Search for latitudinal variation of spectral peak frequencies of low-frequency eigenmodes excited by great earthquakes

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ScienceDirect Polar Science 9 (2015) 17e25 http://ees.elsevier.com/polar/

Search for latitudinal variation of spectral peak frequencies of low-frequency eigenmodes excited by great earthquakes Hironobu Shimizu a, Yoshihiro Hiramatsu b,*, Ichiro Kawasaki c b

a Graduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920-1192, Japan Department of Earth Sciences, College of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920-1192, Japan c Tono Research Institute of Earthquake Science, Yamanouchi 1-63, Meisei, Mizunami, Gifu 509-6132, Japan

Available online 18 July 2014

Abstract Continuous waveform records of STS-1 seismometers of the Incorporated Research Institutions for Seismology (IRIS) and superconducting gravimeters of the Global Geodynamics Project (GGP) obtained during the 2004 Sumatra-Andaman, the 2010 Chile, and the 2011 off the Pacific coast of Tohoku earthquakes are examined to search for latitudinal variations of the spectral peak frequencies of 0S0, 1S0, and 0S2. No latitudinal variation is determined. The observed spectral peak frequencies are identical to those of the Preliminary Reference Earth Model (PREM). © 2014 Elsevier B.V. and NIPR. All rights reserved. Keywords: Free oscillation; D00 layer; Eigenfrequency; Latitudinal variation; Degree-two heterogeneity

1. Introduction The Earth's free oscillations excited by great earthquakes provide a powerful tool for investigating the structure of the Earth's deep interior. The oscillation of the Earth with a finite size can be expressed as the sum of the eigenmodes of spheroidal oscillations (nSm l ) and torsional oscillations (nTm l ), where n, l and m respectively denote the radial, angular and azimuthal orders (see Dahlen and Tromp (1998) for details). Their eigenfrequencies reflect the velocity structure of the Earth's interior. The spherically symmetric Earth causes degeneration by which singlets (m ¼ l,l) sharing the same l and n have the same frequency. The Earth's rotation and the spherical asymmetry of its deep * Corresponding author. E-mail address: [email protected] (Y. Hiramatsu). http://dx.doi.org/10.1016/j.polar.2014.07.002 1873-9652/© 2014 Elsevier B.V. and NIPR. All rights reserved.

structure results in a breakdown of the degeneracy and the spectral peak of a multiplet splits into 2l þ 1 closelyespaced peaks. The coreemantle boundary (CMB) Stoneley modes have an oscillation energy concentrated around the CMB. Inner core modes account for much of the oscillation energy in the inner core. CMB Stoneley modes and inner core modes are generally designated as core modes. Master and Gilbert (1981) first reported that the splitting of spectral peaks of multiplets of core modes is significantly wider than that predicted for the rotation of the spherically symmetric Earth. Spectral splitting of this kind was designated as anomalous splitting. Anomalous splitting has attracted the attention of seismologists. Various deep asymmetrical structure models have been proposed as a cause of the anomalous splitting. Woodhouse et al. (1986) concluded that the

