L self coupling
0
2.50
2.51
Frequency (mHz)
2.52
Phase
(b)Mode 2S0, Station KIP: 5-100 hours.
AC K N OW L E D G M E N T S
0
Amplitude
REFERENCES
0
2.50
2.51
Frequency (mHz)
2.52
Figure 8. Observed and synthetic seismograms for mode 2 S 0 at Station
KIP, time window 5–100 hr for the 1994 June 9 Bolivia event. (a) SC and FC
using the W,G&L model and data from the same event, (b) SC, GC common
modes and FC using the D&R model.
and singlet frequencies of some inner core modes, we conclude that
this approximation is not valid.
The modes that couple with each other are spread over a wide
range of frequencies, in some cases up to 8 mHz. Coupled modes
always differ by an even angular order, as required by the symmetries of the system for even order anisotropy models. The inner core
modes which couple together differ depending on the inner core
anisotropy model used. The WG&L model has the smallest group
and the B&T model the largest group of modes which couple, suggesting that more complex models couple larger groups of modes
together. Inversions which use a GC approximation will clearly produce models which depend on the coupling groups selected. An
avenue for further investigation would be inversion schemes where
the coupling groups are dynamically calculated between iterations
of the inversion. This could considerably reduce the computational
size of the inversion problem whilst retaining the significant interactions present in full-coupling.
The modes which interact with one another are not only those
which would be expected to interact using coupling rules. Coupling
between ‘intermediate’ modes permits some interaction between
modes outside of those which would couple if quasi-degenerate perturbation theory was used. This means that full-coupling, in which
no assumptions are made about the modes which couple, will ensure that unexpected interactions are possible and included in the
calculations.
The differences between the modal eigenfrequencies and attenuations calculated using FC and the SC approximation are sufficiently
large that they can be seen when synthetic seismograms are com-
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We would like to thank Joe Resovsky, Caroline Beghein and Jeannot
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