Observations of the Singlets of Higher-Degree Modes Based on the OSE

0. INTRODUCTION

Earth's rotation, hydrostatic ellipticity, lateral variations in structure, anisotropy and topography on internal discontinuities can remove the degeneracy and lead to the splitting of normal- modes. Therefore normal-mode splitting detection may provide constraints on the large-scale, non-spherically symmetric structure of the entire Earth (He and Tromp, 1996). Previous studies indicate that most of normal mode multiplets below 1 MHz and ultra-low degree modes have been observed. In another aspect, higher-degree normal-modes can yield independent constraints on structure at depth, and are more sensitive to deep structure than fundamental modes at the same frequency. For instance, at long periods, higher-degree modes would enhance resolution in the transition zone and uppermost lower mantle significantly, and at shorter periods they give rise to better constraints on the low velocity zone in the upper mantle (Laske and Widmer-Schnidrig, 2015). Core-sensitive modes, which were first identified by Masters and Gilbert (1981), their splitting ratios are about twice as large as that expected from mantle heterogeneity, are anomalously split modes. Observations of them can provide the first step toward constraining density profiles (Masters and Gubbins, 2003; Kennett, 1998; Widmer-Schnidrig, 1991). The anomalously split modes could also be used to invert for attenuation and anisotropy of the inner core because their elastic and anelastic splitting function coefficients can be robustly measured under the self-coupling approximation (Mäkinen et al., 2014). Hence observing the splitting of isolated higher-degree modes is significant. We will use OSE to detect the splitting singlets of 4 fundamental modes ₀S₇~₀S₁₀ and 2 overtone pairs ₁S₅-₂S₄ and ₂S₅-₁S₆ which are mantle-sensitive (He and Tromp, 1996), and 12 inner-core sensitive modes (₂₅S₂, ₂₇S₂, ₆S₃, ₉S₃, ₁₃S₃, ₁₅S₃, ₁₁S₄, ₁₈S₄, ₈S₅, ₁₁S₅, ₂₃S₅, ₁₆S₆).

In previous studies, there are mainly three stacking methods for stripping and splitting the singlets of Earth's free oscillation normal modes, namely MSE (multi-station experiment), SHS (spherical harmonic stacking), and OSE (optimal sequence estimation). MSE was proposed by Courtier et al. (2000) to search for translational modes of the inner core. In general, MSE is developed based on SHS in the time-domain (Ding and Shen, 2013a; Cummins et al., 1991). SHS was first proposed by Buland et al. (1979) in the frequency domain, and detected the splitting singlets of ₀S₂ and ₀S₃. Chao and Ding (2014) extended SHS method from vertical components to the horizontal components of displacements, then used IRIS datasets to isolate all of the singlets of ₂S₁, ₀S₃, ₂S₂, ₃S₁, ₀T₂, ₀T₃ and the coupled clusters ₂S₂-₁S₃-₃S₁, and first detected the multiplets of both ₀T₂ and ₀T₃. Comparing the multiplets of ₃S₁ after using SHS with MSE, Ding and Shen (2013b) suggested that MSE could strip the triplet of a degree one mode under the condition that only self-coupling is considered, and SHS could strengthen the amplitude of target mode but could not observe the triplet of ₃S₁. OSE (Ding and Shen, 2013b) is developed on the basis of SHS (Buland et al., 1979) and MSE (Courtier et al., 2000), based on the principle of least-squares methods to eliminate the noises-term. Ding and Shen (2013b) identified that OSE and MSE could both search for the Slichter mode (₁S₁) and could isolate singlets of _nS₁ (₂S₁ and ₃S₁), but the singlets could be observed using OSE with higher signal-to-noise ratio (SNR). Ding and Chao (2015a) extended SHS to MSHS (Matrix-SHS), OSE and MSE to horizontal components and arbitrary harmonic degrees, and they detected 8 mantle- sensitive modes ₁S₂, ₀S₄, ₁S₄, ₀S₅, ₀S₆, ₁T₂, ₀T_{4 0}T₆ and 5 inner- core sensitive and anomalous splitting modes ₁₃S₂, ₁₀S₂, ₂S₃, ₃S₂ and ₁₁S₁ by using IRIS seismograms. It is remarkable that the singlet resolutions of ₁T₂, ₀T₄, ₁₀S₂, ₂S₃, ₃S₂, and ₁₁S₁ are reported for the first time (Ding and Chao, 2015a). OSE data stacking scheme is proven to be effective in identifying seismic normal mode singlets (Ding and Chao, 2015b).