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cause of the anomalous splitting was seismic anisotropy in the inner core. Li et al. (1991) pointed out that seismic anisotropy in the inner core failed to explain the anomalous splitting of 3S2 (1.106 mHz, a core mode) and some other core modes. Widmer et al. (1992) claimed that the cause of the anomalous splitting should be in the outer core. Romanowicz and Breger (2000) proposed a polar cap model in which the heterogeneity concentrates around the rotation axis in the outer core. To explain the anomalous splitting of 3S2, Tsuboi and Saito (2002) considered a soft-core splitting model which had a thin layer with small rigidity at the base of the outer core. Stevenson (1987) reported that no lateral perturbation of density larger than 105 in the outer core should be maintained because of hydrostatic considerations. The anomalous splitting of all associated modes, including 3S2, has not yet been modeled and its cause remains an open question. 2. Nonestandard approach for anomalous splitting It seems worthwhile to attempt to introduce a minimum variation to the standard theory if one cannot determine the cause of the anomalous splitting within the framework of the standard theory of free oscillations. In the D00 layer, the ultra-low velocity zone (ULVZ) is distributed above the CMB beneath the central Pacific (e.g. Mori and Helmberger, 1995). Garnero and Jeanloz (2000) pointed out that a perturbation of the density correlates negatively with a perturbation of the velocity in the ULVZ. Elastic constants are the products of the density and the weighted sum of the squares of VP and VS. The equations of motion of an elastic body include only the perturbation of the density if the perturbations of elastic constants are zero as a result of the negative correlation. From this perspective, Kawasaki (2011) proposed a non-standard approach for eigenfrequencies with no perturbation of the elastic constants to introduce latitude-dependent frequencies of eigenmodes, which are designated herein as pseudo-eigenfrequencies. The Appendix presents an outline of the theoretical approach of Kawasaki (2011). Referring to Mori and Helmberger (1995) and Garnero and Jeanloz (2000) who derived the C02 type heterogeneity in the D00 layer, Kawasaki (2011) proposed a “degree-two heterogeneity” in the D00 layer. The reference model is the Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981). The “degree-two heterogeneity” has no

perturbations of the density, VP, or VS above a depth of 300 km above the CMB and in the core. Perturbations of 10%, 5%, and 5% of the density, VP and VS, respectively, are given at the CMB, the bottom of the D00 layer. For the D00 layer, a linear gradient of the perturbation is assumed between the top and the bottom of the D00 layer. All perturbations have a latitudinal dependence of a cos2q-type. In the case of “degree-two heterogeneity”, pseudo-eigenfrequencies of a multiplet vary with cos2q (q is the latitude) of an amplitude of a few percent (Kawasaki, 2011). This variation might yield a scattering of the pseudo-eigenfrequencies leading to the observation of apparently wider splitting of core modes. This has motivated us to seek a latitudinal variation of the pseudo-eigenfrequencies using low-frequency eigenmodes obtained from recent great earthquakes. Furthermore, the development of broadband seismic/geodetic observation networks in high latitude areas, including the polar region, enables us to examine latitudinal variations with accuracy. To search for a cos2q-type latitudinal variation, we specifically examine radial modes of 0S0 (eigenfrequency of 0.816 mHz and eigenperiod of 1228 s by PREM), 1S0 (1.631 mHz and 613 s by PREM) and lowest-frequency eigenmodes of 0S2 (0.309 mHz and 3233 s by PREM). The radial modes of 0S0 and 1S0 have no splitting and are significant in the spectra at many stations for low attenuation (Q of around 5000). Therefore we expect that a latitudinal variation can be found if it exists. Because 0S2 is isolated from nearby eigenmodes, these do not contaminate its spectral peaks of singlets. We do not examine Stoneley-type normal modes (e.g. 1S7e1S15) in this study. One of the reasons is that they have many singlets within a single multiplet and the frequency shifts are difficult to recognize. The other is that their oscillation energy is concentrated below the CMB rather than in the D00 layer. 3. Data and methods We analyze 37-day-long continuous records of the vertical component of STS-1 seismometers at the Incorporated Research Institutions for Seismology (IRIS) stations and superconducting gravimeters at the Global Geodynamics Project (GGP) stations, as obtained from three recent great earthquakes: the 2004 Sumatra-Andaman earthquake (Mw 9.1) of 26 December, 2004; the 2010 Chile earthquake (Mw 8.8) of 27 February, 2010; and the 2011 off the Pacific coast of Tohoku earthquake (hereinafter, the Tohoku earthquake) (Mw 9.0) of 11 March, 2011 (Fig. 1). The

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Table 1 List of stations used for the 2004 Sumatra-Andaman earthquake. A plus denotes that a significant peak is identified. A minus denotes that the peak is unclear. Stations represented by three or four characters are IRIS stations and those represented by two characters are GGP ones. Station Mode

