## 4 Acoustic Field Reconstruction
### Methods for 3D Reconstruction  
The method used for reconstructing acoustic quantities based on acoustic pressure measurements is based on the well-known near-field acoustical holography (NAH) and the statistically optimized NAH (SONAH) method [31; 34; 35; 36]. This method is based on the first step of reconstructing the acoustic field in the free field from acoustic measurements taken in the near field. The reconstruction performed can then be used to define the inverse problem which leads to the localization of the source. In the context of the present study, this methodology is developed to provide a 3D reconstruction of the radiated acoustic field due to squeal noise. The method consists in reconstructing an acoustic field in a \(\Omega\) domain made up of a homogeneous light fluid from pressure measurements. The main assumption is that the reconstructed field can be reconstructed from a series of elementary plane waves noted \(\Psi_{n}\) with \(n=1,\ldots,N\) such that  
\[p_{i}(\mathbf{r}_{i})=\sum_{n=1}^{N}a_{n}\Psi_{n}(\mathbf{r}_{i}), \tag{1}\]  
where \(p_{i}(\mathbf{r}_{i})\) corresponds to the measured pressures. We note that \(i=1,\ldots,I\), \(I\) is the number of available and reliable measurements. Coefficients \(a_{n}\) are the unknowns of the identification problem. This problem can be over-determined or under-determined depending on whether the number of measurements is higher or lower than the number \(N\) of coefficients that have to be identified. In order to better represent all the possibilities of acoustic radiation, it is nevertheless advisable to opt for a high value of \(N\). Equation (1) can be written in matrix form such that  
\[\mathbf{p}=\mathbf{B}\mathbf{a}, \tag{2}\]  
where \(\mathbf{p}\) is the vector of pressures \(\{p_{i}(\mathbf{r}_{i})\}\), \(\mathbf{a}\) is the vector of coefficients \(\{a_{n}\}\), and \(\mathbf{B}\) a matrix containing the values at points \(\mathbf{r}_{i}\) of functions \(\Psi_{n}\):  
\[\mathbf{B}=\left[ \begin{array}{cccc}\Psi_{1}(\mathbf{r}_{1})&\Psi_{2}( \mathbf{r}_{1})&\cdots&\Psi_{N}(\mathbf{r}_{1})\\ \Psi_{1}(\mathbf{r}_{2})&\Psi_{2}(\mathbf{r}_{2})&&\Psi_{N}(\mathbf{r}_{2})\\ \vdots&&\ddots&\vdots\\ \Psi_{1}(\mathbf{r}_{l})&\Psi_{2}(\mathbf{r}_{l})&\cdots&\Psi_{N}(\mathbf{r}_{ l})\end{array} \right]. \tag{3}\]  
The problem defined by Equation (2) has to be reversed. To achieve this, the following equation is used:  
\[\tilde{\mathbf{a}}=\mathbf{B}^{H}\Big{(}\mathbf{B}\mathbf{B}^{H}+\varepsilon \mathbf{I}\Big{)}^{-1}\mathbf{p}, \tag{4}\]  
where symbol \((. )^{H}\) represents the Hermitian transpose and \(\varepsilon\) the Tikhonov regularization coefficient. We consider \(\varepsilon=[\mathbf{A}^{H}\mathbf{A}]_{ii}^{-D/10}\) with \(D=10\), as recommended by [PERSON] [31] and [PERSON] [37]. Obtaining \(\tilde{\mathbf{a}}\) makes it possible to recalculate the pressure field \(\tilde{p}\) everywhere in \(\Omega\):  
\[\tilde{p}(\mathbf{r})=\sum_{n=1}^{N}\tilde{a}_{n}\Psi_{n}(\mathbf{r}). \tag{5}\]  
This equation can also be rewritten in matrix form:  
\[\tilde{p}(\mathbf{r})=\tilde{\mathbf{a}}^{T}\boldsymbol{\alpha}(\mathbf{r}), \quad\boldsymbol{\alpha}(\mathbf{r})=\left\{ \begin{array}{c}\Psi_{1}(\mathbf{ r})\\ \Psi_{2}(\mathbf{r})\\ \vdots\\ \Psi_{N}(\mathbf{r})\end{array} \right\}. \tag{6}\]By combining Equations (4) and (6), the following equation that directly links the calculated pressure in \(\mathbf{r}\) and the measured pressures can be obtained such that  
\[\tilde{p}(\mathbf{r})=\mathbf{p}^{T}\Big{(}\mathbf{A}^{H}\mathbf{A}+\varepsilon \mathbf{I}\Big{)}^{-1}\mathbf{A}^{H}\boldsymbol{\alpha}(\mathbf{r}), \tag{7}\]  
where \(\mathbf{A}=\mathbf{B}^{T}\). Then, it is necessary to specify the shapes of functions \(\Psi\).