With finite SG records, we could hardly detect the stripping and splitting of high-degree and some anomalous splitting modes on the basis of OSE. We conclude that it is necessary to select vertical and horizontal components of seismograms under IRIS station after the 2004 M_w 9.3 Sumatra Earthquake. These observations can provide information related to the Earth structure and new perspectives for constraining the Earth's deep interior (Shen and Luan, 2015; Okal and Stein, 2009; Irving et al., 2008; Stein and Okal, 2007; Park, 2005; Zürn et al., 2000; Widmer-Schnidrig et al., 1992) and explaining the mechanism of large earthquakes.

1. METHOD

1.1 The Principles of OSE

The gravity or vertical-component seismic record ${g_j}\left(t \right)$ of the j-th station can be expressed as (Cummins et al., 1991; Buland et al., 1979)

${\theta _j}$, ${\phi _j}$ are respectively the j-th station's colatitude and longitude, $A_l^m$ represents the amplitude excited by the free oscillation normal modes of angle order l and azimuth order m, ${\omega _m}$ is angular frequency, and ${n_j}\left(t \right)$ is the uncorrelated noise. $Y_l^m\left({{\theta _j}, {\phi _j}} \right)$ is the normalized surface spherical harmonics, which can be expressed as

and $N_l^m = {\left({ - 1} \right)^m}{\left[ {\left({\frac{{2l + 1}}{{4\pi }}} \right)\frac{{\left({l - m} \right)!}}{{\left({l + m} \right)!}}} \right]^{1/2}}$ is the normalizing factor. $P_l^m\left(x \right)$ is the Legendre function, defined as

According to Courtier et al. (2000), considering the degree 1 mode (l=1), for N (> 3), the observed record ${g_j}\left(t \right)$ of j-th station contains three translational mode signals, prograde equatorial, axial, and retrograde equatorial signals, with their corresponding amplitudes ${a_p} = - A_1^{ - 1}N_1^1 = - A_1^{ - 1}\sqrt {3/2\pi } /2$, ${a_\alpha } = A_1^0N_1^0 = A_1^0\sqrt {3/4\pi } $, ${a_\gamma } = A_1^1N_1^1 = A_1^1\sqrt {3/2\pi } /2$ and an-gular frequencies ${\omega _p}$, ${\omega _\alpha }$, ${\omega _\gamma }$ and the uncorrelated noise ${n_j}\left(t \right)$ expressed as

In this study, we applied the optimal sequence estimation (OSE) proposed by Ding and Shen (2013b). Suppose the complex-valued observation Eq. (1) is given, then the three unknown time sequences ${a_p}{e^{i{\omega _p}t}}$, ${a_\alpha }{e^{i{\omega _\alpha }t}}$, ${a_\gamma }{e^{i{\omega _\gamma }t}}$ are esti-mated based on the least squares theory (Ding and Shen, 2013b). Stacking the records from different stations (N > 3), Eq. (4) can be simplified as matrix expression

where

where the superscript "T" denotes matrix transpose, V₁ is the noise term, and B₁ is the coefficient matrix. In this study, we only take self-coupling into consideration, hence the coefficient matrix B does not contain several mutiplets but a target multiplet alone. B₁ and the observation time series matrix G can be respectively written as (Ding and Shen, 2013b):

Equation (5) is the over-determined system when N > 2l+1 and its least-squares solution is (Ding and Shen, 2013b)

where ${P_{ij}} = {\delta _{ij}}{P_j}$ is the corresponding weight matrix of the stations, ${\delta _{ij}}$ is Kronecker symbol: ${\delta _{ij}} = \left\{ { $$\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1&{i = j} \end{array}} \\ {\begin{array}{*{20}{c}} 0&{i \ne j}\end{array}} \end{array}$$ } \right.$. In this experiment, we assume that all of the observations are independent and ${P_j} = 1/{\sigma _j}^2\left({1 \leqslant j \leqslant N} \right)$ is the weight of the ${g_j}\left(t \right)$ with the variance ${\sigma _j}^2$ (Ding and Shen, 2013b).

Concerning degree 2 mode, using the residual gravity time series of different stations (N > 5), the observation equation is expressed as follows (Ding and Shen, 2013b)

where $a_p^{\left({ - 2} \right)}, a_p^{\left({ - 1} \right)}, a_\alpha ^{\left(0 \right)}, a_\gamma ^{\left(1 \right)}, a_\gamma ^{\left(2 \right)}$ and ${\omega _{\left({ - 2} \right)}}, {\omega _{\left({ - 1} \right)}}, {\omega _{\left(0 \right)}}, {\omega _{\left(1 \right)}}, $${\omega _{\left(2 \right)}}$ are the complex amplitudes and frequencies corresponding to the azimuthal number m=-2, -1, 0, +1, +2. The coefficient matrix ${B_{\text{2}}}$ is an $N \times 5$ matrix, written as

and

Then, we have the least squares solution

Concerning degree l mode, we have the solution

where

and the coefficient matrix ${B_l}$ is an $N \times \left({2l + 1} \right)$ matrix, expressed as

