Search for the Slichter modes based on a new method: Optimal sequence estimation

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: SOLID EARTH, VOL., 5 59, doi:./jgrb.53, 3 Search for the Slichter modes based on a new method: Optimal sequence estimation Hao Ding and Wen-Bin Shen, Received February 3; revised August 3; accepted 7 August 3; published September 3. [] The inner core translational mode ( S ), known as the Slichter triplet, can provide important information particularly on the Earth s inner core and its dynamics. A time domain spectral analysis method, the optimal sequence estimation (OSE), is devised and applied to search for the Slichter modes. Applications of the OSE and the multistation experiment (MSE) technique for detection of the singlets of 3 S and S show that both OSE and MSE can isolate their singlets, but OSE provides a higher signal-to-noise ratio. OSE also can completely isolate the modes S and S. Using two superconducting gravimeter data sets from nine Global Geodynamic Project stations before and after the Sumatra earthquake, we search for the possible Slichter triplet via OSE and provide a simple validation using the product spectrum. We set two preliminary criteria in order to identify possible signals; results show three signals as candidates for the Slichter triplet, more likely in the data after the Sumatra earthquake. While choosing the most reasonable criteria needs further studies, we emphasize here the effectiveness of the OSE in searching for the Slichter modes rather than claiming actual detections. Citation: Ding, H., and W.-B. Shen (3), Search for the Slichter modes based on a new method: Optimal sequence estimation, J. Geophys. Res. Solid Earth,, 5 59, doi:./jgrb.53.. Introduction [] The Slichter mode, S,isthefirst overtone of degree seismic normal mode of the Earth, which stems from the gravitational restoring force, with its eigenperiod heavily dependent on the density jump at the inner core boundary (ICB). Due to the rotation and ellipticity of the Earth, the Slichter mode splits into three singlets (triplet). Previous studies demonstrated that observations of the periods of the Slichter triplet may provide further constraints on the density contrast at and the viscosity just above the ICB [e.g., Smylie, 99; Rosat et al., ; Hinderer et al., 7]. However, the Slichter triplet has so far eluded definitive detection despite much effort in doing so by many authors [e.g., Smylie, 99; Crossley et al., 99; Hinderer et al., 995; Smylie and McMillan, 99, ; Courtier et al., ; Rosat et al., 3a; Sun et al., ; Rosat et al., ; Guo et al., 7; Pagiatakis et al., 7; Rosat et al., ; Abd El-Gelil et al., ; Xu et al., ; Rosat and Rogister, ].The main difficulties reside in the following: () the Slichter mode signals are extremely weak, weaker than the background noise of the station; () there are various Earth models predicting disparate Slichter triplet periods; (3) the periods of the Department of Geophysics, School of Geodesy and Geomatics, Key Laboratory of Geospace Environment and Geodesy of the Ministry of Education, Wuhan University, Wuhan, China. State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan, China. Corresponding author: W.-B. Shen, Wuhan University, Luoyu Road 9, Wuhan, Hubei 379, China. (wbshen@sgg.whu.edu.cn) 3. American Geophysical Union. All Rights Reserved /3/./jgrb.53 Slichter modes lie in the subtidal band, so likely to be buried in stronger subtidal signals. [3] There are many methods developed to search for the normal modes of the Earth, either in the frequency domain or time domain. In the frequency domain, there are at least five methods: () the least squares spectral analysis (LSSA) [e.g., Vanícek, 99; Pagiatakis, 999, 7], () the singlet stripping (SS) method [e.g., Gilbert, 97], (3) the product spectrum analysis (PSA) [Smylie, 99], () the crossspectrum analysis [Hinderer et al., 995], and (5) the matrix autoregressive analysis (MARA) [Masters et al., ] as an improvement of the SS method. In addition, Buland et al. [979] proposed the spherical harmonic stacking (SHS) in the frequency domain, and later, Cummins et al. [99] discussed this method in the time domain. In the time domain, Courtier et al. [] introduced the multistation experiment (MSE) technique, and Rosat et al. [] developed a nonlinear damped harmonic analysis method. Almost all these methods have been used to search for the Slichter modes [e.g., Smylie, 99; Smylie et al., 99; Crossley et al., 99; Hinderer et al., 995; Smylie and McMillan, 99, ; Courtier et al., ; Rosat et al., 3a; Sun et al., ; Guo et al., ; Rosat et al., ; Guo et al., 7; Pagiatakis et al., 7; Rosat et al., ; Abd El-Gelil et al., ; Xu et al., ]. [] Generally, the SS, SHS, and MSE methods can isolate and distinguish the singlets of a mode via corresponding spectral peaks, and hence are potentially more effective for identifying the triplet of S, whereas the LSSA, PSA, or the cross-spectrum analysis gives multiplets in the spectrum. However, the SS method needs a priori knowledge of the frequencies of the target modes, although it was improved by MARA [Masters et al., ]. SHS cannot completely isolate 5

