# Precursory Tidal Triggering of the 2011 Mw 9.0 Tohoku Aftershocks
## 2 Data and Methods
### Methods  
Since the aftershock dataset contains a sequence of events with an irregular spatial pattern, it is not suitable for one method to detect periodicities. In this paper, we use the Schuster test and the Schuster spectrum to evaluate the tidal correlation of the aftershocks. The Schuster test is utilized to explore the preliminary tidal correlation, as the existence of clusters may affect the correlation and yield an apparent tidal correlation [14]. Then, the Schuster spectrum method is utilized not only to verify the results obtained by the Schuster test but also to provide further insight.  
The Schuster test consists of calculating the distribution of phase angles of all earthquakes [14; 18; 36]. First, for each earthquake, a tidal phase angle is assigned using a linear scale from \(-180^{\circ}\) to \(180^{\circ}\), according to its time of occurrence and the location of the hypocenter. At this scale, the maximum and minimum tidal loads in the tidal cycle are represented by \(0^{\circ}\) and \(\pm 180^{\circ}\), respectively [17; 18]. If the tidal phase angles for earthquakes concentrate around a particular angle, a dominant phase angle is identified. A \(p\)-value is used to represent the level of significance of the correlation, where a value \(<5\%\) indicates statistical significance [14; 15; 30].  
For further verification, the Schuster spectrum is calculated to evaluate the effect of aftershock clustering on Schuster's \(p\)-value (the level of significance of the correlation). The Schuster spectrum method [34] is based on the Schuster statistical test. The Schuster spectrum is a tool that detects any unknown periodicity in the catalog and analyzes any seismic activity rate [34]. By obtaining the Schuster spectrum of the subsequent seismicity, we can identify the significant periodic components of the aftershocks through a stacking process. To obtain a Schuster spectrum of a nonuniform time series, \(N\) repetitions of the Schuster statistical test are needed. If an earthquake sequence (the signal) has a duration of t and a period of T, the periodic sampling is calculated as [34]:  
\[\Delta T_{i}=\frac{\varepsilon T_{i}^{2}}{t} \tag{1}\]  
where \(\varepsilon\) is a coefficient, and \(\
u=1/T\) is the frequency. The increment of frequency is thus a constant, i.e., \(\Delta\
u=\frac{\varepsilon}{t}\). To calculate a Schuster spectrum between the periods \(T_{\text{min}}\) and \(T_{\text{max}}\), the Schuster test must be run \(N\) times.  
\[N=\frac{t}{\varepsilon}(\frac{1}{T_{\text{min}}}-\frac{1}{T_{\text{max}}}) \approx\frac{t}{\varepsilon T_{\text{min}}} \tag{2}\]  
The value of \(\varepsilon\) determines whether the sampling period is reasonable. An optimal value \(\varepsilon_{0}\) exists for \(\varepsilon\), and _[PERSON] and [PERSON]_[34] suggest \(\varepsilon_{0}\approx 1\). By taking \(\varepsilon\) equal to \(\varepsilon_{0}\), the expected value of the minimum [PERSON]'s \(p\)-value in a spectrum containing \(N\) independent tests is:  
\[\langle\delta_{m}\rangle=\frac{1}{N}\approx\varepsilon_{0}T_{\text{min}}/t \approx T_{\text{min}}/t \tag{3}\]  
It is worth noting that the \(p\)-value in [PERSON]'s statistical test is less than \(5\%\), which indicates that there is a tidal correlation, and the threshold value is fixed. In Equation (3), the expected threshold \(\langle\delta_{m}\rangle\) of the Schuster spectrum related to the selected period of each [PERSON] test (i.e., the expected threshold) is variable. In the analysis of a Schuster spectrum [34], a \(95\%\) confidence level can be used as a criterion.  
In this study, strain tensors caused by solid earth tides and ocean loading are calculated using the GOTIC2 program [37].