# 3D Acoustic Mapping in Automotive Wind Tunnel: Algorithm and Problem Analysis on Simulated Data
## 3 Volumetric Acoustic Mapping Based on IRLS
### IRLS Procedure  
The IRLS procedure to solve the inverse acoustic problem can be formalized with the following expression:  
\[\hat{\mathbf{q}}^{(it)}=F\Big{(}\hat{\mathbf{q}}^{(it-1)},\mathbf{W}^{(it)}, \eta^{2(it)},\mathbf{G}^{(it)},\mathbf{p},p\Big{)}\;. \tag{13}\]  
where the function \(F\) is given by the inverse operator coming from Equation (10). From a Bayesian point of view, this method can be seen as an Expectation-Maximization algorithm which converges to a maximum a posteriori solution [27]. The procedure is the following:  
1. Set the weighting matrix for the current iteration \(\mathbf{W}^{(it)}=\mathbf{W}_{0}\,\mathbf{W}_{sp}^{(it)}\). Calculate \(\mathbf{W}_{sp}^{(it)}\) using Equation (12), with \(\mathbf{W}_{sp}^{(1)}=\mathbf{I}\). Both matrices are normalized, before their multiplication, such that \(\|\mathbf{W}_{0}\|_{\infty}=\|\mathbf{W}_{sp}^{(it)}\|_{\infty}=1\).
2. Estimate the regularization parameter \(\eta^{2\,(it)}\) for the current iteration using Equation (9).
3. Calculate the solution \(\hat{\mathbf{q}}^{(it)}\) and apply a threshold to discard potential sources that do not contribute significantly to the acoustic field. The set of indices \(n\) of potential sources to discard is found using the following criterion: \[\left\{n:10\log_{10}\!\left(\frac{|q_{n}|^{(it)}}{\|\hat{\mathbf{q}}^{(it)} \|_{\infty}}\right)<THR_{dB}\right\}.\] (14) Sources (unknowns) discarded are set to zero in the final solution and the relative columns of direct operator \(\mathbf{G}^{(it)}\) are removed for the next iterations. The same happens with the weighting matrix \(\mathbf{W}^{(it)}\).
4. Evaluate a convergence criterion; if not fulfilled go back to step 1; otherwise, stop the iterative procedure.  
This algorithm returns a sparse approximation of the source field. A priori information introduced with \(\mathbf{W}_{0}\) can be used to penalize regions that are less likely to find sources. [PERSON] et al. [28] proposed using a CB map for similar purposes as it provides a robust estimation of source location, despite the information being rough and smooth. However, any spatial weighting can be adopted as a priori distribution. An advantage of IRLS is that the amount of sparsity enforced on the solution can be finely tuned adjusting \(p\)For example, in the presence of distributed/extended sources, \(p=1\) may be a good choice, while in presence of point sources, \(p=0\) is the best choice. The latter is generally the best choice for source quantification. The discarding of sources is done both to avoid the division by zero in the calculation of weights as well as to speed up the algorithm. The threshold used in this work is \(THR_{dB}=-100\) dB. This value guarantees the discarding of elements that do not contribute significantly to the solution. The convergence criterion adopted here was proposed by [PERSON] et al. in [PERSON] et al. (2014):  
\[\begin{split} MSR=\left\langle\left|\hat{q}_{n}^{(it)}/\hat{q}_{n }^{(it-1)}\right|\right\rangle\\ \epsilon^{(it)}=10\log_{10}\Bigl{(}MSR^{(it)}-\left|\Delta(MSR)^ {(it)}\right|-\left|\Delta^{2}(MSR)^{(it)}\right|\Bigr{)}\end{split} \tag{15}\]  
where \(MSR\) stands for _Mean Source Ratio_, the operator \(\left\langle\cdot\right\rangle\) refers to the spatial average, and the operators \(\Delta(\cdot)\) and \(\Delta^{2}(\cdot)\) are the backward finite differences of first and second order, respectively. This criterion can be evaluated only for \(it>2\) (given the second order finite difference) and requires the solution variation to be small over three last iterations. The algorithm stops when \(\epsilon^{(it)}\geq-0.1\) dB. The IRLS described in this section enables obtaining a volumetric map of sound sources using any array shape (planar, multiple planar, spherical, distributed ([PERSON] et al., 2018), etc.) as it fulfills all requirements of the analysis discussed in the previous section. The characterization of the performance attainable with an inverse method is difficult to assess, as the results are affected by a huge number of factors.