2 the singlets when the number of stations is limited (see later). MSE can effectively isolate all individual singlets of a degree mode without a priori knowledge and, in principle, is quite suitable to search for the S triplet, but still suffers from the dominant error terms [Courtier et al., ; Guo et al., ; also see later]. [5] In this study, we propose a time domain analysis method which improves upon MSE. We call it the optimal sequence estimation (OSE), which applies the least squares principle. Using OSE, individual singlet time sequences, say for the Slichter triplet, can be directly obtained, which only appears in the spectrum of one of the sequences.. Formulation: Optimal Sequence Estimation [] Because the OSE method is closely related to the MSE, we first provide a short review of MSE. According to Courtier et al. [], the observed residual gravity g j (t) ofthejth station comprises three translational mode signals, namely prograde equatorial, axial, and retrograde equatorial signals, with amplitudes a p, a a,anda r, and angular frequencies ω p, ω a,andω r, respectively, as well as the uncorrelated noise n j (t), expressed as g j ðþ¼a t p e i ð ωpt ϕ jþ sinθj þ a a e iωat cosθ j þ a r e i ð ωrtþϕ jþ sinθj þ n j ðþ; t () where θ j and ϕ j are the colatitude and longitude of the jth station, respectively. For N stations, the summation of equation () provides the following expressions [Courtier et al., ]: S ¼ a p e iωpt a a e iωat a r e iωrt T ; (a) V ¼ n j ðþ t N N e iϕ n j j ðþ t n j ðþ t T N N cosθ j N N e iϕ j ; (b) j¼ j¼ and T denotes matrix transpose. [9] From equation (), the following equation can be obtained: j¼ S ¼ B G B V ; (5) where B is the inverse of the matrix B. This procedure is known as MSE. From equation (5), B V is unknown. Neglecting the influences of the noises B V,MSEprovides a unique solution for a p e iω pt, a a e iω at,anda r e iω rt. [] Here we note that the amplitudes a p, a a, and a r given in Courtier et al. [] are complex valued and depend on the latitude, longitude, and seismic moment tensor of an earthquake [Buland et al., 979; Woodhouse and Girnius, 9; Dahlen and Tromp, 99]. Given that the residual gravity time series is real valued, the real part of equation () is used to express the observations [Gilbert, 95; Woodhouse and Girnius, 9]. [] It is easy to find that equation (5) is a unique solution of equation (), neglecting the noise terms n j (t). Moreover, although the effect of the uncorrelated noise n j (t) inthe residual gravity time series can be reduced by the stacking, equations () and (5) show that B V cannot be eliminated. Therefore, using MSE to search for the Slichter triplet, the results will still suffer from the dominant error terms. >< >: N j¼ N j¼ N j¼ g j ðþ t e iϕ j ¼ Na p e iωpt þ a a e iωat N e iϕ j cotθ j þ a r e iωrt N e iϕ j þ N g j ðþ t ¼ a p e iωpt N cosθ j g j ðþ t e iϕ j j¼ j¼ j¼ j¼ e iϕ j tanθ j þ Na a e iωat þ a r e iωrt N e iϕ j tanθ j þ N ¼ a p e iωpt N e iϕ j þ a a e iωat N e iϕ j cotθ j þ Na r e iωrt þ N j¼ j¼ j¼ n j ðþ t e iϕ j ; j¼ n j ðþ t ; cosθ j j¼ n j ðþ t e iϕ j : () [7] Upon writing the coefficient matrix B and time series matrix G as [] Given the limitations of MSE, we here propose a new scheme that branches out from MSE, where the three time B ¼ N N j¼ e iϕ j tanθ j N N e iϕ j j¼ N N e iϕ j cotθ j j¼ N N e iϕ j j¼ N N e iϕ j tanθ j j¼ N N e iϕ j cotθ j j¼ 3 ; G ¼ 7 5 N N j¼ N N N N j¼ g j ðþ t e iϕ j j¼ g j ðþ t cosθ j g j ðþ t e iϕ j 3 : (3) 7 5 [] Equation () becomes where G ¼ BS þ V ; () sequences are estimated directly in an optimal fashion, provided that a linear model and more than three relevant observations are available. In this study, the complex-valued observation equation () is taken, and the three unknown 59

3 time sequences a p e iω pt, a a e iω at, and a r e iω rt are estimated using the least squares method point by time point. With records from N > 3 different stations, equation () can be written as G ¼ BS þ V ; () where the coefficient matrix B and time series matrix G can be respectively written as e iϕ sinθ cosθ e iϕ 3 3 sinθ g ðþ t e iϕ sinθ cosθ e iϕ sinθ B ¼ 7 ; G ¼ g ðþ t e iϕ N sinθn cosθ N e iϕ N sinθn g N ðþ t (7) degree modes than. For instance, the OSE for a degree mode is formulated as follows. [] Considering the degree mode (l = ), for N (>5) residual gravity time series from different stations, the observation equation can be written as g j ðþ¼a t ð p Þ e iω ð Þt sin θ j e iϕ j þ a ð p Þ e iω ð Þt cosθ j e iϕ j þa ð a Þeiω ðþt 3 cos θ j þ a ðþ r e iω ð Þt cosθ j e iϕ j þa ðþ ðþt sin θ j e iϕ j þ n j ðþ; t (9) r e iω where (a ð p Þ, a ð p Þ, a ðþ a, að r Þ, a ð r Þ ) and (ω ( ), ω ( ), ω (), ω (+), ω (+) ) are the complex amplitudes and frequencies corresponding to the azimuthal order number m =,,, +, and +. The coefficient matrix B is an N 5 dimensional matrix, expressed as 3 sin θ e iϕ sinθ cosθ e iϕ 3 cos θ sinθ cosθ e iϕ sin θ e iϕ sin