# Bridge Health Monitoring via Displacement Reconstruction-Based NB-IoT Technology
## 2 Methodology
### Reconstructed Dynamic Displacement Algorithm
\begin{array}{cc}\dot{x}_{i+2}\\ \end{array} & \begin{array}{cc}\dot{x}_{i+1}&\ddot{x}_{i+2}\\ \end{array} & \begin{array}{cc}\dot{x}_{i+2}\\ \end{array} & \begin{array}{cc}\dot{x}_{i+1}&\ddot{x}_{i+2}\\ \end{array} &\begin{array}[]\[u=L_{d}{}^{-1}(v\Delta t+L_{3}a) \tag{8}\] \[u=L_{d}{}^{-1}\Big{[}\big{(}L_{d}{}^{-1}L_{2}a\big{)}\Delta t+L_{3}a \Big{]} \tag{9}\]  
The Thikonov regularization is often used to tackle the ill condition of the matrix inverse problem encountered from the reconstruction process. In this study, instead of using Thikonov regularization, Moore-Penrose pseudo inverse is utilized to handle the inverse of matrix \(L_{d}\). Please note that although Equation (9) yields a set of displacements, only the middle point will be taken as the reconstructed displacement, then the window will continue to slide to obtain another reconstructed displacement. For a given window, the constructed displacement at the middle is the most accurate one among others. To ensure accuracy, a window-sliding technique is utilized wherein each window, only the outcome at the mid-point data is used. That is, for each window operation, a single point is obtained as the reconstructed displacement. Besides acceleration data, another parameter needed for Equation (9) is the window size (\(N_{w}\)). \(N_{w}\) is affected by the frequency and interval time of acceleration records. To determine the size of \(N_{w}\), this study utilizes a metaheuristic optimization technique to determine the appropriate \(N_{w}\) value for each frequency and interval. A single degree of freedom (SDOF) structure with various frequencies from 1-10 Hz with an increment of 1 Hz is generated, in which the accelerations are recorded in different time intervals such as 0.005 s, 0.01 s, and 0.02 s. As a result, 30 different cases are generated, and each SDOF structure is excited by the El Centro earthquake, as shown in Figure 2. Figure 3 shows the ground accelerations of El Centro. Since the structure and excited vibration are given, the exact displacement and acceleration of the structure can be calculated using the numerical integration method. The calculated structural accelerations are then added with noise and transformed to displacements using Equation (9). The added noise is an arbitrary random wave with frequencies between 1 to 200 Hz. In addition, the maximum amplitude of the added noise is twice the absolute mean of structural accelerations. Metaheuristic optimization is performed to find the most suitable \(N_{w}\) value for each case. As illustrated in Figure 2, the objective of the optimization is to minimize root-mean-square error (RMSE) between exact and reconstructed displacements. Symbiotic organism search (SOS) [23] is used in this study. Although metaheuristic optimization probably needs more computational time  
Figure 2: Determination of \(N_{w}\) using optimization and single degree of freedom (SDOF). compared to that of the conventional gradient-based optimization, metaheuristic optimization could avoid to trap at a local minimum point. Table 2 summarizes the metaheuristic optimization used in this study, including the values of parameters adopted. Figure 4 shows the optimization results of using the SOS algorithm. Based on the power regression lines, the formula for the \(\mathbf{N}_{\mathbf{w}}\) value could be formulated as shown in Equation (10), where \(\mathbf{f}\) is the frequency of acceleration and \(\Delta t\) is interval time. \[N_{w}=\frac{1}{\Delta t}f^{-8.9\Delta t-1} \tag{10}\]  
Please note that the value of \(\mathbf{f}\) in Equation (10) only has a moderate influence on the reconstructed displacement. Therefore, an approximate value of the fundamental frequency is often enough. To be specific, it should be acceptable to use a value of \(\mathbf{f}\) with a 10% error. Using the generated 30 cases, the proposed problem is compared to the previous work by [PERSON] et al. Table 3 shows the required \(\mathbf{N}_{\mathbf{w}}\) values for the proposed and the literature methods, respectively.