Equation (14) is used for a spheroidal mode by using vertical components. Ding and Chao(2015a, b) extended OSE from vertical to horizontal components for detecting a sphe-roidal mode. The coefficient matrix $B_l^S$ is an $\left({{N_n} + {N_e}} \right) \times \left({2l + 1} \right)$ matrix, expressed as

and the time-series ${\mathit{\boldsymbol{g}}_j}\left(t \right)$ as

On the other hand, for the horizontal component of a toroidal mode, the unique coefficient matrix $\mathit{\boldsymbol{B}}_l^T$ is an $\left({{N_e} + {N_n}} \right) \times \left({2l + 1} \right)$ matrix, expressed as

where ${\theta _{e/nj}}, {\phi _{e/nj}}, 1 \leqslant j \leqslant N$ are respectively the colatitude and longitude of j-th station for horizontal components.

On the basis of Eqs. (14), (15), and (17), we can use vertical or horizontal components of seismograms to detect the isolated spheroidal or toroidal modes.

1.2 Model-Predicted Singlet Frequency

Based on the perturbation theory, equation for the angular frequency of the m-th singlet of harmonic degree l was established by Dahlen and Sailor (1979).

where a degenerate eigenfrequency ${}_n{\omega _l}$ is split into the $2l + 1$ associated eigenfrequencies ${}_n\omega _l^m$; a, b and c are splitting parameters which can be calculated for a given non- spherical Earth model.

PREM model of Dziewonsky and Anderson (1981) has ocean, while PREM-tidal model (Rogister, 2003) is oceanless. So in this paper we only adapt the PREM-tidal model which is derived by replacing the surficial ocean with a solid crust (Rogister, 2003).

2. DATA AND PREPROCESING

In this study, we selected data sets Ⅰ and Ⅱ.

Data set Ⅰ: 18 minute-interval SG records from 2004 Sumatra Earthquake according to the GGP stations (http://www.eas.slu.edu/GGP/ggphome.html). They are bh (Bad Homburg, Germany), cb (Canberra, Australia), ka (Kamioka, Japan), ma (Matsushiro, Japan), mb (Membach, Belgium), mc (Medicina, Italy), me (Metsahovi, Finland), mo (Moxa, Germany), ny (Ny-Alesund, Norway), st (Strasbourg, France), su (Sutherland, South Africa), vi (Vienna, Austria), tc (TIGO Concepcion, Chile), es (Esashi, Japan), we (Wettzell, Germany). Figure 1 shows the distribution of the SG stations.

Figure  1.  Distribution of stations. Orange triangles denote 15 SG stations; blue inverted triangles denote 69 horizontal BS stations; purple triangles denote 99 vertical BS stations. The red star represents the source position of 2004 Sumatra Earthquake.

DownLoad: Full-Size Img PowerPoint

Data set Ⅱ: 99 vertical components and 69 horizontal components of 10-second-interval seismic data sets from 2004 Sumatra Earthquake which were downloaded from IRIS stations (http://ds.iris.edu/ds/nodes/dmc/). Figure 1 shows the distributions of vertical and horizontal components for the BS (broad-band seismic) stations.

We preprocessed the original gravitational and seismological records before OSE analysis. For SG records, the ETERNA34's analyze package (Wenzel, 1996, 1994) was chosen to calculate each station's tidal factors, then we used T-soft software (Van Camp and Vauterin, 2005) to remove the solid tidal effects. And we also corrected the local atmospheric effects by adopting a nominal admittance of -3.0 nm·s^-2·hPa^-1 (Shen and Ding, 2013). The obtained final residual gravity time series (which is quite small compared to the acceleration gravity on ground, but has the same SI unit as that of the acceleration) were used for observing the target modes. For broad-band seismometers records, we used Rdseed software to transform the data of seed format (seismic format) to the binary sac format, and then we used SAC software to remove means, trend, and instrumental responses and transform the results to accelerations. After obtaining final datasets Ⅰ and Ⅱ, we conducted experiments by applying OSE method.

3. RESULTS AND DISCUSSION

3.1 The Splitting of ₀S₄

We chose the same starting time after the 2004 Sumatra Earthquake 5 hr later with a length of 180 hr in two schemes: (a) using data set Ⅰ; (b) using data set Ⅱ. The reason why we choose 180 hr lies in that it is close to the optimum record length of 1.1Q-cycle (Shen and Luan, 2015; Dahlen, 1982). Observations are presented in Fig. 2. By comparison, Figs. 2a–2b show the differences between SG records and seismic data sets for their OSE results of ₀S₄. Due to the limitation of the number of SG stations, not all of the singlets of ₀S₄ could be observed completely. For instance, in Fig. 2a the singlet m=+1 could not be detected clearly and singlets' SNRs are not as high as the results of Fig. 2b. In contrast, with a great number of seismograms we found that OSE can resolve all singlets of ₀S₄ with high SNR (see Fig. 2b). It seems that using SG records could not detect the splitting multiplets of degree higher than 4.

