Corresponding author: S. Ni, State Key Laboratory of Geodesy and Earth's Dynamics, Institute of Geodesy and Geophysics, Wuhan 430077, China. (email@example.com)
 We report strong short period (0.5–1.2 Hz) precursors to P′P′df at podal to short distances (0–50°) with advance time around 60s. From FK analysis, we find that the precursors at epicentral distance of ∼35° arrive along back-azimuth about +/−120° off the great circle paths (asymmetrical path) with slowness between 2–4 sec/deg, arguing against propagation path in the inner core. Polarization analysis also supports wave propagation in the outer core. Timing and shape of the precursors' waveform envelopes are well matched with synthetic envelopes taking into account the scattering from the rough free surface and volumetric heterogeneities inside the Earth along asymmetrical propagation path. Therefore the precursors are very probably generated with asymmetrical scattering mechanism. This interpretation does not require extra discontinuities or a layer of strong small scale heterogeneities in the upper mantle, which are inferred from P′P′ precursors when only symmetrical scattering is assumed.
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 Short scale heterogeneities in the mantle have significant implications for geodynamical studies. For example, short scale heterogeneities (∼10 km) almost only result from chemical mixing in the Earth, because temperature anomalies are expected to be smooth with rapid thermal diffusions [Ni et al., 2002; Shearer and Earle, 2008]. Short period (∼1 Hz) scattered waves are effective probes to the Earth's structure with scales around 10 km. Among them, precursors to P′P′ have the advantage of less contamination from P coda, and have been applied to studies of scatterers [Earle et al., 2011]. Precursors to P′P′ with advance time less than 70 s have been interpreted as evidence for discontinuities at depths from 50–190 km, assuming that the precursors are associated with symmetric reflection paths (hereafter referred to as the symmetric scattering mechanism) [Adams, 1968]. Alternatively, the propagation path could also be asymmetrical (Figure 1a, dashed lines) [King and Cleary, 1974; Haddon et al., 1977; Vinnik, 1981] (hereafter referred to as the asymmetrical scattering mechanism). At short distances of 30–50°, Earle et al. demonstrated that the P′P′ precursors are from off-azimuth path with fk analysis of waveform data at LASA array. Based on further studies with Monte-Carlo simulations, they proposed that the precursors most likely originate from scattering of PKPbc-to-PKPbc in the Earth's crust and upper mantle.
 But the symmetrical reflection is claimed to be responsible for the strong precursors to P′P′ at near-podal distances (∼7 degrees), and a layer of scatterers at depth of 150–210 km is proposed (Figure 1a) [Tkalcic et al., 2006]. Such a layer of small scale heterogeneity on global scale would have significant geodynamical implications. For example, it may help resolve controversies regarding the 210 discontinuity which is present in the global model of PREM, but has not been observed globally yet [Deuss and Woodhouse, 2004]. However, if the precursors are associated with asymmetrical paths, crustal or shallow mantle heterogeneity may generate the precursors, and this layer is not necessarily required. Therefore, more seismological investigations are needed to resolve which scattering mechanism is involved in generating the precursors at near-podal distances.
 In the following sections, we present observation of strong precursors to P′P′ for almost podal distances (∼1°) up to 50°, then perform array analysis to determine their slowness vector and match observation with synthetic waveform envelopes to demonstrate that the rough free surface or shallow heterogeneities with asymmetrical scattering mechanism is capable of generating the precursors.
2. Data and Analysis
2.1. Observation of P′P′ Precursors
 Following notation by Earle et al. , we label precursors due to propagation along the symmetrical path of reflection at depth d as P′dP′. For example, P′210P′ represents the precursors generated from reflection/scattering at depth of 210 km (Figures 1a and 1b). As for signals due to scattering along the asymmetrical paths (dashed line in Figure 1a), we use “•d•” to represent the scattering at the depth of “d”. The asymmetrical scattering includes the rough free surface scattering (P′•0•P′), and scattering due to volumetric heterogeneity which includes two mechanisms: the back scattering (P′•d•P′) and the forward scattering(P′•d•pP′ and P′p•d•P′) (Figure 1c). The forward scattering P′•d•pP′ represents the PKP scattered by the heterogeneity at depth of d, then up to free surface as “p” and reflected as PKP according to Snell law. There is also the phase of P′p•d•P′ due to reciprocity between source and receiver. The back scattering P′•d•P′(red dashed line) is the PKP wave scattered by heterogeneity at depth d and directly back to the receiver. P′•d•P′ arrives earlier than P′•0•P′, while P′•d•pP′ slight later than P′•0•P′ (Figure 1d). For small d, the phases P′•d•pP′ and P′p•d•P′ arrive almost at the same time as P′•0•P′, so we use P′surfP′ to denote the precursors associated with the three phases because they are all caused by interaction on the free surface.
