## 2 Theoretical Basis
### The Sound Prediction Based on ARM
\tag{17}\]  
where \(\mathbf{W}_{0}\) is the sound power derived from the equivalent sources inside the structure, \(\mathbf{W}_{i}\) is the sound power derived from the image equivalent sources inside the virtual structure, and \(\mathbf{W}_{m1}\) and \(\mathbf{W}_{m2}\) are the sound power components derived from the interactions among the equivalent sources and their images. According to the Helmholtz integral, when the sound source is not inside a closed surface, the integral of the source strength on the surface makes no contribution to the total sound field [1]. It is easy to find that, in Equation (17), \(\mathbf{W}_{m1}\) and \(\mathbf{W}_{m2}\) are the components that make no contribution to the sound radiation. Therefore, when reconstructing or predicting the sound field, only \(\mathbf{W}_{0}\) and \(\mathbf{W}_{i}\) need to be considered. Meanwhile, \(\mathbf{R}_{0}\) is equal to \(\mathbf{R}_{i}\) because of the mirror-image relationship. The strength of equivalent sources could be expanded by the orthogonal basis:  
\[\left\{ \begin{array}{l}\mathbf{q}_{0}+\mathbf{q}_{s}=\mathbf{T}\mathbf{\beta}_{0}=\sum\limits \limits_{i=1}^{N}\mathbf{t}_{i}\mathbf{\beta}_{0i}\\ \mathbf{q}_{R}=\mathbf{T}\mathbf{\beta}_{1}=\sum\limits_{i=1}^{N}\mathbf{t}_{i}\mathbf{\beta}_{1i} \end{array} \right. \tag{18}\]where the vector \(\mathbf{t}_{i}\) is the \(i\)th equivalent source ARM and \(\beta_{0i}\) and \(\beta_{1i}\) are the expansion coefficients for the \(i\)th ARM. When the measurement points are sparse, Equation (18) could be solved with the minimum \(\ell\)1-norm [28]:  
\[\arg_{\beta}\min\lVert\mathbf{\beta}\rVert_{1}\,\ s.t.\ \lVert\mathbf{v}_{ref}-\mathbf{ \
abla}\mathbf{\beta}\rVert_{2}^{2}\leq\varepsilon \tag{19}\]  
where \(\mathbf{\
abla}=\mathbf{DT}\) is the sensing matrix with the dimension of \(L\times N\), and it is possible that \(L\ll N\). Equation (19) is the basic formulation of CMESM; the sparse basis \(\mathbf{T}\) is used here to compress the equivalent source strengths for reinforcing the sparsity of the solution. Thus, the measurement points could be far less than the equivalent sources. In the study [39], [PERSON] defined \(\varepsilon\) as a function of the hologram norm, and \(\varepsilon=\epsilon\lVert\mathbf{v}_{ref}\rVert_{2}\), and the optimal value of \(\varepsilon\) is determined by the cross-validation method. If the structure is located in the half-space, according to the SFS, \(\mathbf{D}\) could be separated into two parts: \(\mathbf{D}_{0}\), which is related to the equivalent sources inside the structure, and \(\mathbf{D}_{1}\), which is related to the mirror equivalent sources. Equation (19) could be rewritten as below:  
\[\arg_{\beta}\min\lVert\mathbf{\beta}\rVert_{1}\,\ s.t.\ \lVert\mathbf{v}_{ref}-\mathbf{ \
abla}^{\prime}\mathbf{\beta}\rVert_{2}^{2}\leq\varepsilon \tag{20}\]  
where \(\mathbf{\
abla}^{\prime}=(\mathbf{D}_{0}+\eta\mathbf{D}_{1})\mathbf{T}\). Equation (20) is a convex optimization problem solved with the CVX package in this paper [40] and \(\varepsilon\) is a parameter related to the sound-noise ratio. The sound pressure in the sound field could be expressed by the ARMs [25]:  
\[p(\mathbf{r})=\lvert\rho_{0}\Omega_{0}g(\mathbf{r}_{i},\mathbf{r})\mathbf{T}\mathbf{\beta} \tag{21}\]  
It should be noted that the ill-condition problem of \(\mathbf{D}_{0}\) and \(\mathbf{D}_{1}\) could be quite severe, which may lead to the ill-condition problem of the sensing matrices. Therefore, even a tiny change of \(\varepsilon\) could lead to obvious variation to the solution of \(\mathbf{\beta}\). [PERSON] solved the problem by arranging the measurement randomly, where blindness exists in engineering applications. Moreover, the ill-condition problem could be improved by increasing the quantity of measurement points, which weakens the advantage of the method [28]. Thus, when the quantity of measurement points is fixed, the measurement arrangement should be optimized to avoid the ill-condition problem of sensing matrices.