Figure  2.  The normalized 9 singlets of the mode ₀S₄ by using OSE, started from 2004 Sumatra Earthquake 5 hr later and with a length of 180 hr. (a) Results of Data set Ⅰ; (b) results of vertical component of Data set Ⅱ; (c) observations and predictions of the splitting frequencies for spherical mode ₀S₄. Grey circles represent results of SG records; inverted light grey triangles denote results of BS data; black dots represent predictions by Eq. (18); red spots denote observations of Shen and Ding (2014) and blue dots denote results of Roult et al. (2006). The same notations in the following figures.

DownLoad: Full-Size Img PowerPoint

The estimated frequencies of ₀S₄ in two schemes are listed in Table 1. In this study we used AR method (Chao and Gilbert, 1980) to estimate the frequencies of corresponding multiplets' peaks, and obtained error bars of the mode frequencies by the bootstrap method (Efron and Tibshirani, 1986), which was suggested by Häfner and Widmer-Schnidrig (2013). From Fig. 2c we can find that the observed frequencies based on BS data are consistent with the predictions of PREM-tidal model and observations of Shen and Ding (2014), it is also the same with the results of SG data except for the singlet m=+1. Shen and Ding (2014) detected ₀S₄ using EEMD (ensemble empirical mode decomposition), but our OSE results have higher resolution. It proves OSE could more efficiently isolate singlets than EEMD. The SNRs based on SG records are low, which might be due to the fact that the SG stations adopted in this paper could not largely excite the singlet and there are insufficient SG records. Observations of ₀S₄ are almost consistent with predictions, implying a weak cross-coupling between ₀S₄ and ₀T₃ (Zürn et al., 2000; Dahlen and Tromp, 1998; Woodhouse, 1980), but we only considered self-coupling in this paper. So further investigations are needed. From the observations presented above, the study shows that high-resolution singlet eigenfrequencies could be obtained by high-quality SG data as well as high- quantity seismograms (Nyman, 1975), and the accuracy of OSE results relies heavily on the number of records. Thus, in the following experiments we select seismic data to observe higher-degree modes.

Table  1.  Model predictions and observations of the splitting frequencies of ₀S₄ (unit: MHz)

Mode	m=-4/+4	m=-3/+3	m=-2/+2	m=-1/+1	m=0
PREM-tidal* (this paper)	0.641 725/ 0.651 213	0.643 257/ 0.650 371	0.644 688/ 0.649 408	0.646 021/ 0.648 385	0.647 253
Roult et al. (2006)	0.641 47±2.562e-4/ 0.651 06±1.908e-4	/	/	/	/
Shen and Ding (2014)	0.641 624±4.5e-5/ 0.651 738±4.0e-5	0.643 336±2.0e-5/ 0.650 723±1.7e-5	0.644 585±1.7e-5/ 0.649 551±1.3e-5	0.645 858±3.8e-5/ 0.647 863±3.1e-5	0.647 057±3.1e-5
This papera	0.641 957±4.6e-6/ 0.651 013±4.6e-6	0.642 936±4.0e-6/ 0.650 857±3.7e-6	0.644 510±3.8e-6/ 0.649 252±4.0e-6	0.645 285±3.2e-6/ 0.643 552±4.0e-6	0.646 912±5.2e-6
This paperb	0.641 371±4.0e-6/ 0.650 979±4.9e-6	0.643 011±4.9e-6/ 0.650 894±4.7e-6	0.644 652±5.0e-6/ 0.649 057±3.3e-6	0.646 231±4.2e-6/ 0.648 624±3.5e-6	0.647 032±4.8e-6
*. PREM-tidal (Rogister, 2003) is the PREM model (Dziewonski and Anderson, 1981) modified by replacing the surficial ocean with a solid crust; a. Results of data set Ⅰ (SG data); b. Results of data set Ⅱ(vertical components of seismic data).

