# Combining Higher-Order Statistics and Array Techniques to Pick Low-Energy P-Seismic Arrivals
## 3 Methodology
### Single-Trace Stage
In the first case, the functions \(h(t)\) show a sharp increase (corresponding to the maxima of \(h^{\prime}(t)\)) at the correct P-onset, and for an emergent arrival, the functions \(h(t)\) increase smoother, reaching smaller maxima values both in \(h(t)\) and \(h^{\prime}(t)\), highlighting an area before the maximum of \(h(t)\), where the onset is likely to be found. Remarkably, the functions \(h^{\prime}(t)\) show multiple relative maxima in the case of an emergent arrival. An even smoother increase in \(h(t)\) and several relative maxima of \(h^{\prime}(t)\) appear in the third case, in which the P-onset is masked by the noise. To account for this variability in \(h(t)\) and \(h^{\prime}(t)\) and thus of the uncertainty of the retrieved P-onset, HOSA employs a multi-threshold criterion and includes an acceptance stage to evaluate the reliability of arrival times. HOSA utilizes a set of thresholds \(\theta_{k,k=1,2,3,4,5}=\{0.1,0.2,0.3,0.4,1.0\}\), which are defined relatively to the maximum of \(h^{\prime}(t)\); in other words, \(\theta_{1}\) corresponds to one-tenth of the maximum of \(h^{\prime}(t)\), and \(\theta_{5}\) corresponds to the value of the maximum of \(h^{\prime}(t)\). Starting from \(\theta_{k}\), we obtain a set of potential onsets \(\{o^{k}\}\) for each trace as follows. We first identify the time corresponding to the maximum of \(h(t)\):  
\[M=\operatorname*{argmax}_{t}h(t). \tag{3}\]  
Figure 3: Examples of vertical components of traces with respective statistics and derivatives. On the left is a typical example of a good-quality seismic signal (station S0809, \(M=0.7\), and epicentral distance = 7.6 km) for which it is possible to visually identify an unambiguous P-onset. In the center is a weak signal (station S0209, \(M=-1.1\), and epicentral distance = 3.6 km) with an emergent P-arrival, which is reflected in an ambiguous P-onset. On the right is a waveform lacking distinguishable events (station S1009, \(M=-0.1\), and epicentral distance = 9.9 km), which implies an unidentifiable P-onset. Dotted gray lines refer to the different threshold levels \(\theta\) of the multi-threshold criterion. We then look for the potential onset \(o^{k}\) for each \(\theta_{k}\), as described in Equation (4) and at the point 1.7 of Algorithm 1:  
\[P^{k} =\{t:h^{\prime}(t)\geq\theta_{k}\ \cdot\max_{t}h^{\prime}(t)\} \cap[M-1s,M+1s],\] \[o^{k} = \begin{cases}\min P^{k}&\text{if}\ \ P^{k}\
eq\emptyset;\\ \text{None}&\text{otherwise.}\end{cases}  \tag{4}\]  
As indicated in Equation (4), the onsets are sought in a 2 s time window centered in M. This is fixed because the functions \(\gamma_{k}\) show a dispersion around \(M\) that is typically in the order of \(\sigma_{M}\simeq 0.3\) s for emergent arrivals (see Figure 3) and is eventually connected to the uncertainty in the P-onset. To account for this uncertainty, we seek the values of \(\{o^{k}\}\) in a time window centered in \(M\) and with a width equal to about \(\pm 3\)\(\sigma_{M}\simeq\pm 1\) s.  
The onset associated with the threshold \(\theta=1\) intercepts, by definition, the main peak of \(h^{\prime}(t)\). Various thresholds identify any potential lower amplitude peaks preceding the main one. We assume that a time consistency among the potential onsets \(o^{k}\) implies a low error definition of the P-onset. In other words, we assume that if \(\{o^{k}\}\) does not change with the threshold, the estimated P-onset is robust. Conversely, if \(\{o^{k}\}\) spans a wide time range, we assume that the estimated P-onset is affected by a large uncertainty. Such uncertainty is quantified as the standard deviation \(\sigma_{o}\) within each set \(\{o^{k}\}\). The user can tune the degree of accuracy of STS by allowing only solutions with \(\sigma_{o}\) lower than a threshold \(\sigma_{\text{max}}\). We will refer to this condition as the multi-threshold criterion. Eventually, from each set \(\{o^{k}\}\) (satisfying the multi-threshold criterion), one final P-onset must be identified.