## 2 Data and Methods
### Physical Mono-Window Model  
We applied a PMW model to the Meteosat time series described in Section 2.1. The PMW model used here is based on radiative transfer runs. Radiative transfer models can be used to estimate the upward and downward atmospheric path radiance (L\({}^{\mathrm{I}}\), L\({}^{\mathrm{I}}\)) and the atmospheric transmittance (\(\uptau\)) in the thermal infrared for a specific atmospheric profile [18]. The downward atmospheric path radiance (L\({}^{\mathrm{I}}\)) is the hemispherically-averaged downward radiance. Approximating the Earth's surface as a Lambertian emitter-reflector and neglecting atmospheric scattering, the radiance L\({}_{\mathrm{c}}\)(\(\uptheta\)), recorded in channel c of a sensor onboard a satellite observing the Earth's surface under view zenith angle \(\uptheta\) may be written as (e.g., [25]):  
\[\mathrm{L_{c}}(\uptheta)=\varepsilon_{\mathrm{c}}\mathrm{B_{c}}(\mathrm{T_{s }})\uptau_{\mathrm{c}}(\uptheta)+\mathrm{L_{c}^{\mathrm{I}}}(\uptheta)+ \mathrm{L_{c}^{\mathrm{I}}}(1-\varepsilon_{\mathrm{c}})\uptau_{\mathrm{c}}( \uptheta) \tag{1}\]  
where \(\varepsilon_{\mathrm{c}}\) and \(\mathrm{T_{s}}\) denote land surface emissivity and LST, respectively. The calibrated Planck function \(\mathrm{B_{c}}(\mathrm{T_{s}})\) provides the radiance emitted by a blackbody at temperature \(\mathrm{T_{s}}\) in channel c. The parameters \(\uptau_{\mathrm{c}}\), \(\mathrm{L_{c}^{\mathrm{I}}}(\uptheta)\) and \(\mathrm{L_{c}^{\mathrm{I}}}\) in Equation (1) are the corresponding surface to top of the atmosphere (TOA) transmittance and the atmospheric upward and downward radiances, respectively. These three parameters can be estimated based on the atmospheric humidity and temperature profiles. For a channel of finite spectral band width, the calibrated Planck function in the frequency domain may be approximated as:_Remote Sens._**2015**, \(7\)  
\[\mathrm{B_{c}(T_{s})}\approx\frac{c_{1}v_{c}^{3}}{\exp\left(\frac{c_{2}v_{c}}{ \alpha T_{s}+\beta}\right)-1} \tag{2}\]  
where c1, c2 are constants and \(\alpha\), \(\beta\) and \(v_{c}\) depend on the spectral characteristics of the channel to be used. Inverting Equations (1) and (2) (e.g., [21,25]), the thermal radiance Lc(\(\theta\)) measured at the sensor level can then be used to estimate LST:  
\[\mathrm{T_{s}}\approx\left(\frac{c_{2}v_{c}}{\ln\left(\frac{c_{1}v_{c}^{2} \tau_{c}(\theta)\varepsilon_{c}}{L_{c}(\theta)-L_{c}^{\dagger}(\theta)-L_{c}^{ \dagger}(1-\varepsilon_{c})\tau_{c}(\theta)}+1\right)}-\beta\right)/\alpha \tag{3}\]  
The PMW LST in this study was calculated with Equation (3) for SEVIRI 10.8-\(\upmu\)m clear-sky TOA brightness temperatures described in Section 2.1, together with surface emissivities (\(\varepsilon_{c}\)) taken from the operational LSA SAF dataset [26]. Values of Lc(\(\theta\)), Lc and \(\tau_{c}(\theta)\) were obtained via the Radiative Transfer for the Television Infrared Observation Satellite Operational Vertical Sounder code (RTTOV, Version 11.2), which is a fast radiative transfer model used operationally at the ECMWF [27]. RTTOV is significantly faster than the commonly-used Moderate Resolution Atmospheric Transmission (MODTRAN) line-by-line radiative transfer code [28]. It uses pre-computed transmittance look-up-tables (LUTs) calculated from a spectroscopic database [29]. PMWs require radiative transfer runs during the satellite data processing. For large data processing, it is hence crucial to run a fast radiative transfer model. [PERSON] [30] has recently compared simulated MODTRAN and RTTOV TOA brightness temperatures and reports an overall bias of about 0.2 K in the SEVIRI spectral range, which is close to the SEVIRI instrumental noise. RTTOV runs performed in this study used atmospheric profiles (temperature and specific humidity) from the ECMWF ERA-Interim reanalysis dataset as input [31], which are available 6-hourly at a spatial resolution of about 75 km.