 | Show Table

DownLoad: CSV

3.2 The Splitting of ₀S₇, ₀S₈, ₀S₉, ₀S₁₀

₀S₇-₀S₁₀ are mantle-sensitive modes which are predominantly sensitive to P velocity in the upper mantle and S velocity in the mid-mantle (He and Tromp, 1996). In this paper, we choose all the length around 1.1Q-cycle (Dahlen, 1982) to improve each mode's SNRs. Figures 3a–3d show that we can clearly observe all splitting singlets of these modes by using OSE. After tidal correction using bandpass filtering in tidal frequency band, the observed frequencies ₀S₇-₀S₉ are basically in line with the predictions of PREM model (Rogister, 2003; Dziwonski and Anderson, 1981), and our results have high SNRs with vertical components, as shown by Figs. 3ai–3di. However, from Fig. 3, we can find that some singlets of ₀S₇-₀S₁₀ are abnormally split, especially for ₀S₁₀, and its observations increase first and then decrease, which are not consistent with the decline trend of PREM-tidal model (Rogister, 2003). Since interactions between ₀S_l and ₀T_l+1 are getting stronger from l=7 to l=10 (Laske and Widmer-Schnidrig, 2015, Fig. 12 therein) whereas predictions are getting worse in increasing l (Figs. 3d–3di), the deviations of ₀S₁₀ from the model predictions might be attributed to the strong coupling with ₀T₁₁.The previous studies also suggested that ₀S₁₀ had a strong cross-coupling with ₀T₁₁ by Coriolis force (Laske and Widmer- Schnidrig, 2015; Zürn et al., 2000; Resovsky and Ritzwoller, 1998; He and Tromp, 1996), and ₀S₈-₀T₉, ₀S₉-₀T₁₀ are weakly coupled multiplets (Ritzwoller et al., 1988). Besides, modes ₀S₇-₂S₃ are strongly coupled and different singlets could hardly be observed (Zürn et al., 2000; He and Tromp, 1996; Giardini et al., 1988). Considering cross-coupling is significant in estimating eigenfrequencies of strongly coupled modes (Laske and Widmer-Schnidrig, 2015). Whereas in this study we only take self-coupling of the modes into account, so the problems are not considered in our present results. Coupling effects will be considered in our future research.

Figure  3.  The normalized OSE results using Data set Ⅱfor 4 high-degree modes, started from 2004 Sumatra Earthquake 5 hr later: (a) ₀S₇, with a length of 85 hr; (b) ₀S₈, with a length of 75 hr; (c) ₀S₉, with a length of 64.4 hr; (d) ₀S₁₀, with a length of 58 hr; (ai)–(di) Observations and predictions of the splitting frequencies for modes ₀S₇~₀S₁₀.

DownLoad: Full-Size Img PowerPoint

Table  2.  Model predictions and observations of the splitting frequencies of ₀S₇ (unit: MHz)

Mode	m=-7/+7	m=-6/+6	m=-5/+5	m=-4/+4	m=-3/+3	m=-2/+2	m=-1/+1	m=0
PREM-tidal* (this paper)	1.228 423/ 1.231 523	1.229 051/ 1.231 698	1.229 618/ 1.231 812	1.230 121/ 1.231 869	1.230 561/ 1.231 867	1.230 936/ 1.231 805	1.231 248/ 1.231 682	1.231 496
This paperb	1.228 039±1.1e-5/ 1.231 296±9.5e-6	1.228 491±1.1e-5/ 1.232 160±1.5e-5	1.229 753±8.1e-6/ 1.232 303±1.1e-5	1.230 172±8.4e-6/ 1.233 044±4.1e-6	1.230 193±7.3e-6/ 1.232 793±9.2e-6	1.230 706±7.0e-6/ 1.232 522±9.2e-6	1.231 005±8.1e-6/ 1.232 596±1.0e-5	1.232 135±1.1e-5
*. PREM-tidal (Rogister, 2003) is the PREM model (Dziewonski and Anderson, 1981) modified by replacing the surficial ocean with a solid crust; b. Results of data set Ⅱ(vertical components of seismic data). The same notations in the following tables.

 | Show Table

DownLoad: CSV

Table  3.  Model predictions and observations of the splitting frequencies of ₀S₈ (unit: MHz)

Mode	m=-8/+8	m=-7/+7	m=-6/+6	m=-5/+5	m=-4/+4	m=-3/+3	m=-2/+2	m=-1/+1	m=0
PREM-tidal* (this paper)	1.410 364/ 1.411 538	1.410 912/ 1.411 981	1.411 341/ 1.412 248	1.411 711/ 1.412 467	1.412 123/ 1.412 638	1.412 378/ 1.412 758	1.412 577/ 1.412 828	1.412 721/ 1.412 846	1.412 810
This paperb	1.410 828±1.8e-5/ 1.411 640±1.5e-5	1.411 388±1.9e-5/ 1.413 730±1.6e-5	1.412 145±1.0e-5/ 1.414 038±1.3e-5	1.412 149±9.4e-6/ 1.414 370±1.4e-5	1.412 408±6.0e-6/ 1.414 578±1.7e-5	1.412 867±7.3e-6/ 1.413 954±1.1e-5	1.413 182±9.3e-6/ 1.414 469±1.5e-5	1.413 783±8.1e-6/ 1.414 063±1.1e-5	1.414 026±1.2e-5

 | Show Table

DownLoad: CSV

Table  4.  Model predictions and observations of the splitting frequencies of ₀S₉ (unit: mHz)

