# Spatiotemporal Variation of Land Surface Temperature Retrieved from FY-3D MERSI-II Data in Pakistan
## 3 Materials and Methods  
### The TFSW Algorithm for LST Retrieval  
The radiance is measured at the top of the atmosphere (TOA) in the TIR bands as follows, using the radiative transfer theory, assuming a cloud-free atmosphere in thermodynamic equilibrium [31]:  
\[\mathrm{B_{i}(T_{i})}=\varepsilon_{i}\mathrm{B_{i}(T_{s})}\tau_{i}+\mathrm{R} \uparrow\mathrm{atm_{i}}+(1-\varepsilon_{i})\mathrm{R}\downarrow\mathrm{atm_{i} }\tau_{i} \tag{1}\]  
where \(\mathrm{B_{i}}\) is the Plank function, \(\mathrm{T_{i}}\) is the brightness temperature (BT), \(\varepsilon_{i}\) is the emissivity, \(\mathrm{B_{i}(T_{s})}\) is the radiance measured when the surface is a blackbody with a surface temperature of \(\mathrm{T_{s}}\), \(\tau_{i}\) is the total atmospheric transmittance (AT), \(\mathrm{R}\uparrow\mathrm{atm_{i}}\) is the upwelling radiance, and \(\mathrm{R}\downarrow\mathrm{atm_{i}}\) is the downwelling radiance. The SW technique represents LST as a straightforward linear or nonlinear combination of the TOA brightness temperatures of two adjacent bands based on differential water vapor absorption in two neighboring TIR bands. This method does not require real-time atmospheric profiles [32]. Due to its practicality and efficiency, the SW algorithm is therefore frequently utilized in LST retrieval. The SW algorithm is one of the most widely utilized delegate algorithms that has produced various satellite LST products [33]. [PERSON] [34] developed a more sophisticated generalized split window approach for the MODIS Collection 6 LST product. [PERSON] et al. [35] developed a TFSW algorithm for LST retrieval from the NOAA-AVHRR Bands 4 and 5, completely based on the mathematical derivation from the thermal radiance transfer equation. The TFSW algorithm uses a series of linear combinations of two apparent temperatures, the corresponding band-averaged emissivity, and transmittance, to estimate the MERSI-II LST as follows:  
\[\mathrm{T_{s}}= A_{0}+A_{1}\mathrm{T_{24}}-A_{2}\mathrm{T_{25}} \tag{2}\] \[A_{0}=\mathrm{a_{24}[D_{25}(1-C_{24}-D_{24})/(C_{24}D_{25}-C_{25}D _{24})]}-\mathrm{a_{25}[D_{24}\;(1-C_{25}-D_{25})/(C_{24}D_{25}-C_{25}D_{24})]}\] (3) \[A_{1}=1+D_{24}/(C_{24}D_{25}-C_{25}D_{24})+\mathrm{b_{24}\;[D_{25} \;(1-C_{24}-D_{24})/(C_{24}D_{25}-C_{25}D_{24})]}\] (4) \[A_{2}=D_{24}/(C_{24}D_{25}-C_{25}D_{24})+\mathrm{b_{25}[D_{24}\;(1 -C_{25}-D_{25})/(C_{24}D_{25}-C_{25}D_{24})]}\] (5) \[\mathrm{C_{24}}=\varepsilon_{24}\tau_{24}\] (6) \[\mathrm{D_{24}}=[1-\tau_{24}]\;[1+(1-\varepsilon_{24})\;\tau_{24}]\] (7) \[\mathrm{C_{25}}=\varepsilon_{25}\tau_{25}\] (8) \[\mathrm{D_{25}}=[1-\tau_{25}]\;[1+(1-\varepsilon_{25})\;\tau_{25}] \tag{9}\]  
where \(\mathrm{T_{s}}\) is the retrieved FY-3D MERSI-II LST (K), \(A_{0}\), \(A_{1}\), and \(A_{2}\) are the internal parameters, \(\mathrm{T_{i}}\) is the brightness temperature (BT) of the FY-3D MERSI-II TIR band i (i = 24 or 25), \(\tau_{i}\) is the AT for band i, \(\varepsilon_{i}\) is Land Surface Emissivity (LSE) for the band i, and \(\mathrm{a_{i}}\), and \(\mathrm{b_{i}}\) are the constants for the band i. The constants for the two FY-3D MERSI-II TIR bands are determined as follows:  
\[\mathrm{L_{i}}=\mathrm{B_{i}\;(T)/[\partial B_{i}\;(T)/\partial T]}\approx \mathrm{B_{i}\;(T)/[[B_{i}\;(T+\Delta T)-B_{i}\;(T)]/\Delta T]} \tag{10}\]  
where \(\mathrm{L_{i}}\) is the derivative of the Planck function \(\mathrm{B_{i}\;(T)}\) with temperature T for the FY-3D MERSI-II TIR band i, which could be approximated as \([\mathrm{B_{i}\;(T+\Delta T)-B_{i}\;(T)]/\Delta T}\) for the determination of the constants as follows:  
\[\mathrm{T_{s}}= A_{0}+A_{1}\mathrm{T_{24}}-A_{2}\mathrm{T_{25}} \tag{2}\]0  
where \(\mathrm{a_{i}}\) and \(\mathrm{b_{i}}\) are the constants for the band i. It assumes \(\mathrm{L_{i}}\) with T from \(-50\)\({}^{\circ}\)C to \(50\)\({}^{\circ}\)C for the two FY-3D MERSI-II TIR bands, and then the constants are finally determined as follows: \(\mathrm{a_{24}}=-53.477\), \(\mathrm{b_{24}}=0.3951\); \(\mathrm{a_{25}}=-57.087\), and \(\mathrm{b_{25}}=0.4292\).