# Evaluation of Orbital Drift Effect on Proba-V Surface Reflectances Time Series
## 3 Data and Methods
### Method for Real Observations Analysis
#### 3.3.2 BRDF Normalization Procedure  
For each Proba-V spectral band and each BELMANIP-2 site, the BRDF parameters of the RTLSR model, provided in Equation (4), were fitted to the cloud-free TOC observations available within a moving window of \(+/-10\) days centered on the considered day. The procedure is repeated for each day of observation. The cloud-free TOC observations are averaged within a box of 5 \(\times\) 5 km around the considered location. The averaging in the spatial domain, which is justified by the homogeneity of the considered sites [34], allows for consolidating the time series, minimizing impact of isolated cloud patches within or nearby the central pixel and smoothing out residual atmospheric contamination in the TOC products at 1 km.  
The gathering of TOC observations during a period of \(N_{days}\) is generally adopted for retrieving the parameters of the BRDF semi-empirical model, under the assumption that surface properties do not change during the gathering period. The choice of the length of this period is ultimately a trade-off between the need of extending the temporal interval, to ensure enough cloud-free observations, and the potential impact of surface variability during the chosen interval. The period considered within this study (\(N_{days}=20\)) is slightly larger than the one used for MODIS Albedo product (16 days) and the assumption of surface stability may fail over dynamic land cover class, notably over cropland.  
The BRDF fitting procedure works as a standard least-square error minimization procedure. The state vector, to be retrieved for each considered spectral band, consists of the triplets of BRDF parameters \([f_{Iso}(\lambda),f_{Vol}(\lambda),f_{Govic}(\lambda)]\). The measurement vector \(y_{j}\) has dimension \(m\), which corresponds to the available cloud-free observations retained in the gathering period of \(N_{days}\). The forward model operator, representing our physical understanding of the measurements, is the RTLSR semi-empirical model of Equation (4). This yields, for a given spectral band \(\lambda\), to the forward model vector: \(R_{j}\), which is function of the angular configuration (\(\phi_{j},\ v_{j},\ \varphi_{j}\)) for the considered cloud-free observation, with \(j=1,\ldots,m\). The inversion process in its linear form consists of minimizing the following cost function, using vector notation: \(J=(y-R)^{T}(y-R)\). A standard least square fit procedure was adopted, using a bounding box of \([0,1]\) for constraining the inversion of BRDF parameters to physically meaningful values. The retrieval is performed only when at least 5 clear observations are available within the gathering period.  
The BRDF-normalized surface reflectances for a given band were estimated using Equation (4) with input the three retrieved parameters and considering a standard observation geometry, defined as nadir-viewing with sun zenith angle at 45\({}^{\circ}\), and a relative azimuth angle of 0\({}^{\circ}\), particularly:  
Figure 3: The figure shows the location of BELMANIP-2 sites [34] used for assessing the multi-temporal consistency of Proba-V TOC products at global scale and for different surface types and environmental conditions. The prevalent global land cover class is also shown with colors indicating the different classes as derived from ESA CCI Land Cover Map V2.0 [35].  
\[R_{Norm}(45^{\circ},\,0^{\circ},\,0^{\circ},\,\lambda)=f_{iso}^{fit}(\lambda)+f_{ Val}^{fit}(\lambda)K_{Val}(45^{\circ},\,0^{\circ},\,0^{\circ},\,\lambda)+f_{ Geo}^{fit}(\lambda)K_{Geo}(45^{\circ},\,0^{\circ},\,0^{\circ},\,\lambda), \tag{5}\]