## 2 Materials and Methods
### Granger Causality  
To apply the GC method, suppose we have two time series: \(x=[x1,x2,...,xN]\) and \(y=[y1,y2,...,yN]\), where \(N\) represents the length of the time series in days. Consider \(x\) the vegetation variable and \(y\) the environmental surface variable. For [PERSON]'s predictive causality, the goal is to predict the value of \(y(t)\) at a specific timestamp \(t\), based on the historical values of \(x\) and \(y\) in previous timestamps [34]. [PERSON] states that the variable \(x\) is Granger caused (G-Caused) by \(y\), if the autoregressive forecast of \(x\) is superior when additional information is provided by the variable \(y\). We used a Vector autoregressive (VAR) model that considers two variables simultaneously and models the relationships between them, investigating the causal links between \(x\) and \(y\). The mathematical calculation is given as follows:  
\[X(t)=\sum_{j=1}^{p}A_{xx,j}X(t-j)+\sum_{j=1}^{p}A_{xy,j}Y(t-j)+E_{1}(t) \tag{1}\]  
\[Y(t)=\sum_{j=1}^{p}A_{yx,j}X(t-j)+\sum_{j=1}^{p}A_{yy,j}Y(t-j)+E_{2}(t) \tag{2}\]  
Figure 1: Flowchart analysis followed throughout this study. where \(p\) is the maximum allowable lagged observations in the model, and \(A\) are matrices containing the influences of each previous observation on the predicted values \(X\) and \(Y\) at a time \(t\). \(E_{1}\) and \(E_{2}\) are the prediction errors. If the variance of the error \(E_{1}\) is reduced by including \(Y\) in the prediction, then it can be stated that \(Y\) is G-causing \(X\)[35]. The null hypothesis \(H_{0}\), stating that \(Y\) does not G-cause \(X\), is rejected when the causality test is statistically significant (\(p<0.05\)). In any other case, \(H_{0}\) is not rejected, and it cannot be concluded that \(X\) is only G-Caused by \(Y\)[42; 43]. Within VAR modeling, a lag refers to the number of previous observations of each variable that are included as explanatory variables when modeling the current value of that variable [44]. This study used 100 days as the lag values. To enable calculating valid results, note that the two time series data streams have to be normalized to avoid seasonal effects. This will remove the trends in the time series that are inherent with the changes in seasons (e.g., the increment of temperature towards summer, decrease in the LAI throughout the winter months). Thus, the GC calculations will solely be performed with the normalized temporal profiles; they only consider the (minor) fluctuations in both the ELV and VP and purely consider the causality from the changes in one to another, avoiding the larger, seasonal changes. Throughout the presented analysis, the first discrete difference normalization method was used. This method was selected due to its simplicity and ease of application in the processing algorithm. This normalization technique obtains each time series value by using the following logic:  
\[y(t)=p(t)-p(t-1) \tag{3}\]  
where \(y\) is the normalized time series value at a time \(t\). \(p(t)\) is the original time series value, and \(p(t-1)\) is the original value one timestep before \(p(t)\). Thus, the series of the differences, \(p(t)-p(t-1)\), for both the ELV and VP will be used to calculate the GC. The F statistic (F stat) of the Granger Causality analysis is obtained from the F-test, which is computed as the ratio between two summed squared residuals from Equations (1) and (2) (\(RSS_{x}\) and \(RSS_{y}\)). It is calculated as follows, with \(n\) data points and \((p_{X}-p_{Y},n-p_{Y})\) degrees of freedom:  
\[F=\frac{\left(\frac{RSS_{X}-RSS_{Y}}{p_{X}-p_{Y}}\right)}{\left(\frac{RSS_{Y}} {n-p_{Y}}\right)} \tag{4}\]  
As is also used by [PERSON] et al. [45], the hereby presented analysis will take into account the F statistic, to evaluate the GC links between different variables. Throughout Section 3.1, global maps representing the F-statistics are shown for various vegetation-environment causality links. Throughout the analysis, the direction of the GC was only investigated on vegetation from the environment (ELV\(\rightarrow\)VP).