SmKS waves provide good resolution of outer-core velocity structure, but are affected by heterogeneity in the D″ region. We have developed an Empirical Transfer Function (ETF) technique that transforms a reference pulse (here, SmKS) into a target waveform (SKKS) by: (1) time-windowing the respective pulses, (2) applying Wiener deconvolution, and (3) convolving the output with a Gaussian waveform. Common source and path effects are implicitly removed by this process. We combine ETFs from 446 broadband seismograms to produce a global stack, from which S3KS-SKKS differential time can be measured accurately. As a result of stacking, the scatter in our measurements (0.43 s) is much less than the 1.29 s scatter in previous compilations. Although our data do not uniquely constrain outermost core velocities, we show that the fit of most standard models can be improved by perturbing the outermost core velocity. Our best-fitting model is formed using IASP91 with PREM-like velocity at the top of the core.
 A number of previous studies have used teleseismic SmKS phases to improve seismic resolution in this region. SmKS waves (Figure 1) propagate as S-waves in the mantle and compressional (K) waves in the core, with m − 1 underside reflections from the core-mantle boundary (CMB). Since SmKS phases turn at relatively shallow depth beneath the CMB, they provide good resolution of outermost core velocity structure [Choy, 1977; Garnero et al., 1993; Tanaka, 2004; Eaton and Kendall, 2006]. In particular, these waves comprise a whispering gallery phase [Aki and Richards, 1980] for which high-order (3 ≤ m < ∞) modes coalesce into a composite, dispersive waveform at the range of periods for which they are typically observed (2–10 s).
 Global compilations of SmKS traveltime residuals along individual paths [Souriau et al., 2003] exhibit a high degree of scatter, likely caused by lateral velocity variations in the D″ region [Garnero et al., 1993]. Tanaka  stacked long-period (T > 10s) waveforms to mitigate this effect, but due to source- and path-dependent waveform variability did not stack shorter period signals, thus sacrificing some fine-scale resolution of outer core structure. Here we introduce an Empirical Transfer Function (ETF) method that, like Receiver Functions [e.g., Langston, 1977; Vinnik, 1977], uses deconvolution to account for waveform differences and thus facilitates stacking of broadband waveforms. The purpose of this paper is to describe the ETF technique and to illustrate its application to the study of outermost core structure using SmKS differential time measurements.
2. SmKS Waves and Empirical Transfer Function Method
 The SmKS pulse contains a superposition of modes for 3 ≤ m < ∞ [Choy, 1977; Eaton and Kendall, 2006]. Relative to the preceding mode, each successive mode contains a time delay, phase shift and amplitude reduction. The phase shift arises from: (1) a Hilbert transform associated with the caustic at the turning point, (2) a small phase rotation, due to post-critical reflection from the base of the CMB, and (3) finite-frequency effects associated with the proximity of the turning point to the CMB [Choy, 1977]. While previous studies have usually incorporated the Hilbert transform [e.g., Souriau et al., 2003; Garnero et al., 1993; Tanaka, 2004], the remaining phase complications have generally been neglected. Here, we introduce ETF deconvolution as a way to simplify the analysis and interpretation of differential times between such closely related teleseismic phases. We define the ETF as a linear deconvolution operator that transforms a reference pulse (here, SmKS) into a target pulse (SKKS). By definition, the ETF incorporates all time shifts, phase rotations and amplitude changes associated with the modal summation process. We use a Wiener deconvolution approach to obtain the ETF, similar in principle to methods used in some surface-wave dispersion studies [e.g., Taylor and Toksöz, 1982] and receiver-function analyses [e.g., Ai et al., 2003].
 Although the characteristics of SmKS waves are mainly sensitive to core velocity structure [Choy, 1977], properties of the D″ region can also influence the observed pulse. First, SmKS arrival time is sensitive to D″ velocity structure along both downgoing and upgoing paths. This is mitigated, however, by the use of differential times, since mantle paths for SKKS and SmKS (m > 2) phases are very similar [e.g., Souriau and Poupinet, 1991]. Here, residual traveltime anomalies are further reduced by stacking waveforms that sample different regions of the lower mantle. In addition, at each underside bounce point, the amplitude and phase of the reflection depends on local properties of both the core and the D″ region. Although pulse shape and amplitude are affected, underside CMB reflectivity will have only a minor (and indirect) influence on arrival times.
