Inner core rotation from event-pair analysis

Earth and Planetary Science Letters 261 (2007) 259 266 www.elsevier.com/locate/epsl Inner core rotation from event-pair analysis Xiaodong Song a,, Georges Poupinet b a Department of Geology, University of Illinois, Urbana, IL 61801, United States b LGIT-CNRS, Universite Joseph Fourier, 38041 Grenoble, France Received 21 April 2007; received in revised form 18 June 2007; accepted 29 June 2007 Available online 19 July 2007 Editor: R.D. van der Hilst Abstract The last decade has witnessed an animated debate on whether the inner core rotation is a fact or an artifact. Here we examine the temporal change of inner core waves using a technique that compares differential travel times at the same station but between two events. The method does not require precise knowledge of earthquake locations and earth models. The pairing of the events creates a large data set for the application of statistical tools. Using measurements from 87 events in the South Sandwich Islands recorded at College, Alaska station, we conclude the temporal change is robust. The estimates of the temporal change range from about 0.07 to 0.10 s/decade over the past 50 yr. If we used only pairs with small inter-event distances, which reduce the influence of mantle heterogeneity, the rates range from 0.084 to 0.098 s/decade, nearly identical to the rate inferred by Zhang et al. [Zhang, J., Song, X.D., Li, Y.C., Richards, P.G., Sun, X.L., Waldhauser, F., Inner core differential motion confirmed by earthquake waveform doublets, Science 309 (5739) (2005) 1357 1360.] from waveform doublets. The rate of the DF change seems to change with time, which may be explained by lateral variation of the inner core structure or the change in rotation rate on decadal time scale. 2007 Elsevier B.V. All rights reserved. Keywords: inner core rotation; PKP differential times; bootstrap 1. Introduction The electromagnetic torque from the geodynamo is likely to drive the solid inner core to rotate relative to the mantle (e.g., Glatzmaier and Roberts, 1995). Since the first observation of the inner core rotation from seismology (Song and Richards, 1996), many studies have given support for an inner core motion, including most recent ones (Zhang et al., 2005; Wen, 2006; Cao et al., 2007). The last decade has also witnessed an animated debate on whether the inner core rotation is real, because Corresponding author. E-mail address: xsong@uiuc.edu (X. Song). of potential biases such as earthquake mislocation (e.g., Poupinet and Souriau, 2001), failure to detect the motion (e.g., Souriau, 1998), and discrepancy in the inferred rotation rate (e.g., Creager, 1997; Vidale et al., 2000; Laske and Masters, 2003). Recent effort attempted to use earthquake waveform doublets to study the inner core rotation, which are pairs of earthquakes that have identical waveforms recorded the same station, indicating close spatial locations of the two events (Li and Richards, 1999; Poupinet et al., 2000). Zhang et al. (2005) has presented perhaps the strongest evidence for inner core rotation so far with a set of 18 high-quality waveform doublets from South Sandwich Islands (SSI) earthquakes. These doublets show systematic shifts in travel times and changes in the 0012-821X/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.epsl.2007.06.034

260 X. Song, G. Poupinet / Earth and Planetary Science Letters 261 (2007) 259 266 waveforms through the inner core. The use of waveform doublets avoids artifacts of earthquake mislocation and contamination from small-scale mantle heterogeneities, and allows them to pin down precisely that the temporal changes must have occurred when the waves travel through the inner core. However, waveform doublets that show clear signals through the inner core are relatively rare. In this study, we examine the temporal change of inner core waves using a technique that compares differential travel times between two events recorded at the same station. Similar to the waveform doublet approach, the method does not require precise knowledge of earthquake locations and earth models. The method avoids the stringent requirement of high waveform similarity for waveform doublets, so the data quality is not comparable to that of waveform doublets and the measurements of the relative-time difference are not as precise. However, the pairing of the events creates a large number of event pairs that we can work with using standard statistics. 