## 5 Results and Discussion
\begin{table}
\begin{tabular}{l c} \hline \hline
**Content** & **Specification** \\ \hline Processor information & Intel(R) Core(TM) i5-8250U@ 1.60 GHz \\ Random access memory & 8.00 GB \\ System type & 64 bit operating system x64 base processor \\ Software & Matlab2020a \\ Operating system & Windows 10 Home Version: 2004 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational environment. Figure 6: Distributions of attenuation rate in the sound source grid. Figure 7 shows that the error of the proposed method is significantly smaller than that of the conventional linear interpolation methods (IDW and TIN interpolation methods). The error for the case with a doubled number of sampling points (\(p=2r\)) in the present method was found to be slightly smaller than that of \(p=r\) in the present method. Although the estimation accuracy is improved by increasing the number of sampling points, its improvement is limited in the present problem setting. The error in the propagation time monotonically decreases with increasing the number of modes from one to 100. Although the error in the attenuation rate also decreases with increasing the number of modes from 1 to 100, it does not significantly decreases after 50 modes. The relationship between the calculation time and estimation error is discussed, and the practicality of this method for the onsite quick check in the aeroacoustic wind tunnel test is evaluated. Although the required calculation time and permissible errors are dependent on the test conditions and test campaign, a quantitative permissive error is defined based on the experience of the present authors. If the error of the propagation time exceeds one corresponding to five degrees in the phase of the target frequency, the error of sound source position that is estimated by beamforming is considered to become large in the present problem setting. These permissible errors are \(1.39\times 10^{-5}\) s, \(1.39\times 10^{-6}\) s, and \(1.39\times 10^{-7}\) s  
Figure 7: Spatial average of relative error for each number of modes. Here, shaded region shows the range from the minimum to maximum values obtained for 50 runs with different random seeds. for the target frequencies of 1000 Hz, 10,000 Hz, and 100,000 Hz, respectively. Figure 8 shows plots of the spatial average of the absolute error and the calculation time for each number of modes for the reconstruction when the number of sampling points is the same as the of modes. The solid line shown in Figure 8 depicts the average of 50 test cases, and the shaded region shows the error bar that is defined by the minimum and maximum values of them, similar to those in Figure 7. The spatial average of the absolute error in each mode for the reconstruction is defined, as follows:  
\[\bar{E}_{\text{abs}}=\sqrt{\frac{1}{N}\|\hat{\mathbf{P}}-\hat{\mathbf{P}}_{\text{ref}} \|_{2}^{2}}, \tag{32}\]  
where \(N\) is the number of dimensions for the reconstructed fields and \(N=40,000\) in this study. Figure 8 illustrates that 10 modes are considered to be an appropriate number of modes. Therefore, the condition with 10 modes is adopted, and the acceleration of the calculation time, which is one of the crucial objectives of the present study, is evaluated. The calculation time for the cases with 10 modes is 0.5936 and 0.5836 s for the propagation time and attenuation rate, respectively. On the other hand, the times for the exhaustive calculation of all 40,000 grid points by the Amiet method [5] are 376.804 and 381.793 s for the propagation time and the attenuation rate, respectively. Thus, the calculation time was greatly reduced by the proposed method as compared with the exhaustive calculation of the steering vector by the Amiet method, as we expected. Table 3 presents the summarized results. Finally, Figures 9-11 show the spatial distribution of the relative error of the proposed method, the IDW interpolation method, and the TIN interpolation method, respectively.