## 3 UVSQ-SAT Data Method and Map Reconstruction of the Variables (Observations and Model)
### Map Reconstruction Method from UVSQ-SAT Observation Time Series  
Time series of the albedo and the OLR at the TOA are obtained from the observations made by the dedicated sensors on-board UVSQ-SAT.  
Physically, each measurement of these time series represents the integral of the signal of interest (OLR, albedo) and depends in practice on a large number of parameters among which the bidirectional reflectance distribution function of the Earth's surface, the opacity of the atmosphere, the spectral and angular sensitivities of the sensors and their FOV. In particular, concerning the opacity, it depends on factors such as aerosol composition, clouds, temperature or pressure.  
In a first approach, we consider that each measurement results from the contribution of a Gaussian distribution \(G(\theta_{i,j})\) of points located at the surface of the Earth. Let the surface of the Earth modeled by a regular grid whose coordinates are expressed with latitude and longitude. Each pixel \((i,j)\) of the grid is defined by its longitude \(\lambda_{i}\) and latitude \(\phi_{j}\) coordinates and by its area \(S_{ij}\) (Equation (5)):  
\[S_{ij}=R^{2}\times\cos(\phi_{j})\ \Delta\lambda_{i}\ \Delta\phi_{j} \tag{5}\]  
where \(\Delta\lambda_{i}\), \(\Delta\phi_{j}\) are size in radians of the pixel at longitude \(\lambda_{i}\) and latitude \(\phi_{j}\).  
Figure 7: Location (latitude and longitude) of UVSQ-SAT observations (red colored dots on the world map) during the month of February 2021.  
The satellite has a location \(\lambda_{sat},\phi_{sat}\) and an altitude \(z_{sat}\). The angle \(\theta_{i,j}\) at the center of the Earth (nadir) between the satellite and the pixel \((i,j)\) is given by Equation (6). The angle \(\alpha_{i,j}\) between the nadir of the satellite and the pixel \((i,j)\) is given by Equation (7). The satellite elevation \(\beta_{i,j}\) (complement of zenith angle) seen by the pixel \((i,j)\) is given by Equation (8).  
\[\theta_{i,j}=\text{acos}[\cos(\phi_{j})\,\cos(\phi_{sat})\,\cos(\lambda_{i}- \lambda_{sat})+\sin(\phi_{j})\,\sin(\phi_{sat})] \tag{6}\]  
\[\alpha_{i,j}=\text{atan}\Bigg{(}\frac{R\,\sin(\theta_{i,j})}{z_{sat}+R\,\left( 1-\text{acos}(\theta_{i,j})\right)}\Bigg{)} \tag{7}\]  
\[\beta_{i,j}=\frac{\pi}{2}-\alpha_{i,j}-\theta_{i,j} \tag{8}\]  
The view angle \(\Omega_{i,j}\) under which the pixel \((i,j)\) is seen by the satellite is given by Equations (9) and (10).  
\[\Omega_{i,j}=S_{ij}\,z^{2}\,\cos(\alpha_{i,j})\,\sin(\beta_{i,j})\qquad\text{ if}\qquad\beta_{i,j}\geq 0 \tag{9}\]  
\[\Omega_{i,j}=0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{ if}\qquad\beta_{i,j}<0 \tag{10}\]  
Figure 8 represents the different angles between the satellite, the nadir, and the observation location. This latter location is seen by the satellite according to a Gaussian distribution \(G(\theta_{i,j})\), which is defined by Equation (11):  
\[G(\theta_{i,j})=\exp\biggl{(}-\frac{\alpha^{2}}{2\sigma^{2}}\biggr{)} \tag{11}\]  
with \(\sigma\), the variance of the Gaussian. \(\sigma\) is related to the FOV via \(\sigma=\text{FOV}/2\). For example, for an average altitude of 540 km and a FOV of 135\({}^{\circ}\), which are the characteristics of UVSQ-SAT, the diameter of the ground track is about 2600 km.  
Figure 8: Visualization of the satellite in orbit, characteristic angles and pixel seen by the satellite.  
The observed flux by UVSQ-SAT \(F_{sat}(\lambda_{sat},\phi_{sat})\) is given by (Equation (12)):  
\[F_{sat}(\lambda_{sat},\phi_{sat})=\frac{\sum_{i,j}\ \Omega_{i,j}\ G_{i,j}\ F_{i,j}}{ \sum_{i,j}\ \Omega_{i,j}\ G_{i,j}} \tag{12}\]  
Finally, to reconstruct the map of what was measured by UVSQ-SAT, i.e., \(F_{sat}(\lambda_{sat},\phi_{sat})\), we calculate the functions \(\Omega_{i,j}\), \(G_{i,j}\) for the satellite track to find the flux \(F_{i,j}\) associated with each pixel.