## 2 Theoretical Basis
### The Sound Prediction Based on ARM
As the CMESM is not able to present the ARM form of the normal velocities, the VR-ESM is put into effect for establishing the ARMs of normal velocity. Meanwhile, the EFI is used for optimization of the measurement system. With Equations (5) and (14), the sound power of could be rewritten as follows:  
\[\mathbf{W}=\mathbf{v}^{H}\mathbf{C}\mathbf{v} \tag{22}\]  
where \(\mathbf{C}=\left(\mathbf{D}^{+}\right)^{H}\mathbf{R}\mathbf{D}^{+}\). For the half-space, the expression of \(\mathbf{C}\) is as follows:  
\[\mathbf{C}=\left[\left(\mathbf{D}_{0}+\eta\mathbf{D}_{1}\right)^{+}\right]^{H}\mathbf{R}(\mathbf{D }_{0}+\eta\mathbf{D}_{1})^{+} \tag{23}\]  
where \(\mathbf{C}\) is a positive definite Hermite matrix whose real part \(\mathbf{R}_{\mathbf{v}}\) is a real symmetric matrix, which could be expressed as the form of eigenvalue decomposition according to the study of [PERSON] et al. [26], \(\mathbf{R}_{\mathbf{v}}=\mathbf{\Phi}^{H}\mathbf{\Lambda}_{\mathbf{v}}\mathbf{\Phi}\). Unitary matrix \(\mathbf{\Phi}\) is composed of a series of orthogonal column vectors, which are regarded as velocity ARMs and sorted by the eigenvalues. Therefore, \(\mathbf{v}\) could be expressed as follows:  
\[\left\{\mathbf{v}\right\}_{L\times 1}=\left[\mathbf{\Phi}\right]_{L\times L}\{\mathbf{c} \}_{L\times 1} \tag{24}\]  
where \(\mathbf{c}\) is the expansion coefficient column vector of ARMs. As Equation (24) has good convergence with the increase of ARM order, it could be rewritten as follows: \(\left\{\mathbf{v}\right\}_{L\times 1}\approx\left[\widehat{\mathbf{\Phi}}\right]_{L \times L1}\left\{\widehat{\mathbf{c}}\right\}_{L1\times 1}\), where \(L1\) is the ARM truncation number. In the process, \(L\geq N\) should be guaranteed. Suppose that only \(L2\) points could be regarded as measurement points from the \(L\) measurement points to be selected, the normal velocities at these points could be expressed as follows: \(\{\mathbf{v}^{\prime}\}_{L2\times 1}=[\widehat{\mathbf{\Phi}}]_{L2\times L1}\{ \widehat{\mathbf{c}}\}_{L1\times 1}\). Generally, the quantity of measurement points should not be less than the truncation number, \(L2>L1\). Thus, the relationship between \(\mathbf{v}\) and \(\mathbf{v}^{\prime}\) could be written as follows:  
\[\mathbf{v}\approx\widehat{\mathbf{\Phi}}\widehat{\mathbf{\Phi}}^{+}\mathbf{v}^{\prime} \tag{25}\]  
It should be noted that the column vectors of \(\widehat{\mathbf{\Phi}}\) are not orthogonal any more. Equation (25) means that total vibration of the structure could be reconstructed by the vibration measured at sparse points. If the total vibration is reconstructed accurately enough, the prediction of sound field is also accurate with the ESM. However, Equation (20) is an inverse problem and the condition number of the sensing matrix \(\widehat{\mathbf{\Phi}}\) should be limited to avoid the ill-condition problem. [PERSON] proposed a method based on the EFI to optimize the measurement point arrangement [31]. A matrix \(\mathbf{E}_{D}\) for measuring the linear-independence contribution of the measurement points to the sensing matrix is defined as follows:  
\[\mathbf{E}_{D}=\widehat{\mathbf{\Phi}}\Big{(}\widehat{\mathbf{\Phi}}^{T}\widehat{\mathbf{\Phi }}\Big{)}^{-1}\widehat{\mathbf{\Phi}}^{T} \tag{26}\]  
Each diagonal entity of \(\mathbf{E}_{D}\) is related to a measurement point on the structural surface. For selecting the optimum locations for measurement points from a set of candidate points to maximize the orthogonality property of \(\widehat{\mathbf{\Phi}}\), an iterative process is established. Within each iteration, \(\mathbf{E}_{D}\) is computed, the measurement point that corresponds to the lowest diagonal value is eliminated from the candidate set, and the corresponding row of \(\widehat{\mathbf{\Phi}}\) is eliminated. The elimination process continues until the number of the remaining measurement points is equal to number of points requested to be identified by the optimum search process.