 Global stations record strong short period P′P′ precursors at epicentral distances from almost-podal distance of ∼1° to short distance of 50° , for almost any deep earthquake stronger than Mw 6 (Figure 2a). The waveform data in Figure 2 are obtained from broadband seismic stations of IRIS DMC and CENC (China Earthquake Networks Center) for earthquakes deeper than 70 km (Table S1 in the auxiliary material). P′P′ and its precursors are usually not identifiable on raw broadband seismograms, but after short period filtering (0.5–1.2 Hz, 4th order Butterworth bandpass) strong signals show up clearly about 50–70 seconds before the theoretical arrival of P′P′df (Figure 2b). As short period seismograms are difficult to model wiggle by wiggle, smoothed envelopes are adopted and displayed in Figure 2c [Shearer and Earle, 2008]. The waveform envelopes of P′P′ precursors are also observed for distances up to 50 degrees, but P′P′df is not readily recognizable, probably due to its low amplitude because it is strongly attenuated at short period after twice traverse in the inner core. Unlike the emergent onset and long duration of the precursors, strong PKKKP (P3KP) is easily identified with sharp onset and short duration (Figure 2c). The strong P3KP wave may present contamination to studies of P′P′ precursors, especially for distances around 18 degrees, where they cross over. Two distance windows can be chosen to avoid the contamination, that is, either for distances larger than 30 degrees where P3KP coda is well ahead of P′P′ precursors, or for distances less than 10 degrees where P3KP arrives much later than P′P′ precursors. At lower frequency (0.5–0.9 Hz), P′P′ precursors are identifiable but with much lower signal noise ratio (SNR). In this band, P′P′df seems to be coherent (Figure S1a) as expected from lower absorption in the inner core at lower frequencies. At higher frequency (1–3 Hz), P′P′ precursors are still observable (Figure S1b). The different frequency dependences between P′P′ precursors and P′P′df suggests that they propagate with different paths. One possible scenario is that the waves associated with the precursors do not propagate in the highly attenuating inner core (Q = 300 − 600). Another scenario is that the scattered waves do not propagate in asthenosphere [Tkalcic et al., 2006]. However, P′P′df travels almost 4800 km in the inner core and only about 200 km in the asthenosphere, so the inner core is more effective in attenuating high frequency waves. Therefore, the precursors are more likely caused by waves not travelling in the inner core.
 The seismogram envelopes in Figure 2c have been aligned after correction of travel time due to different focal depths. The precursors have almost constant travel time of 2350 s after the earthquake origin time for all distances, and are about 60 seconds earlier than P′P′df [Earle, 2002; Rost and Earle, 2010]. In contrast, P′P′df is expected to show substantial move-outs for distances from 0 to 50 degrees (Figure 2c). Different from seismic phase associated with geometrical rays which usually feature impulsive onsets, the envelope of P′P′ precursors behaves like a spindle with emergent onset and slow decaying tail. They rise gradually starting at 2350 sec, reach the peak about 20 sec later, then return to noise level about 60 sec later. The emergent onset is suggestive of wave origin due to scattering [Vidale and Earle, 2000]. These observations are similar with that of Earle et al. .
 According to the symmetrical scattering hypothesis, P′210P′ also arrives at 60s before P′P′df [Tkalcic et al., 2006]. But their study only samples two isolated areas of the Earth, and our dataset consists of global observation of P′P′ precursors at near-podal distances. As the 150–210 km discontinuity or strong heterogeneity layer is not a global structure [Lehmann, 1961; Benz and Vidale, 1993; Rost and Weber, 2001; Deuss and Woodhouse, 2004], our dataset of globally observed precursors are probably not caused by the heterogeneities at depth of 150–210 km.
