Fiber-optic sensing has been a promising field since its appearance. In the past few decades, various fiber optic sensors have been designed and deployed to realize accurate sensing of physical, chemical, or biological parameters.1^–5 Meanwhile, a new fiber sensing concept has appeared that utilizes the fiber network itself as a sensing element.6^,7 Similar to the nervous system of the human body, this concept enables the sensing and communication functions of the fiber network simultaneously. In this way, the network may be turned into a dense sensing array to gather information from a large area. Many sensing schemes have been proposed and can be collectively named distributed fiber-optic sensing (DFOS), which will be a massive uplift for hazard assessment,8^–12 human activity measurement,13^–16 structural health monitoring,17^,18 and earth observation.19^–22

Currently, most DFOS schemes are based on the backscattering of laser pulses that are transmitted in the fiber [Fig. 1(a)]. By analyzing the Rayleigh, Brillouin, and Raman scattering light, the vibration, strain, and temperature along the fiber cable can be sensed and returned.23^–29 For example, the widely used distributed acoustic sensing (DAS) utilizes Rayleigh scattering light and contributes to research progress due to its high sensitivity and spatial resolution. However, because of the backscattering sensing mechanism in Fig. 1(a), there are several problems with DAS: the sensing range and spatial resolution will mutually restrict each other; meanwhile the sensing range will restrict the detectable frequency bandwidth and amplitude dynamic range.29^–35 Other backscattering DFOS schemes have similar trade-offs. Consequently, strict application conditions are required, so these schemes are mostly used on specially deployed dark fibers.21^,22^,36

Recently, another sensing scheme, the forward-transmission fiber sensing based on a continuous wave (CW) laser interferometer, has rapidly developed and received broad attention [Fig. 1(b)]. The scheme can ensure the detection fidelity of wideband vibrations with a large dynamic range over a very long sensing distance.8^,14^,37^–41 However, the scheme faces a key problem: the measured vibrations are integrated over the entire length of the fiber cable. Since the sensing laser travels round trip throughout the fiber, it is influenced by the same vibration twice, as shown in Fig. 1(b). This process induces two similar phase changes, which overlap each other with a slight time delay ${\tau }_0$ and form a new composite phase-changing signal. In general, to localize the vibration source, two detection systems are always used to determine the time delay ${\tau }_0$.8^,14^,37^–39 Moreover, when multiple vibrations occur, this scheme will detect the overall integrated phase changes and cannot localize them. A recent work41 divided the 5800-km submarine cable into more than 100 sections to form a sensing array with a span of 40 to 90 km. However, it faces the same problem: multiple vibrations in one section (40 to 90 km range) are detected as a composite one, and they cannot be precisely distinguished and localized. As a result, the current forward-transmission laser interferometer cannot realize the purpose of distributed sensing.

In this work, we propose a mirror-image correlation (MI-correlation) method and apply this method to the forward laser interferometer to solve this key problem. As shown in Fig. 1(c), from the composite detected signal, the MI-correlation method can be used to extract the true time delay ${\tau }_0$ between two original vibrations. When multiple vibrations happen, the true time delay of each vibration event can also be separately obtained using the MI-correlation method. As a result, the forward-transmission laser interferometer can be used for distributed sensing. In this paper, we first illustrate the principle of the MI-correlation method and demonstrate its feasibility in vibration localizing. Second, an in-lab test is carried out on a 125-km fiber link (round trip: 250 km). Knocking events are applied at three different points simultaneously, and they can be separately localized from the time-overlapped integrated phase-changing signal. Third, we carry out a field test on an urban fiber link. Traffic vibrations occurring at the speed bump of a campus road and an underpass of Beijing Ring Road are localized simultaneously. Finally, we analyze the localizing precision of vibration with different frequencies. The overall precision is better than 25 m for vibrations with a frequency greater than 20 Hz. The MI-correlation-enabled distributed sensing laser (DSL) interferometer system conforms to the trend of low-loss fiber development. It provides a new scheme for sensing complex vibration events along optical fiber links and enjoys great potential in the widespread fiber network.

