## 2 Materials and Methods
These factors, referred to as \"transmission\" and \" upwelling\", approximate the surface radiance leaving the Earth for each pixel within the original cube. By plotting the transmission and upwelling (\(\mathrm{\tau}\), Lu) and generating the cube, the ENVI 5.3 algorithm incorporates this information to perform thermal atmospheric correction using ISAC. In the case of Landsat 5 TM, information regarding the sensor is stored in the Landsat 5 TM metadata file. Assuming the Earth's surface behaves as a blackbody emitter with a \begin{table}
\begin{tabular}{c c} \hline \hline
**Information** & **Value** \\ \hline LMAX\({}_{\lambda}\) & 15.303 \\ LMIN\({}_{\lambda}\) & 1.238 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: LMAX and LMIN information for Landsat 5 TM. spectral radiance, the radiance spectrum obtained from the sensor is then converted into an effective sensor brightness temperature using Equation (4) [25]. \[\mathrm{T_{B}}=\frac{\mathrm{k_{2}}}{\mathrm{ln}\Big{(}\frac{\mathrm{k_{1}}}{ \mathrm{L_{\lambda}}}+1\Big{)}} \tag{4}\]  
In Equation (4), \(\mathrm{T_{B}}\) represents the effective sensor brightness temperature in Kelvin (K), while \(\mathrm{k_{1}}\) and \(\mathrm{k_{2}}\) are calibration constants. For Landsat 5 TM, the values for \(\mathrm{k_{1}}\) and \(\mathrm{k_{2}}\) are set to 607.76 K and 1260.56 K, respectively. It is important to note that the sensor brightness temperature obtained from Equation (4) is related to a black body. To estimate the surface temperature of a gray body, the emissivity of the surface spectrum needs to be improved. This can be achieved by calculating the improved surface temperature emissivity using Equation (5) proposed by [PERSON] and [PERSON] [26]. \[\mathrm{LST}=\frac{\mathrm{T_{B}}}{1+(\lambda\mathrm{T_{B}}/\alpha)\mathrm{ln}\varepsilon} \tag{5}\]  
The surface temperature is typically measured in Kelvin (K). In the context of the equation mentioned above, \(\mathrm{T_{B}}\) represents the effective sensor brightness temperature in K. The variable \(\lambda\) denotes the wavelength of the emitted light, expressed in meters. For the Landsat 5 TM thermal bands, the effective wavelengths are 11.457 \(\mathrm{\SIUnitSymbolMicro m}\), 11.269 \(\mathrm{\SIUnitSymbolMicro m}\), and 10.904 \(\mathrm{\SIUnitSymbolMicro m}\), based on Equation (6) with \(\alpha\) = 1.438 \(\times\) 10\({}^{-2}\) mk. The symbol \(\varepsilon\) represents the surface emissivity. To convert the surface temperature from Kelvin to degrees Celsius (\({}^{\circ}\)C), Equation (6) is applied as follows:  
\[\mathrm{Surface~{}temperature~{}(^{\circ}C)=Surface~{}temperature~{}(K) -273.15} \tag{6}\]  
The thermal surface radiance is captured by the thermal infrared sensor (TIRS) band 10 on Landsat 8 OLI. To eliminate atmospheric effects from the thermal data, preprocessing is conducted. The surface temperature is then derived using the algorithm proposed by [PERSON] et al. [27]. To convert the digital numbers (DN) to spectral radiance L above the atmosphere, Equation (7) is applied as follows:  
\[\mathrm{L_{\lambda}=ML\times Qcal+A_{L}} \tag{7}\]  
In Equation (8), \(\mathrm{L_{\lambda}}\) represents the spectral radiance in units of \(\mathrm{W/(m\cdotsr\cdot\upmu m)^{2}}\). ML corresponds to the rescaled gain specific to a particular band, measured in units of \(\mathrm{W/(m\cdotsr\cdot\upmu m)^{2}}\). \(\mathrm{A_{L}}\) accounts for the band-specific rescaling bias, also measured in units of \(\mathrm{W/(m\cdotsr\cdot\upmu m)^{2}}\). It is important to note that the rescaling bias is applied to a specific band. To extract the sensor brightness temperature Tb in Kelvin (K) from TIRS band 10, similar to the OLI sensor, Equation (8) is utilized as follows:  
\[\mathrm{T_{b}=K_{2}/(\mathrm{ln}(K_{1}/L_{\lambda})+1} \tag{8}\]  
In Equation (9), \(\mathrm{K_{1}}\) and \(\mathrm{K_{2}}\) are predefined constants set to 774.89 (in \(\mathrm{W/(m\cdotsr\cdot\upmu m)^{2}}\)) and 1321.08 (in W), respectively.