Mode	m=-9/+9	m=-8/+8	m=-7/+7	m=-6/+6	m=-5/+5	m=-4/+4	m=-3/+3	m=-2/+2	m=-1/+1	m=0
PREM-tidal* (this paper)	1.575 050/ 1.574 605	1.575 578/ 1.575 121	1.575 855/ 1.575 429	1.576 092/ 1.575 708	1.576 289/ 1.575 955	1.576 443/ 1.576 167	1.576 555/ 1.576 344	1.576 625/ 1.576 481	1.576 652/ 1.576 579	1.576 636
This paperb	1.575 956±4.6e-6/ 1.575 412±1.2e-5	1.576 756±3.1e-6/ 1.577 041±1.3e-5	1.576 955±4.9e-6/ 1.577 094±1.0e-5	1.577 221±8.5e-6/ 1.577 623±1.9e-5	1.577 160±9.0e-6/ 1.577 734±1.3e-5	1.578 251±1.7e-5/ 1.577 969±1.8e-5	1.578 604±2.1e-5/ 1.578 416±1.8e-5	1.578 426±1.8e-5/ 1.578 579±1.6e-5	1.578 672±1.9e-5/ 1.578 970±1.5e-5	1.579 863±1.0e-5

 | Show Table

DownLoad: CSV

Table  5.  Model predictions and observations of the splitting frequencies of ₀S₁₀ (unit: MHz)

Mode	m=-10/+10	m=-9/+9	m=-8/+8	m=-7/+7	m=-6/+6	m=-5/+5	m=-4/+4	m=-3/+3	m=-2/+2	m=-1/+1	m=0
PREM-tidal* (this paper)	1.721 872/ 1.720 846	1.721 989/ 1.720 948	1.721 911/ 1.720 947	1.721 810/ 1.721 000	1.721 773/ 1.721 087	1.721 754/ 1.721 199	1.721 738/ 1.721 319	1.721 716/ 1.721 401	1.726 84/ 1.721 504	1.721 636/ 1.721 594	1.721 572
This paperb	1.720 622±6.4e-6/ 1.723 187±1.5e-5	1.721 929±7.9e-6/ 1.724 154±2.1e-5	1.722 757±9.4e-6/ 1.724 275±1.8e-5	1.723 776±2.7e-5/ 1.724 177±1.9e-5	1.724 196±2.0e-5/ 1.724 577±1.6e-5	1.724 789±9.6e-6/ 1.724 800±2.1e-5	1.724 265±1.9e-5/ 1.725 024±7.2e-6	1.724 158±2.1e-5/ 1.725 482±1.1e-5	1.724 292±2.1e-5/ 1.724 697±1.4e-5	1.723 981±2.0e-5/ 1.724 271±1.6e-5	1.724 602±1.8e-5

 | Show Table

DownLoad: CSV

3.3 The Splitting of ₁S₅-₂S₄, ₂S₅-₁S₆

The pair's ₁S₅-₂S₄ and ₂S₅-₁S₆ may provide constraints on odd-degree aspherical structures (Resovsky and Ritzwoller, 1998, 1995). Masters et al. (2000a) obtained splitting of overlapping modes ₁S₅-₂S₄ for the 1994 Bolivian and Kuril Islands events through receiver strips, and their results suggest that this approach could efficiently separate two multiplets. Then, Masters et al. (2000b) obtained receiver strips by stacking 11 largest events since 1994 for the weakly coupled pair ₁S₅-₂S₄, of which two coupled mutiplets were separated obviously with improved SNRs than Masters et al. (2000a).

