## 3 Vibration Measurement  
As presented in Figure 1, bare fiber with a length of 0.8 m was coiled over a cylindrical PZT at a position of \(\sim\)1.9 km. A sinusoidal electrical signal with a peak-to-peak voltage of 0.8 V and a frequency of 20 Hz from a signal generator was applied on PZT. The left side of Figure 4a,c,e is the 3D plot of spatial-temporal domain demodulated signals after using an adaptive 2-D bilateral filtering algorithm, which is used to remove the undesired noise effectively [22]. The unit of channel was 0.4 m, called as spatial sampling resolution is determined by the DAQ card sampling rate of 250 MS/s. The unit of time was 0.5 ms, which is the reciprocal of the oscillation frequency of the sinusoidal PGC signal. The right side of Figure 4b,d,f is the normalized strain distribution envelope from the 1 st to the  
Figure 3: A schematic of the heterodyne approach for enhanced spatial resolution in the system. Figure 2: Diagram of Raleigh backscattered traces for MI1 (Blue) and MI2 (Red). 180 th channel around 1.9 km along the fiber. The strain distribution envelope was obtained through the superimposition of each trace over a period of time after amplitude normalization. Figure 4a,b are demodulated results related to MI1, Figure 4c,d are relevant to MI2 and Figure 4e,f are the results after the differencing operation. Here, the full width at half maximum (FWHM) of the strain distribution envelope is defined to the ultimate spatial resolution. According to Figure 4b,d, the spatial resolution of MI1-relevant and MI2-relevant were 10.2 m and 9.4 m, respectively. After the differencing approach, an improved spatial resolution of 0.8 m was obtained for the DAS system (shown in Figure 4f). Figure 5 is the demodulated results of the 111 th channel in the time and frequency domains. The red line and green line are related to MI1 and MI2, and the blue dot dash line is the differencing result. It seems that the time and frequency domain information of the MI2-relevant trace and the differencing result are almost the same. This is just a coincidence, because, at this test, the magnitude of the demodulation signal of the MI1-relevant trace happens to be twice as much as that of MI2's. In addition, it should be pointed that the magnitude of the differencing result will meet an inevitable loss, which is originated from the difference algorithm. Particularly, it should be noted that the noise floor of the combined results seems slightly higher in the region where the two different spatial resolution traces were subtracted. The new noise floor comes from the differencing operation and the randomness of RS amplitude along the fiber. Because the two MIs are not perfectly identical, the demodulated phase information of the two MIs is not perfectly decoupled. Furthermore, the normalization operation contributes a lot to the increase of the noise floor. Figure 4: 3D plot of spatial-temporal domain demodulated signals (**a**,**c**,**e**) and strain distribution envelope (**b**,**d**,**f**) from 0 to 180 m positions around 1.9 km along the fiber. MI1: (**a**,**b**); MI2: (**c**,**d**); Heterodyne results: (**e**,**f**). To gain a better understanding of the performance of the DAS system, amplitude-modulation (AM) and frequency-swept (FS) modulation were conducted. Without loss of generality, in the following test, only the differencing area result was investigated. In the AM test, a 100 Hz sinusoidal signal with the voltage varying from 10 mVpp to 1.6 Vpp is applied on the PZT to inspect the strain response of the DAS system. In Figure 6, the square black spots represent the demodulated results of the system, and the red line is the fitting curve, which declares that the measurement data fit excellently with a linearity of 99.94%. An FS test was carried out to study the frequency response of the DAS system. A linear FS sine waveform from 5 to 500 Hz, a 4 s sweep time and 1 Vpp amplitude, were applied to the PZT.