θ e iϕ sinθ cosθ e iϕ 3 cos θ sinθ cosθ e iϕ sin θ e iϕ B ¼ 7 5 : () sin θ N e iϕ N sinθ N cosθ N e iϕ N 3 cos θ N sinθ N cosθ N e iϕ N sin θ N e iϕ N as opposed to the stacked series as in equation (3) of MSE, and V ¼ ½n ðþ t n ðþ t n N ðþ t Š T V. The general least squares solution of equation () is expressed as ^S ¼ B T B PB T PG; () where P ij = δ ij P j is the corresponding weight matrix of the stations assuming that all observations are independent and P j ¼ =σ j ( j N) is the weight of the jth residual gravity time series having variance σ j. [3] The three new sequences a p e iωpt, a a e iωat, and a r e iωrt in equation () thus obtained are time series referred to as the optimal sequences. The scheme proposed here using least squares to search for the S triplet by minimizing noise influences is referred to as the optimal sequence estimation (OSE). [] Based on the above, three spectra can be obtained for degree modes after using OSE, and each of them corresponds to one singlet. Hence, if only one of the spectra has an outstanding peak at one of the frequencies, this peak may correspond to a definite singlet. We have experimentally demonstrated that only those normal mode signals that are associated with equation () have such characteristics. This feature gives OSE and MSE an advantage over other spectral analysis methods (for example, LSSA or PSA) where all three singlets will appear in one spectrum [Smylie, 99; Pagiatakis et al., 7]. By comparison, OSE and MSE have the same starting equation but yield different products: the results of OSE are optimal solutions of equation () and should have a higher SNR than MSE. [5] Note that the frequencies of the Slichter triplet are much lower than mhz, so only the self-coupling and not the cross-coupling effects need considering. In such cases, OSE can be easily generalized to be applicable to higher [7] Finally, we obtain h i T a ð p Þ e iω ð Þt a ð p Þ e iω ð Þt a ðþ a eiω ðþt a ð r Þ e iω ðþt a ðþ r e iω ðþt 3. The Validation of OSE ¼ B T PB B T PG: () [] Before OSE is used to search for the Slichter triplet, we first use it to search for some other degree modes ( S and 3 S ) to verify its usefulness. Although OSE and MSE only consider the self-coupling effect, the first observation of S by Rosat et al. [3b] was exactly based on MSE. That means OSE should also be suitable to search for some normal modes. Since MSE has been frequently used to observe the triplet of some degree modes (such as S, S, and 3 S )[Courtier et al., ; Rosat et al. 3a, 3b; Sun et al., ; Guo et al., ; Rosatetal., ; Shen and Wu, ], to verify the usefulness of OSE, we use both OSE and MSE to detect the singlets of the modes S and 3S, and use OSE to detect the singlets of S,and S. [9] In this study, the superconducting gravimeter (SG) records are chosen as the data sets. The Sumatra earthquake is chosen as an excitation source, because it strongly excites the low-frequency normal modes [e.g., Park et al., 5]. We obtain records after the Sumatra event from SG stations, which are listed in Table. Different group sets of these records will be used to search for the four modes. And for each of the chosen SG records, after removing the effects of tides (solid Earth, ocean loading, and pole) and local atmospheric pressure variations, we obtain the residual gravity time sequences for further use. 3.. Results of Synthetic Data [] In this section, we first conduct a synthetic simulation analysis before analyzing the SG records. According to 5

4 Table. The Latitude, Longitude, and Synthetic Noise Levels of the SG Stations Station Latitude Longitude Synthetic Noise Level Bad Homburg (bh) Canberra (cb) Esashi (es) Kamioka(ka) Matsushiro (ma) Membach (mb) Medicina (mc) Metsahovi (me) Moxa (mo) Ny-Alesund (ny) Strasbourg (st) Sutherland (su) Vienna (vi) Wettzell (we) equation (), we generate synthetic SG records with the same length of 7 h, each synthetic record containing the given triplet signals (see Table ), with the equal amplitude nm/s.for the synthetic noise levels of those stations, we follow the approach presented by Courtier et al. [] as follows. [] We set a common noise power σ ¼ (nm/s ), and for each station, the synthetic noise level can be written as σ i ¼ þ sin θ i =σ ; () where θ i is the colatitude of the ith station. According to equation (), the synthetic noise levels are listed in Table. [] In order to compare with MSE and confirm that SHS could not completely isolate the singlets when the number of stations is limited (as mentioned in section ), OSE, MSE, and SHS are all applied to the synthetic time series. The results are shown in Figure. From Figures a to c, while both OSE and MSE can isolate the three signals, OSE has a higher SNR than MSE. Figures e g clearly show that SHS cannot completely isolate the synthetic singlets. [3] The actual frequencies and attenuations of the three signals, and their corresponding estimated values (using the autoregressive (AR) method, see Chao and Gilbert [9]) are listed in Table. Note that the estimated values are obtained from three new sequences after using OSE. The relative deviations of frequencies and attenuations are only about.