2. Data and method Our data are differential PKP travel-time measurements of SSI events at College, Alaska (COL) station (Fig. 1). Part of the data has been used extensively for studying inner core anisotropy (Creager, 1992; Song and Helmberger, 1998) and inner core rotation (Song and Richards, 1996; Song, 2000; Poupinet et al., 2000; Zhang et al., 2005). The differential travel times involve three branches of the PKP waves, which traverses the inner core (DF), turns at the bottom of the outer core (BC), and turns in mid-outer core (AB), respectively. We select events for which both differential AB BC time and BC DF time can be measured at the same trace (Fig. 1D). We obtain a total of 87 events, spanning 50 yr Fig. 1. SSI events used in this study. (A) Spatial distribution of all SSI events used (a total of 87). Both BC DF and AB BC times at COL have been measured for these events. The tight cluster (box) contains a total of 30 events. (B, C) Latitudes and longitudes plotted as a function of the origin time of the events. The spatial distribution of the events is uniform over time. (D) Example PKP seismograms recorded at COL station. The two events are Mar 15, 1986 (depth 90 km, mb 5.9) and Mar, 26, 1986 (depth 111 km, mb 5.7). The epicentral distances (Δ) to COL are 150.34 and 150.99, respectively. All three branches of PKP are clear in these traces. Differential times between BC and DF and between AB and BC are measured for each trace. In this study, we form the differential travel time differences between a pair of events, i.e. d(bc DF) and d(ab BC). The difference in Δ affects both d(ab BC) and d(bc DF) (more on the former than on the latter), thus we can calibrate the distance difference using d(ab BC) to correct d(bc DF).

X. Song, G. Poupinet / Earth and Planetary Science Letters 261 (2007) 259 266 261 from 1958 to 2007 (Fig. 1A C). The historical data are from various sources, which are described in detail in Song (2000). Differential PKP travel times are measured using waveform cross-correlations for each event. All the seismograms (vertical component) have been converted to the short-period WWSSN instrument and, if helpful, are filtered at frequency band of about 0.6 to 3 Hz to increase signal to noise ratio. The DF and BC phases are correlated directly; the BC and AB phases are correlated after correcting for the Hilbert transform of the AB phase. The cross-correlation coefficients are larger than about 0.50. The median value for the DF and BC correlation is 0.73 and the median value for the BC and AB correlation is 0.70. Our basic method is to compare the difference in the differential BC DF times, d(bc DF), and the difference in the differential AB BC times, d(ab BC), between a pair of events (Fig. 1D). Denoting the subscript 1 for the earlier event, and subscript 2 for the later event, we define dðbc DFÞ ¼ðBC DFÞ 2 ðbc DFÞ 1 ; and dðab BCÞ ¼ðAB BCÞ 2 ðab BCÞ 1 : The difference in the locations between the two events affects both the d(ab BC) and the d(bc DF) values. In fact, it changes the d(ab DF) value more than the d(bc DF) value. Thus we can correct the influence of location difference on d(bc DF) using the d(ab BC) value. The differential time difference ddt (either d(bc DF) or d(ab BC)) is affected by the location (epicentral distance to the station Δ and depth h) difference, heterogeneities of Earth structure, and measurement errors: ddt ¼ f 1 ðddþþf 2 ðdhþþf 3 ðstructureþ þ f 4 ðmeasurement errorsþ: Because these rays are nearly vertical, the depth effect f 2 is much smaller than the distance effect f 1.For the same amount (50 km) of difference in epicentral distance (dδ) and in depth (dh), d(bc DF) would be about 0.36 s and 0.06 s, respectively, and d(ab BC) would be about 0.82 s and 0.20 s, respectively. The SSI earthquakes are relatively shallow. Almost all events are less than 150 km in depth, with the average depth of 80 km and the standard deviation of 45 km. Thus we ignore the f 2 term in our following analysis. We regard the measurement error term f 4 as random noise. The structure term f 3 comes from mantle for both d(bc DF) and d(ab BC) and inner core