Signals from the same source have identical frequencies. If they overlap with a slight time delay ${\tau }_0$, a new composite signal with the same frequency will appear, as shown in Fig. 2(a). In practice, the detected signal is always a composite signal, and people can hardly recognize the original signals hidden in it [original signals in Fig. 2(a)]. Therefore, separating them, obtaining their slight time delay, and localizing the vibration are major challenges in fiber sensing and indoor positioning fields.42^,43 As shown in Fig. 2(a), the MI-correlation method can be used to recover the hidden information: extending the time delay between two original parts step by step, separating them in the time domain, and obtaining their real-time delay ${\tau }_0$. This process is similar to mirroring the two original signals from their composite one, and this is the reason we call it MI-correlation.

The key idea of the MI-correlation is introducing a controllable error mark to analyze the unpredictable composite signal and obtain the time delay ${\tau }_0$ between two parts. In the MI-correlation process, if the estimated time delay $\tau$ is equal to the real-time delay ${\tau }_0$, no error will be induced. If not, an error with known repetition frequency $1/2\tau$ will be enhanced as the error mark to remind us how far the estimated time delay deviates from the real-time delay ${\tau }_0$. To vividly illustrate the mirror-image method, we set the composite signal $S(t)=S_A(t)+S_B(t)$ as the initial signal in Fig. 2(b), where $S_A(t)$ and $S_B(t)$ are two original signals/vibrations shown as two dolls. Caused by the same source, these two parts have identical wave shapes with a slight time delay, and $S_B(t)$ can be written as $S_B(t)=S_A(t+{\tau }_0)$. The mirror-image steps are as follows:

By repeating steps 1 to 4, for the $i$’th-order mirror-image step, we can write the $i$’th-order mirror signal $S_i(t)$ as $S_i(t)=(-1{)}^{i}S(t+i\tau )=S_{iA}(t)+S_{iB}(t)$. Part $S_{iA}(t)$ is employed to eliminate the ($i-1$)’th-order part $S_{(i-1)B}(t)$ and induce residual error $E_i(t)=(-1{)}^{i-1}{S}_A[t+{\tau }_0+(i-1)\tau ]+(-1{)}^i{S}_A(t+i\tau )$. After $m$ times of mirror-image operations, we obtain the output signal $O(t)$ as shown in Fig. 2(b), which can be written as $$\begin{gathered} O(t)=S(t)+\sum _{i=1}^m{S}_i(t)=S_A(t)+S_{mB}(t)+E(t) \\ =S_A(t)+(-1{)}^m{S}_A(t+{\tau }_0+m\tau )+\sum _{i=1}^m{E}_i(t), \end{gathered}$$(5)where $S_A(t)$ is the original part, $S_{mB}(t)=(-1{)}^m{S}_A(t+{\tau }_0+m\tau )$ is the $m$’th-order part, the extended time delay between these two parts is ${\tau }_0+m\tau$, and $E(t)=\sum _{i=1}^m{E}_i(t)$ is the introduced mark error to indicate whether the used time delay $\tau$ is the real one. In the ideal case, when the estimated time delay is $\tau ={\tau }_0$, the induced mark error is $E(t)=0$. As shown in the bottom case of Fig. 2(b), there are no small error dolls. In other cases, the mark error $E(t)$ is induced by a series of operations with period $2\tau$, $$\begin{gathered} E(t)=\sum _{i=1}^m{E}_i(t)=[S_B(t)+S_{1A}(t)]+[S_{1B}(t)+S_{2A}(t)]+\cdots +[S_{(m-1)B}(t)+S_{mA}(t)] \\ =\sum _{i=1}^m(-1{)}^{i-1}\{S_A[t+{\tau }_0+(i-1)\tau ]-S_A(t+i\tau )\}, \end{gathered}$$(6)that is to say, the fundamental nature of MI-correlation operation is to coherently enhance the signal component at the frequency $1/2\tau$. Even in the case of $\tau ={\tau }_0$, the MI-correlation operation will also enhance the signal component at the frequency $1/2{\tau }_0$ (a detailed discussion about the analysis of MI-correlation method in the frequency domain is provided in the Sec. S1 in the Supplementary Material). However, substituting $\tau ={\tau }_0$ into Eq. (6), the mark error will behave as $E(t)=\sum _{i=1}^m(-1{)}^{i-1}[S_A(t+i{\tau }_0)-S_A(t+i{\tau }_0)]=0$ because the induced signals are with the same amplitude but opposite phase. Considering the other two parts $S_A(t)$ and $S_{mB}(t)$ in Eq. (5), their components at the frequency $1/2{\tau }_0$ will also be erased due to the opposite phase. Consequently, for the output signal $O(t)$, its component at the frequency $1/2{\tau }_0$ will turn out to be 0 when $\tau ={\tau }_0$.

