Geophysical Journal International

Transcription

1 Geophysical Journal International Geophys. J. Int. (2014) 196, Advance Access publication 2013 December 13 doi: /gji/ggt468 Observation of spheroidal normal mode multiplets below 1 mhz using ensemble empirical mode decomposition Wen-Bin Shen 1,2 and Hao Ding 1 1 Department of Geophysics, School of Geodesy and Geomatics, Key Laboratory of Geospace Environment and Geodesy of the Ministry of Education, Wuhan University, Wuhan , China. wbshen@sgg.whu.edu.cn 2 State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan , China Accepted 2013 November 19. Received 2013 November 15; in original form 2013 July 10 1 INTRODUCTION Previous studies have demonstrated that the records from sparsely distributed superconducting gravimeters (SGs) deployed under the Global Geodynamics Project (GGP) are extremely sensitive to gravity variations, which are related to various geophysical processes. SGs have high resolution, are very stable and are superior over the best seismometers in the frequency band of 0.3 to 0.8 mhz. Thus, SGs are particularly suitable for the observation of low-frequency seismic modes (e.g. Hinderer et al. 1995; Courtier et al. 2000; Rosat et al. 2003, 2005; Guoet al. 2006; Crossley & Hinderer 2008a,b). Moreover, the splitting of the modes below 1 mhz is highly sensitive to the 3-D density structure of the Earth s mantle and core (e.g. Widmer-Schnidrig 2003). Therefore, numerous studies have focused on splitting detection of some low-frequency seismic modes. The complete sets of the singlets of 0 S 3 and 0 S 2 were first observed by Buland et al. (1979) using seismic data sets from the International Deployment of Accerometers (IDA) network based on the spherical harmonic stacking (SHS) method. In addition, the triplet of 3 S 1 wasfirstobservedbychao&gilbert(1980) using seismic data sets from the IDA network based on the SHS method (Buland et al. 1979); the observed result of the frequency, which corresponds to the singlet m = 0, deviates from the preliminary reference earth model (PREM; Dziewonski & Anderson 1981) prediction of Roult et al. (2006, 2010) that PREM incorporates the SUMMARY Superconducting gravimeter (SG) records after the 2004 Sumatra M w 9.0, 2010 Maule M w 8.8 and 2011 Tohoku M w 9.1 earthquakes are selected to observe the singlets of six spheroidal normal mode multiplets below 1 mhz ( 0 S 2, 2 S 1, 0 S 3, 0 S 4, 1 S 2 and 3 S 1 ). To clearly observe their spectral splitting, the ensemble empirical mode decomposition (EEMD) is applied to SG records as a dyadic filter bank. Comparisons of the product spectra obtained from the direct residual gravity records and those obtained after using EEMD clearly confirm the validity of EEMD. After using EEMD, all of the singlets of the six multiplets, particularly those of 0 S 4 and 1 S 2, are completely observed with high signal-to-noise ratio, whereas some of singlets could not be well resolved without the application of EEMD. This study demonstrates that EEMD may be important in the detection of the splitting of some weak and low-frequency seismic modes. The relevant observation results may improve the Earth s density models. Key words: Time-series analysis; Gravity anomalies and Earth structure; Surface waves and free oscillations. rotation and ellipticity. In this study, for comparison, we use the PREM-re predictions provided by Roult et al. (2010), which were computed using the higher order perturbation theory that was developed successively by different authors (Lognonné 1991; Lognonné &Clévédé 2002; Millot-Langet et al. 2003). The triplet of 3 S 1 was also observed by Roult et al.(2010) using two seismometer records based on the traditional power spectrum estimation method after the 2004 Sumatra earthquake, but their result on the singlet m = 0 had a very low signal-to-noise ratio (SNR). Moreover, Ding & Shen (2013) observed three isolated singlets of 3 S 1 based on the optimal sequence estimation (OSE) method. The triplet of 2 S 1 was first observed by Rosat et al. (2003) who applied multistation experiment method (e.g. Courtier et al. 2000) using five SG records. It was also observed by Hu et al. (2006) using wavelet analysis and Ding & Shen (2013) using OSE. Ritzwoller et al. (1986) obtained nine estimated resonance functions of 0 S 4 by virtue of a technique referred to as singlet stripping (Gilbert 1971), but their results (figs 8b and f of Ritzwoller et al. 1986) show that the estimated resonance functions of the singlets m = 2andm = 3 almost overlapped. In addition, several researchers have calculated the splitting function of 0 S 4 (e.g. He & Tromp 1996; Resovsky & Ritzwoller 1998; Deuss et al. 2011, 2013) based on which the corresponding splitting frequencies can be obtained in theory. However, they did not publish these splitting frequencies. The singlets of 0 S 4 were also studied by Roult et al. (2010), but in their results, only its two singlets (m =±4) were detected. Therefore, we are still uncertain GJI Seismology C The Authors Published by Oxford University Press on behalf of The Royal Astronomical Society. 1631

2 1632 W.-B. Shen and H. Ding on whether all of the singlets of 0 S 4 have been completely observed. Widmer-Schnidrig & Laske (2007) studied the multiplet 1 S 2 using multiplet stripping, but the receiver strips of the singlets m = 1 and m = 2 also overlapped. Deuss et al. (2011) also studied this mode, but they did not provide a spectrum containing all of the five singlets. Furthermore, Roult et al. (2010) used broad-band seismometer records to detect the 1 S 2 multiplet, but the singlet m = 1 of 1 S 2 is difficult to identify in their results. By contrast, Ding & Shen (2013) clearly observed five singlets of 1 S 2 based on the OSE method. Here, we mention that the multiplet 2 S 2 has been investigated by only a few studies (e.g. Fukao & Sudo 1989;Okal&Stein 2009). For instance, Okal & Stein (2009) reviewed early works and concluded that 2 S 2 could not be excited above the noise level; thus, the multiplet 2 S 2 could not be observed. For the multiplet 1 S 3, based on the singlet stripping method, both Ritzwoller et al.