# A New Approach to Defining Uncertainties for MODIS Land Surface Temperature
## 4 Propagation of Uncertainties  
Many applications require LST data with associated uncertainties as gridded products with individual pixel values averaged within each grid-cell (GT RBD). This propagation of uncertainties requires knowledge of the spatial and temporal correlation properties. For each component of the uncertainty budget, it is important to understand to what extent an individual pixel shares properties in common with other pixels, both nearby and distant pixels. The way in which individual components propagate is related to whether pixels are correlated within the output grid cell. For \(N_{\overline{z}}\) clear-sky observations, random components when averaged in a grid-cell are reduced by \(1/\sqrt{N_{\overline{c}}}\). The correlation length scales for locally correlated components are driven by changes primarily in atmospheric water vapour and surface emissivity. These do not reduce by \(1/\sqrt{N_{\overline{c}}}\) when averaged providing the averaging is on a scale within the respective correlation length. The determination of this correlation length itself is subject to uncertainty, and will change dependent on both geographical and seasonal scales. The total cell uncertainty is estimated in Equation (18), and an assumption is made here that all averaging is carried out within the respective length scales of the atmospheric and surface components respectively. \[u_{cell}^{2}=\frac{1}{\sum N_{\overline{c}}}\Bigg{(}\frac{\sum_{i=\overline{ z}}u_{mm}^{2}}{\sum N_{\overline{c}}}\Bigg{)}+\frac{\sum_{i=\overline{z}}u_{loc, atm}^{2}}{\sum N_{\overline{z}}}+\frac{\sum_{i=\overline{z}}u_{loc,sfc}^{2}}{ \sum N_{\overline{c}}}+u_{sys}^{2}+\frac{N_{c}Vart_{LST}}{N_{\overline{c}}+N _{c}-1} \tag{18}\]  
where \(N_{\overline{c}}\) and \(N_{c}\) are the number of clear-sky and cloudy pixels respectively in the grid-cell; the overall total number of pixels being the sum of those pixels where an LST estimate is possible. The individual uncertainty components for the cell are calculated from the clear-sky pixels only. The final term in Equation (18) relates to the grid-cell uncertainty as a result of sub-sampling only clear-sky pixels; as introduced in previous studies of averaging sea surface temperature from satellite [31; 72] to account for such fields not being fully observed and thus how well the calculated average represents the surface temperature of the cell if it was fully sampled. It is assumed this concept is equally valid for spatial and/or temporal averaging. Examples of these gridded uncertainties per cell over a single day of MODIS data are illustrated for scrutiny in Figure 6.  
Figure 6: Aqua MODIS global Level-3 gridded LST data and associated total uncertainty estimates for 1 June 2010: (**a**) daytime LST composite; (**b**) nighttime LST composite; (**c**) total uncertainty for the daytime LST composite; and (**d**) nighttime LST composite. The sub-sampling uncertainty is based on an estimate of the grid-cell variance \(Var_{LST}\). This estimate is taken as the variance of all the clear-sky observations within the grid-cell. For grid-cells where the proportion of clear-sky pixels is high relative to the total number of pixels within the grid-cell the variance is assumed to be representative. Where relatively few clear-sky pixels exist the estimate of \(Var_{LST}\) may become unstable. While it may be possible to overcome this difficulty by deriving a climatology of cell variances this would need to be defined for every possible spatial-temporal framework, thus limiting flexibility. A simple solution is to define a minimum value of \(Var_{LST}\) when the proportion of clear-sky pixels falls below a given threshold value. Rational values for the minimum estimate and threshold value can be obtained empirically through an assessment of completely clear-sky grid-cells. Here this was performed over a sample of a full year of MODIS Aqua data.