As a result, the power of $O(t)$ at the frequency $1/2\tau$ is an appropriate indicator to find ${\tau }_0$. When $\tau ={\tau }_0$, the power will reach the minimum value. When $\tau \ne {\tau }_0$, the power of $O(t)$ at the frequency $1/2\tau$ will be enhanced. We can easily use a bandpass filter and calculate the power of $O(t)$ at the frequency $1/2\tau$ as an indicator, $$P(\tau )=\frac{1}{m\tau }{\int }_0^{m\tau }\{BP[O(t)]{\}}^2\mathrm{d}t,$$(7)where $BP[O(t)]$ is the bandpass operation to extract the $1/2\tau$ frequency component from $O(t)$ and $P(\tau )$ is the power of this component. We can further normalize $P(\tau )$ and define the MI-correlation indicator $M(\tau )$ as $$M(\tau )=10\lg [P(\tau )/P_0(\tau )],$$(8)where $P_0(\tau )$ is the result of Eq. (7) when there is no vibration signal and there is only background noise (a detailed description can be found in Sec. 3.2). Based on Eqs. (7) and (8), when the estimated time delay is $\tau ={\tau }_0$, the $BP[O(t)]\sim 0$, and $P(\tau )\sim P_0(\tau )$, the MI-correlation indicator $M(\tau )$ reaches the minimum. During the MI-correlation process, we should scan the estimated time delay and seek the minimum value of the MI-correlation indicator $M(\tau )$.

To show the detailed MI-correlation process and demonstrate its feasibility, we first carry out a single vibration localizing experiment using the MI-correlation-enabled laser interferometer. To accurately obtain the vibration-induced phase-changing signal, a heterodyne interferometer configuration is employed, as shown in Fig. 3(a). The laser source, NKT BASIK X15, features a line width of $<100\mathrm{Hz}$, enabling long-distance vibration sensing. Theoretically, this laser source allows for the detection of vibrations within the coherence length range (1000 km). However, in practice, the signal-to-noise ratio must be taken into account, to ensure the vibration detection fidelity and support the MI-correlation method for vibration localizing. Based on the previous work,44 the overall fiber link is set as 125 km long (round trip: 250 km) and an acousto-optic modulator (AOM) is used for frequency shifting. The beat note of the interferometer is put into a photodiode (PD), after which the phase-changing signal is extracted and analyzed by a data acquisition module (DAQ). The vibration is applied at point A $\sim 75\mathrm{km}$ away from the far end. It is generated by a fiber stretcher (FST) with a frequency of 20 Hz and an amplitude of $\sim 120\mu \mathrm{m}$ (fiber length change). The corresponding vibration strain is $\sim \pm 12\mu \epsilon$ (considering the 10-m length fiber inside the FST).

As shown in Fig. 3(b), the initially detected phase-changing signal $S(t)$ has a frequency of 20 Hz with a peak-to-peak value of $471\times 2\pi$ rad. In fact, signal $S(t)$ is a composite signal because the sensing light is disturbed by the same vibration twice in a round trip. It consists of two original parts: $S_A(t)$ and $S_B(t)$. However, the time delay ${\tau }_0$ between parts $S_A(t)$ and $S_B(t)$ is only ${\tau }_0\approx \frac{(2\times 75\mathrm{km})·n}{c}\approx 750\mu \mathrm{s}$ ($c$ is the light speed and $n\approx 1.5$ is the refractive index of the fiber). Thus, they overlap each other and cannot be distinguished. In other sensing schemes, extra detection systems or extra sensing signals are employed to obtain the time delay ${\tau }_0$.8^,38^–40

