## 3 Instrument Description, Calibration, and Relevance to ERB Measurement Challenges
### Map Reconstruction Method from Satellites Time Series  
Based on the time-series data, we reconstructed monthly mean maps--or maps at specific dates--of radiative fluxes at the TOA over the entire Earth's surface. These maps have a spatial resolution of \(1^{\circ}\) by \(1^{\circ}\) in latitude and longitude, and they represent the spatial distribution of reflected shortwave (SW) fluxes and emitted longwave (LW) fluxes. Analyzing the temporal evolution of these maps enables the assessment of spatial and temporal variations in OSR and OLR. By also accounting for the variability of incoming solar radiation over time, we can estimate how energy accumulates within the Earth's climate system. The incoming solar radiation is derived from TSI estimates provided by the SATIRE-S model [43], which reconstructs solar irradiance based on variations in solar surface magnetism.  
The method used to generate OSR and OLR maps from Uvsq-Sat data relies on a physically informed modeling of the satellite's observation geometry and radiative transfer conditions [39]. Each measurement represents an integral over the Earth's surface of the quantity of interest (OSR or OLR) and depends on a wide range of factors, including the Bidirectional Reflectance Distribution Function (BRDF) of the surface, the optical thickness  
Figure 3: Spatial distribution of observations from Uvsq-Sat between March 2021 and October 2024.  
of the atmosphere (influenced by clouds, aerosols, temperature, and pressure), and the spectral and angular response of the satellite instruments--particularly their FOV.  
The Earth is modeled as a regular grid in latitude and longitude. Each grid cell \((i,j)\) has coordinates \((\lambda_{i},\varphi_{j})\) and an area, as defined in Equation (1).  
\[S_{ij}=R^{2}\times\cos(\varphi_{j})\times\Delta\lambda_{i}\times\Delta\varphi_{j} \tag{1}\]  
The satellite position is described by its coordinates \((\lambda_{\text{sat}},\varphi_{\text{sat}})\) and its altitude (\(z_{\text{sat}}\)). The angle \(\theta_{ij}\) between the satellite and a surface pixel \((i,j)\), measured from the Earth's center, is computed using spherical trigonometry while accounting for the Earth's shape as an oblate spheroid to improve accuracy. From \(\theta_{ij}\), we derive the off-nadir angle \(\alpha_{ij}\) and the elevation angle \(\beta_{ij}\) of the satellite as seen from the surface. The solid angle \(\Omega_{ij}\) under which the satellite views pixel \((i,j)\) is calculated as shown in Equation (2).  
\[\Omega_{ij}=\begin{cases}\dfrac{S_{ij}}{z^{2}}\times\cos(\alpha_{ij})\times \sin(\beta_{ij})&\text{if }\beta_{ij}\geq 0\\ 0&\text{if }\beta_{ij}<0\end{cases} \tag{2}\]  
Each pixel's contribution is weighted by a Gaussian distribution (Equation (3)), which represents the angular response of the satellite's ERS sensors. The ERS exhibits maximum sensitivity at nadir, with a gradual decrease toward the edges of the field of view.  
\[G(\theta_{ij})=\exp\left(-\dfrac{\alpha_{ij}^{2}}{2\sigma^{2}}\right) \tag{3}\]  
where \(\sigma\) is related to the sensor's Field-Of-View (FOV), as given in Equation (4).  
\[\sigma=\dfrac{\text{FOV}}{2} \tag{4}\]  
For Uvsq-Sat, with an altitude around 540 km and a FOV of 135\({}^{\circ}\), the footprint on the ground spans approximately 2600 km in diameter. At a given satellite position, the flux observed by Uvsq-Sat is determined by a normalized and weighted summation (Equation (5)).  
\[F_{\text{sat}}(\lambda_{\text{sat}},\varphi_{\text{sat}})=\dfrac{\sum_{i,j} \Omega_{ij}\,G_{ij}\,F_{ij}}{\sum_{i,j}\Omega_{ij}\,G_{ij}} \tag{5}\]  
where \(F_{ij}\) represents the radiative flux (either OSR or OLR) at pixel \((i,j)\). This formulation accounts for the observation geometry and the ERS sensor's angular sensitivity, enabling reconstruction of spatial flux distributions along the satellite track.