# A New Approach to Defining Uncertainties for MODIS Land Surface Temperature
## 2 Framework
### Coefficient Generation  
We utilise an approach to coefficient generation, trading off spectral resolution in the radiative transfer model against fast processing of a large number of profiles sufficient to characterise the full range of atmospheric states. The optimum model for this approach is RTTOV (Radiative Transfer Model for TOVS satellite). RTTOV is a fast model for nadir-viewing passive infrared and microwave satellite radiometers, spectrometers and interferometers [42; 43]. A comparison of simulated and observed TOA radiances for a selection of known atmospheric and surface states between the RTTOV model and the full line-by-line Reference Forward Model ([[http://www.atm.ox.ac.uk/RFM/](http://www.atm.ox.ac.uk/RFM/)]([http://www.atm.ox.ac.uk/RFM/](http://www.atm.ox.ac.uk/RFM/))) show very small biases [44]. The version of RTTOV used here is 10.2. This is appropriate even for wide-swath instruments, where radiances can be modelled for varying satellite zenith angles, and has successfully been used as such [45]. In our derivation of the coefficients we stratify based both on water vapour and satellite zenith angle. For the latter case we set different viewing angles in the RTTOV configuration. In other words coefficients are determined by setting different viewing angles in the RTTOV configuration up to 60-65\({}^{\circ}\) for the end of a MODIS scan line. In [45] however, there are higher uncertainties for larger viewing angles and more moist atmospheres. While we attempt to represent this by stratifying the errors in the uncertainty model into in bands of water vapour and satellite zenith angle, we acknowledge there may be greater uncertainty in the quantification of the errors at the extremes. Future work will look into better quantifying any potential increased errors at the larger viewing angles.  
For the generation of the retrieval coefficients for category combination of viewing angle and water vapour, vertical atmospheric profiles of temperature, ozone, and water vapour, surface and near-surface conditions (temperature, wind, humidity), and the surface emissivities are input. The channel spectral response functions of the instrument are also part of the input to the radiative transfer model. ECMWF ERA-Interim [46] daily and invariant fields are used for the atmospheric profile data and surface/near-surface inputs. ERA-Interim uses 60-level atmospheric model data on a T255 resolution grid four times each day. For surface emissivity input to the radiative transfer model the UWIREMIS database is used.  
The distribution of the input profiles should be representative of the range of atmospheres and surfaces encountered globally. Globally, water vapour column amounts will change with season over different latitudinal bands; however, land mass and surface characteristics plus the amount of ocean surface mean that variability is not uniform throughout the tropics, mid-latitudes and polar regions, but is more regionally specific [47; 48]. A uniform random sampling distribution is used to select clear-sky profile data spatially with a temporal sampling strategy to ensure sufficient coverage across seasons and years. This sampling is presented in detail in [1]. Briefly, profiles are selected from the 15 th day of every month from 10 years of data using the two profiles temporally closest to the daytime and nighttime local overpass times of the satellite. Using a land cover map it is ensured that every land cover type across all continents and latitudinal bands are sufficiently represented in the uniform random sampling distribution.  
The process of generating the retrieval coefficients for Equation (1) involves linear regression applied to the simulated BTs and corresponding skin temperatures. A set of coefficients is generated for each combination of satellite zenith angle band and water vapour band.