2.2. Synthetic Envelopes With the Asymmetrical Scattering Mechanism
 Though travel time of P′210P′ is similar to that of P′surfP′ or P′•d•P′ for small d, their slowness and direction of arrival are different. As the rays propagate in the inner core, the slowness of P′210P′ is similar to that of P′P′df and should be less than 1.9 s/deg. In contrast, the slowness of P′surfP′ or P′•d•P′ should be larger because the ray paths only sample the outer core. Moreover, P′210P′ is expected to arrive from the great circle path. So the frequency-wave number analysis (fk-analysis) can be used to resolve where the scattered wave is generated [Rost and Thomas, 2002].
Figure S2 displays fk analysis of P′P′ precursors observed on Yellow Knnife (YK) and Warramunga (WR) arrays (with the Generic Array Processing Code by K. Koper, Source code available on ftp://ftp.eas.slu.edu/pub/koper/gap.1.0.tar.gz). From the fk analysis, it is observed that the precursors arrive with azimuth of about +/−120° off the great circle paths and with slowness larger than 1.9 sec/deg. Ray-tracing with PREM predicts that P′surfP′ or P′•d•P′ arrive with 120 degrees deviation from great circle path at epicentral distances around 30 degrees (Figure S3). These results are direct evidences to support the hypothesis of asymmetrical scattering, and are consistent with Earle et al.'s study. The polarization from three-components waveform records provides extra constraints on the propagation paths of scattered waves. Because of the much smaller ray parameter of the P′210P′ wave at podal distances, the radial component of PKPbc or PKPab is much larger than that of PKPdf at antipodal distances.Figure S4shows that the radial component of P′P′ precursor at near-podal distance is similar to that of PKPbc or PKPab near the caustic distance 145° , consistent with the asymmetrical path of P′surfP′ or P′•d•P′. In contrast, it is much larger than that of PKPdf at antipodal distances, as expected for the symmetric path of P′210P′. This is a direct evidence supporting that the asymmetric scattering mechanism is responsible for the P′P′ precursors.
 The envelope shapes of the precursors provide further evidences for asymmetric scattering mechanism. For example, when seismic waves are generated only with asymmetric scattering and not associated with any geometric arrivals, waveforms show emergent onset and spindle-like envelope shapes [Shearer and Earle, 2008]. Moreover, duration of the waveform envelopes is long for scattering from large area, for example, Vidale and Earle  reported a seismic phase with long duration due to the inner core scattering. The specific behaviors (shown in Figure 1c) of each scattering mechanism will be demonstrated with synthetic enveloped seismograms. Since the crust is the most heterogeneous portion of the earth, for simplicity we just assume heterogeneities in the upper most 10 km crust, though the deeper crust and upper mantle's heterogeneities can also contribute to the scattering.
 We use ray theory to compute synthetic seismograms for scattering from the topography of free surface and the forward and back scattering caused by the volumetric heterogeneities [Scott and Helmberger, 1983; Wu and Aki, 1985; Sato and Fehler, 1998]. We only consider single scattering events, and this might underestimate effects due to multiple scattering which would better be modeled with Monte Carlo methods [Shearer and Earle, 2008]. We use ray tracing for PREM model to calculate the geometric spreading factor and transmission coefficient at core mantle boundary. Combination of PKPab and PKPbc leads to four pairs of scattering paths whose energy contributions are calculated respectively. Our methods are only valid for smooth topography and weak volumetric heterogeneities. The free surface's topography of the Earth is described with Gaussian autocorrelation function (ACF), , where h is the root of mean square (rms) height variation and L is the autocorrelation length. The rms h is larger than 50 m for most free surface's topography and the autocorrelation length ranges from ∼1 km to ∼10 km [Goff et al., 1997]. The upper mantle and crust's heterogeneities is described with exponential ACF, [Sato and Fehler, 1998], where ε is the rms variation of either seismic velocities or densities, and a is the autocorrelation length. The rms ε of S wave perturbation can be up to 10% for shallow crust [Sato, 1984] and a ranges from ∼100 meters to 10 kilometers [Capon, 1974; Wu and Aki, 1985; Yoshimoto et al., 1997]. According to the scattering theory, more scattering is excited in the forward scattering when a increases, so the forward scattering is strong for ak > 1while k is the wave number of incident waves [Wu and Aki, 1985; Sato and Fehler, 1998].
