# A New Approach to Defining Uncertainties for MODIS Land Surface Temperature
## 2 Framework
### Uncertainty Budget
#### 2.5.2 Level-2
We have exploited this by assuming the coefficient of variability at 11.3 \(\upmu\)m would be similar to that for both the MODIS bands centred at 11 \(\upmu\)m and 12 \(\upmu\)m. The pixel-to-pixel variability in the atmospheric state is assumed to be negligible and therefore no random effects associated with the atmosphere are estimated. The estimation of remotely sensed LST data from TOA radiance measurements observed by a satellite instrument is dependent on both atmospheric and surface states which are subject to errors that are in general shared between nearby pixels. The spatial and temporal scale of these error correlations depends on this effect. It is important to take into account the locally correlated errors otherwise there is a general underestimation of the uncertainty [18]. It may be a reasonable assumption that atmospheric fields are correlated on timescales of the order of days in the temporal domain and hundreds of kilometres in the spatial domain, with associated errors also correlated on these scales. For LST retrieval though, the dominant factor is water vapour, which has been argued as being correlated on much smaller scales of a few minutes and a few kilometres [53; 54]. For the derivation of coefficients from radiative transfer for coefficient based retrieval methods, the minimization of the \(\mathrm{l}_{2}\)-norm in the model fitting is a source of uncertainty. The standard deviation of the difference between the input surface temperatures into the model-fitting and the simulated-retrieved output is an estimate of this uncertainty. ECMWF ERA-Interim daily fields are used both as the source of the water vapour band selection in the LST retrieval (Equation (1)), and in the simulated-retrieved output from the model fitting. For the latter perturbations to the water vapour are used to quantify the uncertainty as a result of imperfect knowledge on the water vapour field. These perturbations are performed within error ranges of water vapour determined from comparison of the ECMWF water vapour with respect to in situ measurements from microwave radiometers at six sites of the Atmospheric Radiation Measurement (ARM) network [55]: (i) NSA_C1, Barrow, Alaska; (ii) NSA_C2, Atqasuk, Alaska; (iii) SGP_C1, Lamont, Oklahoma; (iv) TWP_C1, Manus, Papua New Guinea; (v) TWP_C2, Nauru Island; and (vi) TWP_C3, Darwin, Australia. The differences are shown in Figure 1. Thus, the impact of locally correlated error effects from water vapour on the selection of retrieval coefficients are encompassed the simulations. Figure 1: Comparison of European Centre for Medium-Range Weather Forecasts (ECMWF) against water vapour from the Atmospheric Radiation Measurement (ARM) Network for January 2002 to December 2012. The differences are banded into water vapour bands of 0.0–1.5 cm (first row), 1.5–3.0 cm (second row), 3.0–4.5 cm (third row) and 4.5–6.0 cm (fourth row). Since the retrieval coefficients for GT_MYD_2P are categorised into 13 classes of satellite viewing angle and eight classes of water vapour, the locally correlated atmospheric component of uncertainty is stratified into 104 bands. For each range of satellite viewing angle and water vapour, the uncertainty is estimated as:  
\[u_{mn}(x)=\sqrt{u^{2}{}_{mn,T}(x)+u^{2}{}_{nm,\varepsilon}(x)} \tag{5}\]0  
Land surface emissivity is input to the retrieval Equation (1) and therefore errors in the estimation of this data will propagate forward through the LST retrieval. This emissivity can be assumed to be driven by knowledge of the land classification. The CIMSS estimates are physically derived rather than based on surface type. However, it is not the emissivity estimates themselves that are assumed to be correlated, but the error effects. This is a reasonable assumption because a source of error in the estimation of emissivity will be the response of the retrieval to different materials. Surface types sharing similar materials, as can be categorised by land cover classes, present the emissivity retrieval with similar error structures.