(1986) and Widmer-Schnidrig & Laske (2007) obtained seven estimated resonance functions; however, the receiver strips of some singlets overlapped. In addition, Smylie (1992) first claimed, which was later supported by Courtier et al. (2000), that the triplet of 1 S 1 (i.e. Slichter modes) is observed, but various studies (e.g. Crossley et al. 1992; Hinderer et al. 1995; Guo et al. 2006) could not provide positive confirmation. Therefore, concerning the 11 seismic spheroidal mode multiplets below 1 mhz, it appears that the confirmed observations of all of the singlets have been made for only six multiplets: 0S 2, 0 S 3, 1 S 2, 2 S 1, 3 S 1 and 1 S 3. Inasmuch as previous studies have shown that the modes below 1 mhz are very sensitive to density heterogeneities, precise determination of the splitting frequencies of these modes can improve 3-D density models (e.g. Dahlen & Tromp 1998; Widmer-Schnidrig 2003; Rosat et al. 2005). This study focuses on the clear observation of all of the singlets of 0 S 2, 0 S 3, 0 S 4, 1 S 2, 2 S 1 and 3 S 1 based on the ensemble empirical mode decomposition (EEMD) technique (Huang & Wu 2008; Wu & Huang 2009) using SG records. Of the modes concerned, the singlets of 0 S 4 were not yet completed observed by previous studies. The 2004 Sumatra-Andaman (M w 9.0), the 2010 Maule (M w 8.8) and 2011 Tohoku (M w 9.1) earthquakes strongly excited numerous low-frequency Earth oscillation modes (e.g. Park et al. 2005; Rosat et al. 2005;Huet al. 2006;Xuet al. 2008; Roult et al. 2010; Häfner & Widmer-Schnidrig 2013), providing opportunities for detecting those multiplets. We expect that using the three earthquake events can effectively enable the observations of the following six multiplets: 0 S 2, 0 S 3, 1 S 2, 2 S 1, 3 S 1 and 0 S 4. In this study, the EEMD technique is first applied to the SG records as a dyadic filter bank to improve the SNR of those target modes. For comparison, the stacking power spectrum approach (Smylie et al. 1993) is applied to both the direct residual gravity records and those obtained after using EEMD. The results demonstrate that after using EEMD, all of the singlets of the six mentioned modes were completely observed, whereas without using EEMD, some of the modes could not be well resolved. 2 DATA SETS AND METHOD 2.1 Data sets The optimum record length is approximately 1.1Q-cycle to maximize SNR (Dahlen 1982). However, if the SNRs of the modes are significantly excited by an event, for example, the 2004 Sumatra earthquake, a data set with a length of several Q-cycles can be used to increase the frequency resolution (Rosat et al. 2005; Roult et al. 2010). From the minute-interval SG records obtained from the GGP stations after the three earthquakes, we removed the tides, local atmospheric pressure effect and the trend term, obtaining a residual gravity data set {g i (t)} (i = 1, 2,...,13) for further use. Different groups of records after each of the three earthquake events were then chosen to observe the different multiplets. Data selection for detecting each multiplet and relevant results are provided in Section Method EEMD was applied to those residual gravity time sequences g i (t)as a dyadic filter bank. EEMD was proposed to alleviate some unsettled issues of the empirical mode decomposition (EMD, Huang et al. 1998), such as the scale-mixing problem and the end effect (Huang &Wu2008), although EMD has demonstrated its applicability in a wide range of geoscience studies over the last 10 yr (Vasudevan & Cook 2000; Huang et al. 2001; Dwivedi & Mittal 2007; Huang & Wu 2008; Thomas et al. 2009; Jackson & Mound 2010;Chenet al. 2012; Shen&Wu2012). According to Huang & Wu (2008), EEMD is merely an improved EMD; thus, we will introduce the EMD procedure first, and then explain how to implement EEMD. A given time-series x(t) can be decomposed into several intrinsic mode functions (IMFs) using the following steps: (1) All local maxima and local minima of x(t) are identified, and the upper (lower) envelope of x(t) can be formed using a cubic spline line to connect all of the local maxima (minima). (2) h 1 (t) = x(t) m 1 (t) is used to obtain a new series, where m 1 (t) is the mean of the upper and lower envelopes of x(t). Steps 1 and 2 are called sifting procedures. (3) Generally, h 1 (t) is not an IMF, so h 1 (t) is treated as the data given just as x(t). The sifting procedures (steps 1 and 2) are then repeated k times until h k (t) isanimf,thatis,h 1k (t) = h 1(k-1) (t) m 1k (t). Let c 1 (t) = h 1k (t), which is designated as the first IMF. (4) Let r 1 (t) = x(t) c 1 (t), and then the steps 1 3 are repeated to obtain the second IMF, c 2 (t). (5) Steps 1 to 4 are repeated to obtain the j target IMF, c j (t). In these steps, two different iterative loops exist. The first loop is employed to obtain h k (t), in which an IMF must have a specific definition to stop circulating steps 1 and 2. In the second loop, the repeat times j must be a finite number, in which a criterion must be determined to stop the entire sifting process. For the first loop, Huang et al.(1998) limited the size of the standard deviation, SD k = T t=0 m 1k 2 / T t=0 h 1k 2, which was computed from the two consecutive sifting results to stop this loop. SD k can be generally set between 0.2 and 0.3 [the expression for SD in Huang et al. (1998) is not correct; and the correct one can be found in Huang &Wu(2008)]. The second loop can be terminated by either of the two criteria: (1) the component c n or the residue r n becomes so small that it is less than the previously given value of the substantial consequence; or (2) no more IMF can be extracted from r n. The above process is the EMD; more details can be found in Huang et al.(1998). However, given that EMD has the scale-mixing problem and end effects, EEMD was proposed. Based on EMD, EEMD is developed with the following aspects (Huang & Wu 2008): (1) A white noise series is added to the targeted time-series x(t); (2) The series with added white noise is decomposed into IMFs; (3) Steps 1 and 2 are repeated by iteration, but with different white noise series each time;