heterogeneities for d(bc DF). Our earthquake distributions are quite uniform for different time periods (Fig. 1B,C). Our data sample small patches of the mantle and the inner core. The coverage of these patches is quite uniform for event pairs with different time lapses. Thus, we also regard f 3 as random noise. We will discuss further the depth ( f 1 )and mantle ( f 3 ) effects below. When there are enough pairs, the scaling factor from d(ab BC) to d(bc DF) can be calibrated, which is primarily affected by the distance term f 1.Ifthe DF travel time changes with time, the temporal change would affect d(bc DF) but not d(ab BC). Thus we can detect the temporal change by analyzing how d(bc DF), after being corrected by d(ab BC), changes with the time lapse between the two members of the pair. To summarize, our technique involves the following steps. (1) Measure BC DF and AB BC times for each event. (2) Form d(bc DF) and d(ab BC) for pairs of events. (3) Find the scaling factor between d(bc DF) and d(ab BC) using pairs of short time lapses to limit the influence of temporal changes. (4) Correct the d(bc DF) measurement using the d(ab BC) measurement and the scaling factor, i.e., the corrected d(bc DF)=d(BC DF) factor d(ab BC). As discussed above, this step removes the main influence from the location difference (epicentral distance difference) on d(bc DF). (5) Analyze how the corrected d(bc DF) value changes with the time lapse. In our analysis, we use heavily the standard statistical method of bootstrapping (e.g., Efron and Tibshirani, 1993) to estimate the parameters and its errors. When the differential travel times are measured, wrong cycles may be picked in the cross-correlation functions. We corrected or eliminated a few events with apparent cycle-skipping, which would form characteristic bands in the pairing plots. We conduct synthetic tests to demonstrate the methodology and to examine the resolution of the method (Fig. 2). Using the catalog locations of the same events, we calculate the PKP travel times for 1D reference model AK135 (Kennettetal.,1995). We then add a temporal change of 0.008 s/yr to the DF travel times and calculate the differential BC DF and AB BC times for each event. In the first test, we don't include any noise. Using the data with time lapses less than 5 yr, we find the scaling factor to be 0.429+/ 0.005 (1σ) by bootstrapping. Bootstrapping the corrected d(bc DF) values, we find a temporal change that is virtually the same as the input model with a very small error (0.0081+/ 0.0003 s/yr) (Fig. 2A). In the above test (noise free), the full recovery of the temporal change is achieved without considering the

262 X. Song, G. Poupinet / Earth and Planetary Science Letters 261 (2007) 259 266 3. Results Fig. 2. Synthetic tests of the recovery of the imposed temporal trend (0.008 s/yr), with (B) and without (A) noise. See text for description. depth difference of the events. As discussed previously, the depth influence (the f 2 term) is much smaller than the distance influence (the f 1 term). In addition, the random pairing of the events makes the f 2 term to be random as well, as evidenced in the figure. If there were no depth difference for all the pairs, all the synthetic data would fall into the straight line. The scatter (about 0.1 s) can be considered as the limit of the depth term. Thus, the term is expected to contribute to the scatter of the observations but does not affect the rate of the temporal change. In the second test, we add Gaussian noise to both BC DF and AB BC measurements with standard deviations of 0.1 s and 0.2 s, respectively. Typical measurement errors of the differential travel times are less than 0.1 s. The larger errors we impose take into account some influences from the mantle and inner core heterogeneities. The scaling factor in this case differs slightly: 0.407+/ 0.009. The corrected d(bc DF) values show the scatter similar to or perhaps larger than that of our actual data (see below). Despite the large scatter, we can recover the temporal change quite well with a relatively small error bar (0.0077+/ 0.0010 s/yr) (Fig. 2B). We first determine the scaling factor between d(bc DF) and d(ab BC) using all the pairs with time lapses less than 5 yr (total 733 pairs) (Fig. 3). The optimal value is 0.528 +/ 0.013 