Geophysical Journal International

Geophysical Journal International Geophys. J. Int. (2011) 187, 355 370 doi: 10.1111/j.1365-246X.2011.05146.x Global seismic body-wave observations of temporal variations in the Earth s inner core, and implications for its differential rotation Anna M. Mäkinen and Arwen Deuss Bullard Laboratories, Department of Earth Sciences, University of Cambridge, Cambridge CB3 0EZ, UK. E-mail: amm210@cam.ac.uk Accepted 2011 July 9. Received 2011 July 7; in original form 2011 April 26 1 INTRODUCTION The Earth s inner core, discovered now 75 yr ago by Lehmann (1936), continues to present a challenge to our detailed understanding of structures and processes in the Earth s deep interior. Such an understanding would enable us to better constrain models of the formation and evolution of the Earth s inner and outer core, as well as the geodynamo, responsible for the life-sustaining magnetic field. At present, a few facts about the inner core are considered known: that the inner core consists mainly of solidified iron with some light elements, and that it grows by crystallization of the liquid outer core (e.g. Jacobs 1953, 1987; Loper 1984; Buffett et al. 1992, 1996). The heat and light elements released in this crystallization SUMMARY Differential rotation of the Earth s inner core has been predicted in some geodynamo models, and seismic studies over the past 15 yr have resolved rotation rates up to 1 yr 1. Most previous seismic body-wave studies have focussed on South Sandwich Islands events recorded at station COL in Alaska. Here, we present a globally extended study into temporal variations in the inner core over some 25 yr, using PKPbc-PKPdf traveltime residuals. To test for differential rotation of the inner core, displacement of inner-core heterogeneities over time is sought. We introduce a new method of space-flattening to remove the effect of spatial variations on the time variations; this allows for the use of both polar, semi-equatorial and equatorial geometries. First, we reanalyse polar paths from South Sandwich Islands events to stations COL and INK in North America. These stations yield a differential rotation of the inner core at a rate of 0.12 0.38 yr 1 in an eastward direction, in agreement with previous studies. However, station DAWY, which has a very similar path through the inner core as COL, yields at best a westward differential rotation of the inner core. Thus DAWY results are incompatible with the COL/INK inferred rotation. Secondly, earthquakes in the Aleutian Islands region, observed at BOSA and LBTB in southern Africa, exhibit temporal variations that are incompatible with the South Sandwich Islands-COL/INK inferred rotation rate. Thirdly, Kuril Islands events, recorded in South America at station BDF, yield inconclusive results. Finally, our final piece of evidence for the irreconcilability of differential inner-core rotation with global data comes from using earthquakes in the Vanuatu region, recorded at BCAO/BGCA in Central Africa, an equatorial geometry. These residuals resolve a westward inner-core rotation at a rate of 0.14 yr 1, over the same time period that South Sandwich Islands events indicate an eastward rotation. As any rigid-body rotation should yield the same direction and rate independent of where the inner core is sampled, our results allow us to reject previously reported inner-core differential rotation rates of up to 0.1 0.5 yr 1. Instead, our results suggest that structure in either the inner or the outer core is varying with time, over relatively short timescales and in ways that cannot be explained by, and do not support, a differentially rotating inner core. Key words: Magnetic field; Core, outer core and inner core; Body waves; Seismic tomography; Wave propagation. then drive the geodynamo, responsible for the Earth s magnetic field (see e.g. Braginsky 1963; Glatzmaier & Roberts 1995). In recent years, direct observations from seismology have shed light on some intriguing properties of the inner core: we now know that the inner core exhibits cylindrical anisotropy, with a fast axis approximately aligned with the Earth s rotation axis (Poupinet et al. 1983; Morelli et al. 1986; Woodhouse et al. 1986; Shearer & Toy 1991; Creager 1992). It has also been discovered that inner-core structure is hemispherically varying, with a more anisotropic western and a less anisotropic eastern hemisphere (Tanaka & Hamaguchi 1997; Creager 1999; Deuss et al. 2010), and that anisotropy is only present at depth, beneath an isotropic top layer up to 100 km thick (Niu & Wen 2002; Waszek et al. 2011). Regional variations in GJI Seismology C 2011 The Authors 355

356 A. M. Mäkinen and A. Deuss uppermost inner-core structure have also been reported (Stroujkova & Cormier 2004). The presence of a possible differential rotation of the Earth s inner core has been investigated in some detail over the past 15 yr, but remains controversial to this date. The need for a differentially rotating inner core arises in modelling: some models of the geodynamo (Gubbins 1981; Glatzmaier & Roberts 1996) predict that the Earth s inner core rotates differentially with respect to the mantle, at rates as high as 2 3 yr 1, although this view has also been challenged (Aubert et al. 2008; Aubert & Dumberry 2011). Because whether or not the inner core rotates differentially is important for constructing an Earth-like geodynamo model, independent evidence pertaining to the existence of differential motion is necessary to resolve the issue. Seismological evidence for differential rotation has mainly been obtained using body waves. Song & Richards (1996) used PKPbc- PKPdf traveltime differences for individual events to obtain an eastward rotation rate of approximately 1 yr 1. Their analysis, as well as that of Su et al. (1996), which reported eastward rotation