# A New Approach to Defining Uncertainties for MODIS Land Surface Temperature
## 2 Framework
### Uncertainty Budget
#### 2.5.2 Level-2
Thus, for any given land cover class, there may be a mean difference between the assumed and true mean emissivity. This is a locally correlated effect on the scales of variability in the land cover classification. The form of the propagation to L2 uncertainty is estimated as:  
\[u_{mn}(x)=\sqrt{u^{2}{}_{mn,T}(x)+u^{2}{}_{nm,\varepsilon}(x)} \tag{5}\]1  
where \(\mathrm{u}_{\mathrm{loc}}(\epsilon_{\mathrm{c}})\) is the error on the estimation of the surface emissivity per channel. Taking the assumption that the errors are correlated within each land cover class we take as an estimate of these errors the values from [56] who quantified these errors for each land cover class of the IGBP classification system. This length scale based on land cover class is different in space and time to that for the atmospheric effects and thus propagation would be handled differently for each. The emissivity data is temporally interpolated to each date. There may be some additional unknown error that is not accounted due to the temporal estimation, which will be investigated in future work. For estimation of a total locally correlated component, this can be acquired by adding the individual components in quadrature since the different effects from the atmosphere and surface can be assumed to be independent. Large-scale Systematic  
Systematic uncertainties correlated on the global/instrument level directly propagate through the LST retrieval without reduction when averaging. For any given instrument and retrieval algorithm it is assumed that known corrections have been applied by the producers of the satellite-based data product. Corrections can be made at either Level-1 in the calibration or Level-2 in the retrieval process. The unknown remainder is assumed to be an uncertainty in the bias relative to other sources that are more challenging to estimate. One such component is the uncertainty in the radiative transfer model. The basis of this uncertainty is in the ability to accurately simulate radiances for a given channel for a satellite instrument. This could be estimated by performing repeated simulations each with a perturbation applied to a given parameter. Here we have used the estimate from [57], which we assume is of an equivalent magnitude for the radiative transfer model when applied to MODIS filter functions. A further source of error is in lack of knowledge regarding the satellite instrument engineering specifications and calibration. This would be expected to be removed through an instrument calibration model. Calibration results for MODIS from the literature [49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59] indicate bands 31 and 32 are stable with very low systematic errors, particularly for Aqua. Systematic uncertainty can be quantified by way of an estimate over different conditions of the performance with respect to ground-based validation. Other sources of uncertainty not quantified  
In terms of impact on the estimated LST, the largest source of uncertainty can result from misclassification of clear-sky pixels. Here we use the standard cloud flags from the MYD35_L2 product to determine a conservative cloud mask appropriate for LST estimation. In this sense, cloud contamination is defined as the cloudy/part-cloudy pixels which are incorrectly classified as clear-sky. However, this is perhaps the most challenging uncertainty component to quantify and no operational LST algorithm has confronted this to date. A first attempt has been made in [60], which exploits simulated data to estimate an average impact on the LST of misclassification with the probability of any given pixel being misclassified extracted both from previous cloud masking assessment and from the probability of a pixel being subject to cloud contamination. They found there is a dependence on the dominant land cover classification, with uncertainties ranging from 0.09 K for cropland up to 1.95 K over permanent snow and ice. An assumption was made though that different cloud types are both equally likely and have the same impact on the LST.