Analysis of normal-mode amplitudes excited by the Chilean earthquake of 27 February, 2010, shows that theoretical amplitudes computed for the Global Centroid Moment Tensor solution (GCMT) underestimate observation for frequencies below 1 mHz. Data are systematically larger and there is a hint that this ratio increases toward lower frequencies. Amplitude ratios for the three gravest modes, 0S2, 0S3 and 0S4, are 1.36, 1.29 and 1.22 respectively, with standard error (1σ) being about 0.1 for each of them. The most natural explanation for this observation is the occurrence of a slow afterslip. Assuming that this afterslip had occurred on a similar fault with dip angle 18 degrees, the moment of this afterslip is estimated to be 0.26 (26 percent) of the GCMT solution and the rise time of about 170 seconds.
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 The 2010 Mw 8.8 Chilean earthquake that occurred on 27 February was the fifth largest earthquake since 1900. Locations of aftershocks suggest that the main rupture was bounded by the great 1960 Mw 9.5 Chile earthquake to the south and the 1985 Mw 8.0 Valparaiso earthquake to the north. It was apparently a repeat of the 1835 earthquake, reported by Darwin .
 We present the case that the reported Global Centroid Moment Tensor solution (hereafter GCMT) [e.g., Dziewonski and Woodhouse, 1983; Ekström et al., 2005] (also http://www.globalcmt.org/ by Ekström and Nettles) is too small to explain normal-mode amplitudes below 1 mHz. Amplitude ratios between data and theory for GCMT (data/theory) are 1.2–1.4 for low-frequency modes 0S2−0S4 and asymptotically approaches 1 at about 2 mHz. The GCMT solution fits data well above 2 mHz. The amplitude ratio for 0S2 is 1.36 and is smaller than the value (2.5) reported for the 2004 Sumatra-Andaman earthquake [Stein and Okal, 2005; Park et al., 2005; Tsai et al., 2005; Park et al., 2008] but the deviation from 1 is still statistically significant as the standard error (1σ) is about 0.1.
 The most natural explanation for this observation is the occurrence of afterslip. The main subject of this paper will be determination of the size of this afterslip and its duration.
2. Normal Mode Amplitudes
 We computed the Fourier spectral amplitudes of more than 100 Federation of Digital Seismographic Network (FDSN) stations using the time window from 6 hours to 2–3 days after the rupture initiation. Figure 1 shows Fourier spectral amplitudes from stations BFO (Black Forest, Germany) and SUR (Sutherland, South Africa). In each plot, we show vertical-component spectral amplitudes (red) and synthetic spectral amplitudes (blue) computed for the Global Centroid Moment tensor solution (GCMT).
 There are twenty spectral peaks in the frequency band 0.2–1.8 mHz (Figure 1). Some peaks are simple multiplets such as 0S2 and 0S3 and others consist of more than one multiplet such as 1S6 + 2S5. For simplicity, we refer to each peak as a multiplet in this paper.
 If we plot the amplitude ratios for each peak (data/theory) for high-quality data, we get the results in Figure 2 (top). Each circle in Figure 2 represents a ratio for a multiplet measured at a station and plotted at the central frequency of each multiplet. Only the data with the signal-to-noise ratio (S/N) larger than 5 are plotted. A ratio of 1 means that the GCMT solution matches the observed amplitude. For the calculations of the S/N ratios, noise was estimated from the frequency interval between 0.5 and 0.6 mHz where we do not find any significant signals for this event (Figure 1).
 For the computation of modal-amplitude ratios, we integrated each Fourier spectral amplitude within a narrow frequency band for each multiplet and computed a ratio of integrated amplitudes between the observed and theoretical spectra. Specifically, we first applied the Hanning window to the time series to obtain the Fourier spectra F(ω). We then integrated the absolute number of this spectra ∣F(ω)∣ over each narrow frequency band for data and synthetic seismograms before taking the ratio between them. This approach gave us much more stable results than taking the ratios at each frequency and computing the averages of such ratios. Note that the effects from small frequency shifts for modal peaks, which may result from a three-dimensional earth model, can be mostly avoided by this approach.