at a rate as high as 3 yr 1, made use of the assumption that the symmetry axis of the inner-core cylindrical anisotropy is tilted by some 10 with respect to the Earth s rotation axis. Differential rotation of the inner core about the Earth s rotation axis would then lead to different proportions of the cylindrical anisotropy of the inner core being sampled at different times. This idea of an axis tilt predated the discovery of hemispherical anisotropy structure in the inner core (Tanaka & Hamaguchi 1997), and has been abandoned since. Following these studies, individual-event body-wave PKPbc- PKPdf evidence both in favour of an inner-core differential rotation (Creager 1997; Song & Li 2000; Song 2000a,b; Collier & Helffrich 2001; Xu & Song 2003; Li & Richards 2003b; Song & Poupinet 2007), and against it (Souriau et al. 1997; Souriau 1998b; Souriau & Poupinet 2000; Isse & Nakanishi 2002) has been reported, and the issue has been hotly debated (Souriau 1998a; Richards et al. 1998). More recently, body-wave methods have been applied to earthquake waveform doublets, with some studies supporting differential rotation (Li & Richards 2003a; Zhang et al. 2005, 2008), and other studies against it (Poupinet et al. 2000), although the latter has been challenged (Song 2001). Inner-core rotation has also been studied using temporal changes in scattering (Vidale et al. 2000) and changes in PKP coda (Vidale & Earle 2005); these studies generally support the idea that the inner core is rotating with respect to the mantle. Normal-mode studies (Sharrock & Woodhouse 1998; Laske & Masters 1999, 2003), on the other hand, generally fail to resolve any eastward non-zero differential inner-core rotation. The inner-core rotation rate has been estimated as anything between 3 yr 1 eastwards (Su et al. 1996) and zero, with the current fast estimate at approximately 0.3 yr 1 (Zhang et al. 2005). Oscillatory motion (Collier & Helffrich 2001) and time variance of fast differential rotation rates (Lindner et al. 2010) have also been suggested in seismological analyses. More recently, an extremely slow rate of 0.1 1 Myr 1 for an eastward differential rotation of the inner core has been resolved by using a large, global PKPcd-PKPdf data set and the age depth relation in the inner core (Waszek et al. 2011). This result is compatible with the frozen-in, unsmeared hemispherical anisotropic structure of the inner core, unlike the faster rotation rates of previous studies, which are difficult to reconcile with hemispherical structure. It is also in agreement with the geodynamo model of Aubert & Dumberry (2011). However, this slow rotation rate is not in agreement with most previous seismological analyses. Here we revisit this issue of a differentially rotating inner core, using body waves to probe the Earth s deep interior. PKPbc-PKPdf residuals from individual earthquakes are utilized in our quest to understand the dynamics of the inner core, and to reconcile the slow differential rotation observed by Waszek et al. (2011) with the faster rates reported elsewhere. Previous PKPbc-PKPdf and scatterer studies addressing the issue of a fast differential rotation have predominantly utilized the South Sandwich Islands to COL path; in our study, we seek new ray paths to compare and contrast different regions in the inner core, and to extend the analysis to a global scale. This is necessary as truly global conclusions, such as those pertaining to the differential rotation of the entire inner core, cannot be drawn from a single set of ray paths. Furthermore, in addition to polar paths, such as those of South Sandwich Islands events to COL, we seek ray paths close to the equatorial plane. For an inner core rotating differentially as a rigid body about the rotation axis of the Earth, such paths would be expected to exhibit little variation with time, as has been reported for a small number of events (Zhang et al. 2008). Our analysis comprises the commonly used South Sandwich Islands sources to COL, Alaska, receiver path, together with other stations in North America, to be compared with COL. We also utilize new ray paths from the Aleutian Islands region to southern Africa, from the Kuril Islands region to South America, and from the Vanuatu region to Central Africa. Vanuatu events recorded in Central Africa, in particular, enable us to study temporal variations in equatorial ray paths. We show that while temporal variations in PKPbc-PKPdf residuals arise globally, in both polar and equatorial inner-core geometries, results from different regions of the inner core are not in agreement with each other. Data from some regions are consistent with an inner core rotating differentially in an eastward direction, while data from other regions indicate either a westward rotation, or no differential motion of the inner core at all. An inner core rotating as a rigid body should yield the same direction and rate of differential rotation wherever it is probed, contrary to our global observations. Thus we conclude that the observed temporal variations are not reconcilable with a fast differential inner-core rotation, and that other mechanisms to explain these temporal variations should be sought instead. 2 METHODS AND DATA To probe temporal variations of inner-core structure, traveltime differences between two body-wave phases have been used (see Fig. 1). The PKPdf (or PKIKP) phase traverses the crust, mantle, outer core and inner core as a compressional (or P) wave, and is therefore sensitive to inner-core structure. Our reference phase PKPbc traverses a similar path in the crust and mantle, turning at the bottom Figure 1. A cartoon showing the outer-core phase PKPbc and the innercore phase PKPdf used in this study, for a 139-km-deep event at epicentral distance 150.4. IC denotes the Earth s inner core; OC stands for the outer core.

