www.sciencemag.org/cgi/content/full/science.1188596/DC1
Supporting Online Material for
Regional Variation of Inner-Core Anisotropy from Seismic NormalMode Observations
Arwen Deuss,* Jessica C. E. Irving, John H. Woodhouse
*To whom correspondence should be addressed. E-mail: [email protected]
Published 15 April 2010 on Science Express
DOI: 10.1126/science.1188596
This PDF file includes:
SOM Text
Figs. S1 to S7
Tables S1 and S2
References
Regional variation of inner core anisotropy from
seismic normal mode observations
Arwen Deuss, Jessica C. E. Irving, John H. Woodhouse
Supporting online material
Normal modes
Normal modes are standing waves along the surface and radius of the Earth, and thus only
exist for discrete natural frequencies. Spheroidal modes n Sl involve P-SV wave motion and
are characterized by their radial order n and angular order l. Modes that are sensitive
to the inner core typically have large n and small l. Each spheroidal mode multiplet
n Sl consists of 2l + 1 singlets which all have the same frequency (i.e. are degenerate)
for a spherically symmetric, isotropic, non-rotating Earth model such as the Preliminary
Reference Earth Model (PREM) (S1). Rotation, ellipticity, anisotropy and heterogeneous
structure remove the degeneracy, thereby splitting the singlets.
In first approximation, split modes may be treated as isolated; this is the so called
‘self-coupling’ approximation. In self-coupling, a mode is only sensitive to even-degree
structure in the Earth due to symmetry. All currently existing normal mode models
of inner core anisotropy (S2–6) have been made using the self-coupling approximation.
Such models limit the inner core as being cylindrically anisotropic, with no hemispherical
regional variations. If two modes are close in frequency, then self-coupling may not be
valid and ‘cross-coupling’ (i.e. resonance) between the two modes needs to be taken into
account (S7, 8). Cross-coupling between pairs of modes introduces sensitivity to odddegree structure in the Earth. Of particular interest here is the hemispherical structure
found in the inner core, which can be described as odd-degree structure.
Split modes can be observed by measuring spectra for time windows of several tens of
hours in duration for large earthquakes. We recently constructed a new data set containing
95 earthquakes with MW > 7.5 which have occurred since 1976. The data set used in
the paper is a subset of the large data set, as inner core sensitive modes are more easily
1
seen in those deep earthquakes which are sufficiently large to have low noise levels on the
vertical component. Table S1 shows the number of events and spectra used for each mode
pair studied.
Splitting function measurement
The splitting and cross-coupling of pairs of modes can be completely described using the
generalized splitting function approach (S9). Splitting functions linearly depend on the
heterogeneous and anisotropic variations in the Earth, and are used to visualize how a
normal mode sees a depth-averaged Earth structure. Splitting functions cts are measured
by non-linear iterated least squares inversion (S10); s is the angular order and t the
azimuthal order of the spherical harmonic used to describe the structure in the Earth.
The splitting functions linearly depend on the heterogeneous and anisotropic structure in
the Earth δmts (r), where δm are perturbations to elastic moduli (including anisotropy)
and density parametrized by radius r and spherical harmonics of order s, t. The splitting
function coefficients cts are then given by
cts(kk′) =
Z
0
a
δmts (r) · Ms(kk′ ) (r)r 2 dr
(1)
where Ms(kk′) (r) are the sensitivity kernels, a is the radius of the Earth and k denotes
a spheroidal mode with radial and angular order n, l. For self-coupling, k = k ′ and s =
even. Cross-coupling between pairs of normal modes with k 6= k ′ allows sensitivity to
odd-degree structure, if the difference in angular order l − l′ between the two normal
modes is an odd number. We also measure Re(c00 ) and Im(c00 ) which are related to the
shift in center frequency and attenuation factor Q of each mode with respect to the PREM
model.