% and.55%, respectively, very close to the true values. Therefore, we will use the AR method to estimate the frequencies of the chosen modes. 3.. Results for the Modes S and 3 S [] The modes S, S, and 3 S are the first three overtones of the degree modes. If a method is effective in searching for the triplets of S and 3 S, it is reasonable to believe that the same method will also be effective for the S triplet. [5] To observe the triplets of S and 3 S, two data sets ( min sampling rate) obtained after the Sumatra earthquake are chosen. [] Data set I: A total of eight SG records from eight stations (cb, ka, ma, mb, mc, ny, st, su) are used, with each record starting 5 h after the earthquake and spanning h. This data set is used to observe the S triplet. [7] Data set II: records from SG stations (bh, cb, ma, mb, mc, mo, ny, st, su, vi, we) are chosen, each of them starting 5 h after the earthquake and spanning h. This data set is used to observe the 3 S triplet. [] The power spectra of S obtained using the MSE and OSE methods based on data set I are shown in Figure, and the corresponding frequencies are listed in Table 3. Figure clearly shows that both MSE and OSE can isolate the singlet m = +, but the result of OSE has a higher SNR. For the singlet m =, neither of the methods can obtain an observable peak, but both of them isolate the toroidal mode T in the corresponding spectrum with high SNR. For the singlet m =, the SNRs of both the methods are very low. In order to more accurately identify the possible signals, we adopt a rule of thumb suggested by Stuart et al. [97], i.e., a spectral peak is significant if it has a maximum two or three times greater than the surrounding noise level. For the singlet m = of S, the RMS power σ P of the frequency bands mhz and.5.35 mhz (which bracket the target singlet) is computed as a surrounding noise level (similar as Häfner and Widmer-Schnidrig [3]). Obviously, there is a significant peak for the singlet m = using OSE, but none using MSE (though a strange peak located at.757 mhz appears). Careful comparison between Figures a and c suggests that the shapes of those two spectra are almost the same except for the peaks corresponding to the singlet m = ±, which might be caused by the residual error terms in the results of MSE. [9] In addition, our OSE results show that the SNR of T is lower than that of the singlet m = + but higher than that of the singlet m =, consistent with previous studies [see Park et al., 5, Figure ; Rosat et al., 5, Figure ; Hu et al.,, Figure 7; and Roult et al.,, Figure 9]. This indirectly confirms the OSE method proposed in this study. Moreover, our observed result for m = + deviates slightly from the corresponding value given by Deuss et al. [], but it is very close to the results of Rosat et al. [3b, 5]. Because Deuss et al. [] has considered the cross-coupling effect, and the comparison shows that OSE can obtain a sufficiently accurate result for S, even if only the self-coupling effect is considered. [3] The power spectra and estimated frequencies of the multiplet 3 S are shown in Figure 3 and Table. Despite the interference of S 3 being weakened by using a later starting point after the earthquake, the modes S 3 and 3 S are cross-coupled, and the cross-coupling effect may affect the frequencies of the singlets even after one of the modes has attenuated. From Table, for m = ±, results from both OSE and MSE are close to those of previous studies, but for m =, OSE or MSE falls between the results of Chao and Gilbert [9] and other previous studies (see Table ). This may be caused by the cross-coupling effect of S 3. Table. Synthetic Signals and the Corresponding Estimated Values by the AR Method Singlet Method Attenuation m = Synthetic value.39 5.e Estimated value.39 ±.e.95e ±.9e m = Synthetic value. 5.e Estimated value.3999 ±.e 5.e ±.9e m = + Synthetic value. 5.e Estimated value.997 ±.7e 5.e ± 3.99e 5

5 Prograde (m= ) a) Synthetic Power Spectra Prograde (m= ) e) Axial (m=) b) Axial (m=) f) Retrograde (m=+) Product Spectrum 5 c) d) Retrograde (m=+) 5 g) Figure. The power spectra results from the synthetic records from SG stations using different methods. (a c) The power spectra using OSE (red dashed lines) and MSE (black lines). (d) The product spectrum of those records. (e g) The power spectra using SHS. However, there are no previous studies which specifically dealt with the 3 S triplet while considering the cross-coupling or full coupling effect. In light of the result for m = using OSE also with very high SNR, we may conclude that the frequency deviation caused by the cross-coupling is very small. Anyway, Figure 3 clearly shows that both OSE and MSE can isolate the triplet of 3 S, but OSE has a higher SNR than MSE. MSE Power Spectra (nm /s )/mhz a) m=? b) 3 m= T c) m= OSE Power Spectra (nm /s )/mhz d) m= e) 3 m= T f) m= Figure. Power spectra of S using MSE and OSE. (a c) Results obtained by MSE; (d f) results obtained by OSE. From left to right, the spectra correspond to m =,, and +. Arrows indicate the mode frequency values corresponding to the peaks. The results of Deuss et al. [] are denoted by the vertical dashed lines. The RMS power σ P,σ P, and 3σ P of the background frequency bands mhz and.5.35 mhz are indicated respectively by the horizontal solid lines (as the estimate of the background noise level in the target frequency band), dotted lines, and dashed lines. 5