Temporal variations in the Earth s inner core 357 Figure 2. Pairs of seismograms, showing the time evolution of traveltime differences from two event regions. The seismograms have been aligned on the PKPbc phase, using the earlier seismogram in each pair as a reference, indicated by the light blue vertical lines. The dark blue vertical lines indicate the arrival time of the PKPdf phase for the earlier event in each pair. Note that the waveforms do not match between different events, as the source mechanisms are not exactly the same. (A) Two events in the South Sandwich Islands region, recorded at station COL, at almost identical epicentral distance and depth. (B) Two events in the Vanuatu region, recorded at BCAO/BGCA. of the outer core. Epicentral distances lie in the range 147 154 for observing both PKPbc and PKPdf. Using PKPbc-PKPdf traveltime differences and their residuals with respect to a standard 1-D Earth model enables us to avoid contamination by heterogeneities in the mantle and the outer core, as well as problems due to earthquake hypocentre mislocations, which would affect both phases equally. We have measured the arrival times of the PKPbc and PKPdf phases by hand-picking each seismogram. Since the signal-to-noise ratios vary somewhat depending on event magnitude and near-receiver conditions, the phases have been picked on the best-matching early feature between each phase pair. PKPbc and PKPdf are expected to exhibit the same general shape; PKPdf has a smaller amplitude due to being heavily attenuated upon traversing the inner core, and therefore has a wider, more spread-out pulse shape than does PKPbc. Cross-correlation checks have been applied to hand-picks, producing no discernible differences in the overall results obtained. In essence, this approach is similar to previous studies into innercore diffferential rotation (see e.g. Song & Richards 1996; Souriau 1998b), albeit with an extended global coverage of the inner core. Example seismograms from the South Sandwich Islands and Vanuatu regions are shown in Fig. 2. These seismograms have been selected based on similarity in epicentral distance and event depth between events in each pair. Note how, for the South Sandwich Islands events (Fig. 2A), the PKPdf phase for the 1998 event arrives earlier than for the 1983 event. For the Vanuatu events (Fig. 2B), on the other hand, the PKPdf phase for the 2000 event arrives later than for the 1980 event. In our study, all traveltime residuals have been calculated relative to the 1-D reference Earth model AK135 (Kennett et al. 1995). Corrections for the Earth s ellipticity have been applied (Dziewoński & Gilbert 1976). All earthquake hypocentre locations have been taken from the EHB catalogue (Engdahl et al. 1998). Our data are broad-band and short-period digitally recorded vertical channel seismograms, commonly sampled at 20 Hz. The data have been deconvolved with the original station responses, taking into account instrumentation changes over recording time periods. After instrument response removal, the data have been convolved with the WWSSN short-period response for comparability, then bandpass

358 A. M. Mäkinen and A. Deuss Figure 3. Source regions and events used in our PKPbc-PKPdf studies: (A) South Sandwich Islands events, recorded in Alaska and northern Canada at COL, INK and DAWY; (B) Aleutian Islands events, recorded in South Africa and Botswana at BOSA and LBTB; (C) Kuril Islands events, recorded in Brazil at BDF; (D) Vanuatu events, recorded in the Central African Republic at BCAO/BGCA. The filled circles indicate the PKPdf ray turning point in the inner core, projected onto the Earth s surface; the fill colour is determined by the measured PKPbc-PKPdf traveltime residual for each ray. Note the fixed range (B), (C) and (D) are plotted on the same scale, whereas the scale for (A) starts higher. The hemisphere boundaries at 14 E, 151 W (Irving & Deuss 2011) are drawn as solid dark green lines. The orange stars indicate the sources and the green triangles the receivers. filtered using a causal second-order Butterworth filter with corner frequencies 0.5 and 2.0 Hz. We have used earthquakes in four different source regions for a global sampling of the inner core. These regions are (1) South Sandwich Islands (SSI, see Fig. 3A), recorded at the North American stations COL/COLA (College, Alaska, USA, hereafter COL; the code for this station changed in 1996, accompanied by a slight change in station location), INK (Inuvik, Northwest Territories, Canada) and DAWY (Dawson, Yukon, Canada); (2) the Aleutian Islands (AI, see Fig. 3B) region, recorded at the southern African stations BOSA (Boshof, South Africa) and LBTB (Lobatse, Botswana); (3) Kuril Islands (KI, see Fig. 3C), recorded in South America at BDF/BDFB (Brasília, Brazil, hereafter BDF; this station changed code in 1993); (4) the Vanuatu region (see Fig. 3D), recorded in Central Africa at BCAO (Bangui, Central African Republic; data from 1979 to 1990) and BGCA (Bogoin, Central African Republic; data from 1994 to 2002). These source regions and stations have been selected after an extensive search over all suitable stations; the stations are characterized by their long recording time spans and sufficient signal-to-noise ratios. Typically some 5 10 per cent of seismograms available are retained for further analysis. We quote a picking error of 0.1 s due to typical 20 Hz sampling inherent in all our residuals; other errors due to, for example, signal-to-noise ratio are more difficult to quantify, but have been greatly reduced by our rigorous checks of phase waveform similarity within each seismogram and only retaining the very best data available. Further details of the stations and data are given in Table 1. For these stations, data have been recorded over extended periods of time, ranging from 25 yr (1982 2007) at COL for South Sandwich Islands events and 23 yr (1979 2002) at BCAO/BGCA for Vanuatu events to 14 yr (1993 2007) at DAWY for South Sandwich Islands events (see Table 1). Our choice of data is restricted by our wish to analyse digitally recorded seismograms only, unlike in previous studies (e.g. Souriau 1998b; Poupinet et al. 2000; Lindner et al. 2010); this is to avoid any errors associated with varying instrument sensitivity and also additional errors due to digitization

