## 2 Setup and Methods
### Moving Averaging-Moving Differential  
The moving averaging-moving differential (MA-MD) technique has proven itself in time series processing and economics. In addition, attempts have repeatedly been made to process DAS data with it [39]. This method allows one to eliminate the small fluctuations of values and to highlight more significant trends with relatively easy implementation, low computational costs, and acceptable computation time. Another common technology for denoising and classifying DAS events is the neural network. However, this requires significant information and computational resources, at least during the stage of training. In a sense, the MA-MD algorithm can be represented as a bandpass filter, where the set of amplified and suppressed frequencies is determined by the window size of each of the two constituents that make up the method. As a rule, each frequency band requires its own window size for the best gain. In addition, it was found that the most common DAS implementation of the MA-MD method involves the application of both components in a time domain [39]. This generates \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Laser Type**} & \multicolumn{2}{c}{**Event**} & \multicolumn{4}{c}{**SNR, dB**} \\ \cline{3-6}  & \multicolumn{1}{c}{**Frequency, kHz**} & **No Filtering** & **MA-MD** & **AFDA** & **MA-MD and AFDA** \\ \hline \multirow{2}{*}{Commercial} & 0.5 & 18.8 & 27.3 & 22.5 & 29.2 \\  & 8 & 19.6 & 19.6 & 23.3 & 23.4 \\ \hline \multirow{2}{*}{Homemade} & 0.5 & 15.7 & 25.0 & 19.3 & 26.5 \\  & 8 & 17.9 & 19.2 & 21.4 & 22.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the studied algorithms application results. Figure 2: (a) Data obtained with the commercial laser; (b) data obtained with the homemade laser; inserts represent spectra (frequency domain cross sections) at a distance of 3000 m (red lines). a relationship between the quality of the received data, estimated by SNR, and the window size of the constituent parts of the technique--moving averaging and moving differential, respectively. The following equation can be used to calculate an approximately suitable MD window size:  
\[w_{MD}=(2k+1)\text{floor}(\frac{w_{s}}{2k}), \tag{1}\]  
where \(k\) is an integer from 0 to \(\infty\); floor() returns an integer part of a number; \(v_{s}\) is the sensor sampling frequency (i.e., the number of the traces recorded per second); and \(v\) is the center frequency of the band in which the presence of a useful signal is expected. In this case, the MA window size can be taken to be approximately equal to the calculated MD window size. Thus, for noise reduction by the MA-MD method, a window size of 25 counts is suitable for the frequency range of 500 Hz and 1-2 counts are suitable for 8000 Hz band. Some frequency bands (multiples) may feature a window size that provides qualities of processing which are not at the maximum for each, but satisfactory for both of them. It was decided to test these hypotheses in practice. The test utilized the data acquired from the commercial laser. The following formula was applied to its time domain:  
\[ds_{1}^{(w_{MA},w_{MD})}=\frac{1}{w_{MA}}\left(\sum_{i=1+w_{MD}}^{w_{MA}+w_{MD} }s_{i}-\sum_{i=1}^{w_{MA}}s_{i}\right), \tag{2}\]  
where \(ds_{1}^{(w_{MA},w_{MD})}\) is the first processed signal level value, \(w_{MA}\) is the MA window size, \(w_{MD}\) is the MD window size, and \(s_{i}\) is the \(i\)-th signal level value. The obtained results are presented in Figure 3. It can be seen that they are in fairly good agreement with the formula. For the 500 Hz event, the maximum SNR fell on the MA and MD window size values of 23 and 20 or 19 and 24 counts, respectively. And for the 8000 Hz event, they fell on 7 and 1. Fortunately, the latter window sizes also provide the sufficient SNR for the 500 Hz frequency band.