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 The lateral variations in the fluid outer core are believed to be very small from fluid dynamics calculations. Seismological studies on the issue have been limited and controversial. A great challenge is to sort out influences from heterogeneity in the mantle or the inner core. Using high-quality earthquake waveform doublets, we found that waves passing through the fluid core over a few years are significantly more variable than those passing through the mantle only. We interpret the temporal variability as the result of the fluid motion of the heterogeneous materials in the outer core. The level of heterogeneity in the fluid outer core is constrained to be ±0.022 s (95% confidence) in seismic travel times through the core, negligible for most seismological studies. However, the estimated velocity perturbation, about 10−3 for small-scale heterogeneity (10 km) or 10−4 for large-scale heterogeneity (1000 km), borders or exceeds the high-end estimates of the lateral variations that can be supported by dynamic forces within the fluid core. The source of the heterogeneity is not clear at present.
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 The Earth's outer core is made of iron-nickel alloy with some light elements, where thermal and chemical convection has generated and maintained the magnetic field of the Earth. Fluid dynamics calculations suggest lateral variations in the outer core are small [Stevenson, 1987], based on which seismological studies have commonly assumed that the lateral variations are negligible. Direct seismological studies on the issue have been limited and controversial. Some suggest significant lateral variations [Gudmundsson, 1989; Souriau and Poupinet, 1990; Widmer et al., 1992; Tanaka and Hamaguchi, 1993; Romanowicz and Bréger, 2000; Yu et al., 2005], but others suggest little [Souriau and Poupinet, 1991; Ishii and Dziewonski, 2005]. A great challenge is to sort out influences from heterogeneity and anisotropy in the mantle or the inner core on a given seismic observation. Here we report direct evidence for the outer core motion from high-quality earthquake waveform doublets and infer the level of heterogeneity in the fluid outer core.
 We use a similar method that has been used for detecting the rotation of the solid inner core, i.e., comparing seismic waves for a fixed path traversing the Earth's core from repeating sources [Song and Richards, 1996; Zhang et al., 2005]. The basic idea was proposed 25 years ago by Dr. Paul Richards (in a proposal to the U.S. National Science Foundation). He used the analogy that stars “twinkle” as the atmosphere refraction index changes from moving cold and warm air. Thus, as the outer core fluid convects, the changes of the velocity (and density) of the materials crossing the fixed ray path will affect the seismic travel times.
 We use earthquake waveform doublet as repeating sources, i.e., a pair of earthquakes occurring at essentially the same spatial position, as evidenced by their highly similar waveforms at the same station. The existence of such a waveform doublet has been known for a long time [Poupinet et al., 1984]. In recent years, teleseismic waveform doublets, i.e., those with sufficient magnitudes (around 5 and above) to be observable far away, have been discovered. They have been used to detect the inner core rotation [Li and Richards, 2003; Zhang et al., 2005] and the inner core boundary topography [Wen, 2006; Cao et al., 2007; Song and Dai, 2008].
2. Data and Method
 Our teleseismic doublets would not only have to be of the highest quality, as any signals from the outer core are likely to be very small, but also have sufficient data that sample the core and the mantle. We found four best doublets that satisfy our requirements (Table 1). Examples of waveform similarity are shown in Figure 1 and overlays of all the seismograms used in this study are shown in the auxiliary material. The high similarity of the doublet waveforms, including later arrivals and coda waveforms, is critical for precise measurements of relative time shifts. Two doublet pairs are from the South Sandwich Islands (9303 and 9804), which were published by Zhang et al.  and Song and Dai , respectively. The other two pairs are new doublets from the Fiji-Tonga region. The magnitudes (mb) of the events range from 4.7 to 5.6. The time separations between the two doublet events range from 3.5 to 11.8 years. The average cross-correlation coefficients of the doublets range from 0.95 to 0.99 over a time period of 10 s that includes P and its coda. We used the vertical component of broad-band channel (BHZ), the short-period channel (SHZ), or extremely short-period channel (EHZ). For each station, we used the same channel for the two events of the doublet. To increase the signal to noise ratio, we filter all the data using a band-pass filter from 0.6 to 3 Hz. Broad-band data are converted to WWSSN short-period instrument response before the filtering.