This phenomenon clearly appears in its power spectral density (PSD) curve in Fig. 3(e). The left peak corresponds to the 20-Hz vibration, and the right peak corresponds to the mark error of $1/2\tau =497\mathrm{Hz}$. In Fig. 3(d), a real-time delay ${\tau }_0=743\mu \mathrm{s}$ is applied, and the separated parts are recovered with a much lower mark error. As shown in Fig. 3(f), the left peak of the PSD curve corresponds to the 20-Hz vibration, and the right peak corresponds to the mark error of $1/2{\tau }_0=673\mathrm{Hz}$. Therefore, a simple bandpass filter can help us extract the mark error and calculate the MI-correlation indicator $M(\tau )$. For the wrong case $\tau =1007\mu \mathrm{s}$, the MI-correlation indicator is $M(\tau )=12.4\mathrm{dB}$. For the real case ${\tau }_0=743\mu \mathrm{s}$, $M({\tau }_0)=0.6\mathrm{dB}$, which is much smaller than that of the wrong case. When the time delay $\tau$ is scanned from 33 ns to $1250\mu \mathrm{s}$ (33 ns corresponds to the minimum step and $1250\mu \mathrm{s}$ corresponds to the 250 km round-trip time delay), each time delay corresponds to a value of $M(\tau )$, from which the real-time delay ${\tau }_0$ can be obtained at the minimum point of $M(\tau )$. Figure 3(g) shows the $M(\tau )$ curve (blue curve), where the two points when $\tau =1007\mu \mathrm{s}$ and ${\tau }_0=743\mu \mathrm{s}$ are marked.

It is noticeable that the generated vibration is a 20-Hz single-frequency vibration, but we are using mark error with higher-frequency components to calculate $M(\tau )$, such as 497 and 673 Hz. In fact, when vibration happens, the frequency spectrum exhibits a complex composition. In a short time, the vibration event emerges from nothing and causes a nonstationary stage, during which the frequency spectrum is much wider than that of the vibration itself. This is the basis for the MI-correlation method. What we do is enhance those specific frequency components ($1/2\tau =497\mathrm{Hz}$, for instance) and finally form the MI-correlation indicator $M(\tau )$. The minimum value $M({\tau }_0)$ corresponds to the real-time delay and vibration location.

In practice, the mirror-image operation with $m=10$ is enough to introduce mark error for localizing, and there is no need to totally separate the two original parts of the composite signal (using $m=336$ steps). The $M(\tau )$ curve of $m=10$ is shown as the pink curve in Fig. 3(g). Furthermore, we can present the final time-space distribution result of the laser interferometer, shown as the waterfall plot in Fig. 3(h). The $y$ axis is the spatial line of the sensing fiber, and the position scanning step size is 3.3 m, according to the DAQ sampling rate of $30\mathrm{MS}/\mathrm{s}$. The $x$ axis is the timeline and represents the duration of vibration. The $z$ axis describes the effective value of fiber strain by calculating the root mean square of strain in the time window (for single frequency vibration, the effective value is $\sqrt{2}/2$ of the amplitude). In Fig. 3(h), the vibration is localized at 74.3 km from the far end. It starts at 0.21 s and ends at 0.45 s.

Theoretically, only the vibration signal at the beginning and ending segments will have a complex spectral composition and will respond to the MI-correlation method. The FST-induced vibration will turn into a stable stage between the two segments, during which its spectrum narrows back to $\sim 20\mathrm{Hz}$ and lacks the specific frequency component ($1/2\tau =497\mathrm{Hz}$ for instance) for the MI-correlation indicator calculation. Thus, we use the response at the beginning segment as a start indicator of vibration, the response at the ending segment as a stop indicator of vibration, judge the power during the time, and interpolate power peaks.

As mentioned above, when vibration happens, the frequency spectrum exhibits a complex composition. This non-stationary stage will last a short time and then turn into a stationary stage in which the frequency spectrum narrows back. Therefore, the MI-correlation localizes the vibration in a short time. The subsequent vibration part will not respond to MI-correlation, which makes it possible for multivibration localization.

