## 2 Theoretical Analysis
### Algorithm
**Step 3:** Construct an inter-section transfer matrix for bridge displacement. Assume that both acceleration and inclination at cross-sections \(x_{1}\) and \(x_{2}\) were measured, while at \(x_{3}\), only the inclination was measured. According to Equations (23a) and (23b), a surrogate inter  
Figure 3: Flowchart of the algorithm. section transfer matrix for displacement would be constructed with the surrogate displacement mode shape functions following Equation (10) as  
\[T_{sur-dis}= \begin{bmatrix}w_{dis,\,21}&w_{dis,\,2,\,2}\\ w_{dis,\,3,\,1}&w_{dis,\,3,\,2}\end{bmatrix}\begin{bmatrix}w_{dis,\,1,\,1}&w_{ dis,\,1,\,2}\\ w_{dis,\,2,\,1}&w_{dis,\,2,\,2}\end{bmatrix} ^{-1} \tag{25}\]  
Substituting Equation (24a) onto Equation (25), one can obtain  
\[T_{sur-dis}= \begin{bmatrix}\phi_{dis,\,2,\,1}&\phi_{dis,\,2,\,2}\\ \phi_{dis,\,3,\,1}&\phi_{dis,\,3,\,2}\end{bmatrix} T_{S-R}{}^{-1}T_{S-R}{}^{-1} \begin{bmatrix}\phi_{dis,\,1,\,1}&\phi_{dis,\,1,\,2}\\ \phi_{dis,\,2,\,1}&\phi_{dis,\,2,\,2}\end{bmatrix} ^{-1}= \begin{bmatrix}\phi_{ dis,\,2,\,1}&\phi_{dis,\,2,\,2}\\ \phi_{dis,\,3,\,1}&\phi_{dis,\,3,\,2}\end{bmatrix} {}^{ \begin{bmatrix}\phi_{dis,\, 1,\,1}&\phi_{dis,\,1,\,2}\\ \phi_{dis,\,2,\,1}&\phi_{dis,\,2,\,2}\end{bmatrix} ^{-1}=T_{dis} \tag{26}\]  
in which \(T_{dis}\) is the real displacement inter-section transfer matrix. Equation (26) reveals the _third key principle_:  
_Third key principle:The surrogate inter-section transfer matrix for displacement, derived from the integration of the surrogate inclination mode shape functions, is equivalent to the actual inter-section transfer matrix for displacement_. **Step 4:** Since the first key principle (illustrated as Equation (13b)) has shown that \(T_{dis}=T_{acc}\), it can be inferred from Equation (26) that  
\[T_{sur-dis}=T_{acc} \tag{27}\]  
Then, one can estimate the acceleration at \(x_{3}\) using the measured acceleration at \(x_{1}\) and \(x_{2}\):  
\[ \begin{bmatrix}\tilde{u}_{x_{2},t_{0}}&\tilde{u}_{x_{2},t_{1}}&\cdots&\tilde{ u}_{x_{2},t_{N}}\\ \tilde{u}_{x_{3},t_{0}}&\tilde{u}_{x_{3},t_{1}}&\cdots&\tilde{u}_{x_{3},t_{N}} \end{bmatrix} =T_{sur-dis}{}^{ \begin{bmatrix}\tilde{u}_{x_{1},t_{0}}&\tilde{u} _{x_{1},t_{1}}&\cdots&\tilde{u}_{x_{1},t_{N}}\\ \tilde{u}_{x_{2},t_{0}}&\tilde{u}_{x_{2},t_{1}}&\cdots&\tilde{u}_{x_{2},t_{N}} \end{bmatrix} } \tag{28}\]  
Additionally, it is important to note that throughout the entire algorithm, the \(T_{S-R}\) matrix remains unknown and has effectively been eliminated from the process. Consequently, there is no need to determine the explicit form of \(T_{S-R}\). It should be noted that once the \(T_{acc}\) has been determined for a specific scenario under a given traffic loading condition, it should also be applicable to other scenarios with different traffic loadings. This is because, as indicated in Equation (26), the \(T_{acc}\) is solely dependent on the bridge's mode shapes, which are independent of different traffic loadings. This assertion has been substantiated by previous research, and further details can be found in reference [41]. Should a bridge experience degradation that leads to a significant and uneven reduction in its stiffness along its span [46], this would inevitably alter the bridge's mode shapes. Under such circumstances, it is presumed that the \(T_{acc}\) (transfer matrix for acceleration) derived from a pristine bridge would no longer be suitable for the compromised structure. Nevertheless, if monitoring data are gathered from the degraded bridge, our proposed method can be effectively employed to ascertain a new \(T_{acc}\) that aligns with the modified mode shapes of the deteriorated bridge.