*3.1. Vibration Sequences Acquisition and Preparation*

As shown in Figure 12, the regular moving loads caused by subway trains can be regarded as a vibration source. Owing to such excitation, the surface waves propagate omni-directionally on the ground. Because the surface wave couples to the track bed and subway rail, a distributed sensing optic fiber mounted beside the rail along the on-site monitoring area can detect the vibration and can be used to establish the vibration database for each monitoring zone. Figure 13 showed part of the actual engineering scenario in the subway tunnel. The monitoring area of interest covered three underground stations with a total length of nearly three kilometers. According to the spatial resolution of the sensing optic fiber and the on-spot layout of the tunnel structure, more than 500 vibration regions along the track bed can be distinguished based on the interrogated address of the light interference [54]. Here, the repeatability of the demodulator was revealed in [55] and the layout of the monitoring system can refer to [15]. When a train passed, the real-time vibration response triggered in each monitoring zone was fully transmitted back to the platform monitoring center at a sampling rate of 1 kHz and processed by the demodulator and servers. Therefore, the database of vibration response caused by passing train can be established for each monitoring zone and the location code of the monitoring area can be used as a unique label of each vibration sequence database.

**Figure 12.** The processing sketch from ultra-weak fiber Bragg grating (FBG) array to distributed vibration.

**Figure 13.** Field layout of ultra-weak FBG sensing cable used for detecting distributed vibration.

Figure 14 demonstrates the typical vibration responses of a track bed area due to a passing subway train. The triggered vibration responses of each monitoring area automatically recorded due to the

passing of the train were basically within 12 s [15]. The characteristics of the vibration response were mainly composed of pulses with a duration of about 9 s caused by the axle weight. To meet the requirement that the node number of the input samples in the SDAE network must be consistent, the main vibration characteristics caused by the action of the train axle in each sample were retained. The sampling points at both ends of the vibration response were then truncated to match the minimum sequence length of the vibration response. Finally, min-max normalization [56] was used to normalize all the vibration amplitudes to the range of 0~1, which can boost a better learning efficiency for the SDAE network.

**Figure 14.** Typical vibration response of a monitoring zone caused by a passing train.

## *3.2. Result Analysis and Discussion*

Considering the processing power of current experimental hardware that was composed of a graphics processing unit (GPU) core (GTX 1080 Ti) with twelve 2.20 GHz processors (Intel Xeon E5-2650 v4), we collected the distributed vibration responses caused by 100 passing trains within 2 h in the subway, aiming at three randomly selected monitoring zones labeled #130, #135 and #145 to perform similarity measurement through training the SDAE network and searching the optimal constraint bandwidth. The three selected zones belong to the common track bed in the same traveling area and the ultra-weak FBG sensors used to detect vibration were installed with the consistent craft [15]. First of all, all samples were truncated into the sequence with 10,000 dimensionalities and processed by the min-max normalization. Based on the step-by-step parameter tuning, the most appropriate SDAE network structure assessed by the silhouette coefficient is shown in Figure 15, which set the denoising coefficient to 0.2, contained 5 hidden layers and reduced the input length from 10,000 to 600. The constraint bandwidth was then set to 13 by the 1-NN classifier training. For each set of candidate hyperparameters, it took approximately 2–2.5 h to perform the task on feature extraction and bandwidth search of the 100 groups of datasets of the three monitoring zones.

**Figure 15.** Hidden layers schematic of the SDAE network used for processing vibration responses.

After completing the two-step training of feature extraction of the SDAE network and distance measurement of the improved DTW algorithm, the established model can be applied to calculate the similarity of new samples. In another subway operation period, three groups of vibration responses of the #130, #135, and #145 monitoring zones caused by passing trains were collected and used to verify the proposed method of measurement similarity.

Figure 16 shows similarity measures between two vibration sequences related to the monitoring zone and the passing train. The left side of the dotted boundary in the bar graph displays the similarities between each pair of vibration responses of the same monitoring zone under different passing trains, while the right side of the boundary shows the similarities of different monitoring areas passed by the same train. Here, the subscripts A, B, and C of the monitoring area numbers in each bar column indicate the different passing trains, and two related monitoring area labels for measuring distance are connected by the symbol '&'. Obviously, the threshold of the two comparison types represented by the distance derived from the improved DTW algorithm can be identified, and it was about 800 με. The distance unit here depended on the vibration signal denoted by the strain-induced phase variation between two ultra-weak FBGs [10]. Because the results in the left part of Figure 16 are all below the threshold, it is quantitatively revealed that the similarity of the vibration response at the same physical location in the underground structure is significantly higher than the measurement results between different locations. Moreover, by using the mean distance based on the improved DTW algorithm, the similarities for monitoring zones #130, #135 and #145 can be determined as 700 με, 656 με and 756 με, respectively. The quantitative outcomes based on distance measure not only indicated that the similarity of the structural vibration changed with the location of the underground structure but also revealed that the proposed method can effectively quantify and distinguish such similarity difference. Hence, the condition change of the surrounding structure and environment can be tracked by similarities of structural vibration detected by distributed ultra-weak FBG array.

**Figure 16.** Similarity comparison based on the improved DTW distance.

#### **4. Conclusions**

This study proposed a similarity measure method to quantify the distributed vibration responses of underground structures, which involved feature extraction by the SDAE network and distance measurement by the improved DTW algorithm. Combining two datasets of one-dimensional time series from UCR archives, the detailed implementation processes for the similarity measure were introduced, and the advantages of feature extraction and distance measure in the proposed method were revealed according to algorithm comparisons. Considering the current processing capabilities of the experimental hardware, the size of the field dataset used to train the SDAE network was limited, but the subsequent experimental outcomes on distance measure still agreed well with the expected cognition. The prediction results of similarity based on the modeling of 100 groups of vibration sequences in three monitoring zones on the subway site demonstrated that the vibration similarity of the same monitoring zone was significantly higher than that from different ones. Moreover, the

similarities of distributed vibration closely related to the physical location of the underground structure can be distinguished effectively by the improved DTW distance, demonstrating that the proposed method assisted with the distributed vibration detected by the ultra-weak FBG array is promising for quantifying structural status and locating structural anomalies.

**Author Contributions:** Conceptualization, S.L.; Data curation, X.Z.; Funding acquisition, S.L.; Investigation, S.L.; Methodology, X.Z.; Project administration, H.W.; Resources, Z.L.; Software, H.W.; Supervision, L.S.; Validation, Z.L.; Writing—original draft, X.Z.; Writing—review and editing, S.L. and L.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China grant numbers 61875155 and 61735013.

**Acknowledgments:** The research work reported in this paper was supported by the National Engineering Laboratory for Fiber Optic Sensing Technology, Wuhan University of Technology, and the Smart Nanocomposites Laboratory, University of California, Irvine.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


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