# 3D Acoustic Mapping in Automotive Wind Tunnel: Algorithm and Problem Analysis on Simulated Data
## 3 Volumetric Acoustic Mapping Based on IRLS  
### IRLS Solution Strategy  
Inverse problems are generally underdetermined because the number of microphones (i.e., the number of equations) is limited by practical aspects, while the number of potential sources (i.e., the number of unknowns) is often larger, in particular when dealing with three-dimensional volumetric acoustic mapping. Therefore, to find a particular solution, some assumptions on its characteristics must be done. Two kinds of assumptions are discussed hereinafter: solutions with minimum energy (least-square solutions) and solutions with minimum number of non-zero elements (sparse solutions). A straightforward approach for solving Equation (1) is the Moore-Penrose pseudo-inverse, which is a generalization of inverse matrix to rectangular matrices. In the case of an underdetermined system, this inverse operator returns the solution with the smallest \(L_{2}\)-norm among those satisfying the linear equations in Equation (1):  
\[\hat{\mathbf{q}}=\arg\min_{\mathbf{q}}\left(\|\mathbf{q}\|_{2}^{2}\ \text{ subject to }\ \mathbf{G}\mathbf{q}=\mathbf{p}\right). \tag{5}\]  
From a physical point of view, this represents the minimum energy solution that exactly matches the measured pressure data. This is often referred to as a _naive solution_ because errors in the propagation model and/or noise in measured data (providing variations on \(\mathbf{G}\) and \(\mathbf{p}\), respectively) lead to unstable solution. The solution of the inverse problem can be written as  
\[\hat{\mathbf{q}}=\mathbf{G}^{H}(\mathbf{G}\mathbf{G}^{H})^{-1}\mathbf{p}= \mathbf{V}\mathbf{S}^{-1}\mathbf{U}^{H}\mathbf{p}=\sum_{k=1}^{M}\frac{ \mathbf{u}_{k}^{H}\mathbf{p}}{s_{k}}\mathbf{v}_{k} \tag{6}\]  
where the superscript \({}^{H}\) stands for the conjugate transpose. For the last two expressions, Singular Value Decomposition (SVD) of direct operator \(\mathbf{G}=\mathbf{U}\mathbf{S}\mathbf{V}^{H}\) has been used; \(\mathbf{S}\) is a real diagonal matrix containing the singular values \(s_{k}>0\) in decreasing order, while \(\mathbf{U}\) and \(\mathbf{V}\) are two unitary matrix whose columns \(\mathbf{u}_{k}\) and \(\mathbf{v}_{k}\) form two orthonormal bases. Noise in the measured data and propagation modeling errors have an effect on Fourier coefficients, which is amplified when divided by singular values (especially the smallest ones). A common approach to overcome this problem and calculate a regularized solution is the Tikhonov Regularization [22]. This approach consists in the following minimization problem:  
\[\hat{\mathbf{q}}(\eta^{2})=\arg\min_{\mathbf{q}}\left(\|\mathbf{G}\mathbf{q}- \mathbf{p}\|_{2}^{2}+\eta^{2}\|\mathbf{q}\|_{2}^{2}\right). \tag{7}\]This is a joint minimization of the solution norm \(\|\mathbf{q}\|_{2}^{2}\) and residuals norm \(\|\mathbf{G}\mathbf{q}-\mathbf{p}\|_{2}^{2}\), where the regularization parameter \(\eta^{2}\geq 0\) controls the trade-off between the amplitude of the solution and the fitting error. The regularization parameter is used to adjust the cut-off on singular values, preventing the amplification of noise in the inversion operation, thus stabilizing the solution. A more general approach to inverse acoustic problems has been proposed by [PERSON] [23], where Bayesian inference has been exploited for developing a method which is able to (i) identify the optimal basis functions which minimize the reconstruction error, given the topology of the specific acoustic problem; (ii) include a priori information on source distribution to better condition the problem and ease the reconstruction task; and (iii) provide a robust regularization criterion. The idea at the basis of the Bayesian approach for acoustic source reconstruction is to endow all unknown quantities with a probability density function (pdf), instead of considering them as deterministic. Bayesian regularization provides different strategies to estimate \(\eta^{2}\) directly from data, propagator, and a priori information [24].