## 2 Theoretical Analysis
### Algorithm  
In this research, our objective is to estimate the high-frequency acceleration at a specific location on the target bridge, where only an inclinometer is installed. Inclinometers and accelerometers are, however, placed at several other locations. The collected inclination and acceleration data from these locations will be utilized to deduce the unknown acceleration at the target site. The flowchart illustrating the complete algorithm is presented in Figure 3.  
\(\left(\theta_{x_{1}}(t),\theta_{x_{2}}(t)\ldots\theta_{x_{p}}(t)\right)\) denotes the measured inclination time histories at \(p\)-th cross-sections \(x_{1}\), \(x_{2}\), \(\ldots x_{p}\); as an example, it is assumed that at \(x_{1}\) and \(x_{2}\), the acceleration has been measured (as \(\ddot{u}_{x_{1}}\) (\(t\)) and \(\ddot{u}_{x_{2}}\) (\(t\)), respectively), and the accelerations at these two cross-sections are used to estimate the unmeasured acceleration at \(x_{3}\) (i.e., \(\ddot{u}_{x_{3}}\) (\(t\))). **Step 1:** Inclination data from multiple cross-sections are subjected to analysis using the FastICA algorithm to extract the mixing matrix \(W_{mix}\) and the discrete surrogate inclination mode shape functions \(\phi^{\prime}_{incl-1}\) and \(\phi^{\prime}_{incl-2}\). Focusing on the first two modes, we assume the presence of only two independent signal sources, \(s_{1}\) and \(s_{2}\). Should additional modes be taken into account, the number of signal sources would correspondingly expand to match the total count of modes. For each cross-section acting as a signal receiver, there would be two corresponding separated weight parameters, \(w_{1}\) and \(w_{2}\). Assuming that there are \(p\) cross-sections, the mixing matrix would be  
\[W_{mix}= \begin{bmatrix}w_{1,1}&w_{2,1}&\ldots&w_{p,1}\\ w_{1,2}&w_{2,2}&\ldots&w_{p,2}\end{bmatrix} \] (21a) and the discrete first surrogate inclination mode shape function would be the first line of \[W_{mix}\], \[\phi^{\prime}_{incl-1}= \begin{bmatrix}w_{1,1}&w_{2,1}&\ldots&w_{p,1}\end{bmatrix} \] (21b) and similarly, the second surrogate inclination mode shape function would be \[\phi^{\prime}_{incl-2}= \begin{bmatrix}w_{1,2}&w_{2,2}&\ldots&w_{p,2}\end{bmatrix}  \tag{21c}\]  
**Step 2:** Discrete integration of surrogate inclination mode shapes to generate surrogate displacement mode shapes. The nearest two supports are set as location \(x_{0}\) and location \(x_{p+1}\). Coordinates of these \(p+2\) locations are  
\[x= \begin{bmatrix}x_{0}&x_{1}&\dots&x_{p+1}\end{bmatrix}  \tag{22}\]  
Considering that the displacement at the bridge supports is inherently zero, the displacement mode shape coordinates at these supports are accordingly set to zero. To generate the surrogate displacement mode shape function for these intermediate \(p\) coordinates, we employ discrete integration techniques: the Euler method for the endpoints \(x_{1}\) and \(x_{p}\) and Hermite polynomials for the internal range from \(x_{2}\) to \(x_{p-1}\). This approach yields the surrogate displacement mode shape functions at the intermediate \(p\) coordinates as follows:  
\[\phi^{\prime}{}_{dis-1}= \begin{bmatrix}w_{dis,\;1,1}&w_{dis,\;2,1}&\dots&w_{ dis,\;p,1}\end{bmatrix}  \tag{23a}\] \[\phi^{\prime}{}_{dis-2}= \begin{bmatrix}w_{dis,\;1,2}&w_{dis,\;2,2}&\dots&w_{ dis,\;p,2}\end{bmatrix}  \tag{23b}\]  
and \(\phi^{\prime}{}_{dis-1}\) and \(\phi^{\prime}{}_{dis-2}\) form the new mixing matrix for bridge displacement as  
\[W_{dis,mix}= \begin{bmatrix}\phi^{\prime}{}_{dis-1},\phi^{\prime}{}_{dis-2} \end{bmatrix}  \tag{23c}\]  
Additionally, as indicated by Equation (20a), the transfer matrix \(T_{S-R}\) in Equation (20a) is irrelevant to the spatial coordinate \(x\); then, Equation (24a) can be generated as  
\[W_{dis,mix}T_{S-R}=\mathbf{\Phi}_{dis}(x)\] (24a) in which \[\mathbf{\Phi}_{dis}(x)= \begin{bmatrix}\phi_{dis,1}(x),\phi_{dis,2}(x)\end{bmatrix}  \tag{24b}\]  
where \(\phi_{dis,1}(x)\) and \(\phi_{dis,2}(x)\) are the real displacement mode shape functions.