# Combining Higher-Order Statistics and Array Techniques to Pick Low-Energy P-Seismic Arrivals
## 3 Methodology  
We present an algorithm for detecting onsets based on higher-order statistics (HOS), specifically designed for applications to arrays of stations. Hereinafter, we will denote our algorithm as higher-order statistics for arrays (HOSA). As the method aims to reveal P-wave arrival times, we will use only the vertical component of motion, where P-waves are more visible. HOSA consists of two stages: a single-trace stage (STS) and a multi-channel stage (MCS), which are outlined in Figure 2 and described in detail in Algorithm 1. STS receives the vertical components of the seismic waveforms as input. In each trace, it seeks the P-wave arrival time. When the stage successfully identifies an arrival time, it outputs the corresponding value along with a respective weight. MCS receives the arrivals from the previous stage and leverages information about the spatial arrangement of stations. It performs a second selection process, checking the coherence of arrivals within the same array and thus enhancing the reliability of predictions. In the following sections, a detailed description of the two stages is provided. Figure 1: Map of Southern Apennines (Italy) representing the earthquake analyzed in this study (blue dots) and DETECT seismic station locations (red hexagons). ISNet stations are represented by green triangles. Black squares refer to the historical earthquakes that occurred in Irpinia [21], highlighting the significant seismogenic potential of the region [20]. The numbers next to the black squares are the dates of the historical earthquake. The orange rectangles represent the individual seismogenic sources of the 1980 Campania-Lucania earthquakes [30]. The yellow stars represent the position of the earthquakes whose waveforms are utilized as example in Section 5.2. The inset of the figure highlights the study region’s location on the Italian peninsula. ```
Input: Vertical component of waveforms registered by arrays of stations. Part 1: Single-trace stage (STS) for i = 1,..., Number of traces do  1.1. Cut the trace \(u_{i}\): From respective event Origin time to Origin time + \(\overline{t}\) s;  1.2. Apply a band-pass filter \([f_{1},f_{2}]Hz\) to \(u_{i}\);  1.3. Let \(w\) be the time window length on which to compute the HOS; for t = 1,..., \(\text{length}(u_{i})-w\)do  1.4. Compute the \(n-th\) order HOS time-series: \(h_{i}[t]=\gamma_{n}(u_{i}[t:t+w])\); endfor  1.5. Smooth: \(h_{i}[t]=\left(h_{i}[t-1]+h_{i}[t]+h_{i}[t+1]\right)/3\);  1.6. Compute the derivative \(h_{i}^{\prime}=\dfrac{\text{d}}{\text{d}t}(h_{i})\);  1.7. Let \(M_{i}=\text{argmax}(h_{i})\). Search the onsets in the time window \([M_{i}-1 sec,M_{i}+1 sec]\);  1.8. Get multiple onsets for different thresholds \(\theta_{k}\): onset \(o_{i}^{k}\) = first sample of \(h_{i}^{\prime}\) that is \(\geq\theta_{k}\cdot max(h_{i}^{\prime})\);  1.9. Let \(\sigma_{i}\) = StandardDeviation\({}_{k}(o_{i}^{k})\) & \(\sigma_{\text{max}}\) the max allowed value;  1.10. Barrage: the Multi-threshold criterion: if\(\sigma_{i}>\sigma_{\text{max}}\)then  1.11. Reject onset; else  1.12. Adopt as the true onset \(o_{i}=o_{i}^{j}\), where \(j=\text{argmin}_{k}(\theta_{k})\);  1.13. Assign a weight to \(o_{i}\) equal to: \(\text{floor}(3\cdot\sigma_{i}/\sigma_{\text{max}})\). endif endfor
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**Part 2: Multi-channel stage (MCS)**  
_2.1._ Consider the onsets that passed the barrage 1.9; for j = 1,..., Number of events do  2.2. Select the \(j\)-th event; for i = 1,..., Number of arrays do  2.3. Select \(O_{i}=\left\{o_{i1},o_{i2},o_{i3}...\right\}\), the onsets related to \(i\)-th array;  2.4. Let \(N_{i}\) be the total number of onsets for the selected event at \(i\)-th array;  2.5. Let \(D_{i}^{\text{max}}\) be the maximum distance between 2 station for the \(i\)-th array and \(v_{p}\) the P-wave velocity;  2.6. Let \(\delta\) be a tolerance parameter;  2.7. \(\Delta t_{i}^{max}=D_{i}^{\text{max}}/v_{p}+\delta\) is the max allowed time difference between two arrivals for the \(i\)-th array;  2.8.