# 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
We also found that the denser discretized PW directions hardly improved the RME in this case. Additionally, the RMEs of different modes were very different, especially for the extrapolated field points outside the virtual region. This was because samplings were distributed in the local region and these samplings could not obtain sufficient spatial information for some modes. For example, for modes \"No. 3\" and \"No. 8\" of Figure 6b, the virtual region mainly included their nodal planes, so it was difficult for the samplings to capture the effective information of the mode shapes. Therefore, the estimated PWs lead to poor reconstruction accuracy outside the virtual region. Specifically, we considered the mode at approximately 106.5 Hz (closest to 107.5 Hz) as an example. Corresponding desired mode shape is plotted as the 12 th subgraph in Figure 6b. The effects of \(M\) (the number of sample data points) and SNR on the reconstruction accuracy of LS, L1, and L1-parSBL were analyzed. Figure 9a shows that the RMEs inside and outside the virtual region increased as \(M\) decreased for LS, L1, and L1-parSBL, respectively. Although the RMEs inside the region for the three methods were close, L1-parSBL had the lowest reconstruction errors. Both L1 and L1-parSBL could provide acceptable outcomes with \(M=10\) samplings, but according to the error bar curves, specific sampling positions had significant impact. The proposed L1-parSBL achieved much better RMEs than LS and L1 when sufficient number of samples were used for the extrapolated field points. Based on Figure 9b, the RMEs inside and outside the virtual region increased as SNR decreased for LS, L1, and L1-parSBL, respectively. LS and L1 failed to reconstruct the mode shape outside the region (RME \(>0.5\)) when SNR is below 25 dB. However, L1-parSBL still obtained the average RME of 0.5 at 15 dB. For interpolated field points, L1-parSBL could provide RME of 0.34 at 5 dB, whereas the RMEs of LS and L1 were basically more than 0.5 at 10 dB. Although only one mode outcome is illustrated here, similar results were obtained for the other modes as well. Figure 8: (Color online) RME of LS, L1, and L1-parSBL in different modes (—: \(R=400\); - -: \(R=1000\)), (**a**) inside the virtual region and (**b**) outside the virtual region. Figure 10 presents the plots of the estimated PWs of LS, L1, and L1-parSBL under different SNRs for the \"No. 12\" mode at approximately 106.5 Hz by using a set of randomly distributed samplings (\(R=400\)). Note that there are no analytical PWs for this cabin case. The 12 th subgraph in Figure 5(b) reveals that the mode shape had a significant characteristic of sparse PW superposition. There was no change in the modal values along the \(z\)-axis, regardless of the damping floor, indicating that the corresponding PWs of the mode were mainly concentrated on the equator of the unit sphere. It can be seen that LS could not identify sparse PWs. When SNR was 30 dB, the estimated PWs of LS barely exhibited four energy highlands around the equator. For lower SNRs, the estimated PWs distorted severely. A comparison between L1 and L1-parSBL shows that they both estimated PWs well at 30 dB. For lower SNRs, L1 obtained numerous PW artifacts that deviated from the equator. In contrast, the PWs estimated by L1-parSBL under different SNRs were similar; this reflected the robustness of L1-parSBL to some degree. At SNR = 10 dB, the PW estimation error of L1-parSBL was large. This could be partly attributed to the first stage estimate being wrong; thus, Stage 2 achieved no good results. L1-parSBL eliminated many artifacts selected in Stage 1 and significantly improved the reconstruction accuracy. Figure 10: _Cont._

Figure 9: (Color online) RME of LS, L1, and L1-parSBL (—: Inside; - -: Outside ) (**a**) versus \(M\) (the number of samplings) with SNR = 25 dB, and **(b**)** versus SNR with \(M=40\) for the mode at approximately 106.5 Hz with \(R=400\).