## 3 Numerical Simulation and Analysis  
In order to evaluate the efficiency of the MAART and assess the influences of the view angles and numbers, numerical simulations are carried out. Four stationary phantom flames and a square geometry of 8 \(\times\) 8 grids are considered for simultaneous combustion imaging reconstructions. The different flame temperature imaging \(T^{\mathrm{set}}\) and the H\({}_{2}\)O concentration imaging \(C^{\mathrm{set}}\) are generated using a Gaussian function and a Random function as shown in Figures 2 and 3, respectively. Including one flame to four flames, the set temperature distribution \(T^{set}\) are set at 1500 K \(\pm\) 50 K in the high temperature region and 400 K \(\pm\) 50 K in the low temperature region. Typically, the H\({}_{2}\)O concentration distribution is generally well correlated with the temperature distribution in a hydrocarbon flame. Therefore, the set volume fractions of H\({}_{2}\)O \(C^{set}\) in the measurement grids are assumed to be 0.17 \(\pm\) 0.005 and 0.015 \(\pm\) 0.005, correspondingly. It is known that the orientation of the view and the number of rays per view has a significant bearing on the quality of reconstruction [28,29]. Therefore, we assumed four different laser beams distributions to study the influence on the reconstruction in simulations. As shown in Figure 4, the number of laser beams included is 16, 18, 32, and 44 with multi-projection. The integrated absorbance is obtained by simulated line-of-sight integration for two water vapor (H\({}_{2}\)O) absorption lines 7165.82 cm\({}^{-1}\) and 7164.91 cm\({}^{-1}\). Spectroscopic parameters for the absorption transitions such as line-strengths, lower state energy, coefficients for their temperature dependency, and broadening coefficients are selected from the HITRAN (High Resolution Transmission) 2008 database, as shown in Table 1. Meanwhile, in order to select the most suitable spectral line couples, the numerical simulation is carried out using the above database. The two absorption lines simulations are shown in Figure 5, and the known parameter constants as \(P\) = 1 atm, \(L\) = 32 cm, \(T\) = 500 K, \(C\) = 0.2%, and \(T\) = 1500 K, \(C\) = 15%. Simulated line-of-sight data are inverted using the MAART, which is introduced in the above theoretical section. Figure 3: Four different distributions of H\({}_{2}\)O concentration in the simulation. (**a**) One flame, (**b**) Two flames, (**c**) Three flames, and (**d**) Four flames. Figure 2: Four different distributions of temperature phantoms in the simulation. (**a**) One flame, (**b**) Two flames, (**c**) Three flames, and (**d**) Four flames. Furthermore, the tomographic images have been compared with the four simulated flames, which are discretized into \(M\times N\) grids (\(M=N=8\)). The quality of reconstructed results for distributions of both temperature and H\({}_{2}\)O concentration are evaluated by the mean relative error \(e_{T}\) and \(e_{C}\) as Equations (11) and (12), respectively. \[e_{T}=\sum_{i}^{M\times N}\left(\left|T_{i}^{cal}-T_{i}^{set}\right|\right/T_{ i}^{set}\right)/(M\times N) \tag{12}\]  
\[e_{C}=\sum_{i}^{M\times N}\left(\left|C_{i}^{cal}-C_{i}^{set}\right|\right/C_{ i}^{set}\right)/(M\times N) \tag{13}\]  
where \(T^{cal}\) and \(C^{cal}\) denote the calculated temperature and the H\({}_{2}\)O concentration of reconstructed distributions. Similarly, \(T^{set}\) and \(C^{set}\) stand for the set temperature and the H\({}_{2}\)O concentration, respectively. Figure 4: Geometric descriptions of the variety laser beams distribution. **(a)** 16 laser beams, **(b)** 18 laser beams, **(c)** 32 laser beams, and **(d)** 44 laser beams. (Red lines stand for different laser beams). Figure 5: The simulated two H\({}_{2}\)O absorption spectra (\(v_{1}=7164.91\) cm\({}^{-1}\) and \(v_{2}=7165.82\) cm\({}^{-1}\)) at \(T=300\) K, \(x=2\%\) (black dot line) and 1500 K, \(x=15\%\) (red solid line). Figure 6 shows the \(e_{T}\) and \(e_{C}\) variation curves for different simulation flames under four beam distributions.