Inner core anisotropy with cylindrical symmetry (S11) is described by even-degree
structure of order c02 and c04 . We developed new theory to allow this anisotropy to only
exist in one hemisphere (S8). Our forward calculations show that allowing this type of
structure to exist only in one hemisphere with boundaries at 0◦ and 180◦ , results in odddegree splitting functions for Im(c13 ) and Im(c15 ). Moving the boundaries to 14◦ E and
151◦ W (i.e. the boundaries found in short period body wave observations, see Fig. 3 in
the main paper), adds additional coefficients at Re(c13 ) and Re(c15 ).
We identified 3 pairs in the mode catalog up to 10 mHz, which are close in frequency
and follow the angular order selection rules for hemispherical structure; we measured
the cross-coupled odd-degree splitting functions for these three pairs of modes. Several
2
more inner core sensitive modes are sensitive to hemispherical structure, but only the
pairs in this paper are sufficiently close, and dominated by cross-coupling between just
these two modes, to enable us to measure cross-coupled odd-degree splitting functions.
Eigenfunctions for the mode pairs are shown in Fig. S1 and sensitivity kernels in Fig.
J
are
S2. Both the eigenfunctions and sensitivity kernels show that modes 17 S4J and 5 S10
confined to the inner core. They cannot be observed at the Earth’s surface alone, but the
interaction between the observable modes and the inner core confined mode significantly
changes the observed spectra so hemispherical structure (i.e. odd-degree) can be observed.
Table S2 shows the PREM frequency and attenuation factor Q of the observable modes
and our measured values, using self-coupling and after adding cross-coupling. Adding
odd-degree structure using cross-coupling increases the Q values of each of the modes,
suggesting that ignoring cross-coupled has underestimated Q values in previous studies.
Attenuation is inversely related to Q, so our new measurements suggest that the inner
core may be less attenuating than was previously thought.
Model predictions
Hemispherical predictions for odd-degree splitting functions are calculated by assuming
that the anisotropy only exists in the western hemisphere with no anisotropy in the
eastern hemisphere, using the theory we recently developed (S8). Inner core anisotropy
was added using a model made in a previous study by combining body wave and normal
mode measurements (S2). This model is cylindrically symmetric, and only contains zonal
structure at s = 2, t = 0 and s = 4, t = 0. The fast axis of the anisotropy is aligned
with the rotation axis of the Earth. Such a seismic model can be described using three
parameters which are defined in terms of the Love coefficients A, C, L, N, F (S12)
α=
L−N
A − 2N − F
C−A
,β =
,γ =
A0
A0
A0
(2)
where α represents the relative speeds of inner core compressional waves traveling along
and perpendicular to the Earth’s rotational axis, β represent the speeds of inner core
shear waves and γ describes the speeds of waves which travel at intermediate angles. A0
is the value of A at the center of the inner core. Fig. S3A shows the parameters for α, β
and γ used to make splitting function predictions.
This model does not contain hemispheres, but we adjust the model by restricting
the anisotropy to only exist between the two hemisphere boundaries at 151◦W and 14◦ E
3
(i.e. the western hemisphere) and zero anisotropy outside these boundaries (Fig. S3B).
We tried four different models of cylindrically symmetric inner core anisotropy (S2–5),
which all give very similar odd-degree splitting function predictions. The main difference
between the different model predictions is in the amplitude of the anomaly, the pattern
is always the same. We choose to display predictions for this specific model (S2) as the
amplitudes best agreed with the observations.