Temporal variations in the Earth s inner core 359 Table 1. Station information: ζ indicates the angle between the PKPdf ray at its turning point inside the inner core and the Earth s symmetry axis; is the epicentral distance between source and station. Source Station label Latitude Longitude Station operation Average ζ Average region [ ] [ ] dd.mm.yyyy [ ] [ ] SSI COL 64.90 147.79 08.01.1982 22.08.1997 26.12 151.07 COLA 64.87 147.85 14.06.1996 present 26.14 150.65 INK 68.31 133.52 03.06.1992 present 22.62 147.20 DAWY 64.07 139.39 30.09.1992 present 24.51 149.07 AI BOSA 28.61 25.26 26.02.1993 present 48.75 151.37 LBTB 25.01 25.60 17.04.1993 present 50.60 148.90 KI BDF 15.66 47.90 08.06.1982 15.06.1993 59.69 148.83 BDFB 15.64 48.01 22.07.1993 02.12.2002 59.95 148.78 Vanuatu BCAO 4.43 18.54 09.04.1979 11.11.1990 79.59 147.72 BGCA 5.18 18.42 21.07.1994 28.05.2002 79.49 148.07 of analog data. We do not discuss analysis of events beyond 2007 as these have not yet been relocated in the EHB catalogue (Rob van der Hilst, personal communication, 2011); the use of earthquake locations from other catalogues would unnecessarily complicate the analysis by adding the need to scale between catalogues. We first measure the residuals dt(bc DF), given by dt(bc DF) = ( t obs PKPbc t obs PKPdf ) ( t AK135 PKPbc t AK135 PKPdf ). (1) We then probe temporal trends by fitting linear equations of the form dt(bc DF) = at + b, (2) where t is the event date, a is the temporal gradient and b is the temporal intercept. In addition to temporal trends at permanent stations, we have studied the spatial variation of inner-core velocity structure as a function of PKPdf turning longitude. Similar to line-fitting to temporal residuals, we seek linear equations of the form dt(bc DF) = mλ + l, (3) where λ is the PKPdf turning longitude inside the inner core for each ray path yielding a traveltime residual dt(bc DF), m is the spatial gradient and l is the spatial intercept. Studying spatial variations is a necessity for testing the inner-core differential rotation hypothesis: only if adjacent regions in the inner core have smoothly varying velocity structure, can any differential motion of the inner core be observed seismologically. Conversely, if adjacent regions exhibit no variation in velocity structure, rays passing through different parts of the inner core at different times (if the inner core is rotating with respect to the mantle) sample similar structure, and record no temporal variation. Studying regions of the inner core that exhibit linear variations in spatial velocity structure with longitude is particularly well suited for testing the rotation hypothesis, since this rotation is most likely to occur about the Earth s rotation axis (Song 2000a). A linear spatial gradient would then enable us to estimate the direction and rate of any such rotation (see e.g. Song & Richards 1996; Collier & Helffrich 2001) by calculating the ratio a/m. To avoid contamination by temporal effects, we have used data from temporary experiments, spanning just a few years in time, to study the spatial variations. For South Sandwich Islands events, data from the PASSCAL experiment ARCTIC [ Structure and Rotation of the Inner Core, 2004 2007; Lindner et al. (2010)] have been used. For Aleutian Islands events, the PASSCAL experiment SA [ Anatomy of an Archean Craton, South Africa, 1997 1999; Silver et al. (2001)] has provided data. For Kuril Islands events, the experiment BL [Brazilian Lithosphere Seismic Project; 2001 2007] has enabled us to span the inner core. For each experiment, PKPbc- PKPdf residuals for events derived from the same earthquake lists as those used in probing the temporal variations have been analysed at a number of stations. The extended span of the experiments on the Earth s surface has enabled us to pick the same event at several close-lying stations. Combining this with picking of multiple events has resulted in dense and spatially extended sampling of the inner core over time periods of only a few years. For events in the Vanuatu region, we have not been able to obtain independent constraints on spatial structure, and instead use the same set of single-station measurements for both spatial and temporal variations. Having studied temporal and spatial variations, we seek to determine whether the spatial variations alone would be sufficient to explain the variation in traveltime residuals with earthquake date. To investigate whether this is the case, we remove spatial variations from temporal ones. To do this, we project the single-station traveltime residuals to a fixed point in PKPdf turning longitude in a space-flattening procedure. We then re-examine the temporal trends in traveltime residuals and whether they have been obliterated by our removal of spatial effects. The space-flattening procedure is given by dt(bc DF) flat i = dt(bd DF) i m(λ i λ 0 ), (4) where the measured traveltime residual dt(bc DF) i at longitude λ i is projected onto a horizontal line at the value of the reference longitude λ 0 using the spatial gradient m. If the raw residual dt(bc DF) i lay directly on the best-fit longitudinal spatial line, it would be be projected onto the expected residual at λ 0. After space-flattening, the remaining variation in traveltime residuals dt(bc DF) is genuinely temporal, not a result of slight differences in ray path geometry over time. The robustness of both individual temporal and individual spatial fits has been tested by cross-validation: each linear regression fit has been repeated by leaving out 1/20 of all data at a time, and repeating the procedure 20 times. The cross-validation procedure has been applied individually to raw (unnormalized), PKPdf turning point depth-normalized (that is, normalized with respect to epicentral distance for each station), and space-flattened data. The errors quoted for temporal (Table 2) and spatial (Table 3) gradients and rotation rates (Table 4) have then been obtained by considering the maximum ranges (not the standard deviation) from the cross-validation results and the fits to raw, depth-normalized and space-flattened data (temporal trends) or raw and depth-normalized data (spatial trends). Maximum ranges for each variable from such

