# A New Approach to Defining Uncertainties for MODIS Land Surface Temperature
## 2 Framework
### Uncertainty Budget
#### 2.5.2 Level-2  
For Level 2, we present the three-component model in detail, using MODIS as an example to demonstrate each component. Unlike the uncertainty model implemented for SST and Ice Surface Temperature (IST) in the projects introduced above, for LST the locally correlated component is divided into two distinct sub-categories: (i) atmosphere; and (ii) surface. First, we revisit the LST retrieval algorithm (Equation (1)). This can be simplified as:  
\[x=R(X,\,\beta)+\,e \tag{2}\]  
where \(R\) is the retrieval which depends on the state vector \(X(T_{i},\,\varepsilon_{i},\,\theta)\) and \(\beta\) which is the array of retrieval coefficients: \(A_{j},\,B_{j}\) (\(j=1,2,3\)) and \(C\) which depend primarily on the atmospheric state, specifically water vapour (\(wv\)). For an actual retrieval of LST for any given pixel (\(x\)) we do not know the exact values of each state of brightness temperatures (\(T_{i}\)) for each channel \(i\), surface emissivity (\(\varepsilon\)), water vapour (\(wv\)), and satellite zenith angle (\(\theta\)). In most cases, we can make the assumptionthat the satellite zenith viewing angle (\(\theta\)) is a known value for each pixel since any relatively small mis-pointing produces a very small airmass uncertainty; pragmatically, information on any deviation is not generally provided with Level-1 data. In the derivation of LST for any given pixel according to Equation (1) the exact input states, and retrieval coefficients can thus be replaced with the inaccurate input states (\(T_{11}+e_{T11}\), \(T_{12}+e_{T12}\), \(\varepsilon_{11}+e_{\varepsilon 11}\), \(\varepsilon_{12}+e_{\varepsilon 12}\)) and uncertainty in the knowledge of the atmosphere (\(wv+e_{uv}\)). A characterisation of the uncertainty budget for each pixel \(u_{total}(x)\) can thus be given as:  
\[u^{2}{}_{total}(x)=E\left[ \begin{array}{c}(R\Big{(}\begin{array}{c}T_{11} +e_{y_{11}},T_{12}+e_{y_{12}},\ \varepsilon_{11}+e_{\varepsilon_{11}},\ \varepsilon_{12}+e_{ \varepsilon_{12}},\ wv+e_{uv}\end{array}\\ -F\left(\begin{array}{c}T_{11},T_{12},\ \varepsilon_{11},\ \varepsilon_{12},wv \end{array}\right))^{2}\big{|}y_{11},y_{12},\ \varepsilon_{11},\ \varepsilon_{12},wv\end{array} \right] \tag{3}\]  
where given the observations of brightness temperatures (\(y_{11},y_{12}\)), emissivities (\(\varepsilon_{11}\), \(\varepsilon_{12}\)) and water vapour (\(wv\)), \(R\) is the retrieval with the inaccurate input states and \(F\) is the retrieval with the accurate input states. This total pixel uncertainty budget is a combination of the random, locally correlated and (large-scale) systematic components and can be expressed as such. Assuming each component is independent of each other these components can be added together in quadrature:  
\[u_{total}(x)=\sqrt{u^{2}{}_{nm}(x)+u^{2}{}_{loc}(x)+u^{2}{}_{sys}(x)} \tag{4}\]  
The uncertainty analysis thus takes into account the expected performance of the retrieval algorithm under varying surface and atmospheric conditions, with these uncertainties categorized into the three-component model. The derivation of each of these is presented next. Random  
Truly random uncertainties are uncorrelated. Here we identify two distinct random effects: (i) radiometric noise per channel since it is uncorrelated between individual pixels; and (ii) effects from sub-pixel variability in channel surface emissivity. Assuming these are independent of each other, the total random uncertainty per pixel \(u_{mn}(x)\) is:  
\[u_{mn}(x)=\sqrt{u^{2}{}_{mn,T}(x)+u^{2}{}_{nm,\varepsilon}(x)} \tag{5}\]  
where \(u_{mn,T}(x)\) is the random uncertainty due to effects from radiometric noise, and \(u_{mn,\varepsilon}(x)\) is the random uncertainty due to effects from surface emissivity. The random component of the Level-1 channel uncertainty \(u_{nm}(T_{i})\) is the expected radiometric noise for channel \(i\). Taking the MODIS instrument on Aqua as our example, the typical respective NEDTs for the 11-um (band 31) and 12-um (band 32) split-window channels are 0.02 K and 0.03 K, respectively [9].