Fig. 1. Stars show epicenters of the 2004 Sumatra-Andaman earthquake, the 2010 Chile earthquake, and the 2011 off the Pacific coast of Tohoku earthquake. Circles, triangles, and diamonds respectively denote stations used for the 2004 Sumatra earthquake, the 2010 Chile earthquake, and the 2011 off the Pacific coast of Tohoku earthquake.

moment magnitude (Mw) reported by the U.S. Geological Survey is cited. Sampling rates are, respectively, 1 Hz for the STS-1 records, and 1, 2, or 10 Hz for the superconducting gravimeters. The analyzed periods are from 26 December, 2004, to 31 January, 2005, for the 2004 Sumatra-Andaman earthquake, from 27 February, 2010, to 5 April, 2010, for the 2010 Chile earthquake, and from 11 March, 2011, to 18 April, 2011, for the 2011 Tohoku earthquake. Each starting time is set to be the origin time of each earthquake. We do not use stations that cannot provide continuous records throughout the above-indicated periods. Lastly, we use 22 IRIS and seven GGP stations for the 2004 Sumatra-Andaman earthquake, 15 IRIS and one GGP stations for the 2010 Chile earthquake, and 17 IRIS and one GGP stations for the 2011 Tohoku earthquake (Tables 1e3). Fig. 1 shows the distribution of the earthquakes and the stations used in this study. In the case of the STS-1 records, we correct the instrumental response from the original records. For the superconducting gravimeter records, we estimate tidal constants using the Bayesian Tidal Analysis Program-Grouping Model (BAYTAP-G) (Ishiguro et al., 1981; Tamura et al., 1991) and remove the theoretical tide computed with the tidal constants. All records are passed through a band-pass filter of 0.2e10 mHz before spectral analyses. We calculate the power spectrum density for each record using a discrete Fourier transform (DFT). 4. Power spectral densities of 0S0, 1S0, and 0S2 Fig. 2 shows the power spectral density of the superconducting gravimeter record at Syowa station (SY) in Antarctica obtained from the 2011 Tohoku

CMB HOPS MOD ORV OSI GLA ISA DRV HIA KMI COR CTAO LSZ MAJO PAB PMSA QSPA TUC ULN VTS OGS SYO CB ES MC NY SU ST VI

0S0

1S0

Latitude Longitude (degree) (degree) 0S2 (m ¼ 2) 0S2 (m ¼ 2)

þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ   þ  þ þ þ þ 

þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ  þ þ þ   þ   þ  þ 

 þ       þ þ  þ  þ þ   þ           

 þ þ       þ  þ  þ þ   þ þ          

38.03 38.99 41.90 39.55 34.61 33.05 35.66 66.66 49.27 25.12 44.59 20.09 15.28 36.55 39.54 64.77 89.93 32.31 47.87 42.62 27.06 69.01 35.32 39.15 44.52 78.93 32.38 48.62 48.25

120.39 123.07 120.30 121.50 118.72 114.83 118.47 140.00 119.74 102.74 123.30 146.25 28.19 138.20 4.35 64.05 144.44 110.78 107.05 23.23 142.20 39.59 149.01 141.33 11.65 11.87 20.81 7.68 16.36

earthquake. Many peaks correspond to the eigenfrequencies of PREM. Fig. 3 portrays a superposition of power spectral densities at around the eigenfrequencies of 0S0, 1S0, and singlets (m ¼ ±1, ±2) of 0S2 for the 2011 Tohoku earthquake. For 0S0 and 1S0, we recognize a single peak common to power spectral densities at many stations for the three great earthquakes. In fact, for example, it is interesting that the values of the peak frequencies of 0S0 (0.8148 mHz) and 1S0 (1.6314 mHz) estimated from the superconducting gravimeter record of the 2011 Tohoku earthquake are exactly the same as those from the record of the 2010 Chile earthquake at Syowa station (SY). Excluding stations with unclear peaks of 0S0 or S 1 0, we obtain the peak frequency of 0S0 at 25 stations for the 2004 Sumatra-Andaman earthquake, at 16 stations for the 2010 Chile earthquake, and at 17 stations for the 2011 Tohoku earthquake, and that of

