## 3 Line Selection  
The proposal of this study is to measure the temperature using two partially overlapped H\({}_{2}\)O absorption signals within a single scan range. Line selection methods summarized in the study by [PERSON] et al. were applied to measure the optimum temperature [16]. This method can be divided into five criteria, which can be expressed as follows. 1. The two selected absorption lines should have sufficient absorption signals in the temperature range;
2. The two absorption lines are within a single laser scan range and do not overlap with each other in the atmosphere;
3. The \(R\) of the integrated absorbance area should be single-valued with temperature, and the line strengths of the two absorption lines (\(R\) = 0.2-5.0) should be similar;
4. The two absorption lines have sufficiently different lower-state energy E\" to yield an absorption ratio that is sensitive to the probed temperature;
5. There should be no interference from other gas molecules near the two absorption signals. As mentioned above, two suitable partially overlapped H\({}_{2}\)O absorption lines (1.34923 and 1.34927 \(\upmu\)m) were selected meeting the criteria established, except the second one. For the first criterion, [PERSON] et al. described the peak absorption of transition, which includes the gas mole fraction, linestrength, pressure, and optical path length in the measurement environment based on optically thick conditions. The range for the peak absorption of transition was set to be higher than 0.001 and lower than 0.800. To satisfy this range, the H\({}_{2}\)O concentration (range: approximately 10-20%) produced after the premixed CH\({}_{4}\)/air flame reaction was predicted using a chemical equilibrium solver (Chemkin). The linestrength magnitude (cm/molecule, 10\({}^{-22}\)) that did not violate the range for peak absorption of transition in the corresponding H\({}_{2}\)O concentrations was also confirmed [16]. In addition, the optical path length that satisfies the range for the peak absorption of transition was considered. Figure 1 shows a graph of the linestrengths of the two H\({}_{2}\)O absorption lines at 296, 1073, 1473.15, and 1773.15 K using the HITRAN2012 database and compared with the linestrengths of other combustion products [22]. As shown in Figure 1, the linestrengths of the two selected absorption lines between 1073.15 and 1773.15 K had a similar magnitude of 10\({}^{-22}\). It was also confirmed that there was no interference from other combustion products (fifth criterion). Figure 2 is a graph presenting the analysis of the third and fourth criteria. Figure 2a shows the sensitivity of the linestrength ratio for the difference in lower-state energy as a function of temperature. The lower-state energy between the two absorption wavelengths for the fourth criterion must have a sufficient difference as it is related to the sensitivity of the probed temperature. As can be seen from the relative sensitivity equation [18], the large difference in lower-state energy improves the sensitivity to temperature measurement. Most previous studies have found that the difference in lower-state energy at optimal absorption line pair is between 1000 and 2000 cm\({}^{-1}\). At 1.34923 and 1.34927 \(\upmu\)m selected for our study, the lower-state energies were E\({}_{1}\) = 3655.4833 cm\({}^{-1}\) and E\({}_{2}\) = 602.7735 cm\({}^{-1}\), respectively, with a difference of 3052.709 cm\({}^{-1}\). This large difference in lower-state energy can have a high sensitivity for temperature measurement, but it can also have drawbacks depending on the temperature, optical path-length, and concentration in the measurement environment [23]. As shown in temperature equation [24], the temperature is determined by the ratio of the integrated absorbance area. The relationship between the integrated absorbance area and the temperature is determined by the ratio of the integrated absorbance area. The relationship between the integrated absorbance area and the temperature is determined by the ratio of the integrated absorbance area.