# Combining Higher-Order Statistics and Array Techniques to Pick Low-Energy P-Seismic Arrivals
## 5 Results
### Consistency Examination: Semblance  
A quality assessment of the HOSA onsets can be conducted by computing the semblance \(S\), a quantity that can be interpreted as a measure of coherence for multi-channel data [44,45]. To emphasize the source component of the (ground velocity) recordings, for semblance calculations we band-pass filtered the traces in the range of \([5,30]\) Hz, which encloses the typical peak frequency of velocity seismograms produced by microearthquakes (M < 3--[46]). For a fixed event, we defined for the generic \(i\)-th array the value of the semblance \(S_{i}\), computed among the filtered traces \(u_{ij}\) aligned by their respective detected onsets \(o_{ij}\), as explained in Equation (10):  
\[S_{i}=\int_{T_{1}}^{T_{2}}S_{i}^{t}(\tau)\ d\tau=\frac{\int_{T_{1}}^{T_{2}} \left(\sum_{j=1}^{N_{i}}u_{ij}(o_{ij}+\tau)\right)^{2}\!d\tau}{N_{i}\int_{T_{1} }^{T_{2}}\sum_{j=1}^{N_{i}}u_{ij}^{2}(o_{ij}+\tau)\ d\tau}, \tag{10}\]  
where \(N_{i}\) is the total number of traces with a detected onset for the \(i\)-th array, the integrand \(S_{i}^{t}\) is referred to as temporal semblance, and the integrals are computed over \(\tau\), from a \(T_{1}<0\) to a \(T_{2}>0\) (with \(t=0\) corresponding to the P-onset). We computed semblance values both using catalog onsets (\(o_{ij}^{\mathrm{C}}\)) and onsets by HOSA (\(o_{ij}^{H}\)). We fixed the extremes of the integral to the values \(T_{1}=-0.1\) s and \(T_{2}=0.1\) s. This choice allowed us to focus on the first oscillation of the waveforms and to make the semblance sensitive to onset alignments. Semblance assumes values \(0\leq S\leq 1\), and it is independent of individual signal amplitudes [47]. Semblance equals 1 when all the time series consist of identical values, i.e., all the waveforms \(u_{ij}\) are identical and perfectly aligned. In contrast, for uncorrelated noise \(S_{i}\simeq 1/N_{i}\). Examples of the semblance and temporal semblance of Equation (10) are depicted in Figure 8, where traces belonging to two events have been aligned by catalog onsets (top) and HOSA onsets (middle raw), demonstrating higher semblance values when onsets are such that waveforms exhibit higher coherence. We can notice that in the cases presented in Figure 8, the catalog onsets show an error of less than 0.1 s (manual analysis on each trace reveals a typical error of about 0.05 s). Despite the small error, semblance values on these catalog onsets are similar to those computed on noise, indicating the high sensitivity of semblance to small misalignment. It is more straightforward to evaluate the accuracy of the onset \(o_{ij}\) using residual semblance, which is the semblance reduced by the value expected for noise (Equation (11)). \[S_{i}^{\tau}=S_{i}-1/N_{i}. \tag{11}\]  
We computed the residual semblance (Equation (11)) for all traces in each array for all events, employing the HOSA and catalog onsets, respectively. Figure 9 illustrates the distributions of residual semblance using HOSA and catalog onset, respectively. In the upper left panel, the distributions have been computed for all arrays, while the other panels show the value of residual semblance for arrays with a fixed number of detected arrivals. Due to the high sensitivity of semblance to a small misalignment, values of residual semblance slightly higher than zero indicate good alignment among the onsets. Figure 9 shows higher values of residual semblance for HOSA onsets, suggesting improved alignment compared to catalog onsets. Notably, the enhancement in residual semblance values is more pronounced when a larger number of waveforms are employed in semblance computation. In Table 5, we present the percentages of residual semblance that exhibit values greater than 0.1, i.e., significantly higher than zero, indicating a strong alignment among the respective onsets. In each case, HOSA exhibits a percentage of strong alignment greater than that of the catalog and also higher than 30% (except for the scenarioinvolving four detected arrivals per array).