## 6 Results and Discussion
### Estimation of Initial Temperature and the Spatial Resolution Evaluate for CT-TDLAS  
The temperature distribution of a virtual flame was generated by the use of Gaussian function at the temperature range of 280 K to 1200 K. The virtual temperature profiles having three different FWHM (full width at half maximum) values (10 mm, 15 mm, 20 mm) have been generated, and the profiles have been reconstructed by CT-TDLAS using those virtual data. In order to evaluate the initial temperature of the CT-TDLAS, MSE has been tested. Figure 8 shows the result of MSE evaluation for  
Figure 7: Comparisons between the temperatures measured by TDLAS and thermocouple. the case of Table 2 (Case 3). It can be seen that CSLOS method has better performance than SLOS method for initial value calculation. The minimum error value by the CSLOS method was \(3.26\times 10^{-3}\). In order to evaluate the spatial resolution of the CT-TDLAS, three factors have been tested. The first one is the FWHM. Then, the FWHM values of the reconstructed temperature profiles by CT-TDLAS have been compared with those of the virtual temperature profiles. The second one is SSD (sum of squared difference). The SSD value has been defined by Equation (11). If the SSD value is close to \"0\", the two profiles are almost the same patterns. The third one is ZNCC (zero-mean normalized cross-correlation) value shown in Equation (12). If this value is close to \"1\", the virtual data and the data obtained by CT-TDLAS are similar patterns. That is, the correlation between the two values is very high. \[SSD=\sqrt{\frac{\sum\limits_{i=0}^{N-1}\sum\limits_{j=0}^{M-1}\left(\left(T_{i,j}\right)_{virtual}-\left(T_{i,j}\right)_{CT-TDLAS}\right)^{2}}{NM}}/T_{R} \tag{11}\]  
\[ZNCC=\frac{\sum\limits_{i=0}^{N-1}\sum\limits_{j=0}^{M-1}\left\{\left(T_{i,j}- \overline{T}\right)_{virtual}\times\left(T_{i,j}-\overline{T}\right)_{CT-TDLAS} \right\}}{\sqrt{\sum\limits_{i=0}^{N-1}\sum\limits_{j=0}^{M-1}\left(T_{i,j}- \overline{T}\right)_{virtual}\simeq\sum\limits_{i=0}^{N-1}\sum\limits_{j=0}^{M -1}\left(T_{i,j}-\overline{T}\right)_{CT-TDLAS}\text{}^{2}}},\qquad\overline{ T}=\frac{\sum\limits_{i=0}^{N-1}\sum\limits_{j=0}^{M-1}T_{i,j}}{NM} \tag{12}\]  
Here, \(T_{i,j}\) is the temperature at each point in a calculation area, \(T_{R}\) is the representing temperature (the max temperature of virtual data), \(N\) is the total number of meshes along X-axis in the calculation area, and \(M\) is the total number of meshes along Y-axis in the calculation. Table 2 shows the comparison of the above three values between the virtual data and the data obtained by CT-TDLAS for three different cases, of which FWHM is 10 mm, 15 mm, and 20 mm. It can be said that the SSD values for the three cases are less than 5% (13.155 K/280 K) at most. This implies that the discrepancy between the original virtual data and the data obtained by CT-TDLAS calculation is very small, showing that the present calculation approach is appropriate. Figure 9a,b shows the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case No.**} & **FWHM** & **FWHM** & & \\  & **(Virtual)** & **(CT-TDLAS)** & **SSD** & **ZNCC** \\  & **[mm]** & **[mm]** & & \\ \hline Case 1 & 10 & 16.9 & 0.14785 & 0.83801 \\ Case 2 & 15 & 17.7 & 0.05381 & 0.97191 \\ Case 3 & 20 & 18.1 & 0.01962 & 0.99907 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The spatial resolution evaluated for the virtual data. Figure 8: MSE (mean square error) comparison for initial temperature (Case 3). results of the case 3. Figure 9c shows the temperature profiles at the centerline of the calculation area for the virtual one and the CT-TDLAS calculation results. The spatial resolution of CT is related to the spatial density of the calculation grid. The number of the grid in a CT calculation area and the distance between each grid is known as the sampling interval, which is a function of the accuracy of the CT calculation.