360 A. M. Mäkinen and A. Deuss Table 2. Linear regression fits to spatial residual data. Region Experiment Gradient, m Intercept, l [s deg 1 ] 10 2 [s] SSI ARCTIC 6.3 ± 0.5 1.4 ± 0.3 AI SA 1.5 ± 0.4 0.40 ± 0.18 KI BL 0.25 ± 0.04 0.31 ± 0.03 Vanuatu BCAO/BGCA 4.5 ± 0.4 4.7 ± 0.4 Table 3. Linear regression fits to temporal residual data. Source Station Gradient, a Intercept, b region [s yr 1 ] 10 2 [s] SSI COL 0.80 ± 0.34 13 ± 7 INK 2.4 ± 1.2 46 ± 24 DAWY N/A N/A AI BOSA N/A N/A LBTB N/A N/A KI BDF N/A N/A Vanuatu BCAO/BGCA 0.64 ± 0.17 13 ± 3 SSI flat COL 0.73 ± 0.34 11 ± 7 INK 1.4 ± 1.2 26 ± 24 Vanuatu flat BCAO/BGCA 0.53± 0.17 11± 3 SSI COL binned 0.79 ± 0.10 13 ± 2 INK binned 2.3 ± 0.8 45± 17 DAWY binned 4.0 ± 0.7 81± 13 Vanuatu BCAO/BGCA binned 0.72 ± 0.15 15 ± 3 SSI flat COL binned 0.70 ± 0.10 11 ± 2 INK binned 1.5 ± 0.8 28± 17 DAWY binned 4.6 ± 0.7 95± 13 AI flat LBTB binned 1.6 ± 0.3 32± 5 Vanuatu flat BCAO/BGCA binned 0.57 ± 0.15 12 ± 3 considerations of different methods and separate cross-validation of each fit have been taken as a conservative estimate of error in our values. Fits that are not statistically significant have not been included, and are marked as N/A in our tables. Cross-validation has not been performed for temporally binned data owing to the small number of data points; in this case, errors have been estimated from ranges of raw and depth-normalized fits. To account for the mantle and to see whether the observed temporal and spatial variations could be explained by spatial variations in mantle structure alone, we employ two global tomographic mantle models: the P-wave tomographic model MIT-P08 (Li et al. 2008), and the S-wave model S40RTS (Ritsema et al. 2011). For both models, we trace PKPbc and PKPdf rays throughout the mantle. For the S40RTS model, P-wave velocity perturbations are obtained from the S-wave velocity perturbations by depth-dependent scaling: δv s /δv P = R,whereR 0 = 2 at the Earth s surface and R CMB = 3atthe core mantle boundary (CMB). For each model, mantle-corrected predicted PKPbc-PKPdf traveltime differences are subtracted from the observed traveltime differences. These mantle-corrected residuals and their temporal and spatial gradients are then used to calculate inner-core differential rotation rates, in the same way as for the AK135-only-derived residuals. We apply the same space-flattening procedure and error calculations to mantle-corrected results as to the uncorrected ones (except that normalization for depth is not applied to mantle-corrected residuals). Results from MIT-P08 and S40RTS corrections are then compared to those obtained without any corrections for mantle structure. Table 4. Estimated differential inner-core rotation rates. Region Station IC rotation rate [deg yr 1 ] SSI COL 0.13 ± 0.05 INK 0.38 ± 0.19 DAWY N/A SSI flat COL 0.12 ± 0.05 INK 0.23 ± 0.19 DAWY N/A SSI COL binned 0.126 ± 0.018 INK binned 0.37 ± 0.13 DAWY binned 0.63 ± 0.12 SSI flat COL binned 0.111 ± 0.018 INK binned 0.24 ± 0.13 DAWY binned 0.74 ± 0.12 AI flat LBTB binned 1.0 ± 0.3 Vanuatu BCAO/BGCA 0.14 ± 0.04 Vanuatu flat BCAO/BGCA 0.12 ± 0.04 Vanuatu BCAO/BGCA binned 0.16 ± 0.04 Vanuatu flat BCAO/BGCA binned 0.13 ± 0.03 3 RESULTS We shall measure temporal variations in the inner core in four different regions. If the inner core were to rotate differentially as a rigid body, the direction and rate of this rotation should be the same, independent of which region of the inner core is probed. If, however, we resolve different rotation directions and rates depending on where our data sample the inner core, then interpreting the temporal variations in residuals as stemming from differential rigid-body rotation of the inner core is not applicable on a global scale. 3.1 South Sandwich Islands events The South Sandwich Islands to Alaska (COL and Alaska Seismic Network stations) paths are the ones most commonly used to detect differential rotation of the inner core and to determine its rate (see e.g. Song & Richards 1996; Creager 1997; Zhang et al. 2005). Here we introduce the use of measurements at the northern Canadian stations INK and DAWY in addition to re-measured traveltime residuals at COL, and compare the temporal trends at these three stations to each other, making use of independent analysis of the inner-core lateral velocity structure these ray paths sample. If the inner core is rotating differentially, then all three stations should show the same rotation rate, in the same direction. To measure temporal variations and to relate them to differential motion of the inner core, we first need to determine the spatial velocity structure and its gradient sampled by PKPbc-PKPdf ray paths. We then use this lateral gradient as a marker, whose motion we seek to map over time to test the differential inner-core rotation hypothesis. The spatial residuals against the PKPdf turning longitude in the inner core for South Sandwich Islands-North America ray paths are shown in Fig. 4. The data used for spatial gradient calculation are South Sandwich Islands events recorded at ARCTIC experiment stations over 3 yr between 2004 and 2007 (filled blue circles). These ARCTIC data show a clear linear trend with PKPdf turning longitude in the inner core. Data recorded on the permanent Alaska Seismic Network (AK) during 2002 2004, and at COL, INK and

Figure 4. Spatial variation of PKPbc-PKPdf residuals for SSI events at all North American stations. Data are from the temporary PASSCAL experiment ARCTIC (filled blue circles) between 2004 and 2007, the permanent Alaska Seismic Network (AK, green circles) between 2002 and 2007, COL (filled red triangles), INK (filled purple inverted triangles) and DAWY (purple inverted triangles). The black dashed line indicates the linear regression fit to the ARCTIC data, with gradient m and intercept l. DAWY, are shown for comparison, but have not been used to determine the spatial gradient. The epicentral distances, and therefore the turning depths inside the inner core, for ARCTIC data are similar to those for COL, whereas INK and DAWY have shorter epicentral distances. There is no independent constraint for the spatial structure of the inner core for the INK and DAWY paths, as temporary experiments have not been run nearby these stations. However, as Fig. 4 indicates, the spatial structure of deeper parts of the inner core observed for the ARCTIC and COL data can reasonably be extended to shallower layers sampled by the INK and DAWY ray paths. Hence we can apply the same ARCTIC spatial gradient to all three stations. As the temporal extent of the ARCTIC data is small, we are able to attribute variations observed between ray paths to inner-core velocity structure variations in space, not time. We fit eq. (3) to the data and find a sufficiently strong spatial variation of 0.063 ± 0.005 s deg 1 (see Table 2) to allow for mapping out temporal variations in the core, including testing for the possibility of an inner-core differential rotation. The ray paths considered here are polar (i.e. the angle between the PKPdf ray and the Earth s symmetry axis at the PKPdf turning point inside the inner core, ζ,islessthan30 see Table 1). Any rotation of the Earth s inner core with respect to the mantle would be expected, on dynamic and energetic grounds, to be about the rotation axis (i.e. the north south axis). Therefore