Figure 4 shows that we can clearly separate two weakly coupled mutiplets ₁S₅-₂S₄ and ₂S₅-₁S₆ from only one event by using OSE on the basis of self-coupling, combining vertical and horizontal components with their record lengths of around 1.1Q-cycle. The estimations and predictions of these modes are shown in Figs. 4ai–4di. We can see that observations are almost in line with the model predictions of PREM-tidal with high resolutions and high SNRs, except that ₁S₆ has a slight deviation. These indicate that OSE is effective in isolating some weakly coupled multiplets on the basis of self-coupling approximation. Since the quality factors Qs for ₂S₅-₁S₆ are closer than ₁S₅-₂S₄ but the differences between each other are not very big (Q value of ₁S₅ is 291.88, ₂S₄ is 380.61, ₂S₅ is 302.19, ₁S₆ is 345.67) and ₂S₅-₁S₆ is more strongly coupled than ₁S₅-₂S₄ (Ritzwoller et al., 1988), the observed results of ₁S₅-₂S₄ might be more reliable than those of ₂S₅-₁S₆, as shown by Figs. 4a–4d. Here we note that the observed eigenfrequencies of a multiplet poorly fit predictions (based on self-coupling) in the frequency range overlapped with another multiplet, i.e., f > 1.375 MHz of ₁S₅ overlapping with frequencies of ₂S₄, and f < 1.52 MHz of ₁S₆ overlapping with frequencies of ₂S₅, whereas good fits lie in the unoverlapping frequency band. The differences between predictions and observations might be accounted for by cross-coupling effect or by contamination of signals. The differences between predictions and observations might be accounted for by cross-coupling effect or by contamination of signals. Different components of the mentioned coupled multiplets may provide useful constraint on mantle structure. For instance, modes information of the coupling of ₁S₅-₂S₄ and ₁S₆-₂S₅ provide constrain on odd-degree mantle structure (Deuss et al., 2013; Masters et al., 2000b; Resovsky and Ritzwoller, 1998). Further investigations are needed in the future.

Figure  4.  The normalized OSE results using data set Ⅱfor 2 overtone pairs ₁S₅-₂S₄ and ₂S₅-₁S₆, started from 2004 Sumatra Earthquake 5 hr later. (a) ₁S₅, with a length of 65 hr; (b) ₂S₄, with a length of 84.5hr; (c) ₂S₅, with a length of 65 hr; (d) ₁S₆, with a length of 70 hr; (ai)–(di) observations and predictions of the splitting frequencies for modes ₁S₅, ₂S₄, ₂S₅, ₁S₆.

DownLoad: Full-Size Img PowerPoint

3.4 The Splitting of 12 Inner-Core Sensitive Modes (Anomalously-Split Modes)

Masters and Gilbert (1981) first observed that the inner- core sensitive modes ₃S₂, ₁₃S₂ and ₁₈S₄ are anomalously split. Morelli et al. (1986) and Woodhouse et al. (1986) proposed the hypothesis that there exists anisotropy in the inner-core in which waves travel faster along the rotation axis than in the equatorial plane, so that the splitting of most anomalous modes could support for the inner-core anisotropy (Tromp, 1995b, 1993). Ritzwoller et al. (1986) observed the splittings of the anomalously split modes ₀S₆, ₂S₄, ₆S₃ and ₁₁S₄. Widmer-Schnidrig et al. (1992) used singlet stripping technique (Ritzwoller et al., 1986; Buland et al., 1979; Gilbert, 1971) to isolate multiplets of modes ₁₅S₃, ₁₆S₆, ₁₈S₄, ₂₀S₅, ₂₁S₆ and ₂₃S₅. Besides, Tromp (1993) suggested three anomalously split mechanisms and observed and predicted frequencies for the mutiplets of 18 anomalously splitting free oscillations ₁₁S₄, ₁₁S₅, ₁₃S₂, ₁₃S₃, ₁₄S₄, ₁₅S₃, ₁₆S₆, ₁₈S₄, ₂₀S₅, ₂₁S₆, ₂₃S₅, ₂₅S₂ and ₂₇S₂ on the basis of previous studies (Widmer-Schnidrig et al., 1992; Ritzwoller et al., 1988; Giardini et al., 1987; Ritzwoller et al., 1986), which could be explained by the anisotropy of the Earth's inner core (Tromp, 1995a, b, 1993; Woodhouse et al., 1986). By identifying these anomalously split modes, we may better constrain the anisotropy of the inner-core (Tromp, 1993). In this paper, we detected the splitting of 12 inner-core sensitive multiplets ₂₅S₂, ₂₇S₂, ₆S₃, ₉S₃, ₁₃S₃, ₁₅S₃, ₁₁S₄, ₁₈S₄, ₈S₅, ₁₁S₅, ₂₃S₅ and ₁₆S₆ individually. We use vertical components of seismograms only, and most data sets that we used are with the length of around 1.1Q-cycle (Dahlen, 1982). We can find that OSE could isolate all singlets clearly as shown by Fig. 5.

Figure  5.  The normalized OSE results using data set Ⅱfor 12 inner-core sensitive modes, started from 2004 Sumatra Earthquake 5 hr later. (a) ₂₅S₂, with a length of 27 hr; (b) ₂₇S₂, with a length of 25 hr; (c) ₆S₃, with a length of 90 hr; (d) ₉S₃, with a length of 65 hr; (e) ₁₃S₃, with a length of 55 hr; (f) ₁₅S₃, with a length of 40 hr; (g) ₁₁S₄, with a length of 45 hr; (h) ₁₈S₄, with a length of 40 hr; (i) ₈S₅, with a length of 45 hr; (j) ₁₁S₅, with a length of 40 hr; (k) ₂₃S₅, with a length of 30 hr; (l) ₁₆S₆, with a length of 35 hr.