6 Table 3. Model Predictions and Observations of the Splitting Frequencies S (unit: mhz) m = m = m =+ PREM a This paper (OSE).3979 ±.e.5 ± 5.5e 5 This paper (MSE).975 ±.9e 5 Rosat et al. [3b].39 ±.9e.9 ±.e. ±.e Rosat et al. [5].39 ±.e 5. ±.e 5 Hu et al. [].393 ± 5.e 5.5 ± 7.7e 5.35 ±.e 5 Roult et al. [] ±.53e 3.39 ±.35e 3.3 ±.e 3 Deuss et al. [] Rosat et al. [] b.39 ± 9.e. ±.e a From Roult et al. []. b Correction to Rosat et al. [5]. S [3] The observations of S and 3 S using OSE and MSE in this section clearly demonstrate that both the OSE and MSE can strip the triplet of degree mode, but OSE has a higher SNR than MSE, especially for the weaker singlets (such as the singlet m = of S and m = of 3 S excited by the event). These results provide an experimental foundation for searching for the S triplet using OSE. Nevertheless, to further verify the potential that OSE can successfully strip the singlets of a mode, we use OSE to strip the singlets of S and S in the next subsection Results for the Modes S and S [3] In section., OSE has been generalized to detect degree modes. Here, it is used to search for the singlets of S and S. [33] Based on SG records from the bh, cb, ma, mb, mc, mo, ny, su, st, vi, and we stations, with each record starting 5 h after the earthquake and spanning 3 h, the results of S using OSE are shown in Figure. For brevity, the estimated singlet frequencies have not been tabulated, but the arrows indicate the peak values as shown in Figures a e, and the frequency distribution is shown by Figure f. [3] As shown in Figures a e and Figure f, OSE clearly isolates all five singlets of S, and the estimated frequencies of S are very close to the results of previous studies as well as the PREM [Dziewonski and Anderson, 9] predictions [from Roultetal., ]. In particular, our results are very close to those of Deuss et al. [], who have considered cross-coupling. Considering the accuracy of the estimates, the differences between the results of Deuss et al. [] and ours are negligible. This again demonstrates that OSE is suitable for searching for the singlets of a normal mode. [35] Concerning the mode S, we use the same SG records as used for S, except that station ma is being replaced by ka. All records start 5 h after the earthquake and span h. a) b) c) MSE Power Spectra (nm /s )/mhz OSE Power Spectra (nm /s )/mhz m= m= m= d) e) f).9557 m= m= m= Figure 3. Power spectra of 3 S using MSE and OSE. (a c) Results using MSE; (d f) results using OSE. From left to right, the spectra correspond to m =,, and +, respectively. Arrows indicate the mode frequency values corresponding to the peaks. The theoretical frequencies of the preliminary reference Earth model (PREM) are denoted by the vertical dashed lines. 53

7 Table. Model Predictions and Observations of the Splitting Frequencies of 3 S (unit: mhz) m = m = m =+ PREM a This paper (OSE).9557 ±.e ± 5.e ±.e 5 This paper (MSE).95 ± 3.e 5.99 ±.e.9579 ±.7e 5 Chao and Gilbert [9].97 ± 5.5e ± 9.e ±.e 5 Roult et al. [].95 ±.e.99 ± 3.e.9579 ±.93e Shen and Wu [].959 ±.e.93 ±.5e.95 ±.3e a From Roult et al. []. 3S Figure 5 provides the results using OSE (the estimated frequencies also have not been tabulated). All five singlets are isolated and have high SNR, and our results are close to those of Roult et al. [], who did not obtain a clear detection of the singlet m =. [3] Hence, the effectiveness of OSE is further verified by isolating the singlets of S and S.. Search for the Slichter Triplet [37] To search for the Slicther modes, we choose two data sets from nine SG stations (bh, cb, ma, mb, mc, mo, st, su, and we) before and after the Sumatra event, each, h long at h intervals. For each record of data set I, the time span is 3 March 3 to 3 August while each record of data set II starts 5h after the earthquake on December (all chosen records are preprocessed as described in section 3). In this study, we will use h records to search for S, so each record of both data sets is divided into three segments without overlap. In order to make comparison trials, we will adopt the following two different experimental procedures for both of the data sets. [3] The first experimental procedure, for each of the data sets I and II, three equal length subsets are obtained, and every subset consists of nine simultaneously recorded series from different SG stations, but different subsets have different time spans. For every subset, after using OSE, three new sequences s Pi (t), s Ai (t), and s Ri (t) (i =,, 3) can be obtained; then some well-known signals (the