To interpret the principle in detail, we will introduce the MI-correlation results in three stages: before vibration happens, the beginning segment of the vibration, and the stationary segment of vibration. In stage 1, it only has background noise due to the laser source, distributed fiber noise, system noise, and so on. Figure 4(a) shows the MI-correlation result of this stage, which corresponds to the error mark power $P_0(\tau )$ in Eq. (8). We can see $P_0(\tau )$ maintains at a low power level, but there is an obvious minimum point red point in Fig. 4(a). To interpret it, we should first consider the components of the laser phase noise. In our detection system, the interference of the two beams with unequal arm lengths (twice the sensing distance $2L=250\mathrm{km}$) leads to a nonnegligible noise $N(t)$, which can be written as $N(t)=N_1(t)-N_2(t)$. $N_1(t)$ and $N_2(t)$ are phase noises directly from the laser source but with a time delay $N_2(t)=N_1(t-2{\tau }_L)$, where ${\tau }_L=nL/c$. Therefore, the noise $N(t)$ will also be considered a “vibration” by the MI-correlation process and continuously loaded at the midpoint of the sensing distance ($2L/4=62.5\mathrm{km}$). It is the reason why the red point on the $P_0(\tau )$ curve appears at ${\tau }_N\approx 626\mu \mathrm{s}$, which corresponds to the location $L_N\approx 62.6\mathrm{km}$.

In stage 2, the signal has a complex spectrum due to the sudden occurrence of vibration event. As described in Sec. 2, MI-correlation will coherently enhance the $1/2\tau$ frequency component and form the error mark power $P(\tau )$, which is shown in Fig. 4(b). Only when $\tau ={\tau }_0=743\mu \mathrm{s}$, the error mark of $1/2{\tau }_0$ frequency component reaches the minimum. The stage 2 is short in duration (a detailed discussion about the effect of the MI-correlation method in the time domain is provided in Sec. S2 in the Supplementary Material). As mentioned in Sec. 3, mirror-image operation with $m=10$ is enough to introduce a clear error mark for localizing. In our case, the upper limit of the time delay is $1250\mu \mathrm{s}$ (corresponding to a 250 km transfer delay). Therefore, vibration localization can be achieved using the beginning segment of vibration signal whose duration is $<12.5\mathrm{ms}$ ($10\times 1250\mu \mathrm{s}$). In stage 3, the high-frequency spectrum disappears and cannot cover the $1/2\tau$ frequency component. The power of the error mark $P(\tau )$ goes back to a low level as shown in Fig. 4(c). Its characteristic is similar to the background noise in stage 1: there is a minimum point at ${\tau }_N\approx 626\mu \mathrm{s}$. What is more, we can find that the points at ${\tau }_0=743\mu \mathrm{s}$ in both stages 2 and 3 are at the same power level, while other points in stage 3 will not be influenced by the vibration.

In short, during the vibration localization process, the MI-correlation will experience three stages:

The MI-correlation result $P(\tau )$ in stage 3 is similar to $P_0(\tau )$ in stage 1. Therefore, when other vibrations occur in stage 3, their localization results will not be affected by this vibration. It enables the proposed method to locate multiple vibration events. Even if multiple vibrations overlap each other, the MI-correlation method can still be applied for distributed localization, as long as the beginning or ending segments of multiple vibrations differ by more than $\mathrm{\Delta }t=m\times 2nL/c=12.5\mathrm{ms}$. This is the boundary condition of the MI-correlation method for distributed localizing on a 125-km fiber link. Considering vibrations in daily life, this boundary condition can be satisfied in most cases.

To demonstrate its performance in real cases, we first knock a fiber loop with a hammer at point A [$\sim 75\mathrm{km}$ away from the far end in Fig. 3(a)] and analyze the localizing results. As shown in Fig. 5(a), seven knocking events are detected, and the induced phase changing is less than $44\times 2\pi$ rad, which corresponds to $\sim 1.1\mu \epsilon$ strain amplitude. After using the MI-correlation method, the time-space waterfall plot is shown in Fig. 5(b). All seven knocks are localized at point A, $\sim 74.3\mathrm{km}$ away from the far end.