from bootstrapping. The scaling factor is significantly larger than that of 0.43 for model AK135 from our synthetic test. This may be caused by measurement errors but more likely by the departure of the average velocity of the Earth for this path from the global reference model. Possible sources include the inner core anisotropy and the velocity gradients in the lowermost mantle and the lowermost outer core. Using the scaling factor of 0.53, Fig. 4 shows all the corrected d(bc DF) values as a function of time lapse (a total of 3741 pairs). The corrected values show a clear temporal trend with a slope of about 0.0076 s/yr. We explore the robustness of the temporal change of d(bc DF) using bootstrapping (Table 1). We test various groups and subgroups of data sets. For each data set, we run 1000 iterations of bootstrapping by selecting events from the original event list at random, obtain the regression of the corrected d(bc DF) values with respect to the time lapse for each iteration, and finally derive the average and the standard deviation of the regressions. For each data set, we use the scaling factor of 0.53, our optimal value. We also test the scaling factor of 0.43, which is appropriate for AK135 for our data set, to examine the sensitivity of the regression results to the scaling factor. In all cases we have tested, the results do not change significantly with the scaling factor between the two values (well within 1σ) (Table 1). We consider the whole data set as well as various subgroups. (1) The distance (d) between the two events calculated from the catalog locations varies from within 0.2 to within 2. The smaller the distance, the more Fig. 3. All pairs with time lapse less than 5 yr used to find the scaling factor between d(bc DF) and d(ab BC). The optimal value (0.528+/ 0.013) is found from bootstrap regression with 1000 iterations.

X. Song, G. Poupinet / Earth and Planetary Science Letters 261 (2007) 259 266 263 Fig. 4. Corrected d(bc DF) as a function of time lapse (dyr) between the two events. The correction is made using d(ab BC) and the scaling factor of 0.53. The slope of line, obtained from the bootstrap regression of all the data (Table 1, group 6), is the measure of the DF travel time change. likely that the two events are in fact closer, and thus the less bias the Earth's structure heterogeneity has on d(bc DF) and d(ab BC). However, the number of observations decreases with d. (2) We test groups with various d (AB BC) values, essentially using the d(ab BC) as a proxy for the difference in epicentral distances to the station. (3) We test data after 1964, and after 1981, respectively. Between 1958 and 1965, we only have a few events (less than 10%). After 1981, the records became digital, allowing us to obtain differential times more accurately. (4) The epicentral distance (Δ) of our events ranges from 149.7 to 153.3, sampling the top 300 km of the inner core. The inner core rotation may cause slightly different time shifts for waves traveling at different depths and spending different times in the inner core. We consider the data set with Δ between 150 and 152, which contains more than 90% of the total event pairs. (5) More than one third of the events are clustered in a very small area of the study region (Fig. 1A), providing a natural data set without the need to impose other restrictions. Considering that the number of observations varies by an order of magnitude (Table 1), the temporal change in the d(bc DF) value, ranging from 0.071 to 0.098 s/decade (excluding the subgroups using shorter period 1982 2007), is remarkably robust. The standard deviation of the mean depends on the number of observations and the temporal coverage of the data. For pairs with smaller d or d(ab BC) values or in the cluster, which have smaller spatial separation and are less affected by mantle heterogeneity, the inferred rates (0.084 to 0.098 s/decade) are nearly identical to the rate inferred by Zhang et al. (2005) from waveform doublets (0.092+/ 0.004 s/decade). We have assumed that the f 3 term (from mantle heterogeneity) to be random as discussed above. Precise knowledge of the term is not possible as it requires the determination of very fine-scale structure (km to hundredkm scale) of the mantle, particularly the lowermost mantle. However, we