3 Observation of spheroidal normal mode 1633 Figure 1. The synthetic series g(t) (a) and its amplitude spectrum (b), (c) the synthetic series g 1 (t) (black bold curve) and the corresponding IMF3 (red dashed curve) and their amplitude spectra (d), (e) the synthetic series g 2 (t) (black bold curve) and the corresponding IMF2 (red dashed curve) and their amplitude spectra (f), (g) the synthetic series g 3 (t) (black bold curve) and the corresponding IMF1 (red dashed curve) and their amplitude spectra (h). (4) The (ensemble) means of the corresponding IMFs of the decompositions is obtained as the final IMF. In the decomposition using EEMD, the added white noise series cancel each other in the final mean of the corresponding IMFs. The means of the IMFs remains within the natural dyadic filter windows, significantly reducing the possibility of mode mixing and preserving the dyadic property (Huang & Wu 2008). Additional details about EEMD can be found in Huang & Wu (2008) andwu & Huang (2009). Using EEMD, each of the SG residual sequences g i (t) can be decomposed into a finite number of simple IMFs. Different scale signals in an SG residual sequence g i (t) are re-combined by proper IMFs based on the fact that different IMFs have different frequency bands. In addition, EEMD can be used to define the frequency modulation (Huang et al. 1998; Huang & Wu 2008). Thus, EEMD can demodulate a frequency-modulated time-series. Therefore, EEMD can be used as a dyadic filter bank and as a demodulator simultaneously. Given that EEMD has these two advantages, we use it to detect the splitting of the normal modes. First, we used a synthetic series g(t) consisting of three attenuated sine signals, namely, g i (t) = sin(2πf i t) exp( 10 5 t), where f 1 = 0.1 mhz, f 2 = 0.45 mhz and f 3 = 0.8 mhz, to illustrate the function of EEMD. The length of the synthetic series is 60 hr, and the sampling interval is 1 min. After EEMD, 10 IMFs were obtained, but only three of them (IMF1, IMF2 and IMF3) are the actual signals, which correspond to g 3 (t), g 2 (t)andg 1 (t), respectively. Their waveforms and the corresponding amplitude spectra are shown in Fig. 1. The results clearly show that EEMD yields the first three IMFs, which almost coincide with the corresponding original synthetic signals g i (t). To show the differences between the results obtained using EEMD and EMD, the SG record from the Strasbourg station (located at Strasbourg, France) after the 2004 Sumatra earthquake was selected, starting 5 hr after the earthquake with a length of min. This record is the original gravity data, that is, the tides, local atmospheric pressure effect and other factors were not removed. The selected IMFs obtained after using EEMD and EMD are drawn in Fig. 2. The results from EMD showed serious scalemixing (the dotted rectangles) and end effects (the dotted circles). The marked peaks (dotted rectangles) in IMF4 from EMD (Fig. 2c) should appear in IMF3 (Fig. 2b), whereas the marked peaks in IMF5 (Fig. 2d) should appear in IMF4 (Fig. 2c). This phenomenon is just the scale-mixing effects, and obviously, EEMD can effectively reduce such influences. In another example shown in Fig. 3, the SG residual sequence is obtained from the Strasbourg station, starting 5 hr after the 2004 Sumatra earthquake for 200 hr in length. After applying EEMD, 15 IMFs were obtained; only the (linear) power spectra of the first eight IMFs are shown in Fig. 3 because the frequencies of IMF9 to IMF15 are almost equal to zero, and we only consider the frequency band range 0.2 to 1 mhz of interest. For instance, 0 S 2 significantly

4 1634 W.-B. Shen and H. Ding Figure 2. The IMFs after using EEMD and EMD for the SG record obtained from the Strasbourg station after the 2004 Sumatra earthquake. (a) Original SG data without pre-treatment, and (b) (d) selected IMFs showing that EEMD can reduce the scale-mixing problem and end effects existing in the EMD results. The dotted circles indicate the end effects, whereas the dotted rectangles indicate the scale-mixing problem. appears in IMF5 only, 0 S 3 in IMF4 and IMF5 and 3 S 1 in IMF2 and IMF3. Thus, to observe the singlets of 0 S 2, IMF5 suffices as a new sequence, and to observe the singlets of 0 S 3,thesumofIMF4 and IMF5 can be used as a new sequence. In fact, taking a closer look at the IMF5 in Fig. 3, the five singlets of 0 S 2 have already been completely resolved. Note that the characteristics of EEMD are such that the total number of IMFs may be different if a different length of record is chosen (Huang et al. 1998). In practice, with regard to the frequencies of the target modes, certain IMF/IMFs can be selected as a new residual sequence d i (t) for further study. Moreover, we show the effects of EEMD on the amplitudes of a target multiplet. Based on the results in Fig. 3, we selected the multiplet 0 S 2 as an example (Fig. 4). 0 S 2 can be clearly identified in only IMF4 and IMF5, so we selected d(t) = IMF4 + IMF5 as a new series. The amplitude spectra of the st (Strasbourg, France) record (black curve) and d(t) (grey curve) are shown in Fig. 4. The amplitudes of each target peaks are denoted by the black and grey arrows. Results show that the amplitudes of the target peaks after using EEMD have significant differences from those obtained without using EEMD. The biggest deviation reaches up to nm s 2, which is a significant deviation, whereas the background noise level is approximately 0.02 nm s 2. In addition, the amplitudes of m = 0 and m =+1 after using EEMD are larger than those obtained without using EEMD, whereas the three other singlets exhibit opposite results. Therefore, it is worth noting that the procedures adopted in this study are not suitable for amplitude estimation. However, considering that the IMFs can be linearly added to each other, and according to our previous investigations, if four or five adjacent IMFs are selected for a target multiplet (only two IMFs were chosen for the six multiplets in this study), the amplitudes can be more accurately estimated. In this case, however, some kinds of noises, which we tried to remove to detect weak singlets, are simultaneously introduced. For this reason, here we only considered the relative amplitudes of the six target multiplets. Nevertheless, inasmuch as the amplitude information is critical to obtain core and mantle structure insights, the amplitude measurement based on EEMD is also very important. Concerning this interesting topic, the study of Xu et al. (2008) could be a valuable reference, and we will further investigate this subject in our future research work. After showing that the procedures adopted in this study are not suitable for amplitude estimation, we provide a simple example

5 Observation of spheroidal normal mode 1635 Figure 3. Power spectra of the first eight IMFs of the SG residual record obtained at the Strasbourg station after the 2004 Sumatra earthquake. IMF9 IMF15 are not plotted. Arrows indicate the corresponding modes. Figure 4. Amplitude spectra of 0 S 2 from the st record after the 2004 Sumatra earthquake. The black curve denotes the spectrum of the residual gravity series g(t), whereas the grey curve denotes the spectrum of the sum d(t) of the IMF4 and IMF5 modes of the residual gravity series g(t). The amplitudes of the peaks are denoted by the arrows.