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Table 2 List of stations used for the 2010 Chile earthquake. A plus denotes that a significant peak is identified. A minus denotes that the peak is unclear. Stations represented by three or four characters are IRIS stations and those represented by two characters are GGP ones. Station Mode 0S0

CAN INU SSB PAF PPTF RER TAM UNM AAK BFO DGAR FFC KURK PFO TAU SY

1S0

þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ

1S0

þ þ þ þ þ þ þ þ þ þ þ þ þ  þ þ

Latitude Longitude (degree) (degree) 0S2 (m ¼ 1) 0S2 (m ¼ 1) þ þ  þ þ  þ þ þ þ  þ þ  þ 

þ þ þ þ þ  þ þ þ þ  þ þ  þ 

35.32 35.35 45.28 49.35 17.57 21.17 22.79 19.33 42.64 48.33 7.41 54.73 50.72 33.61 42.91 69.01

149.00 137.03 4.54 70.21 149.58 55.74 5.53 99.18 74.49 8.33 72.45 101.98 78.62 116.46 147.32 39.59

at 22 stations for the 2004 Sumatra-Andaman earthquake, at 15 stations for 2010 Chile earthquake, and at 14 stations for the 2011 Tohoku earthquake (Tables 1e3). The power spectral density of 0S2 shows no peak of the singlet of m ¼ 0 but four peaks of the singlets of m ¼ ±1, ±2 for the 2011 Tohoku earthquake (Fig. 3). We can recognize that the peak frequency of the singlet of m ¼ þ1 is 0.3137 mHz and that of m ¼ 1 is 0.3045 mHz at Syowa station from the superconducting gravimeter record of the 2011 Tohoku earthquake. However, the peaks of the singlets of m ¼ ±2 are not significant relative to those of m ¼ ±1. Therefore, we exclude the data of the singlets of m ¼ ±2 for the Tohoku earthquake in the discussion (Table 3). The 2010 Chile earthquake provides significant peaks of the singlets of m ¼ ±1, while the peaks of the singlets of m ¼ ±2 are unclear. We also exclude the data of the singlets of m ¼ ±2 for the 2010 Chile earthquake. On the other hand, the peaks of the singlets of m ¼ ±2 are significant and the data of the peaks of the singlets of m ¼ ±1 are excluded for the 2004 Sumatra-Andaman earthquake. Consequently, we obtain the peak frequency of the singlets of m ¼ 2 at 7 stations and that of m ¼ þ2 at 8 stations for the 2004 Sumatra-Andaman earthquake, that of m ¼ 1 at 11 stations and that of m ¼ þ1 at 12 stations for the 2010 Chile earthquake, and that of m ¼ 1 at 14 stations and that of m ¼ þ1 at 14 stations for the 2011 Tohoku earthquake.

Table 3 List of stations used for the 2011 off the Pacific coast of Tohoku earthquake. A plus denotes that a significant peak is identified. A minus denotes that the peak is unclear. Stations represented by three or four characters are IRIS stations and those represented by two characters are GGP ones. Station Mode

CMB SAO BAR GSC ISA OSI SNCC CAN PPTF RER SPB SSB TAM UNM BFO DGAR GNI SY

0S0

1S0

0S2

þ þ þ þ  þ þ þ þ þ þ þ þ þ þ þ þ þ

þ   þ   þ þ þ þ þ þ þ þ þ þ þ þ

þ   þ   þ þ þ þ þ þ þ þ þ þ þ þ

(m ¼ 1)

0S2

þ   þ   þ þ þ þ þ þ þ þ þ þ þ þ

(m ¼ 1)

Latitude Longitude (degree) (degree) 38.00 36.76 32.68 35.30 35.66 34.61 33.25 35.32 17.57 21.17 23.59 45.28 22.79 19.33 48.33 7.41 40.15 69.01