For mode pair 16 S5 – 17 S4J , which is the only pair sensitive to s = 1 in addition to
s = 3, 5, the odd-degree prediction is dominated by negative splitting function anomalies
near the polar regions in the western hemisphere and positive anomalies in the eastern
hemisphere (Fig. 1E). The other two modes pairs are not sensitive to s = 1 and will
therefore show a splitting function signature with additional alternating bands of positive
and negative anomalies. Fig. S4 shows the predicted odd-degree splitting functions for
J
the mode pairs of Fig. 2 in the main paper. Mode pair 8 S5 – 5 S10
has negative splitting
function anomalies near the polar regions in the western hemisphere (Fig. S4A), similar
to 16 S5 – 17 S4J as both mode pairs have similar sensitivities. Mode pair 14 S4 – 11 S7J has
positive splitting function anomalies near the polar regions in the western hemisphere
(Fig. S4B). This mode pair has quite different sensitivities as both are sensitive to mantle
and core, leading to different splitting function prediction. These variations are seen
in both the observations and the models predictions, and are consistently found for all
four different inner core anisotropy models. Thus, the differences in the hemispherical
and regional structures seen in the observations are similar to the predictions using the
same seismic model structure. Mantle structure is added using shear wave velocity model
S20RTS (S13) and scaling the shear wave velocity perturbations to get compressional
velocity and density perturbations.
Fig. S5 shows the model predictions for hemisphere boundaries at 0◦ and 180◦ (green
diamonds) for mode pair 16 S5 – 17 S4J in comparison with the observed splitting function
(blue diamonds). For this simple model, only Im(c13 ) and Im(c15 ) are significantly different from zero. In the data measurements (blue diamonds), several more coefficients are
significantly different from zero. The values for Re(c13 ) and Re(c15 ) can be explained by
changing the hemisphere boundaries in the model predictions to the body wave boundaries at 14◦ E and 151◦W (red diamonds). Additional structures for the other coefficients
have to be attributed to additional regional variations, which are not present in our model
predictions.
4
Robustness tests
To test for robustness of the cross-coupled odd-degree splitting functions, we run our inversions from four starting models (S2–5) and find consistent results. The cross-coupled
odd-degree splitting functions do not always require a starting model, and the hemispherical patterns can be found starting with zero cross-coupling (i.e. from PREM). We also
find that starting from ‘purely hemispheres’ (i.e. boundaries at 0◦ and 180◦), the inversion moves the boundaries to locations more similar to the body wave observations (i.e.
towards 14◦ E) and makes the strong anisotropy bands narrower. These tests provide evidence for the robustness of the splitting function observations. We find also that allowing
for odd-degree structure consistently lowers the misfit between the data and the measured splitting function (Table S1), providing robust evidence that odd-degree structure
is required to fully explain the observed data.
We inspect the resolution matrix and the covariance matrix at the end of our iterated
inversion. Resolution is generally good for the cross-coupled odd-degree splitting function
(in particular for the anomalous coefficients Im(c13 ) and Im(c15 )).
The diagonal elements of the covariance matrix can be used as a relative measure of
the error in our splitting functions (S14), but this method generally underestimates the
error in the measurements. To improve our error measurements, we used cross-validation
to remeasure the splitting functions, leaving out different events in different runs. Fig. S5
shows the results for this test for the cross-coupled odd-degree splitting function of mode
pair 16 S5 – 17 S4J . The spread in the range of the cross-validation measurements (black
horizontal lines) for each measured splitting function coefficient (blue diamonds) can be
used to estimate the size of the error. Hemispherical variations in inner core anisotropy are
observed at s = 3 and s = 5, which have the smallest error ranges. In particular Im(c13 ),
which is the coefficient that is most strongly anomalous due to hemispherical inner core
anisotropy, has the smallest error boundary. Thus, our measurements for hemispherical
variation in inner core anisotropy are robust and have small enough error boundaries to be
interpreted reliably. Our other mode pairs show similar error ranges for their odd-degree
cross-coupled splitting functions.
5
Compressional body wave measurements
Earthquakes with body wave magnitude Mb > 5.5 for the years 1996–2006 were measured
using cross-correlation between the PKIKP phase (which travels through the crust, mantle, outer and inner core) and either PKPbc or PKPab (which both only travel through
the crust, mantle and outer core) (Fig. S6). The data are filtered between 0.5 and 2 Hz.
Polar paths are defined as those with an angle less than 35◦ between the path of PKIKP
in the inner core and the Earth’s rotation axis. Equatorial paths have an angle larger
than 35◦ .