Table 1. Earthquake Waveform Doublets Used in This Studya
The earthquake locations are from the PDE catalog of USGS.
C.C. is the averaged waveform cross-correlation coefficient of the doublet calculated using a 10-s time window after the onset of P. The number in the parentheses indicates the number of stations used in the calculation.
 The key of our approach is to use relative measurements of all kinds (between events and between phases) to reduce errors. We formulate the relative time shift (ddt) between two phases of the same doublet as follows.
where, for example, T1(p1) is the travel time of event 1 and phase p1, and dt(p1) = T2(p1)-T1(p1) is the travel-time shift of phase p1 between event 1 (the earlier event) and event 2 (the later event) of the doublet. The travel-time shifts are measured using waveform correlation in time domain. The sampling interval is first interpolated to 0.0005 s before cross-correlation. Typical cross-correlation time window is 2-3 s. The relative time shift, dtt(p2-p1) between phase p1 and phase p2, is then calculated using dt(p2)-dt(p1).The ddt formulation removes completely origin time errors. It also removes possible clock errors, unless there is a clock drift in the short duration between phases p1 and p2 for either event.
 For each doublet, we try to get as much data as possible from the global and regional networks. We look for two types of data: a pair of seismic phases that sample the core, and phase pairs that sample the mantle only (Figure 1). (1) We use the outer-core branches of PKP (BC and AB) and define ddt(core) = ddt(AB − BC). We avoid the inner core branches of PKP (DF and CD) because of the known influence from the inner core rotation and topography as mentioned above. Almost all of the BC and AB phases (at distances 146-155°) are well separated (by 2 to 15 s in time) and easy to identify. In a few cases where they do not appear as distinct arrivals, we use predicted travel times as a guide in selecting the time windows for cross-correlation. (2) Any pair of seismic phases that sample the mantle only can be used for mantle reference phases as long as they are energetic enough. Our main reference phases are P, PcP, and pP (Figure 1). All of our first arriving P waves are energetic, impulsive, and easy to identify. Identifying later arriving PcP waves is more difficult. We use predicted relative travel times between P and PcP to guide our selection of time windows. The PcP phase arrives about 259 s after P at our smallest distance for PcP (19.6° for 9303-PMSA) and 5 s after P at our largest distance (82.5° for 9303-SJG) (see auxiliary material). Some of our PcP data appear at the coda of P at distances around 80° (9002-CMB, 9002-PFO, and 9303-SJG). In such a case, we choose the energetic cycle around the predicted time for PcP (relative to first arriving P). Our pP data are from doublet 0004 at distances of 77–80°, where pP arrives at about the same time as PcP and in many cases is affected by P coda. Nevertheless, we call it pP because of the agreement in slowness and the pP phase is probably more energetic than PcP. In these cases, the associated “phases” are not definitive and are affected by the energy from mantle scattering. However, the distinction is not critical as long as the energy comes from the mantle and not from the core. Other mantle phases include Pn and PnPn for two pairs at smallest distances of our data (0004-AFI at distance 7.0° and 0004-RAR at 13.7°). We call these relative time shifts as ddt(mantle). In all, we obtained 51 ddt(core) measurements and 26 ddt(mantle) measurements.
3. Evidence for Outer Core Motion
Figure 2 summarizes our observed relative time shifts for the core phases, ddt(core), and for the mantle phases, ddt(mantle). The relative time shifts for the mantle phases are very small (all within ±0.02 s), which confirms the quality of these waveform doublets and provides a measure of the precision of our relative time measurements. The relative time shifts for the core phases are also very small (within±0.04 s), although the largest time shifts (of about 0.04 s) are visible under careful examination (Figure 1). The standard deviation of the ddt(core) values is 0.0167 s. These numbers put a tight constraint on the upper limit of the temporal variability of the fluid outer core.