120.39 121.45 116.67 116.81 118.47 118.72 119.52 149.00 149.58 55.74 47.43 4.54 5.53 99.18 8.33 72.45 44.74 39.59

5. Latitudinal variation of the peak frequencies of and 0S2

0S0, 1S0,

In Fig. 4, spectral peak frequencies of the three great earthquakes are shown against latitude, excluding those of unclear spectral peaks. The width of an error bar is estimated by a half width of the spectral peaks. Some stations show that this width is equivalent to the resolution of frequency, especially for 0S0, but others do not. Several stations show a two or three times larger width than the resolution of frequency for 1S0 and 0S2. No difference of the observed peak frequency of 0S0, 1S0, and the singlets (m ¼ ±1, ±2) of 0S2 is apparent between those of the three great earthquakes. Fig. 4(a) shows that there is no latitudinal variation of the observed spectral peak frequency of 0S0. All peak frequencies shown are identical to the eigenfrequencies of PREM. In Fig. 4(b), the spectral peak frequencies of 1S0 show a slight variation with the latitude of the stations indicated by the horizontal axis. However, this variation is within the error-bar width. In Fig. 4(c)e(f), spectral peak frequencies for singlets (m ¼ ±1 and ±2) of 0S2 are shown against latitude. The overall variation with latitude is within a width of the error bars around the peak frequency of the singlets, which is similar to the result of Roult et al. (2010) although their latitudinal coverage was somewhat poor for the singlets of m ¼ ±2. We, therefore,

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Fig. 2. Example of the power spectrum density obtained from data of the superconducting gravimeter at Syowa station of the 2011 off the Pacific coast of Tohoku earthquake. Dashed lines show the eigenfrequencies of 0S0, 0S1 and 0S2, the latitudinal dependence of which we analyze. Gray lines show the eigenfrequencies of other modes.

conclude that no significantly discernible latitudinal variation of the peak frequencies of 0S0, 1S0 and 0S2 exists. Red, blue, and green lines in Fig. 4 respectively stand for latitudinal variations of the pseudoeigenfrequency, which show cos2q dependency, for the amplitude of perturbations of 100%, 50% and 10% of [A-15] in the Appendix. These lines intersect with the eigenfrequencies of PREM at ±45 . In lower latitude areas, the pseudo-eigenfrequency is higher than the eigenfrequency of PREM. On the other hand, in higher latitude areas, the pseudo-eigenfrequency is lower than the eigenfrequency of PREM. This variation of the pseudo-eigenfrequency, therefore, indicates the importance of stations in higher latitude areas, such as in Antarctica, to confirm latitudinal variations. As mentioned above, the observed peak frequencies of 0S0, 1S0 and the singlets (m ¼ ±1) of 0S2 from stations in Antarctica show the same values as obtained at stations with latitudes within ±45 . Thus, we consider that the latitudinal variations of the pseudoeigenfrequencies do not fit the observations. A weak latitudinal variation is apparent for the 10% amplitude of the heterogeneity model for 0S0. However, this variation is not only within the error bars but also within the resolution of frequency. We can confirm neither the existence, nor the absence, of a very weak amplitude of frequency shift of 0S0. For 1S0 and 0S2, the latitudinal variation for the 10% amplitude of heterogeneity is negligible. We thereby infer that no degree-two heterogeneity as assumed in [A-15] exists in the D00 layer, as proposed by Kawasaki (2011). Even if it exists, the

Fig. 3. Power spectra of (a) 0S0, (b) 1S0, and (c) 0S2 observed at all stations for the 2011 off the Pacific coast of Tohoku earthquake. Thick lines in (a) and (b), respectively, show the eigenfrequency of 0S0 and 1S0 of PREM. Thick line and thin lines in (c), respectively, show the eigenfrequency of 0S2 and its singlets (m ¼ ±1, ±2) (Roult et al., 2010).

heterogeneity is not so strong as to break the linearity presumed in the standard theory. Kawasaki's (2011) approach reproduced in the Appendix is very rough, ignoring second-order terms attributable to the products of variables and ignoring a perturbation of the eigenfunctions when deriving [A-16]. We would like to develop more proper approaches in future.