6 1636 W.-B. Shen and H. Ding Figure 5. Power spectra of 0 S 3 obtained from a single record at the Sutherland station after the 2004 Sumatra earthquake. Grey area: results obtained using EEMD; black area: results obtained without using EEMD. After applying EEMD, the relative values of the product spectra are selected for comparison because the amplitude of a single IMF is smaller than that of the original residual record. The corresponding theoretical frequencies based on PREM-re are denoted by vertical dashed lines. to show that EEMD may significantly improve the SNR for some weak singlets, which are otherwise difficult to detect by conventional approaches because they are submerged in relatively strong environmental and instrumental noises. The grey and black areas in Fig. 5 indicate the results obtained using EEMD and without using EEMD, respectively. A residual gravity record from the Sutherland station was selected to observe the singlets of 0 S 3, starting from 5 hr after the 2004 Sumatra Figure 6. (a) (c) Product spectra of 0 S 2 obtained from the 2004 Sumatra earthquake (nine SG records), the 2010 Maule earthquake (seven SG records) and the 2011 Tohoku earthquake (seven SG records), respectively; (d) (f) product spectra of 0 S 3 obtained from the 2004 Sumatra earthquake (four SG records), the 2010 Maule earthquake (five SG records) and the 2011 Tohoku earthquake (six SG records), respectively. Grey area: results obtained using EEMD; black area: results obtained without using EEMD. Dashed vertical lines denote the corresponding PREM-re predictions.

7 Observation of spheroidal normal mode 1637 earthquake, with a length of 520 hr. Results show that EEMD significantly enhances the SNRs of the multiplet, particularly for the singlets m = 0andm =±1, demonstrating that EEMD is effective and reliable. In addition to EEMD and in order to further improve the SNR and accuracy of the results, the product spectrum analysis proposed by Smylie et al.(1993) was applied to the data set {d i (t)} obtained after using EEMD and the residual gravity data set {g i (t)} obtained without using EEMD in the frequency-domain. The validity of EEMD was also confirmed by comparing the product spectra of {d i (t)} and {g i (t)} with each other. In addition, we provide the estimated frequencies of all of the observed singlets, although the main objective of this study is to show that EEMD helps in detecting some weak modes. In this study, we used a Lorentz curve to fit the observed singlet spectrum to obtain the corresponding frequency and the bootstrap method (Efron & Tibshirani 1986) to estimate the error bars of the mode frequencies as suggested by Häfner & Widmer- Schnidrig (2013). First, the singlet parameters were estimated using the least square curve fitting. Second, the rms noise amplitude of a target singlet is computed in accordance with the method used by Häfner & Widmer-Schnidrig (2013). Third, the estimated parameters were used to generate a noise-free synthetic series, and then a random noise (whose amplitude is approximately equal to the rms noise amplitude of the target singlet) was added to obtain a noisecontaminated synthetic series. M different synthetic series can be constructed by adding M random noises (in this study, we set M to 300). Finally, each of the M noise-contaminated synthetic series can be used to estimate a singlet frequency, and the standard deviation was taken as the observation error σ of the singlet frequency. More details can be found in Efron & Tibshirani (1986) and Häfner & Widmer-Schnidrig (2013). For a given singlet s i that denotes one splitting singlet of one mode of the six mode sets ( 0 S 2, 2 S 1, 0 S 3, 0 S 4, 1 S 2 and 3 S 1 )after all of the three earthquakes, 1 / σ 2 was used as weight to obtain a weighted average, which was considered as the final estimated frequency based on a set of new M i sequences. Each new sequence consists of one or several IMF series, which may contain the target signals (see earlier description related to Fig. 2) based on an SG record series. In this study, we made the following correspondences: s i (i = 1, 2, 3, 4, 5) correspond to the splitting (m = 2, 1, 0, +1, +2) of 0 S 2, respectively, s i (i = 6, 7, 8) are the (m = 1, 0, +1) of 2 S 1, s i (i = 9, 10,...,15) are the (m = Table 1. Model predictions and observations of the splitting frequencies of 0 S 2 and 1 S 2 (unit: mhz). 3,...,+3) of 0 S 3, s i (i = 16,...,24) are the (m = 4,...,+4) of 0 S 4, s i (i = 25,...,29) are the (m = 2,...,+2) of 1 S 2,and s i (i = 30, 31, 32) are the (m = 1, 0, +1) of 3 S 1. For instance, the singlet s 16 denotes the singlet m = 4 of the mode 0 S 4.The weight for the ith observed singlet frequency from a set of new M i sequences is expressed as P i = (1 / / Mi σ 2 i ) 1 / σ 2 j, (1) j=1 For each singlet s i, M i is the number of the selected new series obtained after using the EEMD filter for the chosen SG records after the three events, σ i is the estimated error of the estimated frequency, and the denominator M i j=1 1/ σ j 2 is applied to make M i j=1 P j = 1. For each singlet s i, we used the following formula (Taylor 1997)to obtain a final estimate of the corresponding frequency, f si, based on a set of new M i sequences M i f si = P j f j, e( f si ) = M i (P j e( f s ( j) i )) 2, (2) j=1 j=1 where e( f si ) is the error of the estimate, P i is the weight given by eq. (1) and f s ( j) i and e( f s ( j) i ) are the estimates of the corresponding frequency value and its corresponding error bar based upon the new sequence d (si ) j( j = 1, 2,...,M i ). 