# A New Approach to Defining Uncertainties for MODIS Land Surface Temperature
## 3 Validation
### Uncertainty Validation  
The uncertainty associated with each pixel is an estimate in its own right and thus should be validated as such. The approach here is to compare the standard deviation of ground-based differences between the satellite-derived LST and in situ measurements, and the total uncertainties of such validation pairs. In other words, we test the goodness-of-fit between the uncertainty from in situ validation (\(\sigma_{sat-ground}\)) and the total satellite product uncertainty for each associated matchup (\(\sigma_{total}\)), where \(\sigma_{total}\) is determined from four components:  
\[\sigma_{total}=\sqrt{\sigma_{sat}^{2}+\sigma_{ground}^{2}+\sigma_{space}^{2}+ \sigma_{time}^{2}} \tag{17}\]  
For each matchup, \(\sigma_{sat}\) is the total LST uncertainty for the satellite pixel (\(u_{LST}\)) from Equation (4); \(\sigma_{ground}\) is the uncertainty associated with the ground-based instrumentation; \(\sigma_{space}\) is the uncertainty associated with matching a satellite and ground observation in a spatial context; and \(\sigma_{time}\) is the uncertainty associated with matching a satellite and ground observation in time. Here we validate the uncertainties for each of the seven sites independently for the full year (2011) of matchups. This is a necessity since the characteristics of the spatial matchups between a 1-km satellite pixel and a point source measurement from the in situ sites vary between sites. The uncertainty associated with the ground-based instrumentation (\(\sigma_{ground}\)) is \(\pm 0.3\) K for the seven SURFAD stations. This is determined firstly from the quoted instrument uncertainty of \(\pm 5.0\) Wm\({}^{-2}\)[70] for measuring the radiance with the upwelling and downwelling pyrygeometers. The second component is the uncertainty for each site, which is composed of the input IR global land surface emissivities [41] uncertainty and the fitting equation uncertainty. [71] gives a standard deviation between 0.005 and 0.02 for the emissivity uncertainty per channel, and an RMSE of 0.005 is given for the emissivity fitting uncertainty [66]. An upper value of 0.01 is estimated from uncertainty propagation of the input emissivity uncertainty and the input fitting uncertainty. For the uncertainty associated with matching a satellite and ground observation in a spatial context (\(\sigma_{space}\)) the standard deviation of the clear-sky LST for the \(5\times 5\) 0.01\({}^{\circ}\) region of interest centred on the validation station is estimated. Since any temporal variability between the satellite data and the in situ measurements is minimised to within one minute for any matchup, the associated uncertainty (\(\sigma_{time}\)) is assumed to be negligible. Figure 5 illustrates the comparison between \(\sigma_{total}\) (\(x\)-axis) and \(\sigma_{sat-ground}\) (\(y\)-axis) for each 0.1 K band of \(\sigma_{total}\). If the uncertainty model is accurate the \(\sigma_{sat-ground}\) estimates should fit within and intersect with the 1:1 dashed lines for each \(\sigma_{total}\) 0.1 K band. Values of \(\sigma_{total}\) outside the dashed lines represent an underestimate of \(\sigma_{total}\), with values inside representing an overestimate. The horizontal section  
of the dashed lines represent the component of \(\sigma_{total}\) that cannot be reduced further, specifically the uncertainty due to the calibration on the in situ instrumentation \(\sigma_{ground}\). In general, this appears to be the case, with the fit remaining relatively stable across the bands of \(\sigma_{\text{total}}\). Nevertheless, differences up to 0.5 K exist between \(\sigma_{\text{sat-ground}}\) and \(\sigma_{total}\). These tend to be for higher values of \(\sigma_{total}\) where there is an overestimate in \(\sigma_{total}\), or for lower values of \(\sigma_{total}\) where there is an underestimate in \(\sigma_{total}\), although the frequency of such matchups is generally much lower. There are several possible explanations for this.