## 3 Spatial-Frequency Model of the Point-Wise FRF  
### Local Spatial Model  
In this section, a spatial model is proposed for point-wise FRF estimation by taking into account local field measurements. The FRF at coordinate \(\mathbf{q}_{0}\) is of interest, and \(\Delta\) is alocal AOI centered around \(\mathbf{q}_{0}\). The AOI (\(\Delta\)) contains numerous pixel measurements (see Figure 1). The AOI is divided into \(2n+1\) parts, and the size of each part is comparable to a miniature accelerometer. Then, the pixel measurements within a part are averaged together, and the averaged time history is assigned to the coordinate \(\mathbf{q}_{l}\), yielding the DFT vector \(\mathbf{Y}_{l}\), \(l\) = -\(n\), \(\cdots\), \(n\). The index \(l\) indicates a relative coordinate with respect to \(\mathbf{q}_{0}\). The bold \(\mathbf{Y}_{l}\) is a vector that contains the DFT of the measurement at \(\mathbf{q}_{l}\), and \(\mathbf{Y}_{l}=\left[\mathbf{Y}_{l}(1),\mathbf{Y}_{l}(2),\cdots,\mathbf{Y}_ {l}(F)\right]^{\mathrm{T}}\) with the superscript \({}^{\mathrm{T}}\) being the transpose operator. For a linear time-invariant structure of structural damping, the FRF at \(\mathbf{q}_{l}\) with the excitation applied at coordinate \(\mathbf{f}\) is  
\[G_{l}(\omega)=\sum_{m=1}^{M}\frac{\psi_{lm}\psi_{fm}}{\lambda_{m}^{2}-\omega^{ 2}} \tag{5}\]  
where the index \(f\) indicates the coordinate \(\mathbf{f}\), \(\lambda\) and \(\psi\) are the complex natural frequency and modal shape, respectively, and \(M\) is the system order. The mode shape is spatially continuous, which can be modeled sufficiently well by using a low-order multivariate polynomial in the local AOI (\(\Delta\)),  
\[\psi_{lm}=\psi_{0m}+\mathbf{\phi}_{l}\mathbf{a}_{m} \tag{6}\]  
where \(\psi_{0m}\) is the \(m\)-th mode shape at \(\mathbf{q}_{0}\), the vector \(\mathbf{a}_{m}=\left[a_{1m},\cdots,a_{n_{p}m}\right]^{\mathrm{T}}\), and \(\phi_{l}\) is the polynomial basis. Let us take a plat surface as an example. \(\phi_{l}=\left[x_{l}-x_{0},\ y_{l}-y_{0}\right]\) for the first-order bivariate polynomial, and (\(x\),\(y\)) are the components of the coordinate. Note that Equation (6) is suitable for modeling curved shapes when a trivariate polynomial is used. Refer to [20] for DIC tests of large-scale structures with complex geometry. By combining Equations (5) and (6), the FRF at coordinate \(\mathbf{q}_{l}\) is derived as  
\[G_{l}(\omega)=G_{0}(\omega)+\sum_{r=1}^{n_{\phi}}g_{r}(\omega)(\mathbf{\phi}_{l})_ {[r]} \tag{7}\]  
where \(G_{0}(\omega)\) is the FRF at \(\mathbf{q}_{0}\), \(\left(\mathbf{\phi}_{l}\right)_{[r]}\) is the \(r\)-th element of \(\mathbf{\phi}_{l}\), and the basis function \(g_{r}(\omega)\) is  
\[g_{r}(\omega)=\sum_{m=1}^{M}\frac{a_{rn}\psi_{fm}}{\lambda_{m}^{2}-\omega^{2}} \tag{8}\]  
Figure 1: Local AOI measurements (\(n\) = 4). (Red diamond indicates the coordinate \(\mathbf{q}_{l}\), which represents the geometric center of each divided part of the AOI). Equation (7) provides a spatially parametric model of the FRF in the local AOI \(\Delta\), which has \(n_{\phi}+1\) unknown functions of frequency. By substituting the expression in Equation (7) into Equation (3), the \(2n+1\) responses in the local AOI are assembled in matrix form  
\[\mathbf{Y=\Phi H+N_{Y}} \tag{9}\]  
where \(\mathbf{H}=\left[\mathbf{G}_{0}^{\mathsf{T}},\mathbf{g}_{1}^{\mathsf{T}}, \cdots,\mathbf{g}_{n_{\phi}}^{\mathsf{T}}\right]^{\mathsf{T}}\), the extended output vector \(\mathbf{Y}=\left[\mathbf{Y}_{-n}^{\mathsf{T}},\cdots,\mathbf{Y}_{0}^{\mathsf{ T}},\cdots,\mathbf{Y}_{n}^{\mathsf{T}}\right]^{\mathsf{T}}\), and  
\[\mathbf{\Phi}= \begin{bmatrix}\langle\mathbf{U}\rangle&\langle\mathbf{U} \rangle(\boldsymbol{\phi}_{-n})_{[1]}&\cdots&\langle\mathbf{U}\rangle( \boldsymbol{\phi}_{-n})_{[n_{\phi}]}\\ \vdots&\vdots&&\vdots\\ \langle\mathbf{U}\rangle&0&\cdots&0\\ \vdots&\vdots&&\vdots\\ \langle\mathbf{U}\rangle&\langle\mathbf{U}\rangle(\boldsymbol{\phi}_{n})_{[1 ]}&\cdots&\langle\mathbf{U}\rangle(\boldsymbol{\phi}_{n})_{[n_{\phi}]}\\ \end{bmatrix}