3 RESULTS AND ANALYSIS 3.1 Splitting of 0 S 2, 0 S 3, 2 S 1 and 3 S 1 To observe the splitting of 0 S 2, the following records were selected: nine records from the bh (Bad Homburg, Germany; two records h1 and h2,), mb (Membach, Belgium), mo (Moxa, Germany; two records m1 and m2), st (Strasbourg, France), vi (Vienna, Austria), and we (Wettzell, Germany; two records w1 and w2) stations after the 2004 event; seven records from the bh (h2), cb (Canberra, Australia), mo (m2),pe (Pecny, Czech Republic), st, su (Sutherland, South Africa) and we (w1) stations after the 2010 event; and seven records from the bh (h3), bf (Schiltach, Germany; record b2 was used), cb, mb, mc (Medicina, Italy), st and tc (Concepcion, Chile) stations after the 2011 event. Each of the records starts 5 hr after the earthquake, with a length of 300 hr. The results are shown in Mode m = 2 m = 1 m = 0 m =+1 m =+2 PREM-re Buland et al. (1979) Rosat et al. (2005) ± 6.3e ± 4.7e ± 6.0e ± 1.1e ± 4.6e-6 Roult et al. (2006) ± 7.4e ± 2.9e ± 1.57e ± 6.9e ± 8.9e-5 0S 2 Rosat et al. (2008) ± 1.5e ± 1.6e ± 2.5e ± 2.6e ± 2.8e-7 Abd El-Gelil et al. (2010) ± 1.2e ± 1.1e ± 1.1e ± 0.5e ± 1.0e-6 Roult et al. (2010) ± 3.313e ± 4.985e ± 3.560e ± 4.480e ± 3.548e-4 Deuss et al. (2011) Rosat et al. (2012) a ± 2.2e ± 5.1e ± 3.3e ± 4.6e ± 2.1e-5 Häfner & Widmer-Schnidrig (2013) ± 9.0e ± 6.0e ± 1.6e ± 5.0e ± 9.0e-6 Ding & Shen (2013) ± 2.0e ± 6.2e ± 3.7e ± 5.5e ± 1.8e-5 This paper (weighted mean) ± 1.1e ± 7.8e ± 9.1e ± 7.7e ± 9.6e-6 PREM-re Roult et al. (2010) ± 3.853e ± 4.703e ± 4.201e ± 4.519e-4 1S 2 Ding & Shen (2013) ± 4.3e ± 1.4e ± 1.7e ± 2.3e ± 5.9e-5 This paper (weighted mean) ± 1.7e ± 9.0e ± 2.3e ± 1.4e ± 1.6e-5 a Correction to Rosat et al. (2005).

8 1638 W.-B. Shen and H. Ding Table 2. Model predictions and observations of the splitting frequencies of 0S3 (unit: mhz). 0S3 m = 3 m = 2 m = 1 m = 0 m =+1 m =+2 m =+3 PREM-re Buland et al. (1979) Rosat et al. (2005) ± 3.0e ± 2.0e ± 1.8e ± 1.1e ± 4.8e ± 2.1e-5 Roult et al. (2006) ± 8.7e ± 7.1e ± 8.3e ± 4.2e ± 5.2e ± 1.33e ± 4.68e-4 Rosat et al. (2008) ± 4.9e ± 4.9e ± 4.6e ± 3.8 e ± 6.9e ± 1.1 e-6 Abd El-Gelil et al. (2010) ± 0.4e ± 0.5e ± 0.2e ± 0.2e ± 0.3e ± 0.7e ± 2.1e-6 Roult et al. (2010) ± 3.515e ± 3.480e ± 4.036e ± 4.220e ± 1.760e ± 4.059e ± 1.786e-4 This paper (weighted mean) ± 1.8e ± 1.1e ± 2.6e ± 3.0e ± 2.2e ± 1.1e ± 1.9e-5 Figs 6(a) (c). Given that all of the six stations have high SNRs after the 2004 event, the results obtained using EEMD (grey area) and those obtained without using EEMD (black area) coincide with each other. However, after the 2010 Maule and 2011 Tohoku events, the singlet m = 0 can be only directly observed at a few SG stations, and m = 0 cannot be observed in the product spectra of the chosen records without using EEMD (Figs 6b and c, black areas). As a contrast, after using EEMD, the SNRs of m = 0 after the two events are significantly enhanced, and can be clearly observed (Figs 6b and c, grey areas). The weighted estimated frequencies of 0 S 2 after all of the three earthquakes are very close to the results of some previous studies and the PREM-re predictions (Roult et al. 2010), as shown in Table 1. To observe the splitting of 0 S 3, four records from cb, ma and su (s1, s2) after the 2004 event, five records from cb, mo (m2), pe, st and we (w2) after the 2010 event, and six records from ap (Apache Point, USA), bh (h3), cb, mc, st and tc after the 2011 event were chosen. Each of the chosen records started 5 hr after the earthquake, with a length of 520 hr. The results of the direct product spectrum of the records after the 2004 event are compatible with those of Rosat et al. (2005). The SNR of the singlet m = 0isverylow(Fig.6d). However, after applying EEMD, all of the seven singlets of 0 S 3 are completely observed with high SNRs, as shown by the grey areas in Fig. 6e. The singlet m = 0 after the 2010 and 2011 events can neither be observed in their corresponding product spectra, but after using the EEMD filter, it is clearly observed in the two corresponding product spectra (Figs 6e and f). The estimated frequencies of these two product spectra are also very close to the results of some previous studies (Table 2). Given that the excited amplitudes of a multiplet depend on the focal mechanism and geographical locations of different earthquakes, the m = 0 singlets of 0 S 2 and 0 S 3 are found in the spectra after the 2004 event, whereas they are not found in the spectra after the 2010 and 2011 events (see Figs 6b c and e f). However, this phenomenon does not mean that no signals for the m = 0 singlets of 0 S 2 and 0 S 3 exist in the SG records after the 2010 and 2011 events; these signals are only seriously submerged in the background noise. Thus, by using a method that can enhance their SNR, such as the EEMD method, such signals may be found in the spectra. In fact, the mentioned signals excited by the 2010 and 2011 events were found in the results of Chao & Ding (2013) based on a new detection method, which also supports our present results. To observe the splitting of 2 S 1, seven records from bh (h1 and h2), cb, mo (m1), st, we (w1 and w2) after the 2004 event, four records from bh (h2), cb, mo (m1), mb after the 2010 event and three records from bh (h5), cb, st after the 2011 event were chosen, and each record starts 5 hr after the earthquake with a length of 168 hr. Figs 7(a) (c) show the spectra obtained after the three events, respectively. After the 2004 event, the SNRs of the singlets m = 1andm = 0are too low to observe without using EEMD (Fig. 7a, black area). This result is consistent with that in previous studies (e.g. Park et al. 2005; Rosat et al. 2005). However, after the EEMD filtering, the SNRs of m = 1andm = 0 are significantly enhanced, and two relatively clear signals are found compared with the surrounding peaks (grey areas in Figs 7b and c). Unlike the 2004 event, both the 2010 and 2011 events excited clear m = 0 signals, but the SNR of m = 1 was low. In addition, by using SG records with EEMD, the splitting of peaks of the toroidal mode 0 T 2 can be more clearly observed (denoted by the vertical arrows in Figs 7a c). Widmer- Schnidrig & Laske (2007, fig. 7) observed at least three 0 T 2 splitting peaks using the strainmeter record at BFO station. Using the broadband seismometer records, Roult et al.(2010, fig. 13) observed five