we are predominantly interested in variation of velocity structure between adjacent regions in the inner core in the longitudinal direction. It is these variations that would act as markers to follow on a differentially rotating inner core. Thus strong longitudinal spatial inner-core velocity structure variations, such as those observed here, are best suited as markers to map out inner-core rotation. We note that South Sandwich Islands-COL paths have been used extensively in previous innercore differential rotation studies (see e.g. Song & Richards 1996; Creager 1997; Song 2000a; Zhang et al. 2005; Song & Poupinet 2007; Zhang et al. 2008; Lindner et al. 2010). Our spatial gradient (see Table 2) of 0.063 s deg 1 eastwards is somewhat larger in magnitude than the spatial gradient of 0.0145 0.0278% deg 1 eastwards reported in (Song 2000a), albeit their study uses COL and Temporal variations in the Earth s inner core 361 AK data, not ARCTIC. ARCTIC data were first used in a smoothing spline analysis by Lindner et al. (2010), who treat time and space simultaneously. Having found such a strong and linear longitudinal spatial gradient, we turn our attention to temporal variation for similar ray paths. To study time variations, we measure PKPbc-PKPdf traveltime residuals for events in the South Sandwich Islands region at three North American stations over time spans of some 25 yr. First, we re-examine an independent data set of South Sandwich Islands events recorded at COL, and their traveltime residuals. Our COL measurements yield a temporal variation of 0.0080 ± 0.0034 s yr 1, as shown in Fig. 5(A) and Table 3. This is in agreement with values reported elsewhere for this station and geometry: 0.011 s yr 1 by Song & Richards (1996) and 0.009 s yr 1 by Zhang et al. (2005). After applying our space-flattening procedure of eq. (4) to South Sandwich Islands-COL residuals, the temporal trend persists (see Fig. 6A), and hence the observed time variation is genuine. The overall space-flattened temporal variation at COL is 0.0073 ± 0.0034 s yr 1. Extending this temporal analysis to two nearby stations, INK and DAWY in northern Canada, we observe further variations of PKPbc-PKPdf traveltime residuals with time. Fig. 5(B) shows that the linearly increasing trend observed at COL is also exhibited by INK data; the residuals themselves have smaller values than do those at COL, as expected from the difference in epicentral distances (ray paths to COL are longer, and so sample more of the inner core than do those to INK). The temporal gradient at INK is 0.024 ± 0.012 s yr 1, which is larger than that at COL. On the other hand, at DAWY (Fig. 5C), the residuals are scattered and do not vary with time in a linear fashion. This is in contrast to what one would expect from the differential rotation hypothesis and the observations at the other two stations, which are close to DAWY. As the ray paths to DAWY sample similar longitudes to AK stations, which in turn fall in the longitudinal trend of the ARCTIC and COL data, the temporal variation at DAWY might be expected to resemble the linear variation at INK. This is especially true as the DAWY ray paths turn at a depth between those to COL and INK in the inner core, and have the same azimuth as ray paths to COL. The absence of a clear linear trend at DAWY thus indicates that a differential rotation of the inner core, even if capable of explaining the results at COL and INK in light of the spatial gradient cannot account for the temporally scattered residuals at DAWY. Applying the space-flattening procedure to INK and DAWY, we find that the linear trend at INK persist, giving a space-flattened gradient of 0.014 ± 0.012 s yr 1 (see Fig. 6B). This is in agreement with the COL gradient within measurement errors. However, no linear trend arises at DAWY even after space-flattening (Fig. 6C). We take this to indicate that the differential rotation hypothesis may not be valid; if it were, the temporal trend at DAWY, both raw and at the very least space-flattened, should follow COL and INK and hence be linear. Further evidence for incompatibility between COL, INK and DAWY is provided by binning the temporal data in approximately 5-yr bins, and then fitting a straight line to these results. This is done to average out the scatter in the data. The temporal gradients at COL (binned raw 0.0079 ± 0.0010 s yr 1, binned space-flattened 0.0070 ± 0.0010 s yr 1 ) and INK (binned raw 0.023 ± 0.008 s yr 1, binned space-flattened 0.015 ± 0.008 s yr 1 ) are robust. At DAWY, binning the originally scattered data results in a linear temporal variation, giving gradients 0.040 ± 0.007 s yr 1 for raw residuals, and 0.046 ± 0.007 s yr 1 for space-flattened ones. These temporal gradients are comparable with those at INK in magnitude, but opposite in direction, contrary to what is expected for rigid-body rotation of the inner core.

362 A. M. Mäkinen and A. Deuss Figure 5. Temporal variations of PKPbc-PKPdf residuals for South Sandwich Islands events, observed at (A) COL, (B) INK and (C) DAWY. The black squares indicate 5-yr binned data averages, with their standard deviations. Blue dashed lines are the linear regression fits of gradient a and intercept b to the data. The grey line in (C) indicates the linear regression fit to binned DAWY data, with gradient a and intercept b. Incompatibility with the differential rotation hypothesis is also evident in examining the inner-core rotation rates inferred at these stations. A simple way to calculate such a rotation rate is to divide the temporal gradient (Table 3) by the spatial one (Table 2); rotation rates, calculated both with and without space-flattening, are given in Table 4. Results at both COL and INK yield negative rotation rates, that is, an eastward differential rotation of the inner core. This Figure 6. Similar to Fig. 5, but with the spatial (longitudinal) gradient in the inner core removed. is in agreement with previous studies (e.g. Zhang et al. 2005). The COL-inferred rotation rate of 0.13 ± 0.05 yr 1 (0.12 ± 0.05 yr 1 space-flattened) is somewhat smaller than that from doublet studies (rate of 0.27 0.53 yr 1 eastwards in Zhang et al. 2005) or the average over 55 yr in a previous COL study (0.39 yr 1 eastwards in Lindner et al. 2010). As the temporal gradient inferred in our study agrees with the previous ones, the most likely source of discrepancy lies in determining the spatial gradient. As temporal variations are observed, it is important to limit the data used in determination of the spatial gradient to seismograms collected over a limited number of years, as has been done in our study.

