## 3 Methodology  
### Methods for Estimating Soil and Vegetation Component Temperatures  
In this study, the separation of the soil and vegetation component temperatures for a pixel is based on the theory of the soil wetness iso-line in the \(T\)-\(\alpha\) space according to the SEB-1S model [25]. Figure 3a illustrates the \(T\)-\(\alpha\) space, and concepts of surface soil moisture isopleths and the temperature and albedo end-members. Figure 3b shows the procedures used to decompose composite temperatures (\(T\)) into soil temperature (\(T_{\mathrm{s}}\)) and vegetation temperature (\(T_{\mathrm{c}}\)). In the \(T\)-\(\alpha\) space, the dry edge (Figure 3, red line) is generally interpreted as representing the limiting conditions of soil moisture or evapotranspiration. In  
Figure 2: Comparison between the modeled and measured land surface parameters, (**a**) is land surface temperature and (**b**) is surface albedo  
contrast, the wet edge (Figure 3, blue line) represents potential evapotranspiration [45]. The study area used to evaluate the _T_-\(\alpha\) space approach is approximately 150 km\({}^{2}\) and is an irrigated cropland (primarily corn) located in a strongly adevective arid region with flat terrain (an artificial oasis). Superimposed in Figure 3a, are the soil wetness iso-lines, and it is assumed that all pixels which pass through the same iso-line have the same _T_s, allowing derivation of _T_c [28, 29, 30]. In the _T_-\(\alpha\) space, the intersection of the iso-line and bare soil line is _T_s, while for _T_c, the value was derived from the intersection of the iso-line and the full vegetation cover line. To determine the position of the _T_-\(\alpha\) space in Figure 3, the value of seven end-members in the _T_-\(\alpha\) space are required: The maximum soil temperature (_T_s,max), the minimum soil temperature (_T_s,min), the maximum vegetation temperature (_T_c,max), the minimum vegetation temperature (_T_c,min), the wet soil albedo (\(\alpha_{\rm s}\)), the green vegetation albedo (\(\alpha_{\rm vg}\)) and the senescent vegetation albedo (\(\alpha_{\rm vs}\)). The study area is irrigated farmland with relatively uniform atmospheric conditions and the images were collected from the early crop growing season through senescence and included DOY 167, 176, 192, 215, 224, 231, 240, 247 and 256. Consequently, this image set can provide estimates of the seven end-members in the application of the T-\(\alpha\) space approach [7, 24, 25, 26]. The end-members of albedo are set to \(\alpha_{\rm s}\) = 0.10, \(\alpha_{\rm vg}\) = 0.17, \(\alpha_{\rm vs}\) = 0.32, and the temperature end-members are estimated by the synergy between _T_-\(\alpha\) and _T_-\(\gamma_{\rm vg}\) spaces based on the procedure developed by [PERSON] _et al._[7, 23, 24]. With the aforementioned analysis, for a given pixel point (_J_) along a random soil wetness iso-line in the _T_-\(\alpha\) space, its soil temperature can be calculated based on the law of tangents:  
Figure 3: A sketch of the polygon _T_-\(\alpha\) space involved in soil and vegetation component temperature separation. Points A and D pass through the dry line, and points B and C pass through the wet line according to SEB-1S model. The lines AB and CD represent the bare soil surface and fully vegetated surfaces, respectively. Slanting straight lines represent superimposed isopleths of soil moisture availability, which increase from dry edge to wet edge. **(a)** illustrates temperature and albedo end-members in the _T_-\(\alpha\) and **(b)** illustrates the separation of soil and vegetation component temperatures from composite temperatures. \[(\alpha_{s}-\alpha_{s})\beta_{J}+T \tag{7}\]  
In Equation (7), \(\alpha_{\rm s}\), \(T_{\rm s}\), \(T\) and \(\omega_{\rm J}\) are the wet soil albedo, soil temperature, surface temperature and albedo of point \(J\), respectively. \(\beta_{\rm J}\) is slope of the iso-line line going across point \(J\), which can be derived by interpolating the slope of the wet edge (\(\beta_{\rm w}\)) and that of the dry edge (\(\beta_{\rm d}\)).