## 3 Methodology
### Derivation and Upscaling of Landsat 8-Based Indices
#### 3.1.2 Landsat 8 Broadband Surface Albedo Estimation  
To date, operational algorithms have not been developed for Landsat 8 OLI broadband surface albedo computation. Compared to its predecessors Landsat TM and ETM+ sensors, Landsat 8 OLI incorporates two additional spectral bands, one deep blue band (0.43-0.45 \(\upmu\)m) and one shortwave infrared band (1.36-1.39 \(\upmu\)m). Besides, reflective wavebands corresponding to the previous Landsat ETM+ bands have narrower waveband widths, _i.e._, different spectral response functions. Therefore, algorithms and parameters developed for albedo calculation for Landsat TM/ETM+ sensors cannot be directly used for the Landsat 8 OLI sensor. In this study, the albedo calculation strategy widely used for SEB models was employed [47], where broadband surface albedo is calculated as the integration of at-surface reflectance across the shortwave spectrum, as shown in Equation (4). \[\alpha=\sum_{b=1}^{7}\left[\rho_{b}\cdot w_{b}\right] \tag{4}\]  
where \(\alpha\) is albedo, \(\rho_{b}\) is surface reflectance for a given band \(b\), which can be derived from any atmospheric correction method such as FLAASH used in this study. \(w_{b}\) is the weighting coefficient representing the surface solar radiation fraction within the spectral range for band \(b\), and is calculated as:  
\[w_{b}=\frac{\int_{LO_{b}}^{UP_{b}}R_{s\lambda}\cdot d\lambda}{\int_{0.3}^{4}R_{ s\lambda}\cdot d\lambda} \tag{5}\]  
where \(R_{s\lambda}\) is at-surface spectral hemisphere solar radiation for wavelength \(\lambda\) (\(\upmu\)m); and \(UP_{b}\) and \(LO_{b}\) are upper- and lower-wavelength bounds assigned to Landsat 8 band \(b\), respectively. Table 3 lists the band settings of Landsat 8 OLI in comparison with Landsat 7 ETM+ sensor. Landsat 8 OLI Cirrus band (band 9: 1.36-1.39 \(\upmu\)m) was not included in calculation due to its high absorption and low reflective characteristics. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Acronym** & **Equation** & **Source** \\ \hline \multirow{4}{*}{Vegetation Greenness Indices} & NDVI & \(\frac{B5-B4}{B5}\) & [PERSON] (1979) [39] \\  & EVI & \(\frac{B5+B4}{B5}\) & [PERSON] _et al._ (2002) [40] \\  & EVI & \(\frac{B5-B4}{B5}\) & [PERSON] (1988) [41] \\  & SAVI & \(1.5\times\frac{B5-B4}{B5}\) & [PERSON] (1988) [41] \\  & MSAVI & \(\frac{2\times B5+1-\sqrt{(2\times B5+1)^{2}-8\times(B5-B4)}}{2}\) & [PERSON] _et al._ (1994) [42] \\ \hline \multirow{4}{*}{Vegetation Water Indices} & NDMI & \(\frac{B5-B6}{B5}\) & Wilson and Sader (2002) [43] \\  & NDWI & \(\frac{B3+B5}{B5}\) & McFeeters (1996) [44] \\  & NDIIb7 & \(\frac{B3+B5}{B5}\) & Hunt and Rock (1989) [45] \\  & & \(1-R^{\prime}=\frac{B6}{(1-c)\times B5+c\times B7}\) & adapted from [PERSON] _et al._ \\  & \(\ast c=\frac{1.609-0.865}{2.201-0.865}\) & \(\ast 0.5569\) & (2003) [46] \\ \hline \hline \end{tabular}  
* The constant \(c\) represents the relative position of band 6 to band 7 and band 5 calculated using the mean wavelength of Landsat bands 5 (0.865 \(\upmu\)m), 6 (1.609 \(\upmu\)m) and 7 (2.201 \(\upmu\)m).  
\end{table} 
Table 2: Landsat 8 vegetation indices evaluated in this study. Landsat 8 bands: B1 = 0.430–0.45 \(\upmu\)m, B2 = 0.45–0.51 \(\upmu\)m, B3 = 0.53–0.59 \(\upmu\)m, B4 = 0.64–0.67 \(\upmu\)m, B5 = 0.85–0.88 \(\upmu\)m, B6 = 1.57–1.65 \(\upmu\)m, B7 = 2.11–2.29 \(\upmu\)m. Vegetation indices: NDVI = Normalized Difference Vegetation Index, EVI = Enhanced Vegetation Index, SAVI = Soil Adjusted Vegetation Index, MSAVI = Modified Soil Adjusted Vegetation Index, NDMI = Normalized Difference Moisture Index, NDWI = Normalized Difference Water Index, NDIIb7 = Normalized Difference Infrared Index\(-\)band 7, D\({}_{1609}\) = Depth of Landsat 8 band 6. Following similar procedures in [47], the Simple Model of the Atmospheric Radiative Transfer of Sunshine (SMARTS) radiative transfer model [48] was used for the calculation of at-surface solar radiation levels for three combinations of land elevations, precipitable water content, and solar angles.