## 1 Introduction  
Distributed fiber optic vibration sensors have gained much attention over the years due to their unique advantages: they are immune to electromagnetic interference, have high detection sensitivity and fast detection response, and can be deployed in harsh conditions at low cost [1, 2]. For distributed vibration sensing based on optical fiber, one particular technique is the use of the phase-sensitive optical time-domain reflectometer (\(\Phi\)-OTDR) [3, 4, 5, 6]. It is attractive because it offers high spatial resolution, broad frequency range, real-time measurement, and a long sensing distance [7, 8, 9]. It can be applied to detect multiple environmental events over long distances, such as pipeline leakages, intrusion detection, perimeter security, acoustic waves, seismic activity, other structural health monitoring applications, etc. [10, 11, 12, 13]. In the \(\Phi\)-OTDR, highly coherent light pulses are launched into a sensing fiber, and the signal traces, which result from the coherent interference of Rayleigh backscattering light originating from multiple scattering centers within the pulse duration, are acquired all along the length of the sensing fiber [12]. In the event of dynamic perturbation such as vibration or intrusion at specific positions along the optical fiber, the amplitude of the backscattering signal modulates in time due to modulation of the length and refractive index profile of the fiber, which leads to localized detection of the event. However, the Rayleigh backscattering (RBS) reflections that make the signal trace of the \(\Phi\)-OTDR are usually very weak, and are accompanied by random noises, such as background and environmental noise, and the noises from the optoelectronic components (e.g., phase noise from the laser source, shot noise from the photodetector) [6]. If the pulse power is set too high to obtain a strong backscattering signal, position-dependent signal fading would be induced [14]. For these many reasons, it is quite difficult to detect the vibration events due to their relatively small signal-to-noise ratio (SNR), which not only limits the performance of the \(\Phi\)-OTDR-based distributed vibration sensor in real-life applications, but also leads to frequent false alarms. To eliminate the random noise from the backscattering signal of the \(\Phi\)-OTDR and to accurately extract and improve the SNR of the dynamic events, much research has been performed. Two approaches have been mainly used. In the first approach, light amplifications are performed using the distributed Brillouin [15], the distributed Raman [16], and the hybrid scheme [17]. Although these schemes can certainly improve the SNR of the vibration event and increase the sensing range of the \(\Phi\)-OTDR, they also add to the complexity and cost of the sensing system. The second approach, due to \(\Phi\)-OTDR being a distributed sensor that produces raw backscattering data in large amounts, relies on the signal processing of the raw data using efficient mathematical algorithms to improve the SNR of the vibration event. More significantly, these signal-processing algorithms enhance the SNR of the event without adding any further optoelectronic components, and therefore greatly simplify the architecture of the sensing system. Several signal-processing algorithms have been demonstrated. [PERSON] et al. [12] demonstrated a distributed fiber intrusion sensor using \(\Phi\)-OTDR based on direct detection that uses a simple averaging method to obtain the location information of intrusion detection with low SNR. [PERSON] et al. [6] proposed a moving average and differential method for a distributed fiber vibration sensor based on coherent detection to obtain both the location information and the frequency measurement. The SNR of the location information has considerably increased by the use of the moving average and differential method. However, the averaging methods have a major drawback: they significantly reduce the bandwidth of the frequency measurement [6].