 Ratios in Figure 2 (top) show some scatter but there is clearly evidence for deviation from 1 for low-frequency modes such as 0S2, 0S3 and 0S4. The mean and error bars for the amplitude ratios are shown in Figure 2 (bottom). The mean and error bars were computed with regular formulas after selection of data by the S/N criterion (larger than 5). The mean for the above three multiplets are 1.36, 1.29 and 1.22 respectively, and there seems to be a trend that these ratios increase toward lower frequencies.
 In Figure 2 (bottom), error bars are typically large for multiplets with frequencies higher than 1 mHz. Three multiplets show particularly large error bars; they are 0S8 (1.40–1.42 mHz), 1S6 + 2S5 (1.51–1.53 mHz), and 0S10 (1.71–1.73 mHz). For other events, we have noted that error bars for the multiplet 0S9 (1.56–1.59 mHz) also become large but that is not obvious for this earthquake.
 Synthetic spectra were computed by using the self-coupling scheme, including the rotation, ellipticity and three-dimensional structure effects through the splitting functions [e.g., Giardini et al., 1988]. Large scatter (Figure 2, top) and large error bars (Figure 2, bottom) are most likely caused by the effects of multiplet-multiplet couplings that are not included in this scheme. In the case of the multiplet pair denoted by 1S6 + 2S5, there is a systematic shift toward a higher ratio. This tendency was also confirmed for data from other earthquakes. As this ratio seems inconsistent with the frequency trend created by other multiplets, we drop this multiplet from further analyses. For this multiplet, there is probably a change in attenuation parameter, possibly caused by coupling to other multiplets. But in this paper we will not investigate further on this point.
 The multiplets that contain 0S8 and 0S10 show large scatter in Figure 2 probably because of the famous toroidal-spheroidal mode couplings due to the Earth's rotation [Masters et al., 1983; Park and Gilbert, 1986; Zürn et al., 2000]. Examination of these modes with other large earthquakes showed similar effects. However, the mean values for these multiplets do not show much deviation from the trend of amplitude as a function of frequency. Therefore, we kept these multiplets in our analysis. We kept most other multiplets in Figure 2 in our analysis except 1S2. This multiplet was dropped because of scarcity of good S/N measurements. In summary, we used amplitude ratios for 18 different multiplet pairs for further analysis, dropping two multiplet pairs 1S6 + 2S5 and 1S2 in Figure 2.
 It has been pointed out that theoretical amplitudes by the self-coupling scheme could deviate from the correct results by a few tens of percent to 50 percent in comparison to more rigorous coupled-mode calculation schemes [Deuss and Woodhouse, 2001, 2004; Resovsky and Ritzwoller, 1998]. We believe this is precisely the reason we see large scatter for modes above 1 mHz in Figure 2 (top). It is important to stress, however, that the estimates for the mean in Figure 2 (bottom) show only small bias for the mean values.
Deuss and Woodhouse  showed an important systematic trend from the along-branch couplings of spheroidal modes. However, they also showed that such effects were not important for low-frequency modes. The frequency range of our analysis happens to be this low-frequency range. This also lends support for using the self-coupling scheme for the frequency range 0.2–1.8 mHz.
 There are a few attractive features in the self-coupling scheme. First, it is relatively simple to code. Second, it is relatively easy to examine many different earth models through different splitting coefficients at a relatively modest computational cost. The fully-coupled calculations are restricted to a chosen earth model, thus costly if multiple models are to be examined. We examined the splitting coefficents obtained by Giardini et al. , He and Tromp , and others summarized by Dr. Gabi Laske in her website (http://igppweb.ucsd.edu/ gabi/ rem.dir/surface/rem.surf.html). While the plots in this paper are the results computed with the coefficients by He and Tromp  (with some modifications to modal Qs according to Laske's table), we confirmed that the main claim of this study, systematically larger amplitude ratios than 1, is independent of a choice of splitting coefficients and attenuation parameters from different models.
3. Search for the Best-Fit Afterslip
 Using the amplitude ratios from 18 multiplets, we seek to obtain an afterslip model that can explain the data in Figure 2. We basically assume two sources, one fast-rupture solution that explains most of the data above 2 mHz and an additional solution that can match normal-mode data when combined with the first source.
