## 2 Problem Statement  
Assuming that multiple sound sources are radiating to the surroundings in space, the sound signals are acquired synchronously using a microphone array consisting of \(M\) channels, \(r_{s},s=1,2,\ldots N\) denoting the discrete grid point position, that is, the potential location of the sound sources, \(r_{i},i=1,2,\ldots M\) denoting the location of the microphones and \(\Gamma\) denoting the spatial domain consisting of discrete sound sources. Considering the relationship between the source distribution \(q(r_{s})\) and the radiated sound pressure as an integral Fredholm form, the sound pressure \(p(r_{i})\) measured by the microphone array can be expressed as:  
\[p(ri)=\int_{\Gamma}G(rirs)q(rs)d\Gamma(rs)+ei,i=1,2,\ldots M,s=1,2,\ldots N \tag{1}\]where \(G(\mathbf{r}_{i}|\mathbf{r}_{s})\) denotes Green's function between the source location \(\mathbf{r}_{s}\) and the measurement point \(\mathbf{r}_{i}\) in the free field, and \(\mathbf{e}_{i}\) denotes the measurement noise received by each microphone. The free-field Green function is calculated as follows:  
\[G(\mathbf{r}_{i}|\mathbf{r}_{s})=\exp(ik|\mathbf{r}_{i}-\mathbf{r}_{s}|)/\big{(}4\pi|\mathbf{r}_{i}- \mathbf{r}_{s}|) \tag{2}\]  
where \(k\) is the wave number. The measurement signals of \(M\) microphones are stacked and converted into column vectors, and the system of equations is expressed in a simple form:  
\[p=\int_{\Gamma}G(rs)q(rs)d\Gamma(rs)+e \tag{3}\]  
The \(i\)-th element of the column vector \(G(\mathbf{r}_{s})\) is \(G(\mathbf{r}_{i}|\mathbf{r}_{s})\), and \(\mathbf{e}\) denotes the column vector of the noise signal. Since measuring the acoustic field over the entire surface surrounding the radiator is impossible, this is an ill-posed problem in the Hadamard sense. To solve this problem, an estimate of the source strength is returned as a linear combination of the measurements [47]. Consider \(L_{snap}\) snapshots and rewrite the Equation (3) in matrix form:  
\[\mathbf{P}=\mathbf{G}\mathbf{Q}+\mathbf{E} \tag{4}\]  
\(\mathbf{P}=[\mathbf{p}_{1},\dots\mathbf{p}_{l},\dots\mathbf{p}_{L_{snap}}]\in\mathbb{C}^{M \times L_{snap}}\), \(\mathbf{p}_{l}=[\mathbf{p}_{1},\dots\mathbf{p}_{i},\dots\mathbf{p}_{M}]\in \mathbb{C}^{M\times 1}\), and \(\mathbf{P}\) represent \(M\) microphones' complex sound pressure matrix under all \(L_{snap}\) snapshots. \(\mathbf{Q}=[\mathbf{q}_{1},\dots\mathbf{q}_{l},\dots\mathbf{q}_{L_{snap}}]\in\mathbb{C}^{N \times L_{snap}}\), \(\mathbf{q}_{l}=[\mathbf{q}_{1},\dots\mathbf{q}_{s},\dots\mathbf{q}_{N}]\in \mathbb{C}^{N\times 1}\), and \(\mathbf{Q}\) represent \(N\) focused grid points' unknown source strength distribution matrix under all \(L_{snap}\) snapshots. \(\mathbf{E}=[\mathbf{e}_{1},\dots\mathbf{e}_{l},\dots\mathbf{e}_{L_{snap}}]\in\mathbb{C}^{M \times L_{snap}}\), \(\mathbf{e}_{l}=[e_{1},\dots\mathbf{e}_{i},\dots\mathbf{e}_{M}]\in\mathbb{C}^{M \times 1}\), and \(\mathbf{E}\) represent the noise matrix of M microphones under all \(L_{snap}\) snapshots. The essence of the source identification problem is to recover the source signal \(\mathbf{Q}\) from the measured value \(\mathbf{P}\), which is an ill-posed problem in the Hadamard sense. However, it is important to note that when the real sound sources are located at \(\mathbf{r}_{source}\) and not on the discrete grid points \(\mathbf{r}_{s}\), it will lead to a mismatch between the distance \(|\mathbf{r}_{i}-\mathbf{r}_{source}|\) from the microphones to the sources and the distance \(|\mathbf{r}_{i}-\mathbf{r}_{s}|\) from the microphones to the discrete grid points, as shown in Figure 1, which will result in the base mismatch error. Therefore, how to improve the accuracy of sound source identification under model base mismatch needs to be investigated. Figure 1: Microphone array measurement model.