2.1. Data Selection
 We used broadband data sampled at 1 s from the Incorporated Research Institutions for Seismology/International Deployment of Accelerometers/U.S. Geological Survey (IRIS/IDA/USGS) network. Our selection criteria for the two data sets (referred to by their reference phase as PP or SS) are shown in Table 1. We use records from shallow events (<75 km) to minimize interference from depth phases. The epicentral distance intervals were chosen to maximize underside reflections from the upper mantle and transition zone discontinuities while minimizing the interference from other phases. Figure 2 shows the global distribution of PP and SS source-receiver midpoints (bounce points). For both data sets, coverage is best in the North Pacific, and for the PP data this extends into China and northern Asia.
Figure 2. Locations of the bounce points (source-receiver mid points) for the traces used in the PP and SS data sets. Our final data sets contain 10,627 and 6138 traces, respectively, for the PP and SS data sets.
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Table 1. Data Selection Parameters for the PP and SS Data Sets
|Data Set (Reference Phase)||PP||SS|
|Event date range||1990–1999||1990–1999|
|Event size mw||5.5–6.5||6.0–7.0|
|Number of events||2425||796|
|Time range from reference phase,a s||−270 to +90||−700 to +300|
|Epicentral distance, deg||80–140||100–160|
 The data were Butterworth band-pass filtered (power of 6) between 8 s and 75 s for PP and 15 s and 75 s for SS. The records were also deconvolved for long-period instrument response. A Hilbert transform was applied to the PP data, and the SS data were rotated to obtain the transverse component.
2.2. Stacking Method
 We enhance arrivals with amplitudes below the noise level of individual records by stacking large numbers of traces in the time-slowness domain. By selecting records that have bounce points in certain regions, it is possible to look at possible lateral variations in discontinuity properties. However, as stacked trace quality increases with the number of traces in the stack, there exists a trade-off between lateral resolution and stack accuracy.
 The initial (predeconvolution) stacking procedure is similar to previous precursor studies [Flanagan and Shearer, 1998; Gu et al., 1998; Flanagan and Shearer, 1999; Deuss and Woodhouse, 2002]. In addition to these techniques we select and align our data on the basis of the cross correlation of the trace with an existing reference pulse, and we remove the sea surface reflection from the PP pulse (see procedure 6). A summary of the stacking procedure is as follows:
 1. We produced average PP and SS reference pulses by stacking ∼1000 handpicked seismograms. These were selected for the clarity of their PP or SS waveforms and were aligned on their maxima before stacking.
 2. We then select data sets by cross correlating the filtered seismograms with the average reference pulses. Only data with a correlation coefficient ≥0.6 and signal-to-noise ratio greater than 3 are used. The signal-to-noise ratio was measured as the ratio of energy in the window (60 s for PP, 100 s for SS) containing the reference phase to that in the window prior to the reference phase.
 3. The arrival time of the surface reflection is measured for each selected trace using the absolute maximum of the cross-correlation function within time widows of 16 s for PP and 32 s for SS around the expected arrival time of the reference phase (from PREM). The sign of the cross-correlation maximum was used to determine the traces polarity, which was corrected before stacking.
 4. The surface reflection energy was determined over time windows of 60 s around the PP pulse and 100 s around SS. This was used to normalize the trace before stacking to prevent large events dominating the stack. In addition, the stacks were weighted by the record's signal-to-noise ratio.
 5. The expected reflection amplitudes were determined, using WKBJ synthetic seismograms [Chapman, 1978] for PREM and applying the same filters as our data for our range of epicentral distances. The amplitudes of the precursor section of each trace were corrected for distance effects using the ratio of the expected reflection amplitudes in the trace to that at the reference distance.
 6. For the PP data with oceanic bounce points the effect of the ocean (ocean function) was removed from the reference phase section of the trace. The ocean function is estimated as two spikes representing P wave reflections from the ocean floor and surface. The spike amplitudes were obtained by deconvolving an oceanic PP pulse (obtained by stacking records with oceanic bounce points) with a continental PP pulse (obtained from a stack of records with continental bounce points). The spike separation was calculated using the local ocean depth in ETOPO5 [National Oceanic and Atmospheric Administration (NOAA), 1988]. Crustal reverberations near the bounce point also distort the reference phases; however, tests with WKBJ synthetics showed the effect of this on our measurement of precursor amplitudes is minor.
 7. Data are aligned on the arrival time of the surface reflection and stacked in the time slowness domain. Each trace in the stack is time shifted by (Δ − Δr) times the slowness (relative to the reference phase), where Δ is the epicentral distance and Δr is a reference distance (110° for PP 130° for SS). A summary stacked trace is made by taking a profile through the time-slowness stack along the expected arrival time and slowness of precursor phases in PREM.
2.3. Accurate Measurement of Precursor Arrival Time and Amplitude
 The trace, dm, is modeled as a convolution of the surface reflection (or reference pulse), gm, with a reflectivity sequence (or deconvolved trace), rn:
Equation (1) can be represented in matrix form:
where the trace vector d is obtained by multiplying the reflectivity sequence vector r by the matrix G. The columns of G will be the reference pulse with a delay appropriate to the corresponding element of r. In principle, it is possible to obtain a deconvolved reflectivity sequence directly by multiplying the data trace by the generalized inverse operator G−g [Menke, 1989]:
However, in practice, best results are obtained by applying equation (3), with a limited number of positions for reflection coefficients rather than parameterizing each time sample as a potential reflector.
 We determine the spike positions in the deconvolved trace by cross correlating the reference pulse with the trace. The spike position is placed at the maximum, and its amplitude is calculated using equation (3). We then derive a model trace by convolving the reference pulse with the spike sequence. This model trace is subtracted from the data trace before subsequent cross correlation with the reference pulse to determine the next spike position. At each step all the reflection coefficients are solved for simultaneously, and the fit of the model trace to the data is measured using an F test on the variance. The process continues until the addition of a spike in the reflectivity sequence fails to improve the fit of the model trace to the data by more than 0.1%.
2.4. Uncertainty in Precursor Amplitude and Arrival Time
 We determine the errors in our stacks using a bootstrap resampling procedure [Efron and Tibshirani, 1991]. For the predeconvolution data stacks the process is straightforward. A bootstrap distribution for each time sample of the stacked trace is derived by making repeated stacks (500) using traces from the original population, randomly selected with replacement. We then estimate the error on the amplitude of each time sample from the standard deviation of the bootstrap distribution.
 The application of the nonlinear deconvolution operator complicates the estimation of the error in the deconvolved stacks. When the deconvolution operator is applied to traces made from bootstrap samples, the position of an arrival will vary, leading to an overestimate of amplitude error at a given time sample if a straightforward bootstrap is applied. To obtain the true bootstrap distribution for the amplitude of an arrival, it is necessary to sum the amplitudes over a time window. The window width is determined by the range for which a discontinuity is seen in the bootstrap samples. Similarly the mean and standard deviation of the arrival time can be calculated from the frequency distribution of the arrival times in the bootstrap data.
 In cases where arrivals in the deconvolved trace are well constrained the technique works well. However, where the data is less robust the arrival positions in the bootstrap stacks can vary greatly which can result in the arrival windows for adjacent reflections overlapping. This makes the method unsuitable for noisy/less reliable stacked traces; in these cases it is only possible to measure P410P and S410S amplitudes from the deconvolved mean trace.