## 2 Materials and Methods
### Calculation  
The FUI was calculated based on the CIE-XYZ standard color system proposed by the International Commission on Illumination (CIE), in which the \(X\), \(Y\), \(Z\) tristimulus values were first proposed. Sentinel-2 has five visible light bands, and the formula for calculating the \(X\), \(Y\), \(Z\) tristimulus values is Equation (3) [27]:  
\[\left\{ \begin{aligned} & X=11.756R(443)+6.423R(490)+53.696R(560)\\ &\qquad+32.028R(665)+0.529R(705)\\ & Y=1.744R(443)+22.289R(490)+65.702R(560)\\ &\qquad+16.808R(665)+0.192R(705)\\ & Z=62.696R(443)+31.101R(490)+1.778R(560)\\ &\qquad+0.015R(665)+0.000R(705)\\ \end{aligned} \right. \tag{3}\]  
The Landsat 8 Operational Land Imager (OLI) has four visible light bands, and the formula is Equation (4):  
\[\left\{ \begin{aligned} & X=11.053R(443)+6.950R(483)+51.135R(561)+34.457R(661 )\\ & Y=1.320R(443)+21.053R(483)+66.023R(561)+18.034R(661)\\ & Z=58.038R(443)+34.931R(483)+2.606R(561)+0.016R(661)\\ \end{aligned} \right. \tag{4}\]  
The relevant information is in Table 2.  
The chromaticity coordinates in the CIE were calculated based on tristimulus values, and the results of the chromaticity coordinates were normalized to 0-1 (Figure 5 and Table 3). Therefore, a new coordinate system was obtained [18] that is described by Equation (5):  
\[\left\{ \begin{aligned} & x=\frac{X}{X+Y+Z}\\ & y=\frac{X}{X+Y+Z}\\ \end{aligned} \right. \tag{5}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Sentinel-2**} & \multicolumn{2}{c}{**Landsat 8 OLI**} \\ \multicolumn{2}{c}{**Central Wavelength (nm)**} & **Bands** & **Central Wavelength (nm)** & **Bands** \\ \hline
443 & Coastal Blue & 443 & Coastal Blue \\
490 & Blue & 483 & Blue \\
560 & Green & 561 & Green \\
665 & Red & 661 & Red \\
705 & Vegetation Red Edge & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Spectral bands for the Sentinel-2 and Landsat 8 OLI  
Figure 4: The result of water extraction. Finally, the hue angle \(\alpha\) was obtained [18] by Equation (6):  
\[\alpha=arctan\left(\frac{y-y_{w}}{x-x_{w}}\right)\times\frac{180}{\pi}+180 \tag{6}\]  
where \(y_{w}\) and \(x_{w}\) are \(\left(\frac{1}{3},\frac{1}{5}\right)\). Figure 5: FUI colors and the subdivision of the FUI from 1 to 21 in CIE chromaticity diagram [28]. The red crosses mark the chromaticity coordinates of the Forel–Ule Index. The hue angle \(\alpha\) is the angle between the vector to a point and the negative x’-axis (at t = 1/3) [18]. \begin{table}
\begin{tabular}{c c c c} \hline
**FUI** & **x** & **y** & **\(\alpha\) (\({}^{\circ}\))** \\ \hline
1 & 0.191363 &  
To eliminate hue angle \(\alpha\) deviations caused by band dispersion and band setting \(\Delta\), we established an error polynomial fitting formula based on the water surface reflectance database and added the system deviation simulated by the polynomial to the hue angle \(\alpha\) calculated from the multi-spectral images. The method had the effect of eliminating bias [29]. The optimal linear relationship between the hue angle \(\alpha\) calculated from the hyperspectral data and that calculated by the satellite sensor was obtained by finely correcting the hue angle \(\alpha\). Specifically, the hue angle \(\alpha\) was corrected by compensating for the linear interpolation between the natural water spectrum and the sensor band. Owing to sensor band limitations, a large offset was not generated randomly. The calibration formula of the hue angle \(\alpha\) based on the Sentinel-2 data is as follows [20]:  
\[\Delta=46.2094a^{5}-412.256a^{4}+1385.5708a^{3}-2128.364a^{2}+1443.7115a-341.6433 \tag{7}\]  
The calibration formula of hue angle \(\alpha\) based on the Landsat 8 OLI is as follows [30]:  
\[\Delta=-52.819a^{5}+334.88a^{4}-758.26a^{3}+746.324a^{2}-315.18a+39.761 \tag{8}\]  
where \(a\) is the hue angle \(\alpha\) divided by 100. By correcting the hue angle \(\alpha\), its value is closer to the actual water color.