Yielding the same overall sense, the rotation rate at INK is 0.38 ± 0.19 yr 1 (0.23 ± 0.19 yr 1 space-flattened), in reasonable agreement with that at COL. Binned data at both COL and INK give similar rotation rates (see Table 4). However, together with a westward (positive) rotation rate of 0.63 ± 0.12 yr 1 (0.74 ± 0.12 yr 1 space-flattened) from binned DAWY data, we infer that a differential rotation of the inner core is not likely to best explain the joint temporal variations at these three North American stations. We note that most previous analyses on this topic have been performed using inner-core PKPdf traveltime normalized residuals, whereas we have quoted results obtained using unnormalized ones. We have chosen to do so since it is not certain how the radially varying anisotropy of the inner core may affect the overall results. Instead, we have decided not to assume a constant velocity structure, with residuals increasing with depth, which would be implied in dividing the observed residuals by the inner-core traveltime inherent in such normalization. Having tested for the effect of normalization on all residuals, we observe that our results presented herein are not affected by this choice. Furthermore, all results discussed earlier are robust with respect to cross-validation and 5-yr binning of both raw and space-flattened data. 3.2 Aleutian Islands events Events in the Aleutian Islands region lie in the correct epicentral range from stations in southern Africa, and the ray paths are semiequatorial (see Table 1). These paths have been used previously by Zhang et al. (2008) in a waveform doublet analysis, but an extended time study on individual events has not been carried out before. We observe and analyse temporal variations in traveltime residuals at two stations, BOSA and LBTB, in southern Africa, and study the inner-core lateral velocity structure using independent data, as for the South Sandwich Islands events discussed earlier. We use data from the temporary experiment SA to constrain the spatial inner-core velocity structure. The outcome of such an analysis is shown in Fig. 7. BOSA and LBTB residuals are shown on the same plot for comparison of ray path geometries. The SA residuals show a linear trend with PKPdf turning longitude in the Figure 7. Spatial variation of PKPbc-PKPdf residuals for Aleutian Islands events at stations in southern Africa. Data are from the temporary experiment SA in the southern part of the African continent (filled blue circles) between 1997 and 1999, BOSA (filled red triangles) and LBTB (filled purple inverted triangles). The black dashed line indicates the linear regression fit to the SA data, with gradient m and intercept l. The BOSA and LBTB data are shown for comparison. Temporal variations in the Earth s inner core 363 Figure 8. Temporal variations of PKPbc-PKPdf residuals for Aleutian Islands events, observed at (A) BOSA and (B) LBTB. The black squares indicate 5-yr binned data averages, with their standard deviations. inner core. Residuals at BOSA and LBTB follow the same trend, which confirms that using SA data to constrain the spatial variations in the inner-core velocity structure to study inner-core rotation at BOSA and LBTB is appropriate. As for South Sandwich Islands events, the variation in spatial structure observed using SA data, 0.015 ± 0.004 s deg 1, is quite strong. These data span the years 1997 1999, a sufficiently short time span to allow us to attribute the variations to space alone. Having found spatially varying structure to use as a marker for inner-core motion, we then study variations in traveltime residuals as a function of time. Temporal variations at BOSA and LBTB are indeed observed, as is evident in Figs 8(A) and (B). Overall the residuals are smaller than those at COL average 0.5 s at BOSA and LBTB compared to 3 s at COL. The Aleutian Islands residuals are also more scattered, spanning a range of 1.2 s compared to 0.5 s at COL; this is in agreement with the hemispherical differences and a more scattering eastern inner-core hemisphere (Cormier 2007). As in evident in Fig. 3(B), the ray paths to BOSA and LBTB cover a longitudinal range significantly wider than the South Sandwich Islands to North America paths discussed earlier. We use our longitudinal space-flattening procedure to correct for this variation. As a result, the scatter in the temporal data is reduced significantly for both stations (see Fig. 9). Following the space-flattening procedure we see that residuals at BOSA do not exhibit a linear trend (Fig. 9A), as would be expected for the differential inner-core rotation hypothesis. Instead, residuals at BOSA appear to increase prior to 2000, then decrease. Dividing the space-flattened BOSA data about the

364 A. M. Mäkinen and A. Deuss and INK for South Sandwich Islands events, although in the same eastward sense. Thus BOSA and LBTB binned space-flattened data are in disagreement; we also note that no oscillation was observed for South Sandwich Islands events. The errors in the gradients are, as before, derived from the maximum ranges of cross-validation and method comparison. Overall, the presence of a strong spatial longitudinal gradient and the absence of mutually consistent linear temporal trends at BOSA and LBTB indicate that the Aleutian Islands results are not compatible with a differentially rotating inner core. Binned space-flattened data at LBTB provide some evidence for an eastward rotation of the inner core, inconsistent with the inner-core oscillation indicated by data at BOSA. Figure 9. PKPbc-PKPdf residuals for Aleutian Islands events observed at (A) BOSA, (B) LBTB, with the spatial (longitudinal) gradient in the inner core removed. The black squares indicate 5-yr binned data averages, with their standard deviations. In (A), the purple dashed line on the right-hand side of the plot marks the linear regression fit, of gradient a and intercept b (on the right in the plot), to the downgoing part of the space-flattened time; the purple dashed line on the left-hand side marks an estimate of a similar fit to the earlier, upgoing, part of the space-flattened time. The grey line in (B) indicates the linear regression fit to space-flattened binned LBTB data, with gradient a and intercept b. year 2000, we fit a straight line according to eq. (2) to the downgoing part of space-flattened temporal residuals, obtaining a temporal gradient of 0.072 s yr 1. A similar line can be estimated to fit the upgoing part of the data, although the small number of data points prior to 2000 prevents the calculation of a linear regression fit. Overall, these fits could be taken as an indication of an oscillating inner core, a type of motion suggested by Collier & Helffrich (2001). This interpretation is purely speculative, as Fig. 9(A) only indicates that the trend in the temporal variation of space-flattened residuals could be divided into two linear sections. Although the BOSA data do appear to suggest an oscillation, a period for such a rigid-body motion cannot