# Temporal Convolutional Network-Based Axle Load Estimation from pavement Vibration Data
## 2 Methods
### Data Acquisition  
In this paper, we utilized a DOVS-based vibration sensing system to acquire the original pavement vibration signal with a higher spatial resolution (see Figure 1: data acquisition and preprocessing). DOVS technology uses the principle of phase optical time-domain reflection (\(\varphi\)-OTDR), which allows a single fiber optic cable to function as a multipoint sensor for measurement and localization. The schematic principle of \(\varphi\)-OTDR is illustrated in Figure 1. During measurement, the measurement device emits pulsed light continuously at one end of the cable, and the light waves reflect the signal continuously to the device along the cable's axis. The position information of the measurement channel can  
Figure 1: The overview of the proposed TCN-based method for axle load estimation.  
be analyzed according to the time difference of each reflected light. When a vibration event disturbs the cable, its mechanical properties in the section change, affecting the scattered amplitude of the reflected light. Therefore, the relative amplitude of the reflected light can be used to analyze the vibration information. Consequently, through the distributed optical vibration sensor, we can measure the vibration along the fiber optic cable at different time instances.  
The fiber optic sensing data typically contain a significant number of redundant signals, which need to be processed before analysis. Moreover, the vibration data must be extracted from the raw data by selecting the appropriate time window for analysis. The vibration data are extracted from raw signals through anomaly detection based on a Gaussian model of the noise signal. Figure 2 shows the vibration signal detection algorithm. First, the sensing data are collected from different channels. Then, the short-time energy of each data frame is calculated as defined by (1). A Gaussian noise model is built based on the short-time energy of a series of noise signal segments collected, according to (2) and (3), to estimate the model parameters. The short-time energy noise signal was tested for Gaussianity using a quantile-quantile plot as depicted in Figure 2 (noise probability models part). The majority of the samples align with the line. When new data are detected, the short-time energy of the signal is calculated and compared with the Gaussian model. If the signal energy surpasses a threshold of 3\(\sigma\), the corresponding signal frame is considered an anomalous segment containing the vibration signal. Consecutive windows containing vibration signals are merged. The decision of whether to record the signal or not is based on whether the number of channels with vibration detection exceeds a ratio of the total number of channels determined by the spatial distribution of signal energy. On the other hand, when a signal frame is detected as noisy, it is incorporated into a series of previous noisy signal frames. The system updates the Gaussian model of noise so that the model can accommodate different noise levels at different times of the day.  
\[E_{i}(n)=\sum_{t=0}^{\infty}\left[x_{i}(t)\omega(t-n)\right]^{2} \tag{1}\]  
\[\omega(t-n)=\begin{cases}1,&n\leq i\leq M+n\\ 0,&\text{otherwise}\end{cases} \tag{2}\]  
\[\mu_{i}=\frac{1}{N}\sum_{n=0}^{N}E_{i}(n)^{2} \tag{3}\]  
\[\sigma_{i}=\frac{1}{N-1}\sum_{n=0}^{N}\left(E_{i}(n)-\mu_{i}\right)^{2} \tag{4}\]  
where \(E_{i}(n)\) is the short-time energy of the \(n\)th frame, \(x_{i}\) is the vibration signals at channel \(I\), \(\omega(t-n)\) is a rectangular windows function expressed as (2), \(\mu_{i}\) is the mean value of the Gaussian model corresponding to the \(i\)th channel, and \(\sigma_{i}\) is the standard deviation value of the Gaussian model corresponding to the \(i\)th channel.