## 3 Methods
### Step 2: Time Series Analysis
#### 3.2.2 Land Surface Phenology for Vegetation Dynamics 2.0
LSP metrics for each year between 1984 and 2006 were then derived from the interpolated time series using a polar transformation-based methodology based on the concept proposed by [PERSON] et al. [33], partially reproduced here for the sake of readability. The time series is transformed into polar space by converting DOY to radians:  
\[r_{t}=DOY_{t}/365.0\cdot 2\pi. \tag{7}\]  
Polar coordinates are then transformed to Cartesian space (Figure 3) using:  
\[ \begin{array}{l}x_{r,f}=f_{t}\cdot\cos r_{t}\\ y_{r,f}=f_{t}\cdot\sin r_{t}\end{array}  \tag{8}\]  
and the long-term polar average vector \([\langle r\rangle,\langle f\rangle]\) is computed from the average Cartesian coordinates \([\langle x\rangle,\langle y\rangle]\) as:  
\[\langle r\rangle=\left\{ \begin{array}{c}atan2(\langle y\rangle,\langle x \rangle)\qquad\quad atan2(\langle y\rangle,\langle x\rangle)>0\\ atan2(\langle y\rangle,\langle x\rangle)+2\pi\qquad\quad\quad otherwise \end{array} \right., \tag{9}\]  
\[\langle f\rangle=\sqrt{\langle x\rangle^{2}+\langle y\rangle^{2}}. \tag{10}\]  
The long-term average vector points to the direction of the central tendency of annual peak cover. As phenological timing can substantially vary from pixel to pixel due to environmental and biophysical characteristics, the typical start of the phenological year \(\langle\theta\rangle\) is computed as the diametric opposite angle of \([\langle r\rangle,\langle f\rangle]\), which points into the direction of least annual cover:  
\[\langle\theta\rangle=\left\{ \begin{array}{c}\langle r\rangle+\pi\qquad \quad\langle r\rangle<\pi\\ \langle r\rangle-\pi\qquad\quad\quad otherwise\end{array} \right. \tag{11}\]  
This long-term measure of the start of the phenological year is used to structure the time series into phenological years, i.e., annual slices that contain a full season. Besides spatial variability, inter-annual variability may be substantial when analyzing highly volatile, precipitation-driven phenologies, such as in the Mediterranean. As such, we further improved on the Brooks method by fine-tuning the start of the phenological year for each season. For this, we use the initial long-term start of the phenological year \(\langle\theta\rangle\), and re-estimate \(\overline{\theta_{s}}\) for each season \(s\) (Equations (9)-(11)). This is achieved by considering \(\overline{\theta_{s}}\) as the diametric opposite angle to the seasonal average vector \([\overline{r_{s}},\overline{f_{s}}]\), which is computedfrom the average of all Cartesian coordinates \([\overline{x_{s}},\overline{y_{s}}]\) in the given season (i.e., the annual slices between \(\langle\theta\rangle\)). The updated values \(\overline{\theta_{s}}\) are then used to restructure the time series into phenological years with a dynamic, yet reasonably constrained, phenological offset. The observations before \(\overline{\theta_{1}}\) and after \(\overline{\theta_{ns}}\) in the first and last season of the time series, respectively, need to be discarded as they do not represent a full phenological cycle; thus resulting in \(ns=ny-1\) seasons, where \(ny\) is the number of years in the time series. If \(\langle\theta\rangle\) is located in the first half of the year, we did not obtain LSP metrics for the last year; or the 1 st year if otherwise. Once the time series was structured into years, the computation of LSP metrics was straightforward, and a multitude of parameters that describe the timing of phenological key events, seasonal variability, seasonal bimodality, etc. were implemented in FORCE (see [[https://force-eo.readthedocs.io/en/latest/components/higher-level/tsa/format.html#phenology](https://force-eo.readthedocs.io/en/latest/components/higher-level/tsa/format.html#phenology)]([https://force-eo.readthedocs.io/en/latest/components/higher-level/tsa/format.html#phenology](https://force-eo.readthedocs.io/en/latest/components/higher-level/tsa/format.html#phenology)), accessed on 18 September 2021, for a complete list). Our reference studies aimed at selecting images at peak vegetation vigor.