DownLoad: Full-Size Img PowerPoint

Figure 6 shows the observations and model predictions of the splitting frequencies for 12 anomalously split modes. Comparing the results with Tromp (1993), we find that our observed frequencies are consistent with previous studies (Tromp, 1993; Widmer-Schnidrig et al., 1992; Li et al., 1991; Giardini et al., 1988; Ritzwoller et al., 1988, 1986) and the model predictions of PREM-tidal. Most previous results are based on the theoretical singlet eigenfrequencies calculated from splitting functions, however our results are completely obtained through the real datasets. In addition, some observed modes of Tromp (1993) have low resolutions, such as ₆S₃, ₉S₃, ₁₃S₃, ₁₅S₃ and ₁₈S₄, while our results have higher resolutions. We note that the results of Tromp (1993) and this study show that some modes have the similar deviation from PREM-tidal predictions, for example ₁₁S₄, ₁₁S₅, ₂₃S₅ and ₁₆S₆. Although we selected different seismic sources, both this study and Tromp (1993) detected the splitting frequencies of the same modes with the similar changing rules, such as ₁₁S₅ and ₁₆S₆ etc. This demonstrates that the 12 inner-core sensitive modes dealt in this paper are anomalously split, and the deviation might be attributed to the intense noise-terms, or it suggests that the PREM-tidal model needs modification. In the future, we will stack more events to further confirm this conclusion.

Figure  6.  Observations and predictions of the splitting frequencies for 12 inner-core sensitive modes. Inverted light grey triangles denote results of BS data; black spots represent the predictions by Eq. (18); dashed line denotes 3th polynomial fitting of observations.

DownLoad: Full-Size Img PowerPoint

4. CONCLUSION

In this paper we applied optimal sequence estimation (OSE) technique and its extension for the horizontal components to detect the splitting of higher-degree modes using records of vertical and horizontal seismograms distributed globally after the 2004 Sumatra Earthquake. Results show that OSE is effective in isolating different singlets of high-degree modes in the case of only considering the self-coupling. We selected all of the data length as around 1.1Q-cycle to more precisely estimate the singlet frequencies of high-degree modes with higher SNRs. We detected obvious splitting of the mantle- sensitive modes ₀S₇-₀S₁₀ for the first time, and obtained the singlets' isolated eigenfrequencies with high resolutions and SNRs. We successfully separated the mutiplets of ₁S₅-₂S₄ by using OSE with fewer records than Masters et al. (2000a), and the resolutions of frequencies are close to Masters et al. (2000b). We obtained the splitting of ₂S₄ and ₂S₅ by using horizontal components, and for first time we completely isolated the singlets of ₂S₅-₁S₆. We presented the multiplets for 12 inner-core sensitive modes clearly, among them the splitting of ₂₅S₂, ₂₇S₂, ₉S₃, ₈S₅ and ₁₁S₅ are for first time observed. Our results not only support the anisotropy of the Earth's inner core by comparing with the modified PREM model, but also provide effective constraints on 3D Earth model.

We also compared the results based on the SG data with those based on the seismograms records. Our observations show that although SG data may provide results with high quality, the number of SG is inadequate so that SG records could not effectively isolate high-degree multiplets. However, we can effectively observe high-degree multiplets using a large number of records from globally distributed seismograms. Hence, records of seismograms under IRIS network are significant for observing especially the high-degree-mode multiplets.

One potential approach is that we stack records after different strong earthquake events to more sufficiently detect complete splitting of higher-degree modes, because some singlets might not be obviously excited by only one event (Nyman, 1975). In future studies, it is necessary to take into account the cross-coupling and stack different seismic sources to isolate the coupled mutiplets, which are closely related to the inner structure of the Earth.

ACKNOWLEDGMENTS

The authors express their sincere thanks to H. Ding and W. Luan for frequent discussions about the contents of this paper. The authors appreciate Yves Rogister for providing the code for computing the theoretical eigensolutions of normal modes. The authors also thank two anonymous reviewers for their valuable comments, suggestions and laborious corrections, which greatly improved the manuscript. This study was supported by the National 973 Project of China (No. 2013CB733305), the NSFC (Nos. 41174011, 41429401, 41574007, 41210006, 41128003, 41021061). The final publication is available at Springer via https://doi.org/10.1007/s12583-017-0810-0.