# Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch
## 3 Proposed Combined Two-Stage Method
### Stage 2: Estimation of PWAs and Directions Using Parametric SBL
#### 3.2.3 Hidden Random Variable Inference
For the same \(R\), L1-parSBL required a slightly longer time as observed from \[\ln\hat{Q}_{j}(\mathcal{H}_{j})=\mathbb{E}_{Q(\mathcal{H}\setminus\mathcal{H}_ {j})}[\ln P(\mathcal{H},\mathbf{\psi})], \tag{13}\]5 This was because in Stage 2, parSBL handled a very low-dimensional dictionary (\(R^{\prime}\ll R\)), which was constructed in Stage 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(R\) & **L1** & **L1-L1** & **L1-parSBL** \\ \hline
400 & 0.95 s & 0.96 s & 1.17 s \\
1000 & 6.23 s & 6.24 s & 6.43 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational times based on 50 independent trials. Figure 4: (Color online) Results of the PW approximation for the (1, 3, 1) mode with \(R=400\). (**a**) Desired mode shape. (**b–d**) Spatial distribution of relative errors of the reconstructed mode when using (**b**) LS, (**c**) L1, and (**d**) L1-parSBL. ### Case: An Aircraft Cabin with Damping Floor

In this subsection, we considered a more realistic enclosure environment for an aircraft cabin, as shown in Figure 6a. The floor was set as a damping boundary with an absorption coefficient \(\alpha=0.1\). In practice, we generally focus more on local regions. Therefore, a virtual parallelepiped of \(4\) m \(\times\)\(2\) m \(\times\)\(0.8\) m with the center at \([4,0,1.4]^{\mathrm{T}}\) was defined as the sampling region. We used the _Eigenfrequency_ study in the _Pressure Acoustics, Frequency Domain_ physics interface in COMSOL Multiphysics for simulation. The vertices of the meshes inside the virtual parallelepiped region were chosen as the interpolated field points for assessment, and those outside the region were chosen as the extrapolated field points. We used 100 sets of \(M=40\) samplings. Zero-mean, additive, Gaussian noise with an SNR of 25 dB was present during each sampling. We examined 20 modes around 107.5 Hz (the first-order blade-passing frequency) whose mode shapes are plotted in Figure 6b. Figure 7 shows the effect of the selection threshold \(\beta\) on the reconstruction RME. We divided the interval \([10^{-3},10^{-0.3}]\) of \(\beta\) into 20 grids which were equally-spaced in a logarithmic scale. It could be observed that the RME increased as \(\beta\) increased, because there were fewer PW directions selected, leading to the lack of some necessary information. When \(\beta\) was less than 0.1, the RME hardly changed with it, for both interpolated or extrapolated field points. However, with smaller \(\beta\), more trivial PW directions were selected. Thus, we selected \(\beta\) to be 0.01 empirically, which meant that the PW directions whose absolute PWAs were less than 1% of the maximum absolute PWA were discarded. Figure 5: (Color online) Comparison between the theoretical and estimated PWs obtained using (**a**) LS (\(R=400\)), (**b**) L1 (\(R=400\)), (**c**) L1-L1 (\(R^{\prime}=23\)), and (**d**) L1-parSBL (\(R^{\prime}=23\)) for the (1, 3, 1) mode. The arrows indicate the PW directions, and their lengths are equal to the absolute PWAs. Figure 6: (Color online) (**a**) Geometry of the enclosure, with the dashed box denoting the virtual parallelepiped and the black dots representing the randomly distributed samplings. (**b**) Desired mode shapes of the 20 modes around 107.5 Hz. Figure 7: (Color online) RME of L1-parSBL versus the selection threshold \(\beta\) in different modes, (**a**) inside the virtual region (i.e., for interpolated field points) and (**b**) outside the virtual region (i.e., for extrapolated field points). Figure 8 shows the RMEs of LS, L1, and L1-parSBL for the 20 modes inside and outside the virtual region, respectively, with \(R=400\) and 1000. We observed that L1-parSBL achieved better RMEs than LS and L1 inside and outside the virtual region for the 20 modes. The RMEs of LS were all greater than 0.5, and L1 could obtain RMEs below 0.3 under four modes for the extrapolated field points outside the region. In contrast, L1-parSBL achieved acceptable reconstruction errors (RME \(<0.3\)) for 13 modes.