 The calculation of the amplitude of short period seismic waves is difficult since the quality factor Q at short period is one of the challenging parameters to be well resolved. But it is fortunate that the ray path of rough surface's scattering is almost the same as the ray path of scattering from shallow heterogeneity. So we assume their attenuation effects are same. We adopt isotropic radiation pattern of large earthquakes at high frequency [Boore, 2003]. We assume h = 100 m and L = 7 km for free surface topography. For the crust, we choose seismic velocities (P and S) and density perturbation of rms ε = 5%, and two correlation length a = 2 km and 13 km are tested respectively. From previous studies, a is taken to be larger than 10 km [Capon, 1974; Wu and Aki, 1985]. Figure 3a illustrates the back and forward scattered energy (black and green line) caused by heterogeneities within the upper most 10 km in the crust and the rough surface (red line). From Figure 3a we observe that the forward scattering is strongest, then the rough surface scattering, while the back scattering is the weakest. The forward scattering is two order of magnitudes stronger than back scattering for a = 13 km. For the case of a = 2 km, the forward scattering is a little bit weaker, but still dozens of time stronger than the back scattering mechanism. So it demonstrates that the main contribution of P′P′ precursor should be from the heterogeneity's forward scattering and the rough free surface scattering but not from the heterogeneity's back scattering.
 The synthetics in Figure 3a do not take earthquake source function into account, therefore they are better named as Green's function. In order to compare the synthetics with observed data, we take the direct P wave's envelope as source time function and convolve it with the Green's function. Figure 3b shows the synthetics (red line) and the observed data (black line) for a deep event (020628). The synthetics' onset times and shapes match the observed data well, both showing rapid increase of amplitude at 2290 s and gradual decay about 30s later. More data and synthetics are displayed in Figure S5, where we just model the P′P′ precursors at distance <10° to avoid the P3KP interference. Therefore, asymmetric scattering from free surface roughness and shallow heterogeneities is capable of explaining both the timing, duration and waveform feature of observed P′P′ precursor. Furthermore, we estimate strength of the asymmetrical (P′surfP′ + P′•d•P′) and symmetrical scattering (P′210P′) based on the mantle Q model by Shearer and Earle  and the core Q model by Li and Cormier . We find that the symmetrical scattering should be two orders of magnitude weaker than the asymmetrical scattering (Figure 3c).
3. Discussion and Conclusion
 Precursors to P′P′ and other multiples phases (SS, PP, etc) are important in studying mantle discontinuities for the case of symmetric paths, and indeed have led to significant discoveries of depth and sharpness of 410 and 660 discontinuities [Shearer and Flanagan, 1999; Xu et al., 2003]. Maybe there are substantial heterogeneities at depths of 150–210 km, and we can not rule out that some components of P′P′ precursors are from the symmetric back scattering of P′210P′. However, the crust as one of the most inhomogeneous portion in the Earth and the free surface as the strongest discontinuity imply that the globally observed short period P′P′ precursors are probably dominated by P′surfP′ and P′•d•P′. The nearly constant onset travel time and deviation of direction of arrival from great circle path with array analysis argue that the asymmetrical scattering P′surfP′ or P′•d•P′ as the source of P′P′ precursor. The polarization analysis provides extra direct evidences of asymmetrical scattering for the near-podal distances. Recent studies also demonstrated that the asymmetrical scattering is responsible for precursors to P′P′ at distances from 30–50 degrees [Earle et al., 2011], consistent with our near-podal analysis. Moreover, our study consists of seismic records from global stations and earthquakes, thus suggesting that the asymmetrical scattering mechanism is applicable for the whole Earth. However, we can not specify whether the rough free surface or the volume heterogeneity inside the Earth is the main contributor because of the limited knowledge of the global crust and upper mantle's heterogeneity, though the topography of free surface is accurately known.
 From quantitative calculation we suggest that P′surfP′ or P′•d•P′ could be from the scattering of rough free surface or the forward scattering of heterogeneity in crust and upper mantle, and the back scattering from these structures could be fairly weak. But maybe at some quiet stations the back scattering of the crust or upper mantle's heterogeneity could be well observed, as the signals are earlier than the observed P′surfP′ [Earle et al., 2011]. If so, the earlier precursors may reveal more information of the crust and upper mantle's heterogeneity, for example the autocorrelation length can be constrained from the ratio of the forward and back scattering energy.
 We are grateful that two reviewers provide constructive comments. Data are provided by IRIS/DMC and Chinese Seismic Network/CSNDMC. Supported by NSFC 40821160549, 41074032.