## 3 Inverse Solving Process
### Sound Source Identification Method Based on Grid Splitting and Refinement  
A sound source identification method based on grid split refinement is proposed in this paper, and the process is shown as follows. The superscript brackets indicate the number of iterations, and the initial value of the iteration number \"_iter_\" is set to 0. **Step 1**: The decay factor \(\delta\), and the iteration stopping threshold \(\sigma\) are set, and the dynamic range is set to \(D_{range}\). These parameter settings can be determined according to the situation. In this paper, \(\sigma\) is set to 0.001 and \(D_{range}\) is set to 30 dB. The selection of the decay factor \(\delta\) will be introduced later. **Step 2**: Collect the snapshot sound signal \(\mathbf{P}^{(iter)}\), discretize the sound source plane into a grid surface with \(A_{s}^{(iter)}\) as the aperture and \(d_{s}^{(iter)}\) as the spacing between adjacent points, and the grid points \([\mathbf{X}^{(iter)},\mathbf{Y}^{(iter)}]\) are used as the scanning point grid. Construct Green's function \(\mathbf{G}^{(iter)}\) as shown in Equation (2). **Step 3**: According to Equations (12)-(18), the initial source strength \(\mathbf{Q}^{(iter)}\) is obtained by sparse Bayesian learning. **Step 4**: The grids are gradually refined and updated near the local maximum of the source strength. For the source strength \(\mathbf{Q}^{(iter)}\), The maximum value of the average of its columns and corresponding coordinate location \(q_{\max}^{(iter)}\,(x_{\max}^{(iter)}\,,y_{\max}^{(iter)})\) is searched. Following this, the set \(\phi_{\max}^{range(iter)}=\left\{q_{\max}^{(iter)},\ldots,q_{\max-D_{range}}^{ (iter)}\right\}\) is determined, consisting of grid positions corresponding to all local maxima within the descending dynamic range \(D_{range}\) of the maximum value \(q_{\max}^{(iter)}\). These positions are selected as the centers of potential grid splits in the next step. **Step 5**: Each element in the set \(\phi_{\max}^{range(iter)}\) is used as the geometric center of grid fission, and the new scan points \([\mathbf{X}^{(iter+1)},\mathbf{Y}^{(iter+1)}]\) are obtained by fission with \(A_{s}^{(iter+1)}=A_{s}^{(iter)}\times\delta^{iter}\) as the aperture of the scanning surface and \(d_{s}^{(iter+1)}=d_{s}^{(iter)}\times\delta^{iter}\) as the interval between adjacent points, where \(\delta^{iter}\) represents \(\delta\) raised to the \"_iter_\" power. Remove the duplicate grid points obtained after fission and construct the transfer model \(\mathbf{G}^{(iter+1)}\) between the new grid points and the measurement surface. **Step 6**: Determine whether \(d_{s}^{(iter+1)}\) obtained in Step 5 reaches the threshold \(\sigma\). If it has reached the threshold, output the source strength \(\mathbf{Q}^{(iter)}\) calculated in Step 3 and the algorithm ends. If the threshold is not reached, combine the \(\mathbf{G}^{(iter)}\) and \(\mathbf{Q}^{(iter)}\) obtained in Step 2 and Step 3, respectively, to form a new sound pressure \(\mathbf{P}^{(iter+1)}=\mathbf{G}^{(iter)}\mathbf{Q}^{(iter)}\), and enter it into the MoG-RPCA model. Replace the new sound pressure with the separated low-rank matrix, that is, \(\mathbf{P}^{(iter+1)}=\mathbf{U}^{(iter+1)}\mathbf{V}^{(iter+1)}\). Then, add one to the number of iterations. **Step 7**: The new transfer model \(\mathbf{G}^{(iter+1)}\) calculated through Step 5 and the new sound pressure \(\mathbf{P}^{(iter+1)}\) calculated through Step 6 are entered into Step 3 to obtain the updated source strength \(\mathbf{Q}^{(iter+1)}\). **Step 8**: Repeat the above Steps (3)-(7) until the iteration is jumped out in the judgment of Step 6 and the final source strength is output. The principle of this method is illustrated in Figure 2.  
The pseudo-code for VG-SBL is given as follows (Algorithm 1). Input: The snapshot sound pressure: \(\mathbf{P}\). 1: Discretize the source plane into a grid surface with \(A_{\text{s}}\) as the aperture and \(d_{\text{s}}\) as the spacing.