Magnetic field models
Models of the radial magnetic field at the Core Mantle Boundary (CMB) have been
produced for different time periods, going back to 5 million years in the past. In the
paper we compare with a model that encompasses the longest time period (S15), and
we find that there are four consistent high latitude flux patches. These flux patches are
also consistently found in other magnetic field models. Magnetic models for the last 400
years have much more resolution that the models for the last 5 million years. Thus,
for comparison we also show the average magnetic field for the last 400 years and for
1990 (Fig. S7) from the gufm1 model (S16), which was obtained using historical records.
Here, the flux patches are visible without the need to remove the dipole component of the
field. The flux locations in the gufm1 model are similar to the 5 million year model (Fig.
4). The sign of the flux patches in the southern hemisphere varies between the different
models, which is because historic and palaeomagnetic field data are more sparse in the
southern hemisphere.
6
A
J
B
J
C
J and (C)
Figure S1. Eigenfunctions for mode pairs (A) 16 S5 - 17 S4J , (B) 8 S5 - 5 S10
14 S4 - 11 S7 .
Solid line denotes U (r) and dashed lines V (r), which are the solutions of the equation of motion
for spheroidal modes. Modes
J
17 S4
J are confined to the inner core and have negligible
and 5 S10
oscillations in the outer core and mantle.
7
A
J
B
J
C
Figure S2. Sensitivity kernels for mode pairs (A)
11 S7 .
16 S5
-
J
17 S4 ,
J and (C)
(B) 8 S5 - 5 S10
14 S4
-
Solid line denotes sensitivity to compressional velocity, dashed line to shear wave velocity
J are confined to the inner
and dashed-dotted line to density perturbations. Modes 17 S4J and 5 S10
core and have negligible sensitivity to mantle and outer core structure.
8
A Model parameters
B Anisotropy model magnitude
1200
radius (km)
1000
800
600
α
400
β
γ
200
0
WEST
EAST
-0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15
amplitude
0.00
0.25
0.50
0.75
Magnitude
1.00
Figure S3. (A) Model parameters α, β, γ for the cylindrically symmetric model made using a
combination of body wave and normal mode observations (S2). (B) Magnitude of the anisotropy
model in terms of hemispherical variations. The eastern hemisphere has no anisotropy (blue
colours, with magnitude zero) and the western hemisphere has anisotropy (red colours, with
scaled magnitude 1). The parameters α, β, γ are multiplied with the lateral magnitude variations
to obtain a model with hemispherical variation.
9
A Predicted odd-degree splitting for inner core
J
8S5-5S10 , s=5
EAST
WEST
B Predicted odd-degree splitting for inner core
14S4-11S7, s=3,5
-20
D Predicted odd-degree splitting for mantle-only
14S4-11S7, s=3,5
EAST
WEST
-40
C Predicted odd-degree splitting for mantle-only
J
8S5-5S10 , s=5
0
µHz
20
40
-40
-20
0
µHz
20
40
Figure S4. Predicted cross-coupled odd-degree splitting functions for mode pairs (A) 8 S5 J
5 S10 ,
and (B)
14 S4
-
11 S7
for anisotropy between hemisphere boundaries at 151◦ W and 14◦ E
in addition to mantle structure. Thick lines denote the hemisphere boundaries found in short
period body wave observations (Fig. 1 main paper). (C),(D) Predicted odd-degree splitting
function for the same mode pairs for mantle-only structure (S13).