atmospheric tidal signals and the nonlinear tides) can be subtracted from those new sequences (after obtaining their complex frequencies and amplitudes). Then the demoded power spectra of each, P Pi (ω), P Ai (ω), and P Ri (ω)(i =,, 3), can be obtained. Finally, the product spectra for different singlets can be obtained by P s ðωþ ¼ 3 P si ðωþ (s = P, A, R). Namely, there i¼ are three demoded product spectra for data set I (also for data set II), which correspond to the singlets m =, and +, respectively. [39] The second experimental procedure, also for the same subsets of data sets I and II, for the ith subset (i =,, 3) from each (I or II), after subtracting some well-known signals from each chosen record, the product spectra can be obtained by Power Spectra (nm /s )/mhz Power Spectra (nm /s )/mhz a) c) e) m= m= m= m= b) d) f) m= x Obeserved Predicted 5 Azimuthal order number m Figure. (a e) Power spectra of S using OSE. (f) The differences between the observed values and the predicted values of PREM. Arrows indicate the mode frequency values corresponding to the peaks. The results of Deuss et al. [] are denoted by the vertical dashed lines. (*Correction to Rosat et al. [5]). 5

8 Power Spectra (nm /s )/mhz Power Spectra (nm /s )/mhz 3 m= a) c) e).739 m= m= m= b) d) f) m= x Obeserved Predicted Roult et al. () This Paper Azimuthal order number m Figure 5. (a e) Power spectra of S using OSE. (f) The differences between the observed values and the corresponding PREM predictions, which are denoted by the vertical dashed lines. Arrows indicate the mode peak frequency values. m= Product in (nm /s /cpd) 3 S 3 S S 5 S S 7 Original MN MS 3 MS Demoded 3 m= Product in (nm /s /cpd) MN MS M SN m=+ Product in (nm /s /cpd) MN MS M SN MS Frequency (cpd) Figure. The original product spectra (green curve) and the demodulated product spectra (red curve) of data set II (after the Sumatra event) after using the first experimental procedure. The red solid curves denote the estimates of the background noise levels of the demoded product spectra, and the red dashed curves show the corresponding 95% CI, while the green solid curves denote the estimates of the background noise levels of the original product spectra, and the green dashed curves show the corresponding 95% CI. The diurnal harmonics of S3 S7 are indicated by the vertical dotted gray lines. Arrows indicate some nonlinear tides. 55

9 m= Product in (nm /s /cpd) 3 a) Dataset I (Before Sumatra) d) Dataset II (After Sumatra) m= Product in (nm /s /cpd) b) e) m=+ Product in (nm /s /cpd) c) f) Frequency (cpd) Frequency (cpd) Figure 7. (a c) The demodulated product spectra of data set I (before the Sumatra event) and (d f) data set II (after the Sumatra event) after applying the first experimental procedure. The black solid curves denote the estimates of the background noise levels, the red and blue dashed curves denote the 95% CI and % CI curves, respectively. The vertical dotted blue lines denote the corresponding peaks that only satisfy criterion, and the vertical dashed red lines denote the corresponding peaks satisfying both of the criteria; the red arrows denote in which spectra the possible signals appear. The vertical solid blue lines and arrows in Figure 7a 7c correspond to the possible signals (denoted by red arrows) shown in Figures 7d 7f. The green dashed lines indicate the results of Smylie [99], and the green arrows denote in which spectra they should appear. " # =9 P i ðωþ ¼ 9 P ij ðωþ (i =,,3),whereP ij(ω) denotes the j¼ demoded power spectra of the jth time series ( j =,,,9)in the ith subset. The final demoded product spectra of each data set (I or II) can be obtained by P F ðωþ ¼ 3 P i ðωþ.fromthe i¼ above description, P F (ω) has the same unit as P s (ω), which facilitates a direct comparison of the different results. [] At first, the power spectra (the log form, hereafter the same) of the data set II upon using OSE are taken as an example in Figure. Note that we use a curve to fit the product spectra to obtain an estimate of the background noise level in this study (similar as, e.g., Smylie [99] and Courtier et al. []). Figure clearly shows that some relatively weak signals are enhanced after subtracting the well-known signals (the atmospheric tidal signals from S 3 to S 5 and the nonlinear tides MN,M, MS,SN,MS ), and confirms that the signals without splitting nature must appear in at least two of the three spectra. [] To judge whether a peak is a possible singlet of S,we can set two criteria based on the characteristics of OSE: () the possible spectral peak P s (ω i )(ω i denoting a fixed frequency) is higher than the 95% confidence interval (CI) curve; and meanwhile, () the other two corresponding spectral peak amplitudes P q (ω i )(q = P, A, R; q s) are lower than the % CI curve. If the two criteria hold, the possible spectral peak is considered as a target singlet. It should be noted that the criterion may be not a very strict judgment condition, but it can reflect the characteristic of the OSE to a certain extent, and taking