 As an initial step in the procedure, we isolate the SKKS and SmKS waveforms by time-windowing the input seismogram using a boxcar function with a Hanning taper at both ends (Figure 2). The start and end of the time window are event-specific parameters selected by the user, in much the same way that time windows are picked for shear-wave splitting analysis [e.g., Silver and Chan, 1991]. Adopting the terminology of Wiener filtering [Wiener, 1964], the Fourier-transformed SmKS and SKKS waveforms are here considered as the received and input signals, denoted by Um(ω) and U2(ω), respectively. These signals are related by a transfer function, F(ω), such that
Although F(ω) may be estimated directly by spectral division, this approach is typically unstable due to spectral holes in Um(ω). A more stable solution may be obtained using Wiener deconvolution. For a given operator length in the time domain, Wiener deconvolution yields the function (ω) that minimizes the least-squares difference between the left and right sides of equation (1). In the time domain, this function is obtained by solving a set of normal equations constructed using the cross-correlation of the two signals and the autocorrelation of the input signal [e.g., Taylor and Toksöz, 1982]. In the frequency domain, (ω) may be expressed as
where C and A are cross- and auto-spectra of the two signals, smoothed in accordance with the desired operator length [Hwang and Mitchell, 1986].
 Here, we define the ETF as the inverse Fourier transform of (ω) and, for display purposes, convolve it with a Gaussian pulse
In practice, we find that a Gaussian pulse with a dominant period of T ∼ 5 s is generally suitable, since it tends to suppress frequencies that are most affected by the windowing process.
 The procedure for measuring differential times using an ETF is illustrated in Figure 2. In the case where the received and input pulses are time-shifted and scaled but otherwise identical signals, the differential time is marked by a peak in the ETF. When the transfer function includes a phase shift, the differential time no longer corresponds to a peak but may still be extracted easily from the ETF. For example, in the case of a Hilbert transform, the differential time is marked by a zero crossing (Figure 2b), even if additional phases are present (Figure 2c). The majority of previously reported single-station SmKS differential times have been measured by cross-correlating the seismogram with its Hilbert transform or with a synthetic seismogram. Since Wiener deconvolution is also based on cross-correlating the two waveforms, picking the zero crossing of an ETF yields an equivalent proxy measurement of differential time. We remark that, for the purpose of determining differential time using a single seismogram, the ETF procedure offers no advantage over cross-correlation methods. The advantage comes, rather, from the ability to stack deconvolved broadband waveforms from different events prior to the determination of differential time, since a waveform-stacking approach is generally more robust than averaging of individual measurements.
 As noted above, full-waveform theory [Choy, 1977] shows that the transfer function between successive SmKS pulses includes other phase shifts in addition to the 90° phase shift from the Hilbert transform. To account for these effects, we compared our results with ETFs obtained from synthetic seismograms computed using the reflectivity method [Fuchs and Müller, 1971], which includes contributions from the complete SmKS series.
 We computed ETFs for 446 radial-component broadband seismograms containing high-quality SmKS observations during the time period 1994–2007. We confined our observations to a distance range of 115–140°, within which the mean turning depth of SKKS and S3KS lies in the top 350 km of the core, where global velocity models exhibit the greatest variability [Eaton and Kendall, 2006]. To the extent possible, broadband seismograms with coherent SmKS phases were selected to include paths that sample high and low-velocity regions of the lowermost mantle, in an attempt to average out the effects of D″ heterogeneity (Figure 3a). We sorted ETFs into bins according to SKKS ray parameter, discarded bins containing 4 or fewer traces, and stacked the results to create a global record section (Figure 3b). Our interpretation of the stacked ETF record section assumes that the outer core is much more homogeneous than the lowermost mantle [e.g., Souriau et al., 2003]. This implies that by stacking a sufficient number of observations, perturbations caused by lower mantle heterogeneity will tend to average out.
S3KS-SKKS differential times were obtained by measuring the corresponding zero crossings. To estimate a S3KS-SKKS differential time function (moveout curve), we computed residual times with respect to the model PREM [Dziewonski and Anderson, 1981] and performed a least-squares linear regression with respect to epicentral distance (Figure 3c). The root-mean-squared (RMS) difference between our measurements and the line of best fit is 0.43 s, compared to a 1.29 s RMS scatter for previously compiled single-station measurements [Souriau et al., 2003] within the same range of epicentral distance. Our data exhibit a trend of increasing residual with distance, a trend also noted by Souriau et al. , albeit with a smaller slope. The positive time residuals are consistent with lower velocities in the outer part of the core, relative to PREM, found in other standard Earth models [Eaton and Kendall, 2006].