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Fig. 4. Observed eigenfrequencies of (a) 0S0, (b) 1S0, and (c, d, e, f) 0S2 (m ¼ ±1, ±2), as a function of the latitude of each station. Solid circles denote the observed eigenfrequencies of the power spectrum density estimated at each station in this study. The error bars show the half width of each peak. A horizontal black line in each panel represents the eigenfrequency of 0S0 and 1S0 of PREM, and of singlets of 0S2 reported by Roult et al. (2010). A red line in each panel shows the cos2q form latitudinal dependence of the eigenfrequency predicted by Kawasaki (2011), and a blue line and a green line in each panel show the cos2q-type latitudinal dependence of which the strength of the heterogeneity is weakened, respectively, by 50% and 10%.

6. Conclusions We have searched for latitudinal variations of the spectral peak frequencies of 0S0, 1S0 and 0S2, using long period records of three recent great earthquakes: the 2004 Sumatra-Andaman, the 2010 Chile, and the 2011 off Pacific coast of Tohoku earthquakes. We have estimated the spectral peak frequency of each eigenmode at each station. No latitudinal variation in the spectral peak frequencies have been found for the eigenmodes analyzed in this study, indicating that no latitudinal degree-two heterogeneity exists in the entire D00 layer, or that the latitudinal degree-two heterogeneity, if it exists at all, is extremely weak. Acknowledgments We used waveform data provided by GGP, IRIS, and the National Institute of Polar Research, and the BAYTAP-G software for analysis. The GMT software

(Wessel and Smith, 1998) was used to draw all figures.

Appendix 1. A-1. Introduction As explained in the text, the cause of anomalous wider splitting of spectral peaks of multiplets of long period eigenmodes of the Earth's free oscillations has persisted as an unresolved question within the framework of standard perturbation theory of free oscillations (Woodhouse and Dahlen, 1978; Dahlen and Tromp, 1998). Following Kawasaki (2011), this Appendix treats an approach based on latitude-dependent pseudo-eigenoscillations with the introduction of a few assumptions into the standard perturbation theory. Our viewpoint is the following. The Earth's rotation results in the splitting width of a multiplet. To measure

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the splitting width of the multiplet from observations, spectral peaks at various stations were shown on a single figure. The fluctuation of pseudo-eigenfrequencies at different latitudes might result in a widening of the distribution of the spectral peaks, which might lead to a misrecognition of anomalous wider splitting.

23

Here, k is a wavenumber, r denotes a distance on the horizontal plane, and u0 is an angular frequency. Substituting [A-6] into [A-5] produces:  r0 u20 S0 ðr; q; f; u0 Þeiðkru0 tÞ ¼ ðH0 ðr; q; f; l0 ; m0 ÞS0 ðr; q; f; u0 ÞÞeiðkru0 tÞ   þ H0 ðr; q; f; l0 ; m0 Þeiðkru0 tÞ S0 ðr; q; f; u0 Þ; ðA  7Þ

A-2. Basics of formulation Following Cauchy (1822), the equations of motion of an elastic body are v2 Uj X vsjp r 2 ¼ ; ðA  1Þ vt vxp p

where S0(r,q,f,u0)j ¼ Sj(r,q,f,u0). Because we can ignore vk/vq and vk/vf for a laterally homogeneous t medium, (H0(r,q,f,l0,m0)ei(kru0)) ¼ 0. Therefore, we obtain the following representation:

where r stands for density, Uj represents displacement, sjp denotes a stress tensor, t signifies time, and xp is a coordinate. Rotation, ellipticity, gravitation, and physical dispersion are ignored for simplicity. Introducing the elasticity tensorlipsq, we have Hooke's law: X vUq sjp ¼ ljpsq : ðA  2Þ vxs s;q

ðA  8Þ

Substituting [A-2] into [A-1] yields 2 v Uj X v X vUq r 2 ¼ ljpsq : vxp s;q vt vxs p

