# Evaluation of Water and Carbon Estimation Models in the Caatinga Biome Based on Remote Sensing
## 2 Material and Methods
### Data Processing and Analysis
#### 2.4.2 ET (SEBAL versus MOD16A2) Estimation  
Energy balance models based on remote sensing data have been widely used to estimate agricultural water use and evapotranspiration and map the spatial distribution of energy fluxes across landscapes [42]. In this sense, the SEBAL algorithm developed for estimating ET using satellite images has been validated in several experimental campaigns throughout the world [43; 44; 45]. To estimate ET on a daily scale, it is assumed that the instantaneous evaporative fraction (estimated from the energy balance components) is constant for 24 h. Refs. [39; 46] stated that the evaporative fraction is a relatively constant indicator of energy partitioning at the surface during daylight hours. It has a minimal diurnal cycle, improving the usefulness of instantaneous satellite imagery for studying the interactions between the land surface and the atmosphere. The energy balance components for the surface were then estimated from:  
\[\mathrm{Rn-G-LE-H=0} \tag{7}\]  
where \(\mathrm{R_{n}}\) is the net radiation (Equation (1)), \(\mathrm{G}\) is the soil heat flux, \(\mathrm{LE}\) is the latent heat flux, and \(\mathrm{H}\) is the sensible heat flux, all in units of W\(\cdot\)m\({}^{-2}\). Soil heat flux (\(\mathrm{G}\)) was estimated according to the equation described in [39]:  
\[\frac{\mathrm{G}}{\mathrm{R_{n}}}=\frac{\mathrm{T_{s}}}{\alpha}\Big{(}0.0038 \alpha+0.0074\alpha^{2}\Big{)}\Big{(}1-0.98\mathrm{NDVI^{4}}\Big{)} \tag{8}\]  
where \(\mathrm{T_{s}}\) is the surface temperature, \(\alpha\) is the surface albedo, \(\mathrm{R_{n}}\) is the net radiation, and \(\mathrm{NDVI}\) is the Normalized Difference Vegetation Index, as described next. For water bodies, the heat flux was taken as 30% of \(\mathrm{R_{n}}\)[47]. The NDVI is the ratio between the differences in the reflectivity of the near-infrared (\(\mathrm{\rho_{IV}}\)) and red (\(\mathrm{\rho_{V}}\)) bands and their sum; for the MODIS/Terra sensor, \(\mathrm{\rho_{IV}}\) corresponds to channel 2 and \(\mathrm{\rho_{V}}\) to channel 1:  
\[\mathrm{NDVI}=\frac{\mathrm{\rho_{IV}-\mathrm{\rho_{V}}}}{\mathrm{\rho_{IV}+ \mathrm{\rho_{V}}}} \tag{9}\]  
Sensible heat flux (\(\mathrm{H}\)) was estimated according to [23]:  
\[\mathrm{H}=\mathrm{\rho_{cp}}\frac{\mathrm{dT}}{\mathrm{r_{ah}}} \tag{10}\]  
where \(\mathrm{\rho}\) corresponds to the air-specific mass, \(\mathrm{c_{p}}\) is the air specific heat at constant pressure (1004 J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\)), dT is the temperature difference between two levels \(Z_{1}\) and \(Z_{2}\), and \(\mathrm{r_{ah}}\) is the aerodynamic resistance to the transport of heat (s\(\cdot\)m\({}^{-1}\)). Based on \(\mathrm{R_{n}}\), \(\mathrm{G}\), \(\mathrm{H}\), and \(\mathrm{LE}\) (W\(\cdot\)m\({}^{-2}\)), the evaporative fraction (\(\Lambda\) is estimated according to:  
\[\Lambda=\frac{\mathrm{LE}}{\mathrm{R_{n}-G}} \tag{11}\]  
The actual daily evapotranspiration (mm\(\cdot\)day\({}^{-1}\)) was estimated using:  
\[\mathrm{ET_{24h}}=\frac{86400\Lambda\mathrm{R_{n24h}}}{\mathrm{\lambda\rho_{ uv}}} \tag{12}\]where \(\lambda\) is the latent heat of water vaporization (J\(\cdot\)kg\({}^{-1}\)), estimated according to [48] through Equation (13); 86,400/\(\lambda\) factor is used to convert ET from W\(\cdot\)m\({}^{-2}\) to mm\(\cdot\)day\({}^{-1}\); \(\rho_{\text{w}}\) is the water density (kg\(\cdot\)m\({}^{-3}\)). \[\lambda=[2.501-0.00236(\text{T}_{\text{a}}-273.16)]*10^{6} \tag{13}\]  
where T\({}_{\text{a}}\) is the air temperature obtained from the MODIS product, in K. \(\lambda\) values vary slightly over normal temperature ranges. A single value can be adopted in many cases, e.g., for T = 20 \({}^{\circ}\)C, \(\lambda\) can be set to 2.45 MJ\(\cdot\)kg\({}^{-1}\)[33]. The evapotranspiration values obtained from the MOD16A2 products were extracted for comparison purposes. The ET was estimated according to the methodology described by [49], which uses satellite-derived energy fluxes and the leaf area index based on the Penman-Monteith method and later improved by [50].