## Materials and Methods
Experimental data have been recorded in two test sessions in order to assess the robustness of the proposed method with respect to modifications in the boundary conditions of the pipeline. In particular, measurements were performed in different periods of the year, characterized by different conditions of the surrounding medium. The first test session (referred to as Test-1) was carried out \begin{table}
\begin{tabular}{c c c} \hline \hline
**Leak** & **Distance [m]** & **Leak Rate [L/h]** \\ \hline L1 & 3 & 200 \\ L2 & 8 & 295 \\ L3 & 26 & 4000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the generated leaks. Figure 2: Sensor setup: (**a**) Schematics; (**b**) close-up of the transducers. during winter after a period characterized by low precipitation. The second test session (referred to as Test-2) was conducted in spring, after a very long period with unusually intense rainfall, during which the test bed remained completely submerged by water. Hence, during Test-1 the soil moisture content was at a minimum, whereas in Test-2 it was extremely high. All the signals were measured simultaneously by means of a LMS SCADAS Recorder SCR-01 (Siemens PLM Software, Plano, Texas, USA, 24 bit \(\Sigma\Delta\) analog-to-digital converter with a 150 dB dynamic range, 8 channels, \(\pm\)10 V input range, 204.8 kHz maximum sampling frequency and analog anti-alias filter), which also provided signal conditioning. The following acquisition parameters were adopted: Sampling frequency, \(F_{s}\) = 5120 Hz; duration, \(T_{s}\) = 10 s. A high-pass filter (cutoff frequency of 10 Hz) was applied to all signals due to the hydrophone frequency response limits. Measurements were performed for both the non-leaking and the leaking conditions to monitor the environmental noise and the vibro-acoustic phenomena related to active leaks, respectively. For the \(k\)-th leak, the following procedure was applied:  
* The leaking condition (Lk) was set by temporarily unfastening the corresponding repair clamp and shifting it to one side of the damage (Figure 1c), i.e., only one active leak at a time was present. * The leaking flow was covered with a plastic sack filled with soil to simulate the condition of water leaking against backfill soil (this solution was adopted to allow for fast removal of the soil, in order to rapidly switch between the different conditions). * Every 30 s, an acquisition was started and recorded, hence collecting a total of six measurements for the condition Lk. * The sack was removed and the repair clamp was reinstalled again to deactivate the leak Lk. * Six measurements in the non-leaking state (referred to as NLk) were carried out after leak deactivation (one acquisition every 30 s). It was shown that the analysis of the autocorrelation function, \(\hat{R}_{xx}(\tau)\), of vibro-acoustic signals may represent a viable tool for reliably detecting leaks. The autocorrelation function, \(R_{xx}(\tau)\), of an infinite continuous function of time, \(x(t)\), is defined as [24]:  
\[R_{xx}(\tau)=\int_{-\infty}^{+\infty}x(t)x^{*}(t-\tau)\mathrm{d}\tau, \tag{1}\]  
where \(\tau\) represents the time lag and the asterisk denotes complex conjugation. For a finite discrete signal, \(\mathbf{x}=\{x_{n}\}\), with \(n=1,\,\ldots,\,N_{S}\), an estimation of the autocorrelation function, \(\hat{R}_{xx}(m)\), can be computed as [25]:  
\[\hat{R}_{xx}(m)=\left\{ \begin{array}{l}\sum\limits_{n=0}^{N_{S}-m-1}x_{n+m} x_{n}^{*},m\geq 0\\ \hat{R}_{xx}^{*}(-m),m<0\end{array} \right., \tag{2}\]  
where the asterisk denotes complex conjugation and  
\[m=\{1;2;\ldots;N_{R}\}-N_{S}, \tag{3}\]  
where \(N_{R}\) = (2\(N_{S}-1\)). The autocorrelation function in the desired form (i.e., in the time lag domain), \(\hat{R}_{xx}(\tau)\), can be obtained from Equation (2) by considering that the lag related to the \(m\)-th value of the autocorrelation function, \(\tau_{m}\), is simply given by:  
\[\tau_{m}=\frac{m}{F_{s}}, \tag{4}\]  
where \(F_{s}\) is the sampling frequency of the signal.