## 5 Results and Discussion  
Figures 5 and 6 show the distributions of the propagation time and the attenuation rate under a certain test condition as an example of the sparse sampling. The test condition is randomly selected within the range of Table 1. Here, the microphone position and Mach number were set to be (\(x\), \(y\), \(z\)) = (1.2589, 1.9246, \(-\)1.4921) and \(M\) = 0.1827, respectively. The theoretical distributions are calculated as distributions of 40,000 points using the relationship between the sound source and microphone position using the Amiet method [5]. Figures 5b and 6b show the estimated distributions when the numbers of the modes \(r\) and the sampling points \(p\) are 10. Circles in those figures show the positions of the selected sampling points. The sampling points that are determined by the sensor selection method [17] are uniformly distributed throughout, and the estimated distributions are close to the original distributions well. Therefore, the proposed method was found to effectively reproduce the distribution with a limited number of sampling points. Table 2 shows the computational environment. The numbers of the modes \(r\) and the sampling points \(p\) for constructing a reduced-order model are significant parameters of the present method, because these values control the degree of freedom for expressing the original large-scale data. In this study, \(\mathbf{\hat{P}}\) was estimated both when the number of sampling points was equal to the number of modes (\(p=r\)) and when it was twice the number of modes (\(p=2r\)). Figure 5: Distributions of propagation time in the sound source grid. The proposed method estimates the amplitude of the modes using the spatial modes and the sampling points that were obtained from the training data, and it reconstructs the distributions. On the other hand, as a more conventional reconstructing method, there is a method of linearly interpolating the values between the sampling points. Therefore, the values of each sampling point selected by the sensor selection method [17] were linearly interpolated and reconstructed, and the distributions of 40,000 points are compared with the proposed method. The inverse distance weighted (IDW) interpolation [29] and triangulated irregular network (TIN) interpolation methods [30] are used for the spatial linear interpolation method. The IDW interpolation method calculates the interpolated values by a linear combination of each sampling point and its weight that reflects the shortest distance between the interpolated point and the sampling point [29]. In addition, the TIN interpolation method divides the space into multiple triangles that are based on the sampling points and calculates the interpolated values using three sampling points at the vertices of each triangle [30]. The estimation accuracy of the proposed method was evaluated using test cases, in which 50 patterns were randomly selected from the range shown in Table 1. The spatial average of relative error in each mode is defined for the quantitative evaluation, as follows:  
\[\bar{E}=\sqrt{\frac{\|\hat{P}-\hat{P}_{\text{ref}}\|_{2}^{2}}{\|\hat{P}_{ \text{ref}}\|_{2}^{2}}}\times 100. \tag{31}\]  
Here, \(\hat{P}\) is a vector of reconstructed results that correspond to the propagation time or the attenuation rate of 40,000 grid points in the source grid. In addition, \(\hat{P}_{\text{ref}}\) is a vector of reference values for those at each test condition. The reference values can be obtained by the exhaustive calculation of the Amiet method [5] without using sparse sampling. Figure 7 shows a comparison of the estimation accuracy of the four methods. The solid line shown in Figure 7 depicts the average of 50 test cases, and the shaded region shows the error bar defined by the minimum and maximum values of them. The calculations of the TIN interpolation methods for one to four modes are not conducted because the number of sampling points that can divide into triangles is five or more in the TIN interpolation method [30].