## 2 Development of LST Retrieval Algorithm
In this study, we developed an SW-type LST retrieval algorithm, because the accuracy and efficiency of SW-type LST algorithms are well known [2, 3, 16, 31, 32, 40]. Various types of linearized SW algorithms have been developed based on the differential absorption in two adjacent split-window channels (10-12.5 \(\upmu\)m) linearizing the radiative transfer equation (e.g., [1, 2, 3, 6, 10]). In general, these algorithms express the LST as a simple linear combination of the impacting factors, such as brightness, temperature difference, SZA and emissivity [1, 2, 3, 6, 10, 16]. However, the linearized SW algorithm results in large errors in LST retrieval under wet and hot atmospheric conditions [42, 43]. To improve the accuracy of LST retrieval, various types of non-linear SW algorithms have been developed [42, 43, 44]. We included a non-linear term (\(\Delta T^{2}\)) for water vapor to account for its non-linear impacts as in [PERSON] and Raissouni [43] and Sun and Pinker [44]. This approach is different from other approaches that separate the retrieval equations according to the atmospheric water vapor content [2, 32]. The equation used for this study was:  
\[[\mathrm{LST}=\mathrm{a}+\mathrm{b}\;T_{IR1}+c\;\Delta T+d\;\Delta T^{2}+e\;( sec\theta-1)+f(1-\bar{\varepsilon})+g\Delta\varepsilon\;] \tag{1}\]  
where T\({}_{IR1}\) is the brightness temperature of IR1, \(\theta\) is the SZA of COMS, \(\Delta\Gamma=T_{IR1}-T_{IR2}\), \(\bar{\varepsilon}=\frac{\varepsilon_{IR1}+\varepsilon_{IR2}}{2}\) and \(\Delta\varepsilon=\varepsilon_{IR1}-\varepsilon_{IR2}\). The coefficients from \(a\) to \(g\) in the CSW algorithm were determined with the simulated brightness temperature data and the prescribed LST data using the statistical regression method, as shown in Equation (2):  
\[ \begin{array}{l}[\mathrm{LST}=29.7890+0.8866\;T_{IR1}+2.1443\Delta T+0.1298 \Delta T^{2}+0.7911(sec\theta-1)+\\ 56.6851(1-\bar{\varepsilon})\;122.172\Delta\varepsilon]\end{array}  \tag{2}\]  
Figure 3 shows a histogram of bias and a scatter plot of the prescribed and estimated LSTs by the COMS Split-Window (CSW) retrieval algorithm for the full range of simulation conditions. In general, the CSW retrieval algorithm slightly overestimated the LST compared to the prescribed LST, except in situations where the LST was between 300 K and 320 K. Although the average estimation capabilities of the CSW retrieval algorithm were very high (averaged bias = 0.00 K, RMSE = 1.41 K, correlation coefficient = 0.99), the bias covered a wide spectrum ranging from \(-4\) K to \(+4\) K. These results suggest \begin{table}
\begin{tabular}{c c} \hline \hline
**Subject** & **Conditions** \\ \hline Atmospheric Profile & 359 profiles of TIGR 2000 database \\ Satellite Zenith Angle & SZA of TIGR point (0\({}^{\circ}\) - 50\({}^{\circ}\)) \\ Land Surface Temperature & Ta-6 K - Ta + 16 K (in steps of 2 K) \\  & \(\varepsilon_{\mathrm{IR1}}\): 0.9478 \(\sim\) 0.9968 (in steps of 0.0049) \\ Emissivity & \(-0.012\leq\Delta\varepsilon\leq 0.012\) (in steps of 0.004) \\  & If \((\varepsilon_{\mathrm{IR2}})>1\), \(\varepsilon_{\mathrm{IR2}}=0.9999\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Atmospheric, surface and sensor-target conditions used in radiative transfer model simulations. SZA, satellite zenith angles. that the current CSW retrieval algorithm can significantly either overestimate or underestimate the LST under certain conditions. The estimation capabilities of the CSW algorithm varied according to the surface lapse rate conditions (_i.e._, the diurnal variation of the difference between LST and air temperature). These data are shown in Figure 4. While the estimation capabilities of the CSW algorithm were clearly impacted by the lapse rate conditions, the algorithm produced more accurate estimations (_i.e._, high correlation, small bias and RMSE) under weak lapse rate conditions.