# Acoustic Rocket Signatures Collected by Smartphones
## 2 Materials and Methods
### Data Alignment  
For each rocket launch, the reported time of the launch and the great circle distances between the source and each active station are used to compute estimates of the signal's arrival time at each station using an estimated speed of sound. There are two likely sources of error in these estimates that we will address. Firstly, the reported launch time is often accurate only to the minute. Secondly, we assumed direct great circle paths from the source to each phone, which, while a reasonable estimate for short propagation distances, may not be an accurate representation of acoustic wave propagation. We expect that error in arrival times arising from simplified propagation paths will increase with distance from the source, while bias error due to imprecisely or inaccurately reported launch times will be independent of distance. For each signal in ASTRA, the effective speed of sound is estimated according to  
\[c_{eff}=331.3\ \frac{m}{s}\cdot\sqrt{1+\frac{\overline{T}}{273.15}}+\overline{v} _{\parallel}, \tag{1}\]  
using the mean temperature \(\overline{T}\) in degrees Celsius and mean parallel (or negative if antiparallel) wind along the great circle path between the source and the station. Temperature and wind values were obtained from Copernicus Climate Change Service's ERA5 [34; 35], and parallel wind is calculated by projecting the wind vector onto a vector representing the estimated speed of sound in the direction of propagation. The magnitude of the parallel or antiparallel wind is then added or subtracted, respectively, from the magnitude of the estimated speed of sound, giving the effective speed of sound along the assumed path. The estimated travel time is simply the assumed propagation distance divided by the effective speed of sound. Figure 1: Bar plots showing the distribution in the dataset of signals (**a**) recorded on different makes and models of smartphones, (**b**) originating from different types of rockets, and (**c**) collected in different range categories. Once the effective sound speed is estimated, the dataset is aligned in two ways, each with a different estimate of the arrival time of the signal. Both estimates of the arrival times are preserved in ASTRA as there are use cases where each estimate may prove more useful than the other. These estimates, their strengths, and their potential weaknesses are outlined in the following paragraphs. The first arrival time estimate is made by simply adding the estimated travel time to the reported launch time. We call data aligned this way \"start-aligned\", and the associated arrival time estimate is labeled as such in ASTRA. The start-aligned estimation is independent of the waveforms themselves and thus unaffected by noise in the data, but it is sensitive to bias error in the reported launch time and assumed propagation path. An example of start-aligned data is shown in Figure 2.  
To calculate the second arrival time estimate, a window from which to select a peak is determined. The window duration is equal to 180 s plus 25% of the estimated propagation time to account for the uncertainty increasing with propagation distance as previously discussed. For each individual launch, estimates are made starting with the closest station to the launch site, for which the first timestamp of the selection window is set to the start-alignment estimate of the arrival time at the same station, minus a 30 s buffer to account for potential error in the reported launch time. For successive stations, the next-closest station's new estimate is used instead of the start-alignment estimate, and an additional buffer term inversely proportional to the difference between the two propagation distances is subtracted. This is represented mathematically in Equation (2). \[t_{i}=t_{i-1}-t_{buffer}-t_{buffer}/(r_{i}-r_{i-1}) \tag{2}\]  
The windowed waveform is then bandpassed.