can now estimate the influence of this term from our tests using different parameters (Table 1). The range of the values (0.71 to 0.98 s/decade) we obtained for pairs in Groups 1, 2, 5, and 6 (Table 1)canbe regarded as the maximum variation caused by mantle heterogeneity, as other factors (measurement errors and differences in the number of pairs) also contribute. The smaller the spatial separation between the two events of the pair is, the less influence the mantle heterogeneity would have. It is encouraging that for pairs with small d or d(ab BC) values or in the cluster, the inferred rates are very similar to the rate inferred from waveform doublets (Zhang et al., 2005). All rate estimates using larger data sets (looser requirement on inter-event distance) are smaller, by 0.05 to 0.027 s/decade or by up to 30%. Estimates of the temporal change using the shorter time period 1982 2007 are consistently smaller than the estimates using the whole data set (Fig. 5). The difference is significant at 68% confidence (1σ) but not significant at 95% confidence (2σ). We have also examined the data of even shorter time period (1992 2007). The average rate using all the data in this time period appears even smaller: 0.031 +/ 0.035 s/decade. However, the uncertainty is very large and the issue requires further investigation. Using various groups and Table 1 Estimates of temporal change of d(bc DF) Group Data selection f=0.53 f=0.43 N 1 Restriction on inter-event distance (d) db0.2 0.090 (0.026) 0.092 (0.025) 371 db0.5 0.074 (0.017) 0.079 (0.016) 979 db1.0 0.071 (0.013) 0.078 (0.013) 1927 db2.0 0.077 (0.012) 0.085 (0.013) 3169 2 Restriction on d(ab BC) value (dab) dabb0.2 s 0.084 (0.019) 0.084 (0.019) 403 dabb0.5 s 0.079 (0.014) 0.079 (0.014) 916 dabb1.0 s 0.077 (0.012) 0.078 (0.012) 1630 dabb3.0 s 0.076 (0.012) 0.081 (0.013) 3323 3 Restriction on origin times of both events 1965 2007 0.080 (0.016) 0.080 (0.022) 3081 1982 2007 0.062 (0.014) 0.076 (0.026) 1431 4 Restriction on epicentral distances (Δ) of both events 150 bδb152 0.073 (0.018) 0.086 (0.017) 2278 5 Pairs with both events in the cluster shown in Fig. 1A Clustered pairs 0.098 (0.024) 0.097 (0.022) 435 6 All data 0.076 (0.012) 0.078 (0.015) 3741 The temporal change and its standard deviation (in parentheses) are in s/decade. The parameter f is the scaling factor from d(ab BC) to d(bc DF), and N is the average number of observations over 1000 iterations of bootstrapping.

264 X. Song, G. Poupinet / Earth and Planetary Science Letters 261 (2007) 259 266 Fig. 5. Estimates of the temporal change of d(bc DF) using various parameters. The solid dots are for data of all time periods (1958 2007) and the open circles are for events in 1982 2007 with modern digital records. The error bar is +/ one standard deviation. Models 1 through 4 are from Group 1 (Table 1) with d from 0.2 to 2, respectively; Models 5 through 8 are from Group 2 with dab from 0.2 s to 3.0 s, respectively; Models 9, 10, 11 correspond to Group 4, 5, and 6, respectively. The scaling factor f=0.53 is used for these calculations. subgroups of the data as in Table 1, the estimated rates range from 0.007 to 0.145. The cause of the change of the temporal rate is not clear. The possibilities include the following. (1) Errors in the estimation of the rate. Both the time lapse and the number of observations decrease. As the time lapse decreases, the amount of the temporal shift also decreases. The total number of pairs (1431) for 1982 2007 is only 38% of that of the whole data set (3741). Thus the estimate of the rate is more subject to errors from measurements and mantle heterogeneity. If we consider only the 1982 2007 time period in the synthetic test (Fig. 2), the rate is also considerably different with a large error (0.0055 +/ 0.0030 s/yr). However, the rate estimate errors for 1982 2007 are comparable to those of the whole data set (Fig. 5). Thus this may not be the main cause. (2) Changes of seismic instrumentation over time. Although the instrument responses to different frequency contents of DF and BC and AB have not been well tested, this may not be the cause either. All the seismograms have been converted to the same narrowband instrument (short-period WWSSN) before the differential travel times are measured using waveform correlation. (3) Sampling