2. State Key Laboratory for Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan 430079, China
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 normalmode splitting detection may provide constraints on the largescale, nonspherically symmetric structure of the entire Earth (He and Tromp, 1996). Previous studies indicate that most of normal mode multiplets below 1 MHz and ultralow degree modes have been observed. In another aspect, higherdegree normalmodes 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, higherdegree 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 WidmerSchnidrig, 2015). Coresensitive 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; WidmerSchnidrig, 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 selfcoupling approximation (Mäkinen et al., 2014). Hence observing the splitting of isolated higherdegree modes is significant. We will use OSE to detect the splitting singlets of 4 fundamental modes _{0}S_{7}~_{0}S_{10} and 2 overtone pairs _{1}S_{5}_{2}S_{4} and _{2}S_{5}_{1}S_{6} which are mantlesensitive (He and Tromp, 1996), and 12 innercore sensitive modes (_{25}S_{2}, _{27}S_{2}, _{6}S_{3}, _{9}S_{3}, _{13}S_{3}, _{15}S_{3}, _{11}S_{4}, _{18}S_{4}, _{8}S_{5}, _{11}S_{5}, _{23}S_{5}, _{16}S_{6}).
In previous studies, there are mainly three stacking methods for stripping and splitting the singlets of Earth's free oscillation normal modes, namely MSE (multistation 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 timedomain (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 _{0}S_{2} and _{0}S_{3}. 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 _{2}S_{1}, _{0}S_{3}, _{2}S_{2}, _{3}S_{1}, _{0}T_{2}, _{0}T_{3} and the coupled clusters _{2}S_{2}_{1}S_{3}_{3}S_{1}, and first detected the multiplets of both _{0}T_{2} and _{0}T_{3}. Comparing the multiplets of _{3}S_{1} 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 selfcoupling is considered, and SHS could strengthen the amplitude of target mode but could not observe the triplet of _{3}S_{1}. 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 leastsquares methods to eliminate the noisesterm. Ding and Shen (2013b) identified that OSE and MSE could both search for the Slichter mode (_{1}S_{1}) and could isolate singlets of _{n}S_{1} (_{2}S_{1} and _{3}S_{1}), but the singlets could be observed using OSE with higher signaltonoise ratio (SNR). Ding and Chao (2015a) extended SHS to MSHS (MatrixSHS), OSE and MSE to horizontal components and arbitrary harmonic degrees, and they detected 8 mantle sensitive modes _{1}S_{2}, _{0}S_{4}, _{1}S_{4}, _{0}S_{5}, _{0}S_{6}, _{1}T_{2}, _{0}T_{4} _{0}T_{6} and 5 inner core sensitive and anomalous splitting modes _{13}S_{2}, _{10}S_{2}, _{2}S_{3}, _{3}S_{2} and _{11}S_{1} by using IRIS seismograms. It is remarkable that the singlet resolutions of _{1}T_{2}, _{0}T_{4}, _{10}S_{2}, _{2}S_{3}, _{3}S_{2}, and _{11}S_{1} 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 highdegree 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; WidmerSchnidrig et al., 1992) and explaining the mechanism of large earthquakes.
