**Sheng Li 1,\*, Xiang Zuo 2, Zhengying Li 2, Honghai Wang <sup>1</sup> and Lizhi Sun <sup>3</sup>**


Received: 23 March 2020; Accepted: 11 April 2020; Published: 12 April 2020

**Abstract:** Quantifying structural status and locating structural anomalies are critical to tracking and safeguarding the safety of long-distance underground structures. Given the dynamic and distributed monitoring capabilities of an ultra-weak fiber Bragg grating (FBG) array, this paper proposes a method combining the stacked denoising autoencoder (SDAE) network and the improved dynamic time wrapping (DTW) algorithm to quantify the similarity of vibration responses. To obtain the dimensionality reduction features that were conducive to distance measurement, the silhouette coefficient was adopted to evaluate the training efficacy of the SDAE network under different hyperparameter settings. To measure the distance based on the improved DTW algorithm, the one nearest neighbor (1-NN) classifier was utilized to search the best constraint bandwidth. Moreover, the study proposed that the performance of different distance metrics used to quantify similarity can be evaluated through the 1-NN classifier. Based on two one-dimensional time-series datasets from the University of California, Riverside (UCR) archives, the detailed implementation process for similarity measure was illustrated. In terms of feature extraction and distance measure of UCR datasets, the proposed integrated approach of similarity measure showed improved performance over other existing algorithms. Finally, the field-vibration responses of the track bed in the subway detected by the ultra-weak FBG array were collected to determine the similarity characteristics of structural vibration among different monitoring zones. The quantitative results indicated that the proposed method can effectively quantify and distinguish the vibration similarity related to the physical location of structures.

**Keywords:** similarity measure; subway tunnel; distributed vibration; feature extraction; autoencoder; ultra-weak FBG

## **1. Introduction**

Over the past decades, with the rapid development of rail transit infrastructure in China, the operation safety and security of subway systems have attracted much attention. According to the recent research progress of distributed optical fiber-sensing technology [1–7], the requirement for time- and space-continuous monitoring for the geotechnical underground structures [8] has gradually become feasible. Comparisons between various commonly used sensors for underground structure monitoring were reported in [9,10], which revealed that the ultra-weak fiber optic Bragg grating (FBG) array [11] can be used for both static and dynamic measurements [12–14]. In the field of dynamic measurement, it was reported that the distributed vibration detected by the ultra-weak FBG array

can be applied to track train and identify incursion [10,15]. Moreover, the change of the structural vibration responses usually reflects the evolution of the structure state to a certain extent. A wide range of research reports concerning the vibration-based structural condition assessment can be found in [16–19]. Compared with ground transportation, the daily operation of underground trains is of obvious regularity. For example, the speed of trains in each travel zone always follows the operation schedule, and the number of passengers does not change suddenly within a certain period due to commuting habits. Moreover, the temperature and humidity fields of underground infrastructure are relatively stable due to the management measures of tunnel ventilation. Therefore, it can be assumed that the structural vibration responses corresponding to the excitation of multiple passing trains in a certain structural state should be stable and similar. With the support of distributed vibration monitoring adapted to the long-distance underground structures, it is possible to quantify the structural status by measuring the similarity of structural vibration responses for a specified monitoring area under different stages and this is the research motivation of the paper.

The vibration responses of subway tunnel structures can be regarded as a collection of typical one-dimensional time-series signals. The similarity measure between time series can often be converted to measure the distance between vectors. The Euclidean distance (ED) [20] and its variants based on common Lp-norm [21] are the most straightforward methods for similarity measures of such one-dimensional time-series. However, there is a slight difference in the length of duration in the vibration responses excited by each train passing through the monitoring area, making the ED and its variants unable to directly perform the similarity measure for unequal-length sequences. Even when dealing with equal-length vibration signals, these methods are susceptible to noise and time misalignment and are unable to deal with local time-shifting. Dynamic time warping (DTW) [22] is an option to overcome time-shifting, which allows a time series to be either stretched or compressed to provide a better match with another time series. Therefore, it can be used to handle similarity measures between inconsistent length sequences. Another group of similarity measures suitable for processing unequal-length time series is developed based on the concept of the edit distance for strings [23]. Compared with DTW which only considers the constrain bandwidth, the similarity measure based on the edit distance requires tuning more parameters [24–26] to find the most similar set of matching patterns. It is reported [15] that the data amount is huge for vibration responses detected by ultra-weak FBG of each monitoring area under the excitation of passing trains. This often results in high time complexity and is expensive in terms of processing and storage costs to directly use the above methods to perform a similarity measure on the raw format of high-dimensional vibration responses of underground structures. Furthermore, it is difficult to completely avoid random outlier interference during data collection and transmission. Therefore, the results of the similarity measure based on any algorithm may significantly deviate from expectations if the raw signals are not carefully wrangled.

Feature extraction should be the most intuitive idea to solve the above problems. It can improve the effectiveness and efficiency of the similarity measure by maintaining the characteristics of the original signal in a smaller dimensionality. Compared with principal component analysis (PCA) [27], linear discriminant analysis (LDA) [28] and other linear feature extraction methods, manifold learning [29], restricted Boltzmann machine (RBM) [30], autoencoder (AE) [31], as typical representatives of non-linear feature extraction methods, can retain much richer sample features of high-dimensional vibration signals. High computational complexity is the bottleneck of manifold learning based on local domain classification and its feature extraction process is sensitive to noise [32]. Therefore, this method is not suitable for extracting the characteristics of the vibration responses of underground structures that cannot avoid noise interference. RBM and its derivative deep belief network [33] use the probability distribution rather than the real-valued sequence to express the characteristics of the hidden layer. These two methods for dimensionality reduction are not suitable for the similarity measure of real-valued sequences. The training of AE resembles that of the RBM. However, models of AE can be easier to train than that of RBM with contrastive divergence and are thus preferred in contexts where RBM training is less effective [34]. Adding a denoising process makes

AE models substantially more robust to input variations or distortion, causing the deep network formed by a stacked denoising autoencoder (SDAE) with higher accuracy than that of the stacked autoencoder (SAE) [35,36]. Thus, the SDAE network is used to achieve feature extraction before the similarity measure in this paper. In the following second section, the implementation process of the proposed similarity measure is introduced in combination with typical one-dimensional datasets in the public UCR (University of California, Riverside) time-series data archives [37]. This part also illustrates the metrics used to evaluate the effectiveness of feature extraction and similarity measure. After that, based on the ultra-weak FBG vibration response of the actual underground structure, the feasibility and significance of the proposed similarity measure method in engineering are discussed.

### **2. Methodology and Implementation of Signal Similarity Measure**