 We adopted the GCMT solution as the fast rupture because it explains higher-frequency surface wave data quite well. In the following analysis, we simply assume that an afterslip immediately followed this fast rupture (Figure 3, top). We define the time duration of this afterslip as τ while we refer to the rupture duration in the GCMT solution as τCMT. We also parameterize the moment of this afterslip by a factor a where a × Mo is the moment of afterslip and Mo is the moment of the GCMT solution (1.84 × 1022N · m). We varied the two parameters a and τ to search for the best fitting afterslip model.
Figure 3 (bottom) shows the variations of misfits (variances) as a function of a (horizontal axis) and τ (vertical axis). The range for a is between 0.1 and 0.4 (unit is Mo, the moment of GCMT) and the range for τ is from 10 to 400 seconds. In a search for wider parameter range, we confirmed that the best fit solution in Figure 3 (bottom) is also the minimum for the range 0 ≤ a ≤ 10 and 10 ≤ τ ≤ 1000. The variance was computed by ∑i (log ξi)2, where ξi is the modal amplitude ratio for the i-th datum (a multiplet for each station).
 In making these plots, we assumed that the afterslip occurred on a plane that is similar to the GCMT solution. The global minimum solution is indicated by the plus sign in Figure 3 (bottom) and is at (a = 0.26, τ = 170 sec). Figure 3 (bottom) shows a permissible range for a and τ from the region of small variances; we summarize that the range for the parameter a is approximately from 0.2 to 0.3 and that for the source duration τ is 140–200 seconds.
 In Figure 4, we show the comparison of amplitude ratios for the GCMT solution (blue) and the two-source solution that includes the afterslip with a = 0.26 and τ = 170 sec (red). For this calculation, normal-mode amplitude ratios were computed for each station and the mean and variances were computed for each multiplet. There are some scatter but the two-source solution shows much better fit than the GCMT solution.
4. Discussion and Conclusion
 The main point of this paper is that the low-frequency normal-mode amplitude data for the 2010 Chilean earthquake show evidence for afterslip. The GCMT solution explains high-frequency surface wave data above 2 mHz quite well but fails to explain normal-mode amplitude data below 2 mHz; there is a systematic trend in this discrepancy that is best explained by an afterslip that has the moment of 0.26 times the moment of the GCMT solution and has a duration of about 170 sec.
 Our claim critically rests on systematic deviation of amplitude ratios, especially for the gravest three modes (0S2, 0S3 and 0S4). If there are instrument response problems at low frequencies, some questions may be raised for the legitimacy of our claim. However, we analyzed both STS-1 and GS-54000 instruments and observed quite consistent behaviors for the amplitude ratios. In terms of good S/N data, we had, 16, 29 and 30 good data for 0S2, 3S3 and 0S4, respectively, and the ratios showed mostly consistent values (Figure 2, top). Therefore, even though our claim is marginally significant at 2-sigma standard error range, it carries some validity. Amplitude ratios for other events about M 8 show much closer numbers to 1. For smaller events, however, we start to lose good signals for 0S2 because of excitation problem and therefore estimates for error bars also become larger.
 Our choice of dip angle (18 degrees) is partly because of the CMT solution but also because the aftershock distribution shows a dip angle very close to it. Choice of dip angle for the afterslip (other than 18 degrees) directly influences the estimate a = 0.26. Seismicity data show that the dip angle of subducting plate increases from 18 degrees at shallow depth to about 30 degrees. If the deeper part of the subduction zone slipped as an afterslip, then the moment could be about 0.16. This is because seismic displacement is approximately proportional to Mosin δ rather than Mo alone, where Mo is the moment and δ is the dip angle [Kanamori and Given, 1981]. However, whether an afterslip occurred at a shallow depth or at a deeper depth cannot be deciphered from normal mode data.
 Uncertainties in attenuation parameters and 3D seismic velocity structure and effects of multiplet-multiplet couplings may cause some modifications to this afterslip model. We believe our analysis should be improved with further analysis. But the systematic deviation in amplitude ratios from 1 by the GCMT solution seems sufficiently robust that more rigorous calculations would not change our main conclusion very much.
 We thank two reviewers, Professor Jefferey Park and an anonymous reviewer, for constructive comments. We also thank Professors Ralph Archuleta and Paul Davis for the discussion. This work would not have been possible without the excellent data provided by IRIS, especially its Global Seismic Network component (GSN) and the Data Management System (DMS). We also appreciate data from other networks that participate in the Federation of the Digital Seismograph Network (FDSN).