into account the fact that the SNR of S must be very low, we adopt these two criteria to show how to search for the possible signals for the S triplet based on OSE. [] The results of data sets I and II after applying the first experimental procedure are shown in Figures 7a 7c and Figures 7d 7f, respectively. Apparently, only one peak from data set I satisfies both of the criteria, while two peaks from data set II satisfy both of the criteria. Note that the possible spectral peak appearing in Figure 7e is the only one that satisfies criterion in that spectrum. The vertical solid blue lines and arrows in Figures 7a 7c correspond to the possible signals found in Figures 7d 7f. Obviously, the possible signals obtained after the event are not significant in the spectra obtained before the event. In addition, there are no significant peaks corresponding to the results given by Smylie [99] (green dashed lines). [3] As Rosat et al. [] have done, in order to give a more rough statistical significance estimate, a Savitzky- Golay smoothing filter [Savitzky and Golay, 9] is used in the frequency domain to smooth the product spectra. The corresponding smoothed spectra of Figure 7 and their CI curves are shown in Figure (only concerning the possible frequency band 3.. cpd). Taking a closer look at the peaks marked by the red circles, we find that the spectral peak in 5

10 m= Product in (nm /s /cpd) 3 m= Product in (nm /s /cpd) 3 m=+ Product in (nm /s /cpd) 3 Dataset I (Before Sumatra) a) b) c) Frequency (cpd) d) e) f) Dataset II (After Sumatra) Frequency (cpd) Figure. The smoothed spectra of Figure 7 (only concerning the possible frequency band 3.. cpd). The indicated peaks are the same as explained in Figure 7. The peaks marked by the red arrows (and circles) are the corresponding spectral peaks of the possible signals in different spectra. Figure b is not lower than the % CI curve any more, while the other five spectral peaks are still lower than their corresponding % CI curves. Namely, the possible signals obtained from data set I may not be a possible singlet of S, while the two possible signals obtained from data set II might still be the possible singlets m = andof S, respectively. According to Figures 7 and, it is clear that there is no possible signal for the singlet m =+of S ; this might be caused by the chosen data sets, the judgment conditions, or not being excited to an observable level. In addition, it is worth noting that, concerning the two possible signals obtained here based on the two chosen criteria, if the criteria are changed, they might no longer be significant. Hence, choosing appropriate criteria is crucial for searching for the S based on the OSE results. This problem needs further investigation, especially for criterion. The estimated frequencies from this study are listed in Table 5 and compared to prior results. [] In order to verify the two possible signals, the demoded product spectra of data sets I and II after applying the second experiment are shown in Figure 9. The red curves are the demoded product spectra, and the black bold solid curves are the smoothed spectra used to judge whether the possible peaks are noise. Apparently, there are still no significant peaks corresponding to the results of Smylie [99] (green solid lines). Figure 9 shows that there are three more significant peaks from data set I. However, there are six more significant peaks from data sets II, and one of them (denoted by the blue arrow) is just the possible signal for the singlet m = obtained in Figure, but there is no significant peak corresponding to the possible signal for the singlet m = obtained in Figure. Without using OSE (as the second experimental procedure), in the spectra (Figure 9) after the event, there are three peaks that are more significant than the possible singlet m = signal, so it is hard to distinguish it from the other possible signals. It is noteworthy that, using PSA, the most significant peak in the spectra after the event is still significant in the spectra before the event (see the peaks marked by the blue circles in Figure 9, located at about 5.7 cpd); hence, it might be considered as a possible singlet (possibly corresponding to m = +, see later). However, concerning the results after using OSE, this possible signal (m = +) did not even satisfy criterion Table 5. The Predicted and Observed Periods of Slichter Modes (Unit: Hour) Model m = m = m =+ Dahlen and Sailor [979] A Smylie [99] A..7.3 CORE observed.5 ± e ± e 3.5 ± e Crossley et al. [99] A Rochester and Peng [993] PREM Peng [997] PREM Rieutord [] A Rogister [3] A PREM This paper(ose) Observed 5.9 ±.7e 3 5. ±.3e 3 This paper(psa) Observed.7 ±.e 3 57

