## 3 Experiment and Results
It can be seen that a weak vibration event was clearly detected with a dominant peak at the location of \(\sim\)1245 m, whose amplitude variation shown in Figure 4b is essential for the measurement of vibration frequency. Figure 5a,b presents the square of the location information for detection of the vibration event of 102 Hz obtained by the moving differential method [6] and the WSVD method, respectively. The performance of the two methods can be estimated using the SNR of the event detection. For a vibration event, if \(V_{s}\) is the voltage of the signal peak and \(V_{n}\) is  
Figure 4: Denoised traces of the Rayleigh backscattering (RBS) reflections obtained after applying the weighted singular value decomposition (WSVD) method for detection of the vibration event of 102 Hz: (**a**) location plot for event detection, and (**b**) density plot for frequency measurement. Figure 3: Singular value distribution of the original spatial-temporal dataset of the phase-sensitive OTDR. the voltage of the background noise, then the SNR of the location information is defined as [19]:  
\[SNR=10 log_{10}\bigg{(}\frac{V_{s}}{V_{n}}\bigg{)} \tag{6}\]  
Here, the voltage amplitude of the denoised backscattering traces is normalized and scaled in the range of 0-1. In Figure 5a, the amplitudes of signal and noise are found to be 1 and 0.275 using the moving differential method, respectively, so the SNR is equal to 5.6 dB. In Figure 5b, the SNR is 10.7 dB by applying the WSVD method. It is evident that the SNR of location information is significantly increased by the application of the proposed WSVD method on the original signal. Figure 6a describes a density plot made by the 800 raw Rayleigh backscattering traces comprising an acoustic event of 525 Hz, which was generated using a sound speaker, as given in Figure 1c. Again, the acoustic signal was very weak and difficult to be observed using our vision in the plot. However, when the raw traces were processed using the WSVD method, as can be seen in Figure 6b, a unique periodic change in voltage amplitude compared with the density plot of 102 Hz (see Figure 4b) was found at the position of 1245 m of the sensing fiber. Figure 5: Denoised traces of the Rayleigh backscattering reflections, showing the detection of the physical event of 102 Hz, are obtained after applying (**a**) the moving differential method and (**b**) the WSVD. Figure 6: Traces of the Rayleigh backscattering reflections, which consist of an acoustic event of 525 Hz, are shown in the density plot: (**a**) before and (**b**) after denoising using the WSVD. The location information yielded after processing by different methods is illustrated in Figure 7. In a similar manner, the noise was removed and the SNR of the vibration event was estimated using two different methods. In Figure 7a, the SNR of the vibration event of 525 Hz is detected to be 5.23 dB by employing the moving differential method, whereas it is obtained to be 12.2 dB by employing the WSVD as shown in Figure 7b. Figure 8 describes the quantitative comparison between the SNR of the vibration event processed by the two different methods. Clearly, the raw backscattering traces processed by the WSVD method show a remarkable \(\sim\)6.5 dB improvement in the SNR of the vibration events of 102 and 525 Hz. To confirm the vibration sensing, Fourier transform was applied on the denoised temporal data at the location of vibration event. Figure 9 shows the normalized power spectra for the vibration events of 102 and 525 Hz at the \(\sim\)1245 m location. The vibration frequencies of the two events were measured with an error \(<\)0.5 Hz. Figure 8: Comparison of the signal-to-noise ratio (SNR) of the location information obtained by using the two signal-denoising methods. Figure 7: Denoised traces of the Rayleigh backscattering reflections, showing the detection of an acoustic event of 525 Hz, obtained after applying (**a**) the moving differential method and (**b**) the WSVD.