 The data from the core are more scattered than those from the mantle. The standard deviation of the ddt(core) values (0.0167 s, above) is more than twice as large as the standard deviation of the ddt(mantle) values (0.0071 s). The difference in the data scatter is clear whether the two events are separated in time by four years or by ten years or more (Figure 2a). Possible sources for the discrepancy include (1) measurement errors, (2) small location difference between the two earthquakes of the waveform doublet, (3) mantle heterogeneity, and (4) the temporal variations of the fluid outer core. Our analyses below suggest that (1) and (2) can be ruled out and that (3) is probably not the cause either.
 One possibility is that the PKP data have a higher noise level. To this end, we compare the ddt values and signal-to-noise (SN) ratios for the mantle and the core phases (Figure 3a). The SN ratio is defined as the peak amplitude of the phase divided by the root-mean-square (rms) amplitude of the background noise preceding the first P or PKP arrival. For each doublet recorded at each station, we obtain four SN ratios (two phases for each of the two events). We then choose the smallest value as the SN ratio corresponding to the ddt value, because the phase with the smallest SN ratio has the greatest impact on the ddt measurement error. We observe that (1) the SN ratios of the core phases are not smaller than those of the mantle phases and (2) at different SN ratios, the scatter of ddt(core) is always larger than that of ddt(mantle)(Figure 3a). We thus rule out measurement errors as the source of the larger scatter for the core phases.
 Although the waveform doublets are of the highest quality, the two events of the doublet may not be at the exactly the same spatial location. We argue that the earthquake location cannot be the cause for the larger scatter of the core data, based directly on the observations of the ddt values as functions of distance and azimuths (Figures 3b and 3c). First, at smaller distances (20–50°), the ray parameter difference between P and PcP is much (3–5 times) greater than that at larger distances or that between PKP(BC) and (AB). Yet, the ddt(PcP-P) values do not increase at these distances (Figure 3b). Second, horizontal location difference would suggest systematic change with azimuth. Yet, there is no obvious change of the ddt(PcP-P) or ddt(AB-BC) values with azimuth (Figure 3c). Furthermore, the scatter of ddt(AB-BC) occurs at similar azimuths, around 300° and 350°, which cannot be explained by horizontal difference in the earthquake location. Thirdly, the core data scatter cannot be explained by depth difference in earthquake location. The ddt(PcP-P) value is 4 times more sensitive and the ddt(pP-P) value is 57 times more sensitive than the ddt(AB-BC) value to the depth difference. Based on these observations, we rule out earthquake location difference as a viable source for the larger scatter of the core data.
 If the locations of the two events of the doublet differ slightly, the ray paths through the mantle would also differ slightly. We argue that the influence from mantle heterogeneity is probably not the cause either. First, the paths of the same core phases from the two events of the doublet would be identical throughout the mantle (less than the location difference between the events or less than 1 km) [Zhang et al., 2005; Wen, 2006], which is much smaller than the wavelength or Fresnel zone at the dominant period of 1 Hz. Thus, it is not clear how much mantle heterogeneity would influence the time shifts. Second, compared with the core phase, the mantle phases sample more diverse regions of the Earth's interior with strong heterogeneity (from the crust and the shallow mantle to the core-mantle boundary). Thus, we would expect that the influence of mantle heterogeneity on the core phases be less than that on the mantle phases. Third, some ddt(PcP-P) measurements and all the ddt(pP-P) measurements were made at distances 75–82°, where PcP or pP arrives in the coda of P. The coda wave trains may be strongly influenced by scattering energy. Mantle heterogeneity should have greater influence on the coda [Snieder et al., 2002] than first arrivals. Thus we'd expect greater mantle influence on the ddt(mantle) measurements than on the ddt(core) measurements.
 In summary, the larger scatter in dtt(core) may not be explained by (1) measurement errors, (2) slight location difference between the two events, or (3) influence from mantle heterogeneity. The most likely source is the motion in the outer core—The larger scatter in the ddt(core) data is the result of the fluid motion of the heterogeneous materials in the outer core.
