## 2 Methodology
### Damage Localization Using RMS Deviation  
Generally, the damage index is vital for the state valuation of the because it focuses on the real part of the vibration sensor's impedance as a sign of the condition, which varies through perceiving the deviations and quantifying damage statistically. The mean function is a supportive means for arranging and examining large datasets. Once the differences in the parameters are positive and negative, as in a sinusoidal wave, the root mean square (RMS) is an active statistical measure for defining the number of variations. The RMS value of various signals can be represented as a sample of equivalent space. The summation of samples is divided by the total of the observations (N), results in the average (mean). In this research work, the root mean square was considered to detect the damage by quantifying the changes in voltage at each sensor with its corresponding position on the  
Figure 2: Frame of proposed approach based on auto encoder.  
tunnel structure and producing a graphic representation related to the damage map for each damage position and size. The subsequent equation outlines the RMS: This method allowed us to calculate the damage location of the structure from the vibration signals of the sensors located on the same structure. The data were acquired from each sensor at a sampling frequency of 1000 Hz, which means 1 sample (x_m (t_n)) every 0.01 s, in fact: t_(n + 1) - t_n = 0.01 s.  
The total scan time for each signal was about 10 s = 0.01 s \(\times\) 1000 samples. Three hundred sensors were placed so the 300 signals were saved as column vectors in a 1000 \(\times\) 300 matrix. We carried out the same procedure for several cases: one undamaged case and 9 damaged cases, obtaining 10 matrices stored in 10.xlsx files. The whole method was implemented in MATLAB.  
N = 1000, n: 1 to N  
M = 300, m: 1 to M  
\[\textbf{Matrix of data}=\begin{pmatrix}\textbf{x}_{1}(\textbf{t}_{1})&\textbf{ x}_{2}(\textbf{t}_{1})&.&.&.&\textbf{x}_{M}(\textbf{t}_{1})\\ \textbf{x}_{1}(\textbf{t}_{2})&\textbf{x}_{2}(\textbf{t}_{2})&.&.&.&.\\.&.&.&.&.&.\\ \textbf{x}_{1}(\textbf{t}_{n})&\textbf{x}_{2}(\textbf{t}_{n})&.&\textbf{x}_{ m}(\textbf{t}_{n})&.&\textbf{x}_{M}(\textbf{t}_{n})\\.&.&.&.&.&.\\ \textbf{x}_{1}(\textbf{t}_{N})&\textbf{x}_{2}(\textbf{t}_{N})&.&.&.&\textbf{x }_{M}(\textbf{t}_{N})\end{pmatrix}\]  
The damage indicator vector represents the relative percentage increase of each sensor signal amid the damaged case (DMG) and the undamaged case (UDG), due to the load. In the end, the damage indicator of each sensor signal is stored in a row vector. Each component of the damage indicator vector was calculated as follows:  
\[\textbf{Damage Indicator (RMSD) (\%)} =\textbf{abs}\Big{(}\frac{\big{(}\sum_{n=1}^{N}(\textbf{x}_{m}( \textbf{t}_{n}))^{2}\big{)}\textbf{UDG}-\big{(}\sum_{n=1}^{N}(\textbf{x}_{m}( \textbf{t}_{n}))^{2}\big{)}\textbf{DMG}}{\big{(}\sum_{n=1}^{N}(\textbf{x}_{m}( \textbf{t}_{n}))^{2}\big{)}\textbf{UDG}}\Big{)}\times\textbf{100}\]  
At this point, we managed to multiply the damage indicator by a scale factor (s = 10), and we added a uniformly distributed white noise to the damage indicator vector in order to simulate an underground tunnel situation for the structure. The percentage of the noise is related to the maximum damage index value vector, as shown in the MATLAB code. Finally, the damage indicator vector (300 \(\times\) 1) was plotted for each case, showing the damage location and amplitude because each component of the vector (1 to 300) is a sensor. For this reason, it is possible to know exactly where the damage is located because the location of each sensor is known from the beginning.