1 METHOD 1.1 The Principles of OSEThe gravity or verticalcomponent seismic record
$ {g_j}\left(t \right) = \sum\limits_{l = 0}^\infty {\sum\limits_{m =  l}^l {A_l^mY_l^m\left({{\theta _j}, {\varphi _j}} \right){e^{i{\omega _m}t}}} } + {n_j}\left(t \right) $  (1) 
$ Y_l^m\left({{\theta _j}, {\phi _j}} \right) = N_l^mP_l^m\left({\cos {\theta _j}} \right){e^{im{\phi _j}}} $  (2) 
and
$ P_l^m\left(x \right) = \left({\frac{{{{\left({1  {x^2}} \right)}^{m/2}}}}{{{2^l}l!}}} \right)\left({\frac{{{d^{l + m}}}}{{d{x^{l + m}}}}{{\left({{x^2}  1} \right)}^l}} \right) $  (3) 
According to Courtier et al. (2000), considering the degree 1 mode (l=1), for N (> 3), the observed record
$ \begin{gathered} {g_j}\left(t \right) = {a_p}{e^{i\left({{\omega _p}t  {\varphi _j}} \right)}}\sin {\theta _j} + {a_\alpha }{e^{i{\omega _\alpha }t}}\cos {\theta _j} + \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_\gamma }{e^{i\left({{\omega _\gamma } + {\varphi _j}} \right)}}\sin {\theta _j} + {n_j}\left(t \right), j = 1, 2, \cdots, N \hfill \\ \end{gathered} $  (4) 
In this study, we applied the optimal sequence estimation (OSE) proposed by Ding and Shen (2013b). Suppose the complexvalued observation Eq. (1) is given, then the three unknown time sequences
$ \mathit{\boldsymbol{G}}{\rm{ = }}{\mathit{\boldsymbol{B}}_{\rm{1}}}{\mathit{\boldsymbol{S}}_{\rm{1}}}{\rm{ + }}{\mathit{\boldsymbol{V}}_{\rm{1}}} $  (5) 
where
$ {\mathit{\boldsymbol{S}}_1} = {\left[ {\begin{array}{*{20}{c}} {{a_p}{e^{i{\omega _p}t}}}&{{a_\alpha }{e^{i{\omega _\alpha }t}}}&{{a_\gamma }{e^{i{\omega _\gamma }t}}} \end{array}} \right]^{\text{T}}} $  (6a) 
$ {\mathit{\boldsymbol{V}}_1} = {\left[ {\begin{array}{*{20}{c}} {{n_1}\left(t \right)}&{{n_2}\left(t \right)}& \cdots &{{n_N}\left(t \right)} \end{array}} \right]^{\text{T}}} $  (6b) 
where the superscript "T" denotes matrix transpose, V_{1} is the noise term, and B_{1} is the coefficient matrix. In this study, we only take selfcoupling into consideration, hence the coefficient matrix B does not contain several mutiplets but a target multiplet alone. B_{1} and the observation time series matrix G can be respectively written as (Ding and Shen, 2013b):
$ {\mathit{\boldsymbol{B}}_1} = \left[ {\begin{array}{*{20}{c}} {{e^{  i{\phi _1}}}\sin {\theta _1}}&{\cos {\theta _1}}&{{e^{i{\phi _1}}}\sin {\theta _1}} \\ {{e^{  i{\phi _2}}}\sin {\theta _2}}&{\cos {\theta _2}}&{{e^{i{\phi _2}}}\sin {\theta _2}} \\ \vdots & \vdots & \vdots \\ {{e^{  i{\phi _N}}}\sin {\theta _N}}&{\cos {\theta _N}}&{{e^{i{\phi _N}}}\sin {\theta _N}} \end{array}} \right], \mathit{\boldsymbol{G}} = \left[ {\begin{array}{*{20}{c}} {{g_1}\left(t \right)} \\ {{g_2}\left(t \right)} \\ \vdots \\ {{g_N}\left(t \right)} \end{array}} \right] $  (6c) 
Equation (5) is the overdetermined system when N > 2l+1 and its leastsquares solution is (Ding and Shen, 2013b)
$ {\mathit{\boldsymbol{\hat S}}_1} = {\left( {{\mathit{\boldsymbol{B}}_1}^{\rm{T}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_1}} \right)^{  1}}{\mathit{\boldsymbol{B}}_1}^{\rm{T}}\mathit{\boldsymbol{PG}} $  (7) 
where
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)
$ \begin{gathered} {g_j}\left(t \right) = a_p^{\left({  2} \right)}{e^{i{\omega _{\left({  2} \right)}}t}}{\sin ^2}{\theta _j}{e^{  i2{\phi _j}}} + {\kern 1pt} a_p^{\left({  1} \right)}{e^{i{\omega _{\left({  1} \right)}}t}}\sin {\theta _j}\cos {\theta _j}{e^{  i{\phi _j}}} + {\kern 1pt} \hfill \\ \quad \quad \quad a_\alpha ^{\left(0 \right)}{e^{i{\omega _{\left(0 \right)}}t}}\left({2{{\cos }^2}{\theta _j}  1} \right) + {\kern 1pt} {\kern 1pt} {\kern 1pt} a_\gamma ^{\left(1 \right)}{e^{  i{\omega _{\left(1 \right)}}t}}\sin {\theta _j}\cos {\theta _j}{e^{i{\phi _j}}} + \hfill \\ \quad \quad \quad {\kern 1pt} a_p^{\left(2 \right)}{e^{  i{\omega _{\left(2 \right)}}t}}{\sin ^2}{\theta _j}{e^{i2{\phi _j}}} + {n_j}\left(t \right) \hfill \\ \end{gathered} $  (8) 