9 Observation of spheroidal normal mode 1639 Figure 7. Product spectra of 2 S 1 (a) (c) and 3 S 1 (d) (f), with the same distribution as those in Fig. 6. Eight and five SG records after the 2004 event, four and five SG records after the 2010 event, and three and nine SG records after the 2011 event were chosen for 2 S 1 and 3 S 1, respectively. Dashed vertical lines denote the corresponding PREM-re predictions of 2 S 1, 3 S 1 and the 0 T 2 mode (m = 0); and the vertical arrows denote the splitting peaks of 0 T 2. 0T 2 singlets. However, detailed detection of the splitting of 0 T 2 is beyond the scope of the current study. For 3 S 1, five records from bh (h1), cb, mo (m1), su (s1 and s2) after the 2004 event, another five records from bh (h1 and h3), cb, mo (m2), st after the 2010 event and nine records from ap, bf (b1 and b2), bh (h3 and h4), cb, mo (m1), st, we (w4) after the 2011 event were chosen. Each of these records starts 50 hr after the earthquake with a length of 680 hr (to weaken the interference of 1 S 3, as suggested by Masters & Gilbert 1983; choosing 50 hr after the event as the starting time is a trade-off between the SNR and the cross-coupling effect). For the 2004 event, only the singlets m =±1 are found in the direct product spectrum of the five records, whereas the singlet m = 0 is submerged in the noise (Fig. 7d, black area). These observations are consistent with previous results (Roult et al. 2010). However, after applying EEMD, the SNR of m = 0 is significantly improved, and all of the three singlets are clearly observed in the product spectrum (Fig. 7d, grey areas). For the 2010 and 2011 events, both of them excited clear triplets of 3 S 1 ; after using EEMD, Table 3. Model predictions and observations of the splitting frequencies of 2 S 1 and 3 S 1 (unit: mhz). Mode m = 1 m = 0 m =+1 PREM-re Rosat et al.(2003) ± 1.9e ± 2.1e ± 1.8e-4 Rosat et al.(2005) ± 6.0e ± 4.2e-5 2S 1 Roult et al. (2006) Roult et al. (2010) ± 2.543e ± 1.352e ± 1.012e-3 Deuss et al.(2011) Ding & Shen (2013) ± 1.2e ± 5.5e-5 This paper (weighted mean) ± 4.3e ± 3.9e ± 1.9e-5 PREM-re Chao & Gilbert (1980) ± 5.5e ± 9.0e ± 4.0e-5 3S 1 Roult et al. (2010) ± 1.241e ± 3.444e ± 1.493e-4 Shen & Wu (2012) ± 4.20e ± 2.65e ± 2.13e-4 Ding & Shen (2013) ± 2.2e ± 5.1e ± 1.8e-5 This paper (weighted mean) ± 1.1e ± 8.7e ± 1.1e-5

10 1640 W.-B. Shen and H. Ding Figure 8. Product spectra of 0 S 4 (a) (c) and 1 S 2 (d) (f), with the same distribution as those in Fig. 6. One and seven SG records after the 2004 event, four and three SG records after the 2010 event, and three and eight SG records after the 2011 event were chosen for 0 S 4 and 1 S 2, respectively. their SNRs are slightly enhanced. The estimated frequencies of the singlets m =±1 are very close to the observations of Roult et al. (2010) and Shen& Wu(2012). However, the estimated frequencies of m = 0 after the three events are slightly larger than the theoretical predictions of PREM-re and the results of some previous studies (Table 3). 3.2 Splitting of 0 S 4 and 1 S 2 For the 0 S 4 multiplet, considering the trade-off between the SNR and the frequency resolution, only one SG record from bh (h2) after the 2004 event, four SG records from bh (h2), cb, mo (m2), we after the 2010 event and three records from bf (b2), Conrad (co), and st after the 2011 event are chosen. Each record starts 5 hr after the earthquake and has a length of 750 hr. The direct product spectra of 0 S 4 (black areas in Figs 8a c) show that some singlets of 0 S 4 are coupled with each other; thus, they cannot be identified easily from the direct product spectra (such as the singlets m =+2, +3 and +4). However, after applying EEMD, nine peaks are completely resolved and clearly observed (grey areas in Figs 8a c) from all of the three events. Although the estimated frequency set is different from the corresponding theoretical predictions of PREM-re and the observed values of Roult et al. (2010) (in their study only the singlets m =±4 were observed based on more than 50 broadband seismometer records after the 2004 Sumatra earthquake), the estimated frequencies are consistent with each other (Figs 8a c), as shown in Table 4. In summary, we suggest that by using the EEMD technique all of the singlets of 0 S 4 are clearly observed. To observe the singlets of 1 S 2, we selected four SG records from cb, bh (h1 and h2), mo (m1), mb, we (w1 and w2) after the 2004 event, three records from bh (h3), cb, we (w1) after the 2010 event and eight records from bf (b1 and b2), bh (h4 and h5), cb, mo (m1), st, we (w4) after the 2011 event. Each of the records starts 5 hr after the earthquake with a length of 350 hr. All of the three events excited the observable singlets of 1 S 2, but the singlets m =+1, +2 after the 2004 earthquake were coupled with each other. Consequently, they Table 4. Model predictions and observations of the splitting frequencies of 0 S 4 (unit: mhz). Modes Singlet (m) PREM-re Roult et al. (2010) This paper (weighted mean) ± 2.562e ± 4.5e ± 2.0e ± 1.7e ± 3.8e-5 0S ± 3.1e ± 3.7e ± 1.3e ± 1.7e ± 1.908e ± 4.0e-5

