# Similarity Analysis for Time Series-Based 2D Temperature Measurement of Engine Exhaust Gas in TDLAT
## 3 Experiment and Tomography Analysis
Figure 4a is the result of averaging each raw signal acquired from the photodetector at the same time when 10 lasers passed through the target area. To convert these signals into absorption spectra, \(\overline{I}_{v}\) and the mean incident signal (\(\overline{I}_{v0}\)) must be known as shown in Equation (4). \(\overline{I}_{v0}\), also known as the baseline, was estimated by PRM, which is a least squares fitting using 5 th order polynomial curve with \(\overline{I}_{v}\)[23; 25]. To facilitate curve fitting, the saw-tooth waveform had a frequency of 1 kHz and a sampling rate of  
Figure 3: Laser paths and measurement cell. Figure 2: Schematic diagram of the experimental setup using tunable diode laser absorption spectroscopy (TDLAS). 2 us. All measured absorption spectra generated from \(\overline{I}_{v}\) and \(\overline{I}_{v0}\) is shown in Figure 4b. These spectra consisted of 311 wavenumbers. Laser group 0 and 1 represent a group of vertical and horizontal lasers, respectively. The spectra pattern can be identified by different laser numbers. As shown in Figure 1, the measured absorption spectra are determined by the temperature of the measurement area, so the reconstructed temperature distribution in measurement zone can be traced by a tomographic reconstruction method from the information of each absorption spectrum. The determination method of cycle and temperature is illustrated in Figure 5. First, raw absorption signals, reference time \(i\) and candidate cycle \(j\) are input. \(i\) denotes analysis time point at engine running time. A total of 990 absorption signals were used in 0.1 s from 1.0 to 9.9 s. \(j\) denotes the number of cycles of absorption signal to be averaged. Eighteen candidate cycles were chosen to evaluate the analytical performance in various cycles (31, 51, 71, 91, 111, 151, 201, 251, 301, 351, 401, 451, 501, 601, 701, 801, 901, and 1001 cycles). Figure 4: Results of mean transmission signals and mean absorption spectra of 10 laser beams: (**a**) 10 mean transmitted signals using 201 cycles and (**b**) 10 mean absorption spectra using 201 cycles. Figure 5: Flowchart for tracking cycle and reconstructed temperature using TDLAT. In the second step, the PRM and \(\delta\) value are combined to obtain mean absorption spectra, described in more detail above. This technique was able to remove noise more effectively than the conventional PRM. Reconstruction calculation consisting of 9 \(\times\) 9 array was calculated using mean absorption spectra. The reconstructed temperature distribution was tracked by repeated calculations until errors in the experimental and theoretical absorption spectra were minimized by the multiplicative algebraic reconstruction technique (MART) algorithm [24]. \[\alpha_{v(n,m)}{}^{(k)}=\alpha_{v(n,m)}{}^{(k-1)}\times\left[\frac{A_{v}}{L_{( n,m)}\sum_{m=1}^{M}\sum_{n=1}^{N}\alpha_{v(n,m)}{}^{(k-1)}}\right]^{\beta L_{(n,m)}}, \tag{6}\]  
where, the superscript \(k\) represents the number of iteration and the subscript (n, m) is the gird number. \(\beta\) is the relaxation parameter that affects the rate of convergence and was set to 0.1. The initial values of \(\alpha_{v(n,m)}\) at each grid are an important factor for fast convergence calculation, which were approximated to target values by adopting the multiplication line of sight (MLOS) method [26]. The reconstruction calculation was terminated when difference of \(\alpha_{v(n,m)}{}^{(k)}\) and \(\alpha_{v(n,m)}{}^{(k-1)}\) was less than \(1\times 10^{-6}\). Figure 6 shows a comparison of the measured absorption spectra with the calculated absorption spectra in the engine running time of 5.0 s. The average root-mean-square error (RMSE) at 311 wavenumber were \(9.766\times 10^{-4}\), \(4.485\times 10^{-5}\), \(1.273\times 10^{-3}\), and \(2.144\times 10^{-3}\), respectively, in Figure 6a-d.