# 3D Acoustic Mapping in Automotive Wind Tunnel: Algorithm and Problem Analysis on Simulated Data
## 5 Results of Parametric Study  
Time data have been processed to estimate the CSM using [PERSON]'s method (block size: 1024 samples, overlap: 50%, window: Hanning). The number of eigenmodes used for each analysis conducted in this work is \(M_{0}=10\). The maximum sparsity constraint is imposed by setting \(p=0\). The reference point \(\tau_{0}\) for the acoustic transfer function is the origin of the coordinate system. The acoustic maps represented in the figures reported in this paper are given in terms of pressure generated by each monopole at the reference point with a dynamics of 40 dB. Source strengths are also calculated from the acoustic maps, in terms of volume acceleration (m\({}^{3}\)/s\({}^{2}\)). They are retrieved by integrating the equivalent source strengths over a sphere of radius \(\lambda\) centered in the source position. \(\lambda\) is the wavelength of  
Figure 3: Source–microphone masking effect due to the model. Green continuous lines: propagation paths in the line-of-sight condition. Red dotted lines: propagation paths in the non-line-of-sight condition. (**a**) Left side mirror. (**b**) Right side mirror. (**c**) Front left tire. (**d**) Rear left tire. (**e**) Front right tire. (**f**) Rear right tire.  
sound corresponding to the frequency at which the calculation is performed (or the center frequency of the band analyzed).  
In order to make an objective comparison among different cases/parameters configured in the simulation, a set of scores has been defined to evaluate the localization and quantification performance. The localization scores are as follows:  
1. _Identified sources_. A source can be labeled as \"identified\" if there exists at least one equivalent source with non-null strength in its zone of interest. The set of indices \(n\) of equivalent sources to determine if a source is identified can be formalized as \[I_{s}=\{n:\|\mathbf{r}_{n}-\mathbf{r}_{s}\|_{2}\leq\lambda\ \wedge\ q_{n}\
eq 0\},\quad s=1,\ldots,S\,.\] (20)
2. _Number of sources identified_ (\(S_{id}\)). This simple score is formalized as the cardinality of the set that includes all indices \(s\) corresponding to non-empty sets \(I_{s}\): \[S_{id}=card(\{s:I_{s}\
eq\varnothing\})\,.\] (21)
3. _Identified source position_ (\(\mathbf{r}_{s,id}\)). This indicator is the centroid (weighted with source powers) of the cloud of equivalent sources for each \(I_{s}\): \[\mathbf{r}_{s,id}=\frac{\sum_{n\in I_{s}}Q_{n}\mathbf{r}_{n}}{\sum_{n\in I_{s }}Q_{n}}\] (22) where \(Q_{n}=q_{n}q_{n}^{*}\) is the power associated to each equivalent source in the map.
4. _Non-dimensional distance error_ (\(D_{s}\)). This score is defined as the geometric distance between the identified source position and the real one, normalized by the wavelength: \[D_{s}=\frac{\|\mathbf{r}_{s,id}-\mathbf{r}_{s}\|_{2}}{\lambda}\,.\] (23) The quantification scores are as follows:  
1. _Power concentration_ (\(W_{\%}\)). This score indicates the sum of the power of identified sources with respect to the total power of map. It is calculated as \[W_{\%}=\frac{\sum_{s=1}^{S_{id}}\sum_{n\in I_{s}}Q_{n}}{\sum_{n=1}^{N}Q_{n}}\,.\] (24)
2. _Source power ratio_ (\(W_{s}\)). It indicates the ratio of the power associated to each identified source with respect to the real source power. \[W_{s}=log\frac{\sum_{n\in I_{s}}Q_{n}}{Q_{s,exact}}\] (25)  
Power concentration is useful because it can be calculated in real applications, while the source power ratio is useful in this context to evaluate the quantification accuracy as the real power of simulated sources is known. Both localization and quantification scores must be calculated on the total map (energetic sum of all component maps). They can be calculated for each spectral line or for the whole band of analysis. In the latter case, the wavelength used is the one associated to the center frequency of the band.