Furthermore, we knock points A, B, and C simultaneously (points A, B, and C are marked in Fig. 3(a), point B $\sim 25\mathrm{km}$ away from the far end, point C $\sim 124\mathrm{km}$ away from the far end). Figure 5(c) shows the detected vibration signal in the time domain. Vibrations at three points (A, B, C) overlap each other and can hardly be recognized. When the MI-correlation method is used, a time-space waterfall plot [Fig. 5(d)] can be obtained to distinguish these vibrations. We can find that vibrations at different points have different characteristics because they are knocked by different people. During the 2-s detection time, four vibration events are localized at point A (74.3 km away from the far end), and six vibration events are localized at points B and C (24.8 and 123.8 km away from the far end), respectively. This demonstrates that the MI-correlation method can distinguish and localize multiple vibration events from the integrated detected signal.

To further demonstrate the performance of the MI-correlation method, we carry out a field test on a $\sim 32\mathrm{km}$ urban fiber link together with a $\sim 50\mathrm{km}$ round-trip fiber spool in the lab. The urban fiber link passes through Tsinghua University campus, Olympic center, green parks, and education zones, which is the same as in our former work.16 Along this urban link, we can detect two main vibration sources caused by passing vehicles. One is the 9-m fiber cable buried under the road speed bump on campus, corresponding to the red mark in Fig. 6(a) ($\sim 65\mathrm{km}$ away from the far end). The other is a section of urban cable above the pedestrian underpass at the fourth ring road in Beijing, corresponding to the green mark in Fig. 6(a) ($\sim 81\mathrm{km}$ away from the far end). When vehicles pass through these two positions, noticeable vibrations will be generated and detected.16

Here, we show the vibration detection when there are vehicles driving across the speed bump in campus. Consequently, there are two different scenarios. The first is not many vehicles drive above the underpass at the fourth ring road, and traffic vibrations at two positions are observed separately. Figure 6(b) shows the vibration signals, among which the traffic vibrations at the speed bump ($50\pi \text{-}500\pi$ rad peak-to-peak) are marked. In detail, vibration envelopes correspond to the front and rear wheels of passing vehicle. The insets at the top of Fig. 6(b) extract these vibrations and show their detailed waveform and occurrence time. Two vehicles pass by during the observation, and four vibration envelopes are recorded. When applying the MI-correlation method, the waterfall plot Fig. 6(d) can be obtained. In the first 3 s, the localizing results of vibrations at the speed bump are shown. The location peaks of the vehicle’s front and rear wheels are, respectively, marked by two red/pink rectangles, and the location is $\sim 81.29\mathrm{km}$ from the far end. At the end of this observation, vibrations at the underpass emerge and lead to two localizing peaks at $\sim 65.27\mathrm{km}$ from the far end.

The second scenario is that there is dense traffic flow at the underpass of the fourth ring road; thus, vibrations at two positions are detected simultaneously, which is shown in Fig. 6(c). The continuous traffic vibration signals are generated at the underpass ($400\pi \text{-}1000\pi$ rad peak-to-peak) and overwhelm the vibration signals at the speed bump. As a result, we cannot distinguish these two kinds of vibrations from the time-domain signals. Based on our video record, we know there are vehicles at the speed bump during the observation. After using the MI-correlation method, the waterfall plot in Fig. 6(e) proves this statement: even with the influence of dense traffic at the underpass, the MI-correlation method can still successfully localize the vibrations at the speed bump. Different colored rectangles in Fig. 6(e) represents the localization results for different vehicles, and the localizing peaks are all at a distance of $\sim 81.29\mathrm{km}$ away from the far end. For those vibrations at the underpass, the passing vehicle stream generates continuous vibrations and leads to multiple localizing peaks. These results consistently point to the specific pedestrian underpass along the urban link at a distance of $\sim 65.27\mathrm{km}$ from the far end.

