## 2 Theoretical Analysis
### FastICA for Surrogate Inclination Mode Shape Function  
The FastICA algorithm is a prevalent technique for addressing the BSS problem, which involves estimating the unknown mixing matrix \(\mathbf{A}\) and \(m\) independent signal sources \(s_{m}\) that combine to produce the measured signal matrix \(\mathbf{z}\), as depicted in Equation (14a,b) [45]:  
\[\mathbf{z}=\mathbf{As}\] (14a) in which \[\mathbf{s}=(s_{1},s_{2},\dots,s_{m})^{T} \tag{14b}\]  
The expansion of Equation (14a) is  
\[ \begin{bmatrix}z_{1,t_{0}}&z_{1,t_{1}}&\dots&z_{1,t_{N}}\\ z_{2,t_{0}}&z_{2,t_{1}}&\dots&z_{2,t_{N}}\\ \vdots&\vdots&\vdots&\vdots\\ z_{O,t_{0}}&z_{O,t_{1}}&\dots&z_{O,t_{N}}\\ \end{bmatrix} = \begin{bmatrix}A_{1,1}&A_{1,2}&\dots&A_{1,m}\\ A_{2,1}&A_{2,2}&\dots&A_{2,m}\\ \vdots&\vdots&\vdots&\vdots\\ A_{O,1}&A_{O,2}&\dots&A_{O,m}\\ \end{bmatrix}\begin{bmatrix}s_{1,t_{0}}&s_{1,t_{1}}&\dots&s_{1,t_{N}}\\ s_{2,t_{0}}&s_{2,t_{1}}&\dots&s_{2,t_{N}}\\ \vdots&\vdots&\vdots&\vdots\\ s_{m,t_{0}}&s_{m,t_{1}}&\dots&s_{m,t_{N}}\\ \end{bmatrix}  \tag{15}\]  
in which the subscript \(1\), \(2\), \(\dots\) O denote O-th signal receivers. The resemblance between structural dynamics and the BSS problem can be discerned by examining the parallels between Equations (3) and (14a), as well as Equations (6) and (15). In both cases, the measured signals \(\mathbf{z}\) and the displacement (also for inclination and acceleration) at multiple cross-sections are the result of a weighting matrix (\(\mathbf{A}\) and \(\mathbf{\Phi}(\mathbf{x})\)) multiplying a source signal matrix (\(\mathbf{s}\) and \(\mathbf{Y}(\mathbf{t})\)). Consequently, in this study, we have harnessed the FastICA algorithm [40] to derive a surrogate inclination mode shape function from the dynamic inclinations measured at multiple cross-sections. However, the FastICA ensures only the statistical independence of the separated signal sources in \(\mathbf{s}\). The separated matrices \(\mathbf{A}\) and \(\mathbf{s}\) may not correspond directly to the physical model of inclination shapes and their associated time functions. For instance, if the measured inclination at \(x_{1}\) is mainly contributed by the first two vibration modes, then the inclination at \(x_{1}\) can be formulated as  
\[\mathbf{z}_{incl}(x_{1},t)=\phi_{incl,1}(x_{1})\ \cdot Y_{incl,1}\ (t)+\phi_{incl,2}(x_{1})\ \cdot Y_{incl,2}\ (t) \tag{16}\]  
where \(\phi_{incl,1}(x_{1})\) and \(\phi_{incl,2}(x_{1})\) denote the value of the inclination mode shape functions \(\phi_{incl,1}(x)\) and \(\phi_{incl,2}(x)\) at \(x_{1}\), while the FastICA separation result might be  
\[\mathbf{z}_{incl}(x_{1},t)=\phi^{\prime}_{1}(x_{1})\cdot s_{1}(t)+\phi^{ \prime}_{2}(x_{1})\cdot s_{2}(t) \tag{17}\]  
for which  
\[s_{1}(t) =a\cdot Y_{incl,1}(t)+b\cdot Y_{incl,2}(t) \tag{18a}\] \[s_{2}(t) =c\cdot Y_{incl,1}(t)+d\cdot Y_{incl,2}(t) \tag{18b}\]where \(a\), \(b\), \(c\), and \(d\), are unknown weights. \(\phi^{\prime}_{1}(x_{1})\) and \(\phi^{\prime}_{2}(x_{1})\) are vectors in \(\mathbf{A}\). These two vectors are used as surrogate mode shape functions. Also, \(s_{1}(t)\) and \(s_{2}(t)\) are separated independent signal series. Furthermore, a relationship can be established between the actual and surrogate mode shape coordinates as follows:  
\[\left(\phi^{\prime}_{1}(x_{1}),\phi^{\prime}_{2}(x_{1})\right)T_{S-R}=\left( \phi_{inc,1}(x_{1}),\phi_{inc,2}(x_{1})\right)\] (19a) where \[T_{S-R}\] is the transfer matrix between the surrogate mode shape vector to the real mode shape vector, \[T_{S-R}= \begin{bmatrix}a&b\\ c&d\end{bmatrix}  \tag{19b}\]  
Moreover, \(s_{1}(t)\) and \(s_{2}(t)\) are consistent across all the cross-sections, ensuring that Equations (18a) and (18b) hold true for each cross-section.