# Similarity Analysis for Time Series-Based 2D Temperature Measurement of Engine Exhaust Gas in TDLAT
## 3 Experiment and Tomography Analysis
Measured spectra were in good agreement with spectra by the theoretical calculation, but the measured spectra for the 51 and 201 cycles were slightly different from each other, and their reconstruction calculations were also differed slightly. This implies that the number of mean absorption cycles is an important parameter for the reconstruction results. In the third step, the reconstructed temperatures, \(T_{Rec(i,j)}\), calculated for each candidate cycle and time are output. To estimate the \(T_{Rec(i,j)}\) from the finally converged \(\alpha_{v(n,m)}\) at all grids, the theoretical absorption coefficients at the temperature has already been calculated using the HITRAN database for the mole fraction 1.0. The correlation between the theoretical absorption coefficient, \(\alpha_{v,the}\), and the reconstructed absorption coefficient, \(\alpha_{v}\), was evaluated to find the value with the most similar pattern, i.e., the nearest correlation coefficient to 1.0, and to obtain \(T_{Rec(i,j)}\). The correlation coefficient, \(C_{n,m}\), at each grid is described as follows. \[C_{n,m}=\frac{\sum_{v=1}^{n}(\alpha_{v}-\overline{\alpha})\Big{(}\alpha_{v, the}-\overline{\alpha}_{the}\Big{)}}{\sqrt{\sum_{v=1}^{n}(\alpha_{v}-\overline{ \alpha})^{2}\sum_{v=1}^{n}\Big{(}\alpha_{v,the}-\overline{\alpha}_{the}\Big{)} ^{2}}}, \tag{7}\]  
where \(\overline{\alpha}\) and \(\overline{\alpha}_{the}\) denote the average value each of the theoretical and reconstructed absorption coefficient. Figure 7 shows the results of 2D reconstructed temperature distribution using 51 and 201 cycles at the engine running time of 5.0 s. Comparing Figure 7a,b, the temperature difference at the center point was 12.35 K and the RMSE at all grids was 5.87 K. This is an example of the highest temperature difference compared to other engine running times. The characteristics of the temperature distribution for each time were similar, but the temperature fluctuation was shown differently. From these results, it was found that the number of mean cycles affected the results of CT analysis. Additionally, the output \(\overline{T}_{Rec}\) was set to the average value of nine grids located in the center of the measurement area as the TC was installed in the center of the exhaust pipe and the temperature was measured at one point. After all, the output \(T_{Rec}\) consists of a set of elements \(\overline{T}_{Rec}\) for each i and j. The number of \(T_{Rec}\) was 17820 in the i \(\times\) j array (990 \(\times\) 18). In the fourth step, the statistical-based analysis was performed using \(T_{Rec}\) and \(T_{Tlic}\), which means temperature measured by TC and was used to determine the cycle and \(T_{Rec}\) most similar to \(T_{Tlic}\). Further details are described in Section 4. The computation time of each step on an Intel Core i7 2600K CPU with 16 GB RAM was as follows. The CPU time of Process I to convert signal data into mean absorption spectra was 1.38 s and 1.90 s (311 wavenumbers) respectively at 51 and 201 cycles. As the number of mean cycles increased, the CPU time increased linearly. The CPU time of Process II was scored in under 1.79 s (311 wavenumbers), mostly converging on less than 150 iterations. Figure 1:0 The spectra of reconstructed and measured absorption data at the engine running time 5.0 s: (**a**) 51 cycles, boundary; (**b**) 201 cycles, boundary; (**c**) 51 cycles, center; and (**d**) 201 cycles, center. Figure 1:1 2D reconstructed temperature distributions at the engine running time 5.0 s: (**a**) 51 cycles and (**b**) 201 cycles.