# A New Approach to Defining Uncertainties for MODIS Land Surface Temperature
## 2 Framework
### Uncertainty Budget
#### 2.5.2 Level-2
The effect of the NEDTs combined across all channels relevant to the retrieval needs to be appropriately propagated to give the uncertainty contribution from random effects due to radiometric noise. Assuming the radiometric noise is Gaussian the propagation can be expressed as:  
\[u_{mn,T}(x)=\sqrt{\sum_{i=1}^{n}\biggl{(}\frac{\partial R}{\partial T_{i}}u_{ mn}(T_{i})\biggr{)}^{2}} \tag{6}\]  
where the partial derivatives of the retrieval algorithm (Equation (1)) for each channel with respect to \(T_{11}\) and \(T_{12}\) are respectively given by Equations (7) and (8):  
\[\frac{\partial R}{\partial T_{11}}=\frac{(A_{1}+B_{1})}{2}+\frac{(A_{2}+B_{2} )(1-\varepsilon)}{2\varepsilon}+\frac{(A_{3}+B_{3})\Delta\varepsilon}{2 \varepsilon^{2}} \tag{7}\]\[\frac{\partial R}{\partial T_{12}}=\frac{(A_{1}-B_{1})}{2}+\frac{(A_{2}-B_{2})(1- \varepsilon)}{2\varepsilon}+\frac{(A_{3}-B_{3})\Delta\varepsilon}{2\varepsilon^ {2}} \tag{8}\]  
where \(\varepsilon\) is the mean land surface emissivity of the two channels ((\(\varepsilon_{11}-\varepsilon_{12}\))/2) and \(\Delta\varepsilon\) the emissivity difference between the channels (\(\varepsilon_{11}-\varepsilon_{12}\)). The second source of random effects relates to the sub-pixel variations in emissivity, which are not fully captured in the auxiliary emissivity data. In other words, this variation is due to sub-pixel variability more representative of the actual variability of emissivity on the ground. Assuming the effects due to sub-pixel variability are Gaussian the propagation through the retrieval can be expressed as:  
\[u_{nn,\varepsilon}(x)=\sqrt{\sum_{c=1}^{n}\!\left(\frac{\partial R}{ \partial\varepsilon_{c}}u_{nn}(\varepsilon_{c})\right)^{2}} \tag{9}\]  
where the partial derivatives of the retrieval algorithm (Equation (1)) for each channel with respect to \(\varepsilon_{11}\) and \(\varepsilon_{12}\) are respectively given by Equations (10) and (11):  
\[\frac{\partial R}{\partial\varepsilon_{11}}=-\frac{A_{2}(T_{11}+T_{12})}{( \varepsilon_{11}+\varepsilon_{12})^{2}}+\frac{2A_{3}(T_{11}+T_{12})(3 \varepsilon_{12}-\varepsilon_{11})}{(\varepsilon_{11}+\varepsilon_{12})^{3 }}-\frac{B_{2}(T_{11}-T_{12})}{(\varepsilon_{11}+\varepsilon_{12})^{2}}+ \frac{2B_{3}(T_{11}-T_{12})(3\varepsilon_{12}-\varepsilon_{11})}{( \varepsilon_{11}+\varepsilon_{12})^{3}} \tag{10}\]  
\[\frac{\partial R}{\partial\varepsilon_{12}}=-\frac{A_{2}(T_{11}+T_{12})}{( \varepsilon_{11}+\varepsilon_{12})^{2}}+\frac{2A_{3}(T_{11}+T_{12})( \varepsilon_{12}-3\varepsilon_{11})}{(\varepsilon_{11}+\varepsilon_{12})^{3 }}-\frac{B_{2}(T_{11}-T_{12})}{(\varepsilon_{11}+\varepsilon_{12})^{2}}+ \frac{2B_{3}(T_{11}-T_{12})(\varepsilon_{12}-3\varepsilon_{11})}{( \varepsilon_{11}+\varepsilon_{12})^{3}} \tag{11}\]  
To quantify \(u_{nn,\varepsilon}(x)\) some estimate of the effects due to sub-pixel emissivity variability per channel is required (\(u_{nn}(\varepsilon_{c})\)). Pixel-to-pixel scale emissivity variability is assumed to be categorised by the different land cover types. An estimate of the magnitude of these effects is therefore determined for each land cover class. For land cover classification in the MODIS example, we use the MCD12C1 MODIS land cover product [50; 51] which applies the International Geosphere-Biosphere Programme (IGBP) land cover classification system of 17 land cover types. To quantify this sub-pixel scale emissivity variability we utilised knowledge of the emissivity at finer spatial resolution. In this case, the Global Emissivity Database (GED), derived from ASTER data at 100 m [52] was used to characterise the mean standard deviation within a 1-km MODIS pixel for each IGBP class. We use band 14 of ASTER, centred at 11.3 \(\upmu\)m, to determine our statistics. For many materials the emissivity spectra is much flatter in the traditional split-window windows around 11 \(\upmu\)m and 12 \(\upmu\)m than in the 8-9 \(\upmu\)m region.