where
$ \begin{gathered} {\mathit{\boldsymbol{B}}_2} = \left[ {\begin{array}{*{20}{c}} {{{\sin }^2}{\theta _1}{e^{  i2{\phi _1}}}}&{\sin {\theta _1}\cos {\theta _1}{e^{  i{\phi _1}}}} \\ {{{\sin }^2}{\theta _2}{e^{  i2{\phi _2}}}}&{\sin {\theta _2}\cos {\theta _2}{e^{  i{\phi _2}}}} \\ \vdots & \vdots \\ {{{\sin }^2}{\theta _N}{e^{  i2{\phi _N}}}}&{\sin {\theta _N}\cos {\theta _N}{e^{  i{\phi _N}}}} \end{array}} \right. \hfill \\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} {2{{\cos }^2}{\theta _1}  1}&{\sin {\theta _1}\cos {\theta _1}{e^{i{\phi _1}}}}&{{{\sin }^2}{\theta _1}{e^{i2{\phi _1}}}} \\ {2{{\cos }^2}{\theta _2}  1}&{\sin {\theta _2}\cos {\theta _2}{e^{i{\phi _2}}}}&{{{\sin }^2}{\theta _2}{e^{i2{\phi _2}}}} \\ \vdots & \vdots & \vdots \\ {2{{\cos }^2}{\theta _N}  1}&{\sin {\theta _N}\cos {\theta _N}{e^{i{\phi _N}}}}&{{{\sin }^2}{\theta _N}{e^{i2{\phi _N}}}} \end{array}} \right] \hfill \\ \end{gathered} $  (9) 
and
$ \begin{gathered} {\mathit{\boldsymbol{S}}_2} = \left[ {\begin{array}{*{20}{c}} {a_p^{\left({  2} \right)}{e^{i{\omega _{\left({  2} \right)}}t}}}&{a_p^{\left({  1} \right)}{e^{i{\omega _{\left({  1} \right)}}t}}} \end{array}} \right. \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\left. {\begin{array}{*{20}{c}} {a_\alpha ^{\left(0 \right)}{e^{i{\omega _{\left(0 \right)}}t}}}&{a_\gamma ^{\left(1 \right)}{e^{  i{\omega _{\left(1 \right)}}t}}}&{a_\gamma ^{\left(2 \right)}{e^{  i{\omega _{\left(2 \right)}}t}}} \end{array}} \right]^{\text{T}}} \hfill \\ \end{gathered} $  (10) 
Then, we have the least squares solution
$ {\mathit{\boldsymbol{\hat S}}_2} = {\left( {{\mathit{\boldsymbol{B}}_2}^{\rm{T}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_2}} \right)^{  1}}{\mathit{\boldsymbol{B}}_2}^{\rm{T}}\mathit{\boldsymbol{PG}} $  (11) 
Concerning degree l mode, we have the solution
$ {\mathit{\boldsymbol{\hat S}}_l} = {\left( {{\mathit{\boldsymbol{B}}_l}^{\rm{T}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_l}} \right)^{  1}}{\mathit{\boldsymbol{B}}_l}^{\rm{T}}\mathit{\boldsymbol{PG}} $  (12) 
where
$ \begin{gathered} {\mathit{\boldsymbol{\hat S}}_l} = \left[ {\begin{array}{*{20}{c}} {\hat a_p^{\left({  l} \right)}{e^{i{\omega _{\left({  l} \right)}}t}}}&{\hat a_p^{\left({  \left({l  1} \right)} \right)}{e^{i{\omega _{\left({  \left({l  1} \right)} \right)}}t}}}& \cdots \end{array}} \right. \hfill \\ {\left. {\begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat a_\alpha ^{\left(0 \right)}{e^{i{\omega _{\left(0 \right)}}t}}}& \cdots &{\hat a_\gamma ^{\left({l  1} \right)}{e^{i{\omega _{\left({l  1} \right)}}t}}}&{\hat a_\gamma ^{\left(l \right)}{e^{i{\omega _{\left(l \right)}}t}}} \end{array}} \right]^T} \hfill \\ \end{gathered} $  (13) 
and the coefficient matrix