To explore the localizing precision of MI-correlation method, we carry out in-lab experiments on the 125-km sensing system [Fig. 3(a)]. Three vibrations are simultaneously applied at points A, B, and C using different FSTs with different frequency components to demonstrate the localizing precision of vibrations in the 20 to 50 kHz range. Figure 7(a) shows the detected vibration signal. Vibrations at each point are overlapped and form a complicated waveform.

With the help of the MI-correlation method, we can localize them respectively. The frequency spectrum of the vibration at point A is shown in Fig. 7(b), and it is localized 74,282.2 m from the far end [Fig. 7(c)] with a 24.9 m standard deviation (SD). The frequency spectrum of the vibration at point B is shown in Fig. 7(d), and the average localization result is 24,782.3 m from the far end with an SD of 17.6 m [Fig. 7(e)]. Figure 7(f) shows the frequency spectrum of the vibration at point C. Its localization result is 123,790.6 m from the far end with the best SD of 3.7 m [Fig. 7(g)]. Actually, the vibration parameters at points A, B, and C are set to represent multiple scenarios: normal vibration (20-Hz vibration at point A); irregular and strong vibration (vibration at point B with an amplitude around $100\mu \epsilon$ and frequency range of 100 to 300 Hz); and high-frequency vibration (50-kHz vibration at point C with an amplitude around $0.4\mu \epsilon$), respectively.

In addition, we find that vibrations with a higher frequency will lead to more precise localization. For the 50-kHz vibration, the localizing precision nearly reaches the step size of 3.3 m, which is limited by the sampling rate of $30\mathrm{MS}/\mathrm{s}$. For low-frequency vibrations of $\sim 20\mathrm{Hz}$, the localizing precision of $\sim 25\mathrm{m}$ still satisfies the requirements of DFOS, such as seismic monitoring, border intrusion warning, bridge health measurement, seafloor imaging, and submarine volcanism study.

We proposed an MI-correlation method that can obtain the real-time delay between two original signals from a composite signal. The key idea is to introduce a controllable error mark indicator $M(\tau )$ to analyze the unpredictable composite signal. Even if we cannot know the wave shape of the original signals and cannot detect their time delay directly, the MI-correlation method can be used to extract the error mark to find the hidden information. It is effective for distinguishing overlapped signals from the same source, which promises to be useful in fiber sensing and indoor positioning fields.

To demonstrate its practicability, we used the proposed MI-correlation-enabled DSL system to detect multiple hammer-knocking events and performed a field test on an urban fiber link. We also investigated the localizing precision of the system on different frequency bands. In conclusion, the DSL system enjoys the advantages of superior performance and simple structure. It also conforms to the trend of low-loss fiber development, which makes it possible to form a widespread fiber-sensing network.

Zhongwang Pang received his BS degree from Tsinghua University, Beijing, China, in 2021, where he is currently pursuing his PhD in instruments science and technology. His current research focuses on fiber network sensing.

GuanWang received his BS degree from Tsinghua University, Beijing, China, in 2019, where he received his PhD in instruments science and technology in 2024. His research interests include fiber network sensing and time-frequency synchronization.

Fangmin Wang received his BS degree from Shanxi University, Taiyuan, China, in 2017. He received his PhD in instrument science and technology from Tsinghua University, Beijing, China, in 2024. His research focuses on distributed timekeeping networks.

Hongfei Dai received his BS degree from Tsinghua University, Beijing, China, in 2020, where he is currently pursuing his PhD in instrument science and technology. His current research focuses on fiber-based time-frequency synchronization.

Wenlin Li received his BS degree from China University of Geosciences, Beijing, China, in 2022. He is currently pursuing his PhD in instrument science and technology from Tsinghua University, Beijing, China. His current research focuses on phase noise analysis and fiber-based time-frequency synchronization.

Bo Wang received his PhD in optics from Shanxi University, Taiyuan, China, in 2007. From 2007 to 2010, he was a postdoctoral researcher with the Max-Planck Institute for the Science of Light, Erlangen, Germany. He is currently a tenured associate professor with the Department of Precision Instruments, at Tsinghua University, Beijing, China. His current research interests include space-time and standards technology and fiber network sensing. He is a senior member of IEEE and OPTICA.