## 3 Methods
### Coefficients Generation
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Mono-Window Algorithms**} \\ \hline
**Method** & **Formula** & **Notes** \\ \hline Price 1983 [9] & \(LST=\frac{p_{0}+\epsilon_{4}}{p_{1}}(p_{2}t_{4}+p_{3})\) & (1) & Developed for AVHRR/1 \\ \hline \multirow{4}{*}{[PERSON] et al. 2001 [11]} & \(LST=\frac{1}{2}(\frac{1}{2}(\frac{1}{2}+\frac{1}{2})+\frac{1}{2})\Delta\) & (2) & \\  & & \\  & & \\  & & \\  & & \\ \hline \multirow{4}{*}{[PERSON] 2003 [12]} & \multirow{4}{*}{\(LST=\gamma\Big{(}\frac{1}{2}(\frac{1}{2}+\frac{1}{2})+\frac{1}{2})\Delta\)} & \multirow{4}{*}{(6)} & Developed for LANDSAT TM \\  & & & \\  & & & \\  & & & \\  & & & \\  & & \\ \hline \multirow{4}{*}{Linear} & \multirow{4}{*}{\(LST=\frac{\Delta t+p_{1}}{t^{4}}\)} & \multirow{4}{*}{(7)} & Linear expression, developed for \\  & & & this assessment \\ \hline \multicolumn{2}{c}{**Split-Window Algorithms**} \\ \hline
**Method** & **Formula** & **Notes** \\ \hline Price 1984 (as & \multirow{4}{*}{\(LST=(4+p_{0}(4t-5))\Big{(}\frac{p_{0}-\varepsilon}{p_{2}}+p_{3}5 da\Big{)}\)} & \multirow{4}{*}{(8)} & Developed for AVHRR/2 \\  & & & \\  & & & \\  & & \\  & & \\  & & \\ \hline \multirow{4}{*}{Becker and Li 1990 [14]} & \(\begin{array}{c}a=1+p_{1}\frac{1-\varepsilon+\varepsilon}{\tau}+p_{2}\frac{da}{ \tau}\\ \beta=p_{3}+p_{4}\frac{1-\varepsilon}{\tau}+p_{5}\frac{da}{\tau}\\ LST=p_{0}+\alpha\Big{(}\frac{da+5}{\tau}\Big{)}+\beta\Big{(}\frac{da+5}{ \tau}\Big{)}\end{array}\) & (10) & Developed for AVHRR/2 \\ \hline [PERSON] et al. & \multirow{4}{*}{\(LST=4+p_{0}(4t-5)+p_{1}(1-\varepsilon)-p_{2}da\)} & \multirow{4}{*}{(12)} & Developed for AVHRR/2 \\  & & & \\  & & \\  & & \\ \hline \multirow{4}{*}{[PERSON] and [PERSON] 1996 [16]} & \(LST=\Big{(}p_{0}+p_{1}\frac{1-\varepsilon+\varepsilon}{\tau}+p_{3}\frac{da}{\tau} +p_{5}\frac{da}{\tau}-2\Big{)}+p_{6}\) & (13) & Developed for MODIS \\  & & & \\ \hline \multirow{4}{*}{[PERSON] et al. 2008 [3]} & \(LST=4+p_{0}(4t-5)+p_{1}(1-\varepsilon)-p_{2}da\) & \multirow{4}{*}{(14)} & Developed for AVHRR/2 \\  & & & \\  & & & and AVHRR/3 \\ \hline \multirow{4}{*}{[PERSON] 2008 [3]} & \(LST=4+p_{1}(4t-5)+p_{2}(4t-5)^{2}+p_{0}\) & \multirow{4}{*}{(15)} & Developed for ATSR2, ATSR, \\  & & & \\  & & & MODIS, AVHRR/3, \\  & & & \\  & & & \\  & & & \\  & & & \\ \hline \multirow{4}{*}{[PERSON] 2008 [3]} & \(LST=4+p_{1}(4t-5)+p_{2}(4t-5)^{2}+p_{0}\) & \multirow{4}{*}{(15)} & Developed for ATSR2, ATSR, \\  & & & & MODIS, AVHRR/3, \\  & & & MODIS, AVHRR/3, \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\ \hline \multirow{4}{*}{[PERSON] and [PERSON] 2008 [3]} & \(LST=4+p_{0}(4t-5)+p_{1}(1-\varepsilon)-p_{2}da\) & \multirow{4}{*}{(13)} & Developed for MODIS \\  & & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\ \hline \multirow{4}{*}{[PERSON] and [PERSON] 2008 [3]} & \multirow{4}{*}{(6)} & Developed for LANDSAT TM \\  & & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\ \hline \end{tabular}
\end{table} 
Table 4: Formulations of the tested mono- and split-window methods. The coefficient estimation was done for three different kinds of input cases, whenever possible:  
* Case A: The coefficients were retrieved for each class of sensor view angle (VA). * Case B: The coefficients were retrieved for each class of sensor view angle, as well as for 8 ranges of columnar water vapor (VA/CVV). * Case C: The coefficients were retrieved for each class of sensor view angle, for 8 ranges of columnar water vapor and for 4 classes of surface temperature (VA/CVV/LST). Using the [PERSON] and [PERSON] 2008 algorithm and [PERSON] et al., 2001 algorithm, the coefficients were retrieved for sensor view angle and surface temperature (VA/LST). The [PERSON] et al., 2008 algorithm uses ranges of columnar water vapour and surface temperature only (CVV/LST). Case C has for each algorithm the maximal surface ranges necessary. The hypotesis is that Case C will give the highest accuracies, while Case A shows the highest errors.