of laterally-varying small-scale inner core structures, which have been suggested (Vidale and Earle, 2000; Cormier and Li, 2002; Koper et al., 2004). If the inner core rotates at 0.3 to 0.5 /yr (Zhang et al., 2005), significant lateral variation over 100 200 km distance in the inner core would explain the change in the d(bc DF) temporal shift. (4) A change in the inner core rotation rate. If the lateral structure of the inner core at this region varies smoothly (linearly), a Fig. 6. Probability density distribution of raw data and the means. (A) Distribution of the corrected d(bc DF) value, i.e., d(bc DF) 0.53 d(ab BC), for pairs with different time lapses. The time lapses are within 5 yr for each group, except the top group with largest time lapses, which include data with time lapses between 40 and 48 yr. The number of pairs in each group is labeled at the left. (B) Distribution of the means of the corrected d(bc DF) value for groups with different time lapses. A sliding time-lapse window of 5 yr and a sliding step of 2.5 yr are used to form the groups, except for the last group (top), which has the time window of 6.3 yr. The means are calculated using 5000 bootstrapping iterations. The dashed line is bootstrap result using all the data (Table 1) with a slope of 0.0076 +/ 0.0012 s/yr.

X. Song, G. Poupinet / Earth and Planetary Science Letters 261 (2007) 259 266 265 change of rotation rate could cause the change in the DF travel time. Cases 3 and 4 cannot be distinguished with the data at hand. The robust temporal change is demonstrated graphically in Fig. 6. It is clearly visible in the distribution of the raw data when they are divided according to the time lapse (Fig. 6A). It is even clearer in the distribution of the means from boot-strapping (Fig. 6B). The means generally follow a linear trend with time lapse along the average rate with some fluctuations, particularly at larger time lapses when the number of observations decreases. 4. Conclusion We propose a technique that compares differential PKP travel time measurements between event pairs to detect the temporal change of the inner core. Using 87 SSI events recorded at COL station, we estimate the average temporal change in the DF times to be about 0.07 to 0.10 s/decade over the past 50 yr. For pairs that are close in location (as indicated by smaller d, d(ab BC) values, or in the cluster), the inferred rates (0.084 to 0.098 s/decade) are nearly identical to the rate inferred by Zhang et al. (2005) from waveform doublets (0.092 +/ 0.004 s/decade). Bootstrapping analyses of various groups and subgroups of the data set suggest the temporal trend is very robust. The rate of the DF change seems to change with time, which may be explained by lateral variation of the inner core structure or the change in rotation rate in decadal time scale. If the rotation rate change is true, it may explain the discrepancy inferred from the data of different time periods (e.g., Song, 2000; Vidale et al., 2000; Laske and Masters, 2003). The most appealing aspect of the new method is the minimum assumptions required. We rely solely on a small initial data set (174 differential time measurements from 87 events) and do not need precise knowledge of earthquake locations and earth models. The calibration for location difference is purely based on the original data set. Another aspect of the method is the large data set created by the pairing of events, which allow us to use statistical tools to find coherent signals and reduce the influence of random noises (from measurement errors or from structure variations). Our synthetic tests suggest that the temporal rate can be well resolved if the noise is well behaved (random) and the signal to noise ratio is high enough. The technique provides an alternative when perfect doublets are hard to find. It can be similarly applied to other regions and other seismic phases and may be applicable to other processes (such as temporal changes in crustal properties and in magma chambers). Acknowledgements Historical data at COL station were acquired with the help of many people (Song, 2000). Recent digital data were obtained from the IRIS DMC. We thank Ping Ma for comments on statistics and John Vidale for helpful review. 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