ðA  3Þ

A4 Therein, (U0(r,q,f,t))j ¼ Uj(r,q,f,t). The subscript denotes quantities in a laterally homogeneous medium. 0 We can rewrite [A-4] in symbolic form as. v2 U0 ðr; q; f; tÞ ¼ H0 ðr; q; f; l0 ; m0 ÞU0 ðr; q; f; tÞ; vt2 ðA  5Þ

where H0 stands for H0 ¼ ðl0 þ m0 Þgrad div  m0 V , which we call a Hermitian operator. We express the displacement Uj(r,q,f,t) in the following form: 2

Uj ðr; q; f; tÞ ¼ Sj ðr; q; f; u0 Þeiðkru0 tÞ :

We redefine the density, Lame's constants, and the Hermitian operator as: rðr; q; fÞ ¼ r0 ðrÞ þ 〈drðr; q; fÞ〉; lðr; q; fÞ ¼ l0 ðrÞ þ 〈dlðr; q; fÞ〉; mðr; q; fÞ ¼ m0 ðrÞ þ 〈dmðr; q; fÞ〉;

ðA  9Þ

Hðr;q;f;l;mÞ¼H0 ðr;q;f;l0 ;m0 Þ þ〈dHðr;q;f;〈dlðr;q;fÞ〉;〈dmðr;q;fÞ〉Þ〉;

Introducing Lame's constants l0 and m0 and neglecting their spatial derivatives, [A-3] can be rewritten in the following vector form in spherical coordinates: v r0 ðr Þ 2 U0 ðr; q; f; tÞ ¼ ðl0 þ m0 Þgrad divU0 ðr; q; f; tÞ vt  m0 V2 U0 ðr; q; f; tÞ

r0 ðrÞ

r0 u20 S0 ðr; q; f; u0 Þ ¼ H0 ðr; q; f; l0 ; m0 ÞS0 ðr; q; f; u0 Þ:

ðA  6Þ

where 〈dr(r,q,f)〉, 〈dl(r,q,f)〉 and 〈dm(r,q,f)〉 are perturbations, which can be expanded as X drts ðrÞYst ðq; fÞ; 〈drðr; q; fÞ〉 ¼ s;t

〈dlðr; q; fÞ〉 ¼

X

dlts ðrÞYst ðq; fÞ;

s;t

〈dmðr; q; fÞ〉 ¼

X

dmts ðrÞYst ðq; fÞ:

s;t

We specifically examine a single multiplet of angular order l to neglect coupling between multiplets. We introduce the pseudo-eigenfrequency of uHl(q,f) and pseudo-eigenfunction of SHl(r,q,f) for the laterally homogeneous Earth as: þ 〈duml 〉; umHl ðq; fÞ ¼ u0lX SHl ðr; q; fÞ ¼ Clp Spl ðr; q; fÞ;

ðA  10Þ

p

 1 where Spl ðr; q; fÞ ¼ pffiffiffiffiffiffiffiffiffi y1l ðr ÞPpl ðqÞeipf ; lðlþ1Þ

vPp ðqÞ veipf þ Þ : y3l ðr Þ l eipf ; y3l ðr ÞPpl ðqÞ vq vf A superscript þ denotes a transposed matrix.

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Substituting [A-10] into [A-8] and replacing r0(r), l0(r) and m0(r) with r(r,q,f), l(r,q,f) and m(r,q,f) in [A-9], we produce the equation shown below:

 ðr0 ðr Þ þ 〈drðr; q; fÞ〉Þ



X

2 u0l þ 〈duml 〉

dVP/VP0 ¼ 10% to 20% and dVS/VS0 ¼ 10% to 50% with reference to PREM (Dziewonski and Anderson, 1981). Referring to their conclusion, we

