## 2 Virtual SMA Extrapolation Method  
Based on the inverse problem theory of sound field reconstruction, it is assumed that there is a spherical envelope outside the SMA (with the same spherical center as the SMA), there are several virtual sources on the envelope surface, and the data received by the SMA element is assumed to be the result of the radiation superposition of the virtual sound sources on the envelope surface. The extrapolation of the virtual SMA is realized through the optimal solution of the virtual source strength on the envelope surface and the sound field transmission relationship between the virtual source and any point in the envelope surface [13; 14; 15]. The extrapolation diagram of spherical array is shown in Figure 1.  
Assuming that there is a virtual source on the envelope surface \(S\) radiating from the external actual sound source to the envelope surface, the sound pressure \(p\) of the virtual sound source at \(S\) at any point \(r\) satisfies the improved Helmholtz equation [5]:  
\[\
abla^{2}p_{r}(r)+k^{2}p_{r^{\prime}}(r)=\delta(r-r^{\prime}) \tag{1}\]  
where \(k\) is wavenumber, and \(k=\omega/c\), \(\omega\) represents angular frequency, \(c\) is sound speed in the air. Then it can be obtained that the sound field of the virtual source on the envelope \(S\) and inside \(S\) is  
\[p(r)=\int\limits_{\partial\Lambda}G(r,r^{\prime})q(r^{\prime})dS(r^{\prime}) \tag{2}\]  
where \(q(r^{\prime})\) is virtual sources strengths, \(G(r,r^{\prime})\) is the free field Green function, \(p(r)\) is the sound pressure. In the actual calculation, it is assumed that \(M\) virtual sources are uniformly distributed on the envelope surface \(S\), and the integral formula in Equation (2) is discretized and written in matrix form:  
\[\mathbf{p}(\mathbf{r}_{l})=\mathbf{G}(\mathbf{r}_{l},\mathbf{r^{\prime}}_{m}) \mathbf{q}(\mathbf{r^{\prime}}_{m}) \tag{3}\]  
where \(\mathbf{p}(\mathbf{r}_{l})\) represents the received signal vector of the spherical array element, and \(\mathbf{G}(\mathbf{r}_{l},\mathbf{r^{\prime}}_{m})\) represents the transfer function between the virtual source and the SMA element. The source strength of the virtual source can be obtained by inverting the above equation:  
\[\mathbf{q}= \mathbf{G}^{-1}\mathbf{p} \tag{4}\]  
In the actual inversion calculation process, usually, the number of array elements does not match the number of virtual sources on the envelope surface. \(\mathbf{G}\) is a non-square matrix, and the direct inversion result is unstable. The result will have a serious deviation from the real value. A regularization method is needed to obtain a stable solution of the above equation in the solution process. Tikhonov regularization is based on the idea of minimizing the joint weighting between the residual parametrization and the solution parametrization of the above equation, i.e.,  
\[\mathbf{q}_{reg}=\arg\min\Bigl{\{}\|\mathbf{p}-\mathbf{G}\mathbf{q}\|_{2}^{2} +\lambda^{2}\mathbf{\Omega}(\mathbf{q})^{2}\Bigr{\}} \tag{5}\]  
where \(\mathbf{\Omega}(\mathbf{q})\)=\(\|\mathbf{L}\mathbf{q}\|^{2}\) is the discrete smoothing norm, which constrains the smoothness of the undetermined solution. \(\mathbf{L}\) is the penalty matrix, when \(\mathbf{L}\) is the unit matrix, this method is called the standard Tikhonov regularization method, and \(\lambda\) is the regularization parameter. In general, the traditional regularization methods tend to produce an overly smooth solution. However, in practice, the solution is likely to have the property of discontinuity or sharp corners and other unsmooth properties, which will lead to the loss of part of the information, and thus the accurate regularization solution cannot be obtained. To overcome the shortcomings of traditional regularization and obtain a stable solution,  
Figure 1: Schematic diagram of virtual spherical array extrapolation.