11 cannot be identified from the direct product spectra (Fig. 8d, black areas). In addition, by product spectrum, the singlet m = 0 after the 2010 event cannot be found (Fig. 8e, black areas). However, after applying EEMD, all of the five singlets are clearly observed (grey area in Figs 8d f). The estimated frequencies are very close to the results of Roult et al. (2010) and Ding & Shen (2013). By comparing Figs 6 8 with Tables 1 4, we found that the SNR of a singlet correlates negatively with the error of the corresponding estimated frequency, which is consistent with the conclusion of Häfner & Widmer-Schnidrig (2013). In addition, some distinct differences are found between our new result sets (estimated frequencies) of the six modes and their corresponding PREM-re predictions. These deserve further confirmation upon using other data sets or methods. 4 CONCLUSION Comparisons of the product spectrum results obtained from direct residual gravity records and those obtained after using EEMD show that EEMD can significantly enhance the SNR of some weak lowfrequency seismic modes. By applying EEMD, all of the singlets of 0S 2, 0 S 3, 0 S 4, 1 S 2, 2 S 1 and 3 S 1 are completely resolved with high SNRs. Except for the frequency estimates of 0 S 4, the estimates for the five other multiplets show no significant differences from the PREM-re predictions and the results of previous studies, because the eigenfrequencies of those multiplets of the real Earth possess unique values that do not depend on different seismic events. However, our results, wherein all of the singlets of the six target normal mode multiplets are identified, are obtained based on fewer records. Though the classic multiplet stripping method can provide both frequency and amplitude information, it strongly depends on source receiver geometry and numerous records from many stations; it may also require multiple events to resolve singlets for multiplets of low angular orders. Thus, although the proposed process based on the EEMD method may not be suitable in estimating the amplitudes of the multiplets, the proposed method may be effective in estimating the multiplet frequencies. Combining EEMD with other methods to determine the amplitudes of some weak signals needs further study. In view of the improved resolutions of the six modes obtained in this study, the EEMD method may be important in the detection of the splitting of some weak and low-frequency seismic modes, and it can be also a potential technique in detecting many other spheroidal modes. ACKNOWLEDGEMENTS The authors are grateful to the GGP station managers for releasing high-quality SG records, as well as to the editor-in-chief Jeannot Trampert, the reviewer Joseph Resovsky, and an anonymous reviewer for their valuable comments and suggestions, which significantly improved the manuscript. The authors also sincerely thank Jim Ray, who smoothed the English expressions. This study is supported by the NSFC (Grant No ), the 973 Project (Grant No. 2013CB733305), the NSFC (Grant Nos , , , and ) and the Fundamental Research Funds for the Central Universities (Grant No ). REFERENCES Abd El-Gelil, M., Pagiatakis, S. & El-Rabbany, A., Normal mode detection and splitting after Sumatra Andaman earthquake, J. Geodyn., 50, Observation of spheroidal normal mode 1641 Buland, R., Berger, J. & Gilbert, F., Observations from the IDA network of attenuation and splitting during a recent earthquake, Nature, 277, Chao, B.F. & Ding, H., Fourier-transformed autoregressive spectrum: in search of harmonic signals in a time series, submitted. Chao, B.F. & Gilbert, F., Autoregressive estimation of complex eigenfrequencies in low frequency seismic spectra, Geophys. J. R. Astron. Soc., 63, Chen, J., Heincke, B., Jegen, M. & Moorkamp, M., Using empirical mode decomposition to process marine magnetotelluric data, Geophys. J. Int., 190(1), Courtier, N. et al., Global superconducting gravimeter observations and the search for the translational modes of the inner core, Phys. Earth planet. Inter., 117, Crossley, D. & Hinderer, J., 2008a. Report of GGP activities to commission 3, completing 10 years for the worldwide network of superconducting gravimeters (A), in International Association of Geodesy Symposia: Observing Our Changing Earth, Vol. 133(3), pp , ed. Sideris, M.G., Springer-Verlag. Crossley, D. & Hinderer, J., 2008b. The contribution of GGP superconducting gravimeters to GGOS (A), in International Association of Geodesy Symposia: Observing Our Changing Earth, Vol. 133(3), pp , ed. Sideris, M.G., Springer-Verlag. Crossley, D., Rochester, M.G. & Peng, Z.R., Slichter modes and Love numbers, Geophys. Res. Lett., 91, Dahlen, F.A., The effect of data windows on the estimation of free oscillations parameters, Geophys. J. R. Astron. Soc., 69, Dahlen, F.A. & Tromp, J., Theoretical Global Seismology, Princeton Univ. Press, pp Deuss, A., Ritsema, J. & Heijst, H., Splitting function measurements for Earth s longest period normal modes using recent large earthquakes, Geophys. Res. Lett., 38, L04303, doi: /2010gl Deuss, A., Ritsema, J. & van Heijst, H., A new catalogue of normalmode splitting function measurements up to 10 mhz, Geophys. J. Int., 193, Ding, H. & Shen, W.B., Search for the Slichter modes based on a new method: optimal sequence estimation, J. geophys. Res., 118, doi: /jgrb Dwivedi, S. & Mittal, A.K., Forecasting the duration of active and break spells in intrinsic mode functions of Indian monsoon intraseasonal oscillations, Geophys. Res. Lett., 34, L16827, doi: /2007gl Dziewonski, A.M. & Anderson, D.L., Preliminary reference earth model, Phys. Earth planet. Inter., 