## 4 Discussion  
According to the simulations in Section 3.2, the effect of EFI on improving the prediction accuracy of VR-ESM is obvious. Meanwhile, whether the optimal measurement system obtained with the EFI is appropriate for the CMESM is of interest to be discussed. In the following simulation, the measurement systems are the same as Figures 12(b) and 13(b), the sound predictions are completed by CMESM, and other simulation parameters remain unchanged. The results are shown in Figure 16.  
Figure 16: The sound prediction errors of CMESM on the prediction plane at the frequency of 71 Hz when 90 measurement points are used (optimized with EFI). **(a)** Free field (upper: \(p_{E}\); lower: \(\varphi_{E}\), \(L1=66\) ); **(b)**\(h_{U}=0.2\) m (upper: \(p_{E}\); lower: \(\varphi_{E}\), \(L1=75\) ). \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Uniform Measurement (Without EFI)** & **Optimal Measurement (With EFI)** \\ \hline Free field & 5783.4 (\(L1=66\)) & 2.749 (\(L1=42\)) \\ Half-space & 582.3 (\(L1=75\)) & 3.24 (\(L1=35\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Different values of \(\gamma^{\prime}\) in vibration reconstruction equivalent source method (VR-ESM) with 90 measurement points. EFI, efficient independence. Figure 15: The sound prediction errors of VR-ESM on the prediction plane at the frequency of 71 Hz when 90 uniformly arranged measurement points are used. **(a)** Free field (upper: \(p_{E}\); lower: \(\varphi_{E}\), \(L1=66\) ); **(b)**\(h_{U}=0.2\) m (upper: \(p_{E}\); lower: \(\varphi_{E}\), \(L1=75\) ). From Figure 16, it is interesting to find that, with the optimal measurement system obtained with the EFI, the prediction results of CMESM are also acceptable. It could also be explained by the ill-condition problem of the sensing matrices. The values of \(\gamma\) with different types of measurement systems are given in Table 6.  
Generally, for structures with regular shapes, the velocity ARMs usually have good spatial symmetry, which will reinforce the ill-condition problem of sensing matrices for uniform measurement point arrangement [30]. Though the optimal measurement system obtained with the VR-ESM and EFI is not necessarily optimal for the CMESM, the ill-condition problem of sensing matrices could also be limited efficiently, which allows more accurate and robust solutions of \(\beta\). Thus, the quality of sound predictions could also be improved. From the above analyses, the sound prediction based on sparse vibration measurement has been proven to be efficient. However, there still exist some factors that could affect the prediction, such as the submerged depth \(h_{U}\) and the signal-noise ratio. Meanwhile, whether the method could be applied for other compute frequencies has not been discussed yet. Thus, the above factors will be discussed in the following content. To save space, when evaluating the prediction accuracy, define the average prediction errors of sound pressure amplitude and phase as follows: \(\epsilon_{p}=\sum_{i=1}^{N_{p}}\left(\left(\|p_{i}-p_{i}^{\prime}\|_{2}/\|p_{i }^{\prime}\|_{2}\right)/N_{p}\right]\times 100\%\), \(\epsilon_{\varphi}=\sum_{i=1}^{N_{p}}\left[\left(\left|\varphi_{i}-q_{i}^{ \prime}\right|\right)/N_{p}\right]\), where \(N_{p}\) is the quantity of field points on the plane. A Sound Prediction with Different Frequencies  
Repeat the simulation in Section 3.1, \(h_{U}=0.2\) m, and select 21 frequency points ranging from 200 Hz to 500 Hz. The measurement systems are optimized with the EFI and the quantity of measurement points is 90. The prediction errors versus frequency are shown in Figure 17.  
From Figure 17, it could be seen that, at different frequencies, both the CMESM and the VR-ESM could be well applied for the sound prediction. If the measurement system is designed with the SVM, the scale of the measurement system will definitely change with the variation of the SVM.