11 Product in (nm /s /cpd) 3 Product in (nm /s /cpd) 3 Before Sumatra After Sumatra Frequency (cpd) Figure 9. The demodulated product spectra of data set I (before the Sumatra event) and data set II (after the Sumatra event) after applying the second experimental procedure. The red curves are the demoded product spectra, and the black thin solid curves are estimates of their background noise levels, while the dashed curves are their corresponding 95% C.I. curves. The black bold solid curves are the corresponding smoothed spectra. Every black dashed line denotes two corresponding peaks before and after the event in which at least one of them is over the 95% CI (from left to right, the peaks indicated by the sixth and seventh lines are overlapped, i.e., they can be considered as the same peaks). The vertical red dashed lines denote the two possible signals obtained as shown in Figure. The green solid lines denote the results of Smylie [99]. The blue arrow means the possible signal found in Figure, which is still significant in the spectra obtained from the PSA method without using OSE. The peaks marked by the blue circles have almost the same frequency, and they are both significant in their corresponding spectra. (see Figures 7 and ). So the results are confusing. But we at least demonstrate that the possible singlet m = based on OSE is still significant in the results using only PSA. [5] To the present, two possible singlets (m = : 5.9 ±.7 3 h, m =: 5.±.3 3 h) based on OSE, and one possible singlet (.7 ±. 3 h) based on PSA, are obtained. According to the theoretical periods listed in Table 5, we suggest that the possible singlet based on PSA may correspond to m = +. Hence, based on two different experiments, we may cautiously suggest that the three candidate signals might be the S triplet, and the relative period deviations among those candidates and the corresponding theoretical values provided by Rochester and Peng [993], Peng [997], and Rogister [3] are only about.% (see Table 5). However, according to Rosat and Rogister [], if S can be excited by an earthquake to a detectable level, the earthquake must be larger than Mw 9.7. If we accept their statement, we could not conclude that those candidates are the possible Slichter triplet. 5. Discussions and Conclusions [] This study proposed the OSE method, which is based on the least squares (LS) principle, to search for the Slichter triplet. Theoretically, if we only consider the self-coupling effect, OSE can be generalized to be applicable to an arbitrary degree mode, and in this study, we provided the formulas for degrees and. To confirm the effectiveness of the OSE and ensure that it can successfully strip a given multiplet, we used this method to detect the singlets of S, 3 S, S,and S based on SG records. All of the singlets were completely resolved and isolated, except for the mode S where only the singlets of m = andm = + were resolved but not the singlet m = which presumably was not excited by the event to an observable level. Furthermore, we clearly observed the mode T in the spectra corresponding to the singlet m =. As a result, there are no significant differences between our results and previous studies which considered the cross-coupling effects. Therefore, we may conclude that OSE is effective for the detection of the normal modes, and hence potentially useful for the search for the Slichter triplet. [7] Based on two SG data sets from nine Global Geodynamic Project stations before and after the Sumatra earthquake, after using OSE and demodulating some known signals, we set two criteria to identify whether a spectral peak is a possible signal of interest. And after smoothing the spectra to further improve the statistical significance, two significant peaks (m = and ) of the Slichter triplet obtained from the data set after the event have been detected. Based on the same two data sets, we found a very significant possible signal upon using PSA, but it cannot even satisfy criterion in the spectra using OSE. Although the results are confusing, we suggest this signal may be the possible singlet m =+. [] Although our results seem to support the appearance of the S triplet in the data set after the Sumatra earthquake, Rosat and Rogister [] concluded that the Sumatra earthquake might be not large enough to excite an observable S signal. In addition, these candidates are obtained based on the two adopted criteria which might not be very reasonable, especially criterion. Therefore, the detected signals might not be the true Slichter triplet. Setting reasonable criteria to search for the S triplet using OSE needs further investigations. 5

12 [9] In this study, we prefer to test procedures using OSE to search for the possible Slichter triplet rather than to emphasize the suggested candidates. Nevertheless, based on the results for the modes S, 3 S, S,and S, we suggest that combining OSE and PSA and using more SG data sets might be a potential way of finally confirming the existence of the Slichter triplet. [5] Acknowledgments. The authors would like to express their sincere thanks to the Associate Editor and the two anonymous Reviewers for their valuable comments and suggestions, which greatly improved the manuscript. We also thank Jim Ray for the corrections and for smoothing the English expressions of the manuscript. 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