4. Level of Outer Core Heterogeneity
 The evidence for outer core motion can be used to estimate the level of the heterogeneity in the outer core. The ddt(core) consists of AB and BC time shifts from the core as well as time shifts from outside the core (sources 1 to 3 above). Assuming all the contributions are independent, the variance of ddt(core): Var(ddt(core)) = Var(AB) + Var(BC) + Var(outsidethecore). The travel times of AB and BC through the outer core are similar, about 530 and 650 s (at the average distance of 151°), respectively. If we assume that the variances from the core contributions Var(AB) and Var(BC) are the same and we further assume that the contribution from outside the core is similar to that from ddt(mantle), i.e. Var(outside the core) = Var(ddt(mantle)), we derive the standard deviation of the AB or BC temporal travel-time shifts to be 0.011 s. We take this value to be the level of heterogeneity in the fluid outer core, i.e., ±0.022 s (95% confidence) or about 0.004% of the total travel time of the compressional wave through the outer core. This level of heterogeneity is insignificant for most seismological studies and can indeed be safely ignored.
 Estimates on velocity perturbation depend on the scale length l of the heterogeneity in the outer core. As the total travel-time perturbation is the sum of the perturbations over the ray path, the level of heterogeneity is inversely proportional to , assuming random motion or random distribution of the heterogeneous materials. For l = 1000, 100, and 10 km, the standard deviation of the velocity perturbation is estimated to be 4 × 10−5, 1 × 10−4, and 4 × 10−4, respectively. Thus the limits of the velocity perturbation are about ±10−4 for l = 1000 km and ±10−3 for l = 10 km at 95% confidence. Both positive and negative travel time perturbations occur along similar paths (Figure 2b) and over a few years (Figure 2a), suggesting small-scale heterogeneity. This favors the high-end estimate of velocity perturbation (on the order of 10−3).
 The data that sample the outer core polar region inside the tangent cylinder (from Fiji-Tonga to Europe) seem to indicate greater variability than the data that sample the outside equatorial region (from South Sandwich Islands to Alaska and East Asia) (Figure 2b). The standard deviations are 0.0120 s and 0.0197 s for the equatorial and polar samples, respectively. The standard deviations for the mantle phases from the South Sandwich Islands doublets and the Fiji-Tonga doublets differ only slightly (0.0051 s and 0.0076 s, respectively). The observation may indicate greater level of heterogeneity inside the tangent cylinder of the fluid core than outside.
5. Conclusion and Discussion
 We report direct evidence for the outer core motion from high-quality earthquake waveform doublets–The observed larger scatter in the ddt(core) data is the result of the fluid motion of the heterogeneous materials in the outer core. The level of heterogeneity in the fluid outer core is constrained to be ±0.022 s (95% confidence) in seismic travel times through the core, which is indeed negligible for most seismological studies. The velocity perturbation is about 10−3 for small-scale heterogeneity (10 km) or 10−4 for large-scale heterogeneity (1000 km).
 Our estimate of the overall velocity perturbation (10−4 to 10−3) borders or exceeds the high-end estimates of the lateral variations (10−8 to 10−4) that can be supported by dynamic forces within the fluid core [Stevenson, 1987]. The source of the observed outer core variability is not known at present. Gravitational forcing from external mass anomalies (in the mantle or the inner core or from the core-mantle boundary topography or the inner core boundary topography) can cause lateral structure inside the fluid core [Wahr and de Vries, 1989]. However, such lateral variations do not generate fluid vorticity and thus cannot be a source of the observed time variability (D. Stevenson, personal communication, 2008).
 We are grateful to Dave Stevenson and Barbara Romanowicz for their insightful comments. We obtained the waveform data from IRIS DMC, GRSN data center, Orfeus data center, Alaska Seismic Network Data Center, SCEC Data Center, and Northern California Earthquake Data Center. We used data from GSN and some 45 global and regional seismic networks that contributed to these data centers. This work was supported by NSF-0330749.