$ \begin{gathered} {\mathit{\boldsymbol{B}}_l} = \left[ {\begin{array}{*{20}{c}} {P_l^{  l}\left({\cos {\theta _1}} \right)}&{P_l^{  \left({l  1} \right)}\left({\cos {\theta _1}} \right)}& \cdots \\ {P_l^{  l}\left({\cos {\theta _2}} \right)}&{P_l^{  \left({l  1} \right)}\left({\cos {\theta _2}} \right)}& \cdots \\ \vdots & \vdots & \vdots \\ {P_l^{  l}\left({\cos {\theta _N}} \right)}&{P_l^{  \left({l  1} \right)}\left({\cos {\theta _N}} \right)}& \cdots \end{array}} \right. \hfill \\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} {P_l^0\left({\cos {\theta _1}} \right)}& \cdots &{P_l^{\left({l  1} \right)}\left({\cos {\theta _1}} \right)}&{P_l^l\left({\cos {\theta _1}} \right)} \\ {P_l^0\left({\cos {\theta _2}} \right)}& \cdots &{P_l^{\left({l  1} \right)}\left({\cos {\theta _2}} \right)}&{P_l^l\left({\cos {\theta _2}} \right)} \\ \vdots & \vdots & \vdots & \vdots \\ {P_l^0\left({\cos {\theta _N}} \right)}& \cdots &{P_l^{\left({l  1} \right)}\left({\cos {\theta _N}} \right)}&{P_l^l\left({\cos {\theta _N}} \right)} \end{array}} \right] \hfill \\ \end{gathered} $  (14) 
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 spheroidal mode. The coefficient matrix
$ \mathit{\boldsymbol{B}}_l^S = \left[ {\begin{array}{*{20}{c}} {  \frac{{\partial P_l^m\left({\cos {\theta _{nj}}} \right)}}{{\partial {\theta _{nj}}}}} \\ {\frac{{\partial P_l^m\left({\cos {\theta _{ej}}} \right)}}{{\sin {\theta _{ej}}\partial {\phi _{ej}}}}} \end{array}} \right], {\kern 1pt} {\kern 1pt} {\kern 1pt} nj = 1, \cdots, {N_n};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ej = 1, \cdots, {N_e};{\kern 1pt} {\kern 1pt} {\kern 1pt}  l \leqslant m \leqslant l $  (15) 
and the timeseries
$ {\mathit{\boldsymbol{g}}_j}\left(t \right) = {\left[ {\begin{array}{*{20}{c}} {{g_{nj}}\left(t \right)}&{{g_{ej}}\left(t \right)} \end{array}} \right]^{\text{T}}} $  (16) 
On the other hand, for the horizontal component of a toroidal mode, the unique coefficient matrix
$ \mathit{\boldsymbol{B}}_l^T = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial P_l^m\left({\cos {\theta _{ej}}} \right)}}{{\partial {\theta _{ej}}}}} \\ {\frac{{\partial P_l^m\left({\cos {\theta _{nj}}} \right)}}{{\sin {\theta _{nj}}\partial {\phi _{nj}}}}} \end{array}} \right], {\kern 1pt} {\kern 1pt} nj = 1, \cdots, {N_n};{\kern 1pt} {\kern 1pt} {\kern 1pt} ej = 1, \cdots, {N_e};{\kern 1pt} {\kern 1pt}  l \leqslant m \leqslant l $  (17) 
where
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 ModelPredicted Singlet FrequencyBased on the perturbation theory, equation for the angular frequency of the mth singlet of harmonic degree l was established by Dahlen and Sailor (1979).
$ {}_n\omega _l^m = {}_n{\omega _l}\left({1 + a + bm + c{m^2}} \right),  l \leqslant m \leqslant l $  (18) 
where a degenerate eigenfrequency
PREM model of Dziewonsky and Anderson (1981) has ocean, while PREMtidal model (Rogister, 2003) is oceanless. So in this paper we only adapt the PREMtidal model which is derived by replacing the surficial ocean with a solid crust (Rogister, 2003).
2 DATA AND PREPROCESINGIn this study, we selected data sets Ⅰ and Ⅱ.
Data set Ⅰ: 18 minuteinterval 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 (NyAlesund, 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.
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Data set Ⅱ: 99 vertical components and 69 horizontal components of 10secondinterval 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 (broadband 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 Tsoft 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 broadband 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 _{0}S_{4}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 1.1Qcycle (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 _{0}S_{4}. Due to the limitation of the number of SG stations, not all of the singlets of _{0}S_{4} 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 _{0}S_{4} with high SNR (see Fig. 2b). It seems that using SG records could not detect the splitting multiplets of degree higher than 4.
