## 3 UVSQ-SAT Data Method and Map Reconstruction of the Variables (Observations and Model)  
### UVSQ-SAT Data Processing to Obtain Observation Time Series  
#### 3.1.1 Instrumental Equations of the Earth's Radiative Sensors  
The UVSQ-SAT scientific payload ([PERSON] et al., 2016) consists of twelve miniaturized Earth's radiative sensors (ERS) based on thermopiles for monitoring the incoming solar radiation and the outgoing terrestrial radiation. Each face of the UVSQ-SAT satellite has two ERS sensors, which have different optical coatings (carbon nanotubes or optical solar reflector). The properties of the coatings were characterized in lab (solar absorption, hemispherical emissivity, bidirectional reflectance distribution function (BRDF)). The principle of these ERS sensors with their associated coatings is to convert thermal energy into electrical energy. The output voltage (\(V\)) is passively induced from the thermopile proportional to the heat flux (Wm\({}^{-2}\)) through the sensor or similarly the temperature gradient across the thin-film substrate and number of thermocouple junction pairs. The output voltage is conditioned by an electronic unit and converted into analog-to-digital units (ADU) and is read by the UVSQ-SAT onboard computer (OBC) thanks to a serial peripheral interface (SPI) bus, and stored by the OBC. Once data are retrieved, the ADU signals \(S_{ADU}\) are \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Requirements** & **Scientific Relevance** & & & \\ \hline – & Absolute uncertainty & Stability per decade & Spatial resolution & Time resolution \\ \hline TSI & \(\pm 0.54\) Wm\({}^{-2}\) at 1\(\sigma\) & \(\pm 0.14\) Wm\({}^{-2}\) at 1\(\sigma\) & – & 24 h \\ OSR & \(\pm 1.00\) Wm\({}^{-2}\) at 1\(\sigma\) & \(\pm 0.10\) Wm\({}^{-2}\) at 1\(\sigma\) & 10–100 km & Diurnal cycle \\ OLR & \(\pm 1.00\) Wm\({}^{-2}\) at 1\(\sigma\) & \(\pm 0.10\) Wm\({}^{-2}\) at 1\(\sigma\) & 10–100 km & Diurnal cycle \\ \hline \hline
**Requirements** & **UVSQ-SAT Performances** & & & \\ \hline – & Absolute uncertainty & Stability per year & Spatial resolution & Time resolution \\ \hline TSI & – & – & – & – \\ OSR & \(\pm 10\) Wm\({}^{-2}\) at 1\(\sigma\) & \(\pm 5\) Wm\({}^{-2}\) at 1\(\sigma\) & 1000 km & 15 days \\ OLR & \(\pm 10\) Wm\({}^{-2}\) at 1\(\sigma\) & \(\pm 1\) Wm\({}^{-2}\) at 1\(\sigma\) & 1000 km & 15 days \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Scientific requirements for the components of the Earth’s radiation budget. High scientific relevance for a satellites constellation (Terra-F) and UVSQ-SAT expected performances. converted into physical units to express the incident flux measurements \(\Phi_{in}\) on an ERS sensor using the transfer function (Equation (1)):  
\[\Phi_{in}=\left(\frac{S_{ADU}(T_{s},T_{b})-V_{ADU}^{ref}(T_{b})}{N_{samp}}+C_{1}( T_{s},T_{b})\right)\times\frac{1}{G(T_{b})}\times\frac{1}{\text{Sens}(\text{T}_{s})}+C_{2}(T _{s}) \tag{1}\]  
where \(V_{ADU}^{ref}\) is a reference voltage, \(N_{samp}\) is the number of samples for a measurement, \(G(T_{b})\) is the gain of the electronic unit, Sens is the sensitivity of the sensor (calibration in lab), \(C_{1}\) and \(C_{2}\) are corrective offsets, which depend on the temperature of the sensor. In the case of thermopiles, \(C_{2}=-e\sigma T_{s}^{4}\). \(\epsilon\) is the hemispherical emissivity of the thermopile coating and \(T_{s}\) is the ERS sensor temperature. \(T_{b}\) represents the temperature of the sensors electronic board. The total incident flux (\(\Phi_{in}\)) on each ERS sensor is the sum of the solar incident flux, the planetary incident flux (\(\Phi p_{in}\)), and the albedo incident flux (\(\Phi a_{in}\)). The incident solar flux is considered to be known (TSI is measured precisely elsewhere).