## 2 Methodology
### Thermal Radiance Estimation of Objects  
The thermal radiance of objects composed of several MODIS TR pixels was estimated in two steps: We identified the MODIS TR pixels that were totally or partly covered by the object, as shown in Figure 4, mainly because the area of the objects generated from the high-resolution multispectral data may not have exactly overlaid several MODIS pixels entirely. Then, the TR of the object (\(R_{\text{obj}}\)) was calculated with the MODIS pixels through Equation (4):  
\[R_{\text{obj}}=\frac{\sum\limits_{i=1}^{M}R_{\text{mod},i}S_{\text{mod}}+\sum \limits_{j=1}^{N}R_{\text{mod},j}S_{\text{mod}}W_{j}}{S_{\text{obj}}} \tag{4}\]  
where \(R_{\text{mod},i}\) is the TR of the MODIS pixel, \(i\), that is totally covered by the object; \(S_{\text{mod}}\) is the area of the MODIS pixel, \(i\), that is totally covered by the object; \(M\) is the number of MODIS pixels that are totally covered by the object; \(R_{\text{mod},j}\) is the TR of the MODIS pixel,  
Figure 3: Experimental procedure for MODIS data.  
\(j\), that is partly covered by the object; \(N\) is the number of MODIS pixels that are partly covered by the object; \(W_{j}\) is the weight of the MODIS TR pixels that are partly covered by the object; and \(S_{\text{obj}}\) is the area of the object.  
The thermal radiance conservation condition is expressed by Equation (5):  
\[\sum_{j=1}^{n}W_{j}=1 \tag{5}\]  
where \(W_{j}\) are the weights of the MODIS TR pixels that are partly covered by different objects, and \(n\) is the number of parts that are decomposed by different objects.  
The weight of the MODIS TR pixels that are partly covered by the object is needed for the computation of thermal radiance for each subpixel in the decomposed LST image. [PERSON] et al. [38] gave the weight calculation method based on the initial decomposed LST that is first estimated by the assistant high-spatial-resolution data. If the initial decomposed temperature (\(T_{ini\_k}\)) and emissivity (\(\epsilon_{ini\_k}\)) for each subpixel of the decomposed LST image at the finer resolution could be estimated, the TR of each subpixel at a given wavelength could calculated by the Planck equation formula for radiance through Equation (6):  
\[R_{ini\_k}=\epsilon_{ini\_k}R_{b}(T_{ini\_k})=\frac{\epsilon_{ini\_k}K_{1}}{ \exp(K_{2}/T_{ini\_k})-1} \tag{6}\]  
where \(R_{ini\_k}\) is the estimated TR for subpixel \(k\), and \(T_{ini\_k}\) and \(\epsilon_{ini\_k}\) are the estimated initial LST and emissivity of the subpixel \(\mathbf{k}\). Then, the total thermal radiance of the partly covered MODIS TR and its parts can be estimated from the pixels that are partly or totally covered by the target object through Equation (7):  
\[R_{tot}=\sum_{k=1}^{m}R_{k}S_{tar}/S_{\text{mod}} \tag{7}\]  
\[R_{p}=\sum_{k=1}^{n}R_{k}S_{tar}/S_{p} \tag{8}\]  
where \(R_{tot}\) is the total thermal radiance of the MODIS TR for partly covered pixels, \(S_{tar}\) is the area of decomposed pixels at the target resolution, \(m\) is the number of pixels decomposed  
Figure 4: The main process of the TR estimation of objects.  
from the MODIS TR at the target resolution, \(R_{p}\) is the thermal radiance of partly covered pixels for the MODIS TR, and \(n\) is the number of pixels decomposed from the part at the target resolution, as shown in Equation (9) below:  
\[W_{j}=R_{p}/R_{tot}=S_{\text{mod}}\sum_{k=1}^{n}R_{k}/S_{p}\sum_{k=1}^{\overline {m}}R_{k} \tag{9}\]  
Therefore, we should obtain the objects and the estimated initial temperature and emissivity of subpixels at the target resolution to estimate the thermal radiance of the objects.