! Clp Spl ðr; q; fÞ

p

¼ ðH0 ðr; q; f; l0 ; m0 Þ þ 〈dHðr; q; f; 〈dlðr; q; fÞ〉; 〈dmðr; q; fÞ〉Þ〉Þ

X

! Clp Spl ðr; q; fÞ

ðA  11Þ ;

p

Neglecting second-order terms and subtracting [A-8], [A-11] reduces to ! X p p 2  〈drðr; qÞ〉ðu0l Þ Cl Sl ðr; q; fÞ p

X p p    r0 r 2u0l 〈duml 〉 Cl Sl ðr; q; fÞ

!

introduce perturbations, 〈dr(r,q)〉, 〈dVP(r,q)〉, and 〈dVS(r,q)〉, in the following simple form. 〈drðr; qÞ〉 ¼ 〈dr02 ðrÞ〉cos2q; r0 ðrÞ 〈dVP ðr; qÞ〉 〈dVS ðr; qÞ〉 1 ¼ ¼  〈dr02 ðrÞ〉cos2q: VP0 ðrÞ VS0 ðrÞ 2

¼ 0:

p

ðA  12Þ Multiplying the left-hand side of [A-12] by ðr; q; fÞ* and using the orthogonality condition. Sm n Z   * r0 r Sml ðr; q; fÞ Stl ðr; q; fÞdV ¼ 0 if mst; ðA  13Þ we have the perturbation of eigenfrequency Z * 〈drðr;qÞ〉Sml ðr;q;fÞ Sml ðr;q;fÞdV u0l m Z 〈dul 〉¼ ; 2 * r0 ðr ÞSml ðr;q;fÞ Sml ðr;q;fÞdV ðA14Þ where a superscript * denotes a complex conjugate of the transposed matrix. A-3. Introduction of degree-two heterogeneity Garnero and Jeanloz (2000) compared the relative amplitudes of S waves reflected, and not reflected, at the CMB during passage through the ultra-low velocity zone (ULVZ). They concluded that perturbations between the density, dr, and the velocity,dVP and dVS, in the ULVZ are negatively correlated as dr/r0 ¼ þ60%,

ðA  15Þ They give rise to 〈dlðr; q; fÞ〉=l0 ðrÞ ¼ 〈dmðr; q; fÞ〉=m0 ðrÞ ¼ 0. Subsequently a perturbation of 〈dH(r,q,f,〈dl(r,q,f)〉,〈dm(r,q,f)〉)〉 in [A-11] disappears, which makes the formulation simpler than that of the standard theory. We designate [A-15] as degreetwo heterogeneity in the ULVZ. Now, we imagine two extremities of the CMB Stoneley waves traveling along latitudinal and longitudinal lines radiating from an epicenter on the equator, with wave energy trapped in the ULVZ. The propagation velocity of the CMB Stoneley waves along the latitudinal circle (the equator) could be lower than that along the circle of longitude if the heterogeneity [A-15] exists in the ULVZ. In this case, the propagation velocities and the corresponding apparent eigenfrequencies could mutually differ. We designate them as pseudo-eigenfrequencies. This has motivated us to propose an unconventional assumption: a latitude-dependent frequency shift of 〈dum l 〉. Although this breaks “separation of variables”, we ignore second-order terms arising from products of variables to prevent the formulation from becoming complicated. Ignoring a perturbation of the eigenfunctions and substituting [A-15] into [A-14], we obtain:

H. Shimizu et al. / Polar Science 9 (2015) 17e25

25

Z

    * r0 r 〈dr02 r 〉Sml ðr; q; fÞ Sml ðr; q; fÞdr 0 cos 2 q   u Z 〈duml q 〉 ¼  l :   2 * r0 r Sml ðr; q; fÞ Sml ðr; q; fÞdr

In actuality, 〈dum l ðqÞ〉 is a rough approximation to the latitude-dependent frequency shift due to degreetwo heterogeneity. Our formulation includes much simplification. We therefore do not claim that it is better than the standard perturbation theory. However, we feel that it seems worthwhile to introduce a minimum variation to the standard theory if we cannot determine the cause of the anomalous splitting within the framework of the standard perturbation theory.

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ðA  16Þ

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