## Appendix A Derivation of the Transfer Matrix of Equation (12) [19]  
\[\begin{array}{c}-\begin{bmatrix}sin\frac{\pi x_{1}}{L}&sin\frac{2\pi x_{1}}{L}& \cdots&sin\frac{\pi nx_{1}}{L}\\ sin\frac{\pi nx_{2}}{L}&sin\frac{2\pi nx_{2}}{L}&\cdots&sin\frac{\pi nx_{2}}{L} \\ sin\frac{\pi nx_{3}}{L}&sin\frac{2\pi nx_{3}}{L}&\cdots&sin\frac{\pi nx_{3}}{L} \\ \end{bmatrix}\begin{bmatrix}\frac{\Delta_{\alpha,1}}{1-S_{1}^{2}}S_{1}sin \omega_{b,1}t_{0}&\frac{\Delta_{\alpha,1}}{1-S_{1}^{2}}S_{1}sin\omega_{b,1}t_{ 1}&\cdots&\frac{\Delta_{\alpha,1}}{1-S_{1}^{2}}S_{1}sin\omega_{b,1}t_{N}\\ \frac{\Delta_{\alpha,2}}{1-S_{2}^{2}}S_{2}sin\omega_{b,2}t_{0}&\frac{\Delta_{ \alpha,2}}{1-S_{2}^{2}}S_{2}sin\omega_{b,2}t_{1}&\cdots&\frac{\Delta_{\alpha,2} }{1-S_{2}^{2}}S_{2}sin\omega_{b,2}t_{N}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{\Delta_{\alpha,\alpha}}{1-S_{2}^{2}}S_{n}sin\omega_{b,n}t_{0}&\frac{ \Delta_{\alpha,\alpha}}{1-S_{n}^{2}}S_{n}sin\omega_{b,n}t_{1}&\cdots&\frac{ \Delta_{\alpha,\alpha}}{1-S_{n}^{2}}S_{n}sin\omega_{b,n}t_{N}\\ \end{bmatrix}\\ =\begin{bmatrix}u_{x_{1},t_{0}}&u_{x_{1},t_{1}}&\cdots&u_{x_{1},t_{N}}\\ u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ u_{x_{3},t_{0}}&u_{x_{3},t_{1}}&\cdots&u_{x_{3},t_{N}}\\ \end{bmatrix}\\ =\begin{bmatrix}u_{x_{1},t_{0}}&u_{x_{1},t_{1}}&\cdots&u_{x_{1},t_{N}}\\ \frac{\Delta_{\alpha,2}}{1-S_{2}^{2}}S_{2}sin\omega_{b,2}t_{0}&\frac{\Delta_{ \alpha,2}}{1-S_{2}^{2}}S_{2}sin\omega_{b,2}t_{1}&\cdots&\frac{\Delta_{\alpha, 2}}{1-S_{2}^{2}}S_{2}sin\omega_{b,2}t_{N}\\ \vdots&\vdots&\vdots&\vdots\\ \frac{\Delta_{\alpha,\alpha}}{1-S_{n}^{2}}S_{n}sin\omega_{b,n}t_{0}&\frac{ \Delta_{\alpha,\alpha}}{1-S_{n}^{2}}S_{n}sin\omega_{b,n}t_{1}&\cdots&\frac{ \Delta_{\alpha,\alpha}}{1-S_{n}^{2}}S_{n}sin\omega_{b,n}t_{N}\\ \end{bmatrix}\\ =\begin{bmatrix}u_{x_{1},t_{0}}&u_{x_{1},t_{1}}&\cdots&u_{x_{1},t_{N}}\\ u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ \end{bmatrix}\\ =\begin{bmatrix}u_{x_{1},t_{0}}&u_{x_{1},t_{1}}&\cdots&u_{x_{1},t_{N}}\\ u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ \end{bmatrix}\\ =\begin{bmatrix}u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ u_{x_{3},t_{0}}&u_{x_{3},t_{1}}&\cdots&u_{x_{3},t_{N}}\\ \end{bmatrix}\\ \end{array} \tag{12a}\] \[\begin{bmatrix}u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ u_{x_{3},t_{0}}&u_{x_{3},t_{1}}&\cdots&u_{x_{3},t_{N}}\\ \end{bmatrix}\\ \end{array} \tag{12b}\]  
Equations (12a) and (12b) can be rewritten as Equations (12a) and (12b), respectively:  
\[\begin{array}{c}-\begin{bmatrix}sin\frac{\pi x_{1}}{L}&sin\frac{2\pi x_{1}}{L} \\ sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2}}{L}\\ \end{bmatrix}\begin{bmatrix}\frac{\Delta_{\alpha,1}}{1-S_{1}^{2}}S_{1}sin\omega_{ b,1}t_{0}&\frac{\Delta_{\alpha,1}}{1-S_{1}^{2}}S_{1}sin\omega_{b,1}t_{1}&\cdots&\frac{ \Delta_{\alpha,1}}{1-S_{1}^{2}}S_{1}sin\omega_{  
High-frequency responses at \(x_{1}\) and \(x_{2}\) are utilized to estimate the responses at \(x_{3}\) as follows:  
\[\begin{bmatrix}u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\\ u_{x_{3},t_{0}}&u_{x_{3},t_{1}}&\cdots&u_{x_{3},t_{N}}\end{bmatrix}=\mathbf{T} \begin{bmatrix}u_{x_{1},t_{0}}&u_{x_{1},t_{1}}&\cdots&u_{x_{1},t_{N}}\\ u_{x_{2},t_{0}}&u_{x_{2},t_{1}}&\cdots&u_{x_{2},t_{N}}\end{bmatrix} \tag{10}\]  
Then, the transfer matrix \(\mathbf{T}\) can be derived from Equations (11a) and (11b):  
\[\mathbf{T}=\begin{bmatrix}sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\end{bmatrix}\begin{bmatrix}sin \frac{\pi x_{1}}{L}&sin\frac{2\pi x_{1}}{L}\\ sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2}}{L}\end{bmatrix}^{-1} \tag{11}\]