10
Angular order s=1
Angular order s=3
30
5
20
0
10
-5
0
-10
-10
Cst (micro Hz)
10
-15
1
2
3
4
coefficient index
-20
1
2
3
4
5
6
coefficient index
7
8
Angular order s=5
Cst (micro Hz)
20
15
10
5
0
-5
-10
1
2
3
4
5
6
7
coefficient index
8
9
10
11
12
Figure S5. Results from cross-validation test for the cross-coupled odd-degree splitting function of mode pair
16 S5
-
J
17 S4 .
For s=1, the index (1,2,3) relates to c01 , Re(c11 ), Im (c11 ). For
s=3 the index (1,2,3,4,5,...) relates to c03 , Re(c13 ), Im (c13 ), Re(c23 ), Im (c23 ), ... etc. The blue
diamonds indicate the measurements for the whole data set (as plotted in Fig. 1D in the main
paper) and the black horizontal lines are the measurements for ten cross-validation runs. The
green diamonds denote model predictions for hemispherical boundaries at 0◦ and 180◦ and the
red diamonds for the hemisphere boundaries at 14◦ E and 151◦ W, obtained from body wave
measurements.
11
PKPab
PKPbc
PKIKP
MANTLE
OUTER
CORE
INNER
CORE
Figure S6. Raypaths for the body waves used to study hemispherical variations. PKIKP
travels through the mantle (denoted P), outer core (K) and inner core (I). PKPbc and PKPab
only travel through the mantle and outer core. The three main layers in the Earth are labeled.
The star indicates the earthquake location and the triangle the seismic station.
12
A Magnetic field at CMB
gufm1, 400-year average
WEST
-300000
0
nT
B Magnetic field at CMB
gufm1, current field in 1990
EAST
300000
WEST
600000
-300000
0
nT
EAST
300000
600000
Figure S7. Current day (i.e. 1990) and average magnetic field for the last 400 years from the
gufm1 model (S16). Four concentrated flux patches are visible at high longitudes.
13
Mode pair
Freq. (mHz)
events
spectra
J
- 5 S10
J
16 S5 - 17 S4
14 S4 - 11 S7
4.157-4.166
6.836-6.845
5.541-5.563
75
41
32
1078
515
433
8 S5
even
misfit
0.35
0.34
0.46
even+odd
misfit
0.31
0.31
0.42
Table S1. Number of events, frequency range, data segments (spectra) and final data misfit
for each mode pair.
Mode
8 S5
16 S5
14 S4
PREM
freq. (µHz)
4166.2
6836.4
5541.8
even
freq. (µHz)
4165.0
6830.5
5542.2
even+odd PREM
freq. (µHz)
Q
4165.2
611
6830.8
581
5542.0
742
even even+odd
Q
Q
665
757
512
549
686
693
Table S2. Center frequencies and Q measurements for the modes used in this study. Even
denotes the measurements using self-coupling only, even+odd are the measurements with crosscoupling included.
14
References and Notes
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S4. J. Tromp, Nature 366, 678 (1993).
S5. J. H. Woodhouse, D. Giardini, X. D. Li, Geophys. Res. Lett. 13, 1549 (1986).
S6. M. Ishii, J. Tromp, A. M. Dziewonski, G. Ekström, J. Geophys. Res. 107, 2379,
doi:10.1029/2001JB000712 (2002).
S7. A. Deuss, J. H. Woodhouse, Geophys. J. Int. 146, 833 (2001).
S8. J. C. E. Irving, A. Deuss, J. H. Woodhouse, Geophys. J. Int. 178, 962 (2009).
S9. J. S. Resovsky, M. H. Ritzwoller, J. Geophys. Res. 103, 783 (1998).
S10. A. Tarantola, B. Valette, Rev. Geophys. 20(2), 213 (1982).
S11. F. A. Dahlen, J. Tromp, Theoretical Global Seismology (Princeton University Press,
New Jersey, 1998).
S12. A. E. H. Love, A treatise on the mathematical theory of elasticity (Cambridge University Press, Cambridge, 1927).
S13. J. Ritsema, H. van Heijst, J. Woodhouse, Science 286, 1925 (1999).
S14. X. He, J. Tromp, J. Geophys. Res. 101, 20053 (1996).
S15. P. Kelly, D. Gubbins, Geophys. J. Int. 128, 315 (1997).
S16. A. Jackson, A. R. T. Jonkers, M. R. Walker, Phil. Trans. R. Soc. Lond. A. 358, 957
(2000).
15