be inferred as the data time span is insufficient (only part of a possible oscillation is seen). These results are robust with respect to cross-validation and 5-yr binning of both raw and space-flattened data, as for the South Sandwich Islands events. We note that while eq. (2) cannot be fitted to raw or normalized residuals at LBTB, or to space-flattened data at this station, a straight line fits the binned space-flattened data at LBTB (see Fig. 9B and Table 3). The inner-core differential rotation rate resolved from these binned data is 1.0 ± 0.3 yr 1 (Table 4). This is an order of magnitude larger than the rates measured at COL 3.3 Kuril Islands events The paths from Kuril Islands to recording station BDF in Brazil are semi-equatorial (see Table 1). As for the South Sandwich Islands and Aleutian Islands events, we start our study of events in the Kuril Islands region by seeking temporary experiments to independently constrain the spatial velocity structure of the inner core. To this effect, we use data from the BL experiment, between years 2001 and 2006, with most of the events we use in 2003. We then proceed to analyse observations over a longer period of time to constrain temporal variations in traveltime residuals and to test the differential inner-core rotation hypothesis. Our inner-core spatial structure analysis for sources in the Kuril Islands region reveals a spatial longitudinal gradient of 0.0025 ± 0.0004 s deg 1, as shown in Fig. 10. Here we remark that this longitudinal gradient is an order of magnitude smaller than those for South Sandwich Islands-North America and Aleutian Islandssouthern Africa (see Table 2). As the spatial variation in innercore structure is small, also the temporal variations expected for a differentially rotating inner core are small, and thus more difficult to detect. Our observations at the station BDF show no clear temporal trend (see Fig. 11). As earlier, these results have been tested using cross-validation, comparison of raw, normalized and space-flattened residuals, and binning the data in approximately 5-yr bins. As for our South Sandwich Islands study, the PKPdf rays traverse the western hemisphere of the inner core (see Fig. 3C). However, for Figure 10. Spatial variation of PKPbc-PKPdf residuals for Kuril Islands events. Data are from the temporary experiment BL (filled blue circles) between 2001 and 2006, with most data from events in 2003. The black dashed line indicates the linear regression fit to the data, with gradient m and intercept l.

Temporal variations in the Earth s inner core 365 Figure 11. Temporal variations of PKPbc-PKPdf residuals for Kuril Islands events, observed at BDF. The black squares indicate 5-yr binned data averages, with their standard deviations. the Kuril Islands events the traveltime residuals are generally small, making the detection of the linear increase that would be required to agree with the South Sandwich Islands-COL inner-core rotation rate rather challenging. Indeed, utilizing the COL-inferred inner-core rotation rate we find that, given the observed spatial variation shown in Fig. 10, the expected temporal variation would be somewhere on the order of 0.0003 s yr 1, or 0.003 s per decade. This is an order of magnitude smaller than at the stations used for South Sandwich Islands and Aleutian Islands events, and so small as to not be resolvable using our current method. Hence our results using the Kuril Islands data remain inconclusive, though small variations with time are clear in the PKPbc-PKPdf residuals. 3.4 Vanuatu region events Finally, we study events in the Vanuatu region to further elucidate the trends in temporal variations in the inner core and their possible interpretations. All events in the Vanuatu region have been observed at the stations BCAO and BGCA in the Central African Republic. These ray paths are nearly purely equatorial, with ζ approximately 80 (see Table 1). Station BCAO has been operational for the time period 1979 1990, whereas station BGCA has records for 1994 2002. The separation between these two stations, some 74 km on the Earth s surface, is greater than for COL and COLA or BDF and BDFB, which are essentially at the same site. However, we note that the trends observed in our analysis arise independently for both stations, and are in agreement. The residuals have been calculated using actual station coordinates, and the correspondence between BCAO and BGCA appears sufficient to warrant treating these two stations the same way as COL and COLA or BDF and BDFB. As the station location is seismologically somewhat isolated, we have not been able to collect independent data to constrain spatial variations in inner-core velocity structure. However, as the rays travel almost parallel to the equatorial plane, the effect of ray-lateral variations on traveltime residuals is easy to remove by space-flattening in latitude. Additionally, one would expect little temporal variation in such strongly equatorial paths due to innercore rotation, about the north south axis, even in the presence of a strong longitudinal gradient, as the longitudinal range of different paths is small. To study the spatial variation of velocity structure, we start by extracting the longitudinal gradient from the PKPbc-PKPdf residu- Figure 12. Longitudinal spatial variations of PKPbc-PKPdf residuals for Vanuatu events, observed at BCAO and BGCA. The black dashed line is the linear regression fit of gradient m and intercept l to combined BCAO/BGCA data. Figure 13. Temporal variations of PKPbc-PKPdf residuals for Vanuatu events, observed at BCAO and BGCA. The black squares indicate 5-yr binned data averages, with their standard deviations. The blue dashed line is the linear regression fit of gradient a and intercept b to combined BCAO/BGCA data. als at BCAO/BGCA (Fig. 12), obtaining 0.045 ± 0.004 s deg 1. Again having found a strong spatial longitudinal gradient, we turn our attention to the temporal variations of PKPbc-PKPdf residuals. These variations with time are shown in Fig. 13. The time variations are of the same order of magnitude as those observed at COL for South Sandwich Islands events, 0.0064 ± 0.0017 s yr 1,but here decreasing with event date (also see Table 3). We note that as the spatial and temporal data here are not independent, but derived from the same data set, the rotation rate derived from these data is not as robust as those from South Sandwich Islands-COL and South Sandwich Islands-INK data. Nonetheless, particularly with the space-flattening procedure to account for structure variation with latitude, the rotation rate derived later has some significance. We apply the space-flattening procedure in the latitudinal direction, instead of longitude as for other source regions. This is to account for the variation in PKPdf turning latitude inside the inner core, given the nearly purely equatorial ray paths. Since no independent spatial data exist, the latitudinal gradient is determined by plotting the BCAO/BGCA traveltime residuals as a function of