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The estimated frequencies of _{0}S_{4} 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 WidmerSchnidrig (2013). From Fig. 2c we can find that the observed frequencies based on BS data are consistent with the predictions of PREMtidal 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 _{0}S_{4} 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 _{0}S_{4} are almost consistent with predictions, implying a weak crosscoupling between _{0}S_{4} and _{0}T_{3} (Zürn et al., 2000; Dahlen and Tromp, 1998; Woodhouse, 1980), but we only considered selfcoupling in this paper. So further investigations are needed. From the observations presented above, the study shows that highresolution singlet eigenfrequencies could be obtained by highquality 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 higherdegree modes.
_{0}S_{7}_{0}S_{10} are mantlesensitive modes which are predominantly sensitive to P velocity in the upper mantle and S velocity in the midmantle (He and Tromp, 1996). In this paper, we choose all the length around 1.1Qcycle (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 _{0}S_{7}_{0}S_{9} 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 _{0}S_{7}_{0}S_{10} are abnormally split, especially for _{0}S_{10}, and its observations increase first and then decrease, which are not consistent with the decline trend of PREMtidal model (Rogister, 2003). Since interactions between _{0}S_{l} and _{0}T_{l}_{+1} are getting stronger from l=7 to l=10 (Laske and WidmerSchnidrig, 2015, Fig. 12 therein) whereas predictions are getting worse in increasing l (Figs. 3d–3di), the deviations of _{0}S_{10} from the model predictions might be attributed to the strong coupling with _{0}T_{11}.The previous studies also suggested that _{0}S_{10} had a strong crosscoupling with _{0}T_{11} by Coriolis force (Laske and Widmer Schnidrig, 2015; Zürn et al., 2000; Resovsky and Ritzwoller, 1998; He and Tromp, 1996), and _{0}S_{8}_{0}T_{9}, _{0}S_{9}_{0}T_{10} are weakly coupled multiplets (Ritzwoller et al., 1988). Besides, modes _{0}S_{7}_{2}S_{3} are strongly coupled and different singlets could hardly be observed (Zürn et al., 2000; He and Tromp, 1996; Giardini et al., 1988). Considering crosscoupling is significant in estimating eigenfrequencies of strongly coupled modes (Laske and WidmerSchnidrig, 2015). Whereas in this study we only take selfcoupling of the modes into account, so the problems are not considered in our present results. Coupling effects will be considered in our future research.
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The pair's _{1}S_{5}_{2}S_{4} and _{2}S_{5}_{1}S_{6} may provide constraints on odddegree aspherical structures (Resovsky and Ritzwoller, 1998, 1995). Masters et al. (2000a) obtained splitting of overlapping modes _{1}S_{5}_{2}S_{4} 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 _{1}S_{5}_{2}S_{4}, 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 _{1}S_{5}_{2}S_{4} and _{2}S_{5}_{1}S_{6} from only one event by using OSE on the basis of selfcoupling, combining vertical and horizontal components with their record lengths of around 1.1Qcycle. 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 PREMtidal with high resolutions and high SNRs, except that _{1}S_{6} has a slight deviation. These indicate that OSE is effective in isolating some weakly coupled multiplets on the basis of selfcoupling approximation. Since the quality factors Qs for _{2}S_{5}_{1}S_{6} are closer than _{1}S_{5}_{2}S_{4} but the differences between each other are not very big (Q value of _{1}S_{5} is 291.88, _{2}S_{4} is 380.61, _{2}S_{5} is 302.19, _{1}S_{6} is 345.67) and _{2}S_{5}_{1}S_{6} is more strongly coupled than _{1}S_{5}_{2}S_{4} (Ritzwoller et al., 1988), the observed results of _{1}S_{5}_{2}S_{4} might be more reliable than those of _{2}S_{5}_{1}S_{6}, as shown by Figs. 4a–4d. Here we note that the observed eigenfrequencies of a multiplet poorly fit predictions (based on selfcoupling) in the frequency range overlapped with another multiplet, i.e., f > 1.375 MHz of _{1}S_{5} overlapping with frequencies of _{2}S_{4}, and f < 1.52 MHz of _{1}S_{6} overlapping with frequencies of _{2}S_{5}, whereas good fits lie in the unoverlapping frequency band. The differences between predictions and observations might be accounted for by crosscoupling effect or by contamination of signals. The differences between predictions and observations might be accounted for by crosscoupling 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 _{1}S_{5}_{2}S_{4} and _{1}S_{6}_{2}S_{5} provide constrain on odddegree mantle structure (Deuss et al., 2013; Masters et al., 2000b; Resovsky and Ritzwoller, 1998). Further investigations are needed in the future.
