Journal of Earth Science  2018, Vol. 29 Issue (6): 1398-1408   PDF    
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Observations of the Singlets of Higher-Degree Modes Based on the OSE
Shi-Yu Zeng1, Wen-Bin Shen1,2    
1. School of Geodesy and Geomatics/Key Laboratory of the Geospace Environment and Geodesy, Wuhan University, Wuhan 430079, China;
2. State Key Laboratory for Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan 430079, China
ABSTRACT: In this study, we selected 18 SG (superconducting gravimeter) records from 15 GGP stations with 99 vertical and 69 horizontal components of IRIS broad-band seismograms during 2004 Sumatra Earthquake to detect the splitting of higher-degree Earth's free oscillations modes (0S4, 0S7~0S10, 2S4, 1S5, 2S5, 1S6) and 12 inner-core sensitive modes (25S2, 27S2, 6S3, 9S3, 13S3, 15S3, 11S4, 18S4, 8S5, 11S5, 23S5, 16S6) by using OSE (optimal sequence estimation) method which only considers self-coupling. Results indicate that OSE can completely isolate singlets of high-degree modes in time-domain, effectively resolve the coupled multiplets independently, and reduce the possibility of mode mixing and end effect, showing that OSE could improve some signals' signal-to-noise ratio. Comparing the results of SG records with seismic data sets suggests that the number of SG records is inadequate to detect all singlets of higher modes. Hence we mainly selected plentiful seismograms of IRIS to observe the multiplets of higher modes. We estimate frequencies of the singlets using AR method and evaluate the measurement error using bootstrap method. Besides, we compared the observa-tions with the predictions of PREM-tidal model. This study demonstrates that OSE is effective in isolating singlets of Earth's free oscillations with higher modes. The experimental results may provide constraints to the construction of 3D Earth model.
KEY WORDS: free oscillation    mode splitting detection    high-degree modes    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, 0T4 0T6 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 Mw 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)

$ {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)

${\theta _j}$, ${\phi _j}$ 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

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

$ 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 ${g_j}\left(t \right)$ 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

$ \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 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

$ \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, 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):

$ {\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 over-determined system when N > 2l+1 and its least-squares 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 ${P_{ij}} = {\delta _{ij}}{P_j}$ 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)

$ \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 $a_p^{\left({ - 2} \right)}, a_p^{\left({ - 1} \right)}, a_\alpha ^{\left(0 \right)}, a_\gamma ^{\left(1 \right)}, a_\gamma ^{\left(2 \right)}$ 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

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

$ \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 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

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

$ {\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 $\mathit{\boldsymbol{B}}_l^T$ is an $\left({{N_e} + {N_n}} \right) \times \left({2l + 1} \right)$ matrix, expressed as

$ \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 ${\theta _{e/nj}}, {\phi _{e/nj}}, 1 \leqslant j \leqslant N$ 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).

$ {}_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 ${}_n{\omega _l}$ 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.

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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.

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. 2a2b 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.

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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.

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)
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 3a3d 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. 3ai3di. 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 0Sl and 0Tl+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. 3d3di), 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.

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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.
Table 2 Model predictions and observations of the splitting frequencies of 0S7 (unit: MHz)
Table 3 Model predictions and observations of the splitting frequencies of 0S8 (unit: MHz)
Table 4 Model predictions and observations of the splitting frequencies of 0S9 (unit: mHz)
Table 5 Model predictions and observations of the splitting frequencies of 0S10 (unit: MHz)
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. 4ai4di. 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. 4a4d. 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.

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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.
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.

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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.

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.

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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.
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.


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