25, Efron, B. & Tibshirani, R., Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Stat. Sci., 1, Fukao, Y. & Sudo, N., Core modes of the Earth s free oscillations and structure of the inner core, Geophys. Res. Lett., 16(5), Gilbert, F., The diagonal sum rule and averaged eigenfrequencies, Geophys. J. R. Astron. Soc., 23, Guo, J.Y., Dierks, O., Neumeyer, J. & Shum, C.K., Weighting algorithms to stack superconducting gravimeter data for the potential detection of the Slichter modes, J. Geodyn., 41, Häfner, R. & Widmer-Schnidrig, R., Signature of 3-D density structure in spectra of the spheroidal free oscillation 0 S 2, Geophys. J. Int., 192(1), He, X. & Tromp, J., Normal-mode constraints on the structure of the Earth, J. Geophys. Earth, 110(B9), Hinderer, J., Crossley, D. & Jensen, O., A search for the Slichter triplet in superconducting gravimeter data, Phys. Earth planet. Inter., 90, Hu, X.G., Liu, L.T., Hinderer, J., Hsu, H.T. & Sun, H.P., Wavelet filter analysis of atmospheric pressure effects in the long-period seismic mode band, Phys. Earth planet. Inter., 154, Huang, N.E. & Wu, Z., A review on Hilbert-Huang transform: method and its applications to geophysical studies, Rev. Geophys., 46, RG2006, doi: /2007rg

12 1642 W.-B. Shen and H. Ding Huang, N.E. et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. Roy. Soc. Lond., A454, Huang, N.E., Chern, C.C., Huang, K., Long, L.W.S.S.R. & Fan, K.L., A new spectral representation of earth-quake data: Hilbert spectral analysis of station TCU129, Chi-Chi, Taiwan, 21 September 1999, Bull. seism. Soc. Am., 91, Jackson, L.P. & Mound, J.E., Geomagnetic variation on decadal time scales: what can we learn from empirical mode decomposition? Geophys. Res. Lett., 37, L14307, doi: /2010gl Lognonné, P., Normal modes and seismograms in an anelastic rotating Earth, J. geophys. Res., 96, Lognonné, P. & Clévédé, E., Normal modes of the earth and planets, in International Geophysical Series 81A: Handbook on Earthquakes and Engineering Seismology, pp , eds Kanamori, H., Jennings, P. & Lee, W., Boulder. Masters, G. & Gilbert, F., Attenuation in the earth at low frequencies, Phil. Trans. R. Soc. Lond., A308, Millot-Langet, R., Clévédé, E. & Lognonné, P., Normal modes and long period seismograms in a 3D anelastic elliptical rotating Earth, Geophys. Res. Lett., 30, Okal, E.A. & Stein, S., Observations of ultra-long period normal modes from the 2004 Sumatra-Andaman earthquake, Phys. Earth planet. Inter., 175, Park, J. et al., Earth s free oscillations excited by the 26 December 2004 Sumatra-Andaman earthquake, Science, 308, Resovsky, J.S. & Ritzwoller, M.H., New and refined constraints on 3-D earth structure from normal modes below 3 mhz, J. geophys. Res., 103, Ritzwoller, M., Masters, G. & Gilbert, F., Observations of anomalous splitting and their interpretation in terms of aspherical structure, J. geophys. Res., 91(B10), Rosat, S., Hinderer, J. & Rivera, L., First observation of 2 S 1 and study of the splitting of the football mode 0 S 2 after the June 2001 Peru earthquake of magnitude 8.4, Geophys. Res. Lett., 30, Rosat, S., Sato, T., Imanishi, Y., Hinderer, J., Tamura, Y., McQueen, H. & Ohashi, M., High resolution analysis of the gravest seismic normal modes after the 2004 M w = 9 Sumatra earthquake using superconducting gravimeter data, Geophys. Res. Lett., 32, L13304, doi: /2005gl Rosat, S., Fukushima, T., Sato, T. & Tamura, Y., Application of a non-linear damped harmonic analysis method to the normal modes of the Earth, J. Geodyn., 45, Rosat, S., Sato, T., Imanishi, Y., Hinderer, J., Tamura, Y., McQueen, H. & Ohashi, M., Correction to High-resolution analysis of the gravest seismic normal modes after the 2004 M w = 9 Sumatra earthquake using superconducting gravimeter data, Geophys. Res. Lett., 39, L22601, doi: /2012gl Roult, G., Rosat, S., Clévédé, E., Millot-Langet, R. & Hinderer, J., New determinations of Q quality factors and eigenfrequencies for the whole set of singlets of the Earth s normal modes 0 S 0, 0 S 2, 0 S 3 and 2S 1 using superconducting gravimeter data from the GGP network, J. Geodyn., 41, Roult, G., Roch, J. & Clévédé, E., Observation of split modes from the 26th December 2004 Sumatra-Andaman mega-event, Phys. Earth planet. Inter., 179, Shen, W.B. & Wu, B., A case study of detecting the triplet of 3 S 1 using superconducting gravimeter records with an alternative data preprocessing technique, Ann. Geophys., 55, Smylie, D.E., The inner core translational triplet and the density near Earth s center, Science, 255, Smylie, D.E., Hinderer, J., Richter, B. & Ducarme, B., The product spectra of gravity and barometric pressure in Europe, Phys. Earth planet. Inter., 80, Taylor, J.R., An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd edn, University Science Books, pp Thomas, E.R., Dennis, P.F., Bracegirdle, T.J. & Franzke, C., Ice core evidence for significant 100-year regional warming on the Antarctic Peninsula, Geophys. Res. Lett., 36, L20704, doi: / 2009GL Vasudevan, K. & Cook, F.A., Empirical mode skeletonization of deep crustal seismic data: theory and applications, J. geophys. Res., 105, Widmer-Schnidrig, R., What can superconducting gravimeters contribute to normal mode seismology? Bull. seism. Soc. Am., 93(3), Widmer-Schnidrig, R. & Laske, G., Theory and observations normal modes and surface wave measurements, in Treatise on Geophysics, Vol. 3, pp , ed. Schubert, G., Elsevier. Wu, Z.H. & Huang, N.E., Ensemble empirical mode decomposition: a noise-assisted data analysis method, Adv. Adapt. Data Anal., 1(1), Xu, Y., Crossley, D. & Herrmann, R.B., Amplitude and Q of 0 S 0 from the Sumatra earthquake as recorded on superconducting gravimeters and seismometers, Seis. Res. Lett., 79(6),