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Masters and Gilbert (1981) first observed that the inner core sensitive modes _{3}S_{2}, _{13}S_{2} and _{18}S_{4} are anomalously split. Morelli et al. (1986) and Woodhouse et al. (1986) proposed the hypothesis that there exists anisotropy in the innercore 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 innercore anisotropy (Tromp, 1995b, 1993). Ritzwoller et al. (1986) observed the splittings of the anomalously split modes _{0}S_{6}, _{2}S_{4}, _{6}S_{3} and _{11}S_{4}. WidmerSchnidrig et al. (1992) used singlet stripping technique (Ritzwoller et al., 1986; Buland et al., 1979; Gilbert, 1971) to isolate multiplets of modes _{15}S_{3}, _{16}S_{6}, _{18}S_{4}, _{20}S_{5}, _{21}S_{6} and _{23}S_{5}. Besides, Tromp (1993) suggested three anomalously split mechanisms and observed and predicted frequencies for the mutiplets of 18 anomalously splitting free oscillations _{11}S_{4}, _{11}S_{5}, _{13}S_{2}, _{13}S_{3}, _{14}S_{4}, _{15}S_{3}, _{16}S_{6}, _{18}S_{4}, _{20}S_{5}, _{21}S_{6}, _{23}S_{5}, _{25}S_{2} and _{27}S_{2} on the basis of previous studies (WidmerSchnidrig 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 innercore (Tromp, 1993). In this paper, we detected the splitting of 12 innercore sensitive multiplets _{25}S_{2}, _{27}S_{2}, _{6}S_{3}, _{9}S_{3}, _{13}S_{3}, _{15}S_{3}, _{11}S_{4}, _{18}S_{4}, _{8}S_{5}, _{11}S_{5}, _{23}S_{5} and _{16}S_{6} individually. We use vertical components of seismograms only, and most data sets that we used are with the length of around 1.1Qcycle (Dahlen, 1982). We can find that OSE could isolate all singlets clearly as shown by Fig. 5.
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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; WidmerSchnidrig et al., 1992; Li et al., 1991; Giardini et al., 1988; Ritzwoller et al., 1988, 1986) and the model predictions of PREMtidal. 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 _{6}S_{3}, _{9}S_{3}, _{13}S_{3}, _{15}S_{3} and _{18}S_{4}, 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 PREMtidal predictions, for example _{11}S_{4}, _{11}S_{5}, _{23}S_{5} and _{16}S_{6}. 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 _{11}S_{5} and _{16}S_{6} etc. This demonstrates that the 12 innercore sensitive modes dealt in this paper are anomalously split, and the deviation might be attributed to the intense noiseterms, or it suggests that the PREMtidal model needs modification. In the future, we will stack more events to further confirm this conclusion.
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In this paper we applied optimal sequence estimation (OSE) technique and its extension for the horizontal components to detect the splitting of higherdegree 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 highdegree modes in the case of only considering the selfcoupling. We selected all of the data length as around 1.1Qcycle to more precisely estimate the singlet frequencies of highdegree modes with higher SNRs. We detected obvious splitting of the mantle sensitive modes _{0}S_{7}_{0}S_{10} for the first time, and obtained the singlets' isolated eigenfrequencies with high resolutions and SNRs. We successfully separated the mutiplets of _{1}S_{5}_{2}S_{4} 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 _{2}S_{4} and _{2}S_{5} by using horizontal components, and for first time we completely isolated the singlets of _{2}S_{5}_{1}S_{6}. We presented the multiplets for 12 innercore sensitive modes clearly, among them the splitting of _{25}S_{2}, _{27}S_{2}, _{9}S_{3}, _{8}S_{5} and _{11}S_{5} 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 highdegree multiplets. However, we can effectively observe highdegree multiplets using a large number of records from globally distributed seismograms. Hence, records of seismograms under IRIS network are significant for observing especially the highdegreemode multiplets.
One potential approach is that we stack records after different strong earthquake events to more sufficiently detect complete splitting of higherdegree 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 crosscoupling and stack different seismic sources to isolate the coupled mutiplets, which are closely related to the inner structure of the Earth.
ACKNOWLEDGMENTSThe 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/s1258301708100.
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