## 5 Method Validation  
### Simulated Example  
The proposed method is illustrated using the simulation data of a square plate which is simply supported. The side length of the plate is 1 m, and the thickness is 0.002 m. The material properties of the plate are as follows: elastic modulus of 200 GPa, density of 7850 kg/m\({}^{3}\), Poisson's ratio of 0.3. The plate is excited at (0.3 m, 0.4 m) by an impulsive signal. The point-wise FRF of interest is located at \(\mathbf{q}_{0}=\) (0.2 m, 0.7 m) from the left-bottom corner.  
The response of the plate is calculated using the finite element method based on the mode superposition principle, where the viscous damping is considered. The simulatedresponse is then disturbed by a zero-mean band-limited Gaussian noise. The signal-to-noise ratio of the response is set to be 10 dB. The sampling frequency is 1024 Hz, the frequency band for identification spans from 1 Hz to 140 Hz, and \(F=274\). The second-order bivariate polynomial is used for the spatial-frequency model presented in Section 3.1.  
A classical spectral H\({}_{1}\) estimator, which is used as a reference for comparison purposes, is computed as  
\[\hat{G}_{SA}(\omega_{k})=\frac{1}{2n+1}\sum_{l=-n}^{n}\frac{Y_{l}(k)}{U(k)} \tag{40}\]  
where the spectral estimate \(\hat{G}_{SA}(\omega_{k})\) uses AOI averaging to mitigate the effect of measurement noise on FRF estimation.  
The estimated FRF is affected by the estimation variance and bias. For a fixed order of the spatial model, the size of the local AOI (\(d\), see Figure 1) plays an important role in finding an optimal trade-off between estimation uncertainty and bias. The root mean square error (RMSE) is therefore introduced to investigate the size effect on estimation accuracy, which is defined via the Monte Carlo simulation  
\[\text{RMSE}(\omega_{k})=\sqrt{\frac{1}{M_{c}}\sum_{m=1}^{M_{c}}\left|\hat{G}^{ [m]}(\omega_{k})-G^{0}(\omega_{k})\right|^{2}} \tag{41}\]  
where \(\hat{G}^{[m]}\) is the FRF estimate obtained at the \(m\)-th Monte Carlo run, \(G^{0}\) stands for the true FRF, and \(M_{c}=100\). By definition, the RMSE metric can quantify both estimation uncertainty and bias error.  
When the AOI size (\(d\)) increases, more data are used, and, accordingly, the estimation uncertainty is reduced. However, the increase in size can lead to a bias increment, as seen in Figure 2. An optimal trade-off between estimation uncertainty and bias error is achieved when \(d=10.6\) mm. This is justified by Figure 2, which shows that the proposed method can improve the accuracy of FRF measurement in an effective manner, and globally outperforms the classical spectral analysis method for a wide range of local area sizes.  
The RMSE computed with the proposed method is found to be more pronounced for the first three modes, in which it is basically dominated by the bias error. This is because the implemented DC kernel is incapable of fully representing the complex behavior of the lightly damped structure. The bias error can be decreased by resorting to a multiple-kernel-based regularization [28], which provides more flexibility in model design to describe the spatial-frequency property of the FRF.  
With the optimal local area size \(d=10.6\times 10^{-3}\) m, the FRF's MAP is displayed together with its standard deviation in Figure 3. In the whole frequency band, the predicted standard deviation agrees well with the sample standard deviation estimated over Monte Carlo runs. The residual error between the MAP and the true FRF falls below the standard deviation almost everywhere.  
Figure 2: Size effect of the local AOI on FRF estimation.