## 2 Materials and Methods
### Random Forest Model  
To establish quantitative relationships between the baseline sensor (L8) and peer sensors (S2A and ASTER) NDVI, we constructed an RF as a multivariate non-parametric regression method [70; 71], using the peer sensor's NDVI values, unique common geographic location (Northing (m) and Easting (m)) and month of the year from NDVI time series as predictor variables and baseline sensor's NDVI values as target variable.  
The RF algorithm builds many (typically between 100 and 1000) regression trees based on random subsamples, with replacement, from the training data set. This process is known as bootstrap aggregation, or bagging, and is used to train the individual trees with the results of the ensemble produced by computing the mean prediction (for regression) or the majority vote (for classification) [71; 72]. At each node in a tree, a subset of predictor variables is selected at random and the optimal binary split is computed using the training data subsample and a metric known as \"purity.\" During this procedure, all candidate splits are evaluated to determine which one maximizes the purity of the resulting branch. Residual sum of squares (RSS), shown in Equation (2) is used as the splitting criteria for regression trees.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sensor** & **Subsystem** & **Band** & **Spectral Range (\(\mu\)m)** &
\begin{tabular}{c} **Signal to Noise** \\ **Ratio (SNR)** \\ \end{tabular} & **Spatial Resolution (m)** & **Swath Width (km)** \\ \hline Landsat 8 & NIR & 5 & 0.85–0.87 & 204 & 30 & 185 \\ OLI & Red & 4 & 0.63–0.68 & 227 & 20 & \\ Sentinel-2A & NIR & 8A & 0.86–0.88 & 72 & 20 & \\ MSI & Red & 4 & 0.65–0.68 & 142 & 10 & 290 \\ ASTER & NIR & 3N & 0.78–0.86 & 202 & 15 & 60 \\ Red & 2 & 0.63–0.69 & 306 & 15 & 60 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of bands for NDVI computation from Landsat 8, Sentinel-2A, and ASTER.  
\[\text{RSS}=\sum left{(y_{i}-y_{L})}^{2}+\sum right{(y_{i}-y_{R})}^{2} \tag{2}\]  
where, \(\sum left{(y_{i}-y_{L})}^{2}\) and \(\sum right{(y_{i}-y_{R})}^{2}\) refer to the left and right nodes of the binary split, respectively.  
The classification and regression trees in an RF grow to their maximum purity although they are commonly constrained by a maximum depth parameter. A key element in an RF is that each individual tree is exposed to only a randomly selected subset of the training data and must produce a prediction based on this limited information. Thus, it takes many trees to assemble a functional RF which also makes it resistant to overfitting. The number of trees is usually determined by an optimization procedure designed to calculate the minimum number of trees (for computational speed and efficiency) to achieve a desired result. RF is appealing in this application because of its generally accurate predictions and its calculation  
Figure 4: Schematic flowchart of the virtual constellation process.  
of feature importance that can act as a means for evaluating the influence of each feature on the algorithm [72]. Furthermore, RF models provide the user with additional randomness and computational efficiency during the bagging process. Splits within standard decision trees are typically determined by choosing an optimal split from all predictor variables. In contrast, an RF randomly selects a subset of the predictor variables, the number of which can be set as a parameter by the user. Subsequently, the best possible split is determined and used for that node. Further details regarding RF can be found in [71]. We also suggest that readers look at the figures of [54] to see an illustration of an example ensemble containing three trees and also a detail of one tree from the ensemble.