 Since any realistic radially symmetric Earth model predicts a smoothly varying S3KS-SKKS differential time function, we assumed that the scatter in our measured arrival times results from heterogeneous Earth structure, especially in the D″ region, which is not entirely averaged out by the stacking process. We used the regression line to obtain an inferred moveout curve, by adding it to the predicted differential times from PREM. This S3KS-SKKS moveout curve provides a new kinematic constraint against which velocity models for the core may be tested. To search the parameter space in the neighborhood of existing standard Earth models (which satisfy many other constraints beyond those considered here), we constructed trial velocity models by perturbing the models PREM [Dziewonski and Anderson, 1981], IASP91 [Kennett and Engdahl, 1991] and AK135 [Kennett et al., 1995]. Trial models were created by: (1) assigning a P-wave velocity, Vp to the top of the core; (2) requiring Vp to be continuous with the starting model at 200 km depth below the CMB; and (3) using a quadratic function to smoothly join the top of the core with the velocity at depth. We sampled a Vp range at the CMB of 7.7 – 8.4 km/s, sampling every ∼0.02 km/s (Figure 4). To evaluate the acceptability of each trial model, we computed synthetic seismograms using the reflectivity method [Fuchs and Müller, 1971], used the synthetic traces to construct an ETF record section, measured the zero-crossing times and computed the RMS difference with respect to our moveout curve.
 In cases where the RMS misfit between the modeled times and the moveout curve exceeded the observed scatter (0.43 s), we deemed a trial Earth model to be unacceptable. Using this criterion, we found that PREM, IASP91 and AK135 (RMS misfit of 0.48 s, 0.52 s and 0.49 s, respectively) are all marginally unacceptable. Slight perturbations to IASP91 using our simple search strategy yielded models that fit our acceptance criterion (Figure 4). Of all models tested, the one with the lowest RMS misfit (0.27 s) used IASP91 as a base model, with a PREM-like velocity of 8.040 km/s at the top of the core. Relative to PREM, this model contains a velocity “sag” in the outer 200 km of the core, consistent with the conclusions of most previous studies of SmKS [Souriau and Poupinet, 1991; Garnero et al., 1993; Garnero and Helmberger, 1995; Souriau et al., 2003; Tanaka, 2004; Eaton and Kendall, 2006; Tanaka, 2007].
 We also considered whether models containing a thin, high-velocity layer beneath the CMB can be excluded based on our kinematic criteria. To evaluate this possibility, we superimposed a constant-velocity layer 10 km thick at the top of our best fitting model, and tested velocities of 8.23 and 8.36 km/s. These velocity values correspond with possible values for a light S-rich, immiscible fluid layer proposed by Helffrich and Kaneshima  for core temperatures of 3500 and 4300 K, respectively. Although the addition of such a layer increases the misfit from the best-fitting model, these two models still meet our acceptance criteria since the misfit (0.28 s and 0.30 s) is still below the scatter in the observed data.
 The Empirical Transfer Function method proposed here is a new method for measuring the differential time between a reference phase (in this case, SmKS) and a target phase (SKKS) that facilitates stacking of waveforms from different paths. Using this method, we have constructed a global stacked record section, from which the S3KS-SKKS differential time can be measured directly. This approach yields significantly less scatter (0.43 s) than previous single-station and array-analysis methods (1.29 s). By perturbing standard Earth models, we have identified a range of velocity models that fit the observed data to within the scatter in our observations. Our best fitting model is derived from IASP91, but has a PREM-like velocity at the top of the core. On the basis of our results, we cannot rule out the possibility of a thin, high velocity layer at the top of the core as proposed by Helffrich and Kaneshima  and Eaton and Kendall . As more data becomes available, the stacked global reference section should improve, resulting in less scatter. When sufficient data becomes available, this should provide a strong constraint on the velocity structure of the outermost core.
 The authors are grateful to S. Tanaka, who provided a preprint of his 2007 manuscript. GMT software [Wessel and Smith, 1998] was used to produce parts of Figure 3. This work was funded by the Natural Sciences and Engineering Research Council of Canada and by an Ontario Graduate Scholarship in Science and Technology to C.A. Data used for this study were obtained from the waveform archives of IRIS and POLARIS.