# Bridge Health Monitoring via Displacement Reconstruction-Based NB-IoT Technology
## 2 Methodology
### Reconstructed Dynamic Displacement Algorithm  
The objective of the reconstructed dynamic displacement algorithm is to translate measured accelerations into dynamic displacements, initially introduced by [PERSON] et al. [23]. The main concept is to minimize the difference between two accelerations: one is from the records, and the other one is transferred from the assumed displacements. Figure 1 illustrates the procedure of the algorithm. As shown in Figure 1, noise-contaminated acceleration data are the input, and the reconstructed displacement is the output. To be specific, at each time of reconstruction, a window length (\(N_{w}\)) of acceleration are chosen, and only a single displacement is reconstructed. The window slides from left to right with the same increment of the acceleration data. Taking window \(i=1\) as an example, as shown in Figure 1, acceleration data inside window \(i=1\) are the input for the reconstruction algorithm, and the output is a single displacement termed as \(d_{i=1}\). Such a process is repeatedly performed until window \(i=j\). The length of the window (\(N_{w}\)) is determined by an optimization process; details are explained later. [PERSON] et al. [22] adopted the central difference method to transfer the assumed displacement to acceleration, in which the reconstructed displacement does not contain noise. Instead of using the central difference method, the Newmark-beta method is utilized in this study to explore the possibility of enhancing the accuracy in displacement reconstruction. Equations (1) and (2) show the Newmark-beta equations, where \(x_{i}\), \(\dot{x}_{i}\), and \(\ddot{x}_{i}\) are the displacement, velocity, and acceleration,  
Figure 1: Illustration of displacement reconstruction scheme. respectively. \(\Delta t\) refers to the time interval, \(\gamma\) and \(\beta\) are the Newmark constants. In this study, \(\gamma\) and \(\beta\) are taken as 1/2 and 1/6, respectively. Such assumption represents a linear acceleration process:  
\[\dot{x}_{i+1}=\dot{x}_{i}+\Delta t\Big{[}(1-\gamma)\ddot{x}_{i}+(\gamma)\ddot{x }_{i+1}\Big{]} \tag{1}\]  
\[x_{i+1}=x_{i}+(\Delta t)\dot{x}_{i}+\Delta t^{2}\Big{[}(0.5-\beta)\ddot{x}_{i} +(\beta)\ddot{x}_{i+1}\Big{]} \tag{2}\]  
Equation (1) could be rewritten in a matrix form, as indicated in Equation (3). As shown in Equation (3), matrix \(\mathbf{a}\) is a set of acceleration data, and in the proposed algorithm, matrix \(\mathbf{a}\) is the measured acceleration while \(\mathbf{L_{d}}\) and \(\mathbf{L_{2}}\) are the constant. The dimension of the matrix \(\mathbf{a}\) is adjusted to be the same length as each window (from \(\mathbf{i}=1\) to \(\mathbf{N_{w}}\)). Thus, the size of matrix \(L_{d}\) and matrix \(\mathbf{L_{2}}\) are \(\mathbf{N_{w}}\times(\mathbf{N_{w}}+1)\) and \(\mathbf{N_{w}}\times\mathbf{N_{w}}\), respectively. Matrix \(\mathbf{v}\) in Equation (3) can be considered as the reconstructed velocity. However, it is understood that the exact solution for \(\mathbf{v}\) is not possible. \[L_{d}\mathbf{v}-L_{2}\mathbf{a}=0 \tag{3}\]  
where:  
\[L_{d}=\left[\begin{array}{ccccc}-1&1&&&&\\ &-1&1&&0&\\ &0&&\cdots\cdots&&-1&1\\ &&&-1&1&\\ &&&&v=\Big{[} \begin{array}{cc}\dot{x}_{i}&\dot{x}_{i+1}&\dot{x}_{i+2}\\ \end{array} & \begin{array}{cc}\dot{x}_{i+1}&\dot{x}_{i+2}\\ \end{array} & \begin{array}{cc}\dot{x}_{i}&\dot{x}_{i+1}&\ddot{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}&\ddot{x}_{i+1}&\ddot{x}_{i+2}\\ \end{array} & \begin{array}{cc}\dot{x}_{i+2}\\ \end{array} & \begin{array}{cc}\dot{x}_{i}&\ddot{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}{cc}\dot{x}_{i+2}\\ \end{array} & \begin{array}{cc}\dot{x}_{i}&\ddot{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} &