## 3 Methods
### Step 2: Time Series Analysis
#### 3.2.1 Spectral Unmixing  
For each acquisition date--the full dataset for VD 2.0, the reduced dataset for VD 1.0--an ecologically reasonable index for representing vegetation cover was computed based on the bottom-of-atmosphere reflectance spectra. Spectral Mixture Analysis (SMA, see for example, [PERSON] et al. (2018) for an overview) is superior compared to vegetation indices in quantifying sparse vegetation cover over bright background materials ([PERSON] et al., 2018; 83). For comparability with [PERSON] et al. (2019), and as we deem conventional SMA a straightforward means to generate consistent vegetation fractions for long and dense time series, we employed a linear unmixing approach following:  
\[\rho_{b}=\sum_{i=1}^{m}F_{i}\cdot\rho_{i,b}+E_{b}, \tag{1}\]  
wherein band-wise reflectance \(\rho_{\text{b}}\) is explained as the summarized product of \(m\) endmember reflectance spectra \(\rho_{\text{i,b}}\) with their respective fractions \(F_{\text{i}}\). A least-squares optimization was  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Target Date** & \(\Delta\)**D = 0** & \(\Delta\)**D = \(-\)9** & \(\Delta\)**D = \(-\)1** & \(\Delta\)**D = +7** & \(\Delta\)**D = Other** \\ \hline
**Unit** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** \\ \hline
1984-06-03 & 66.60 & 5.37 & 0.00 & 28.02 & 0.02 \\
1986-05-24 & 69.45 & 0.00 & 0.00 & 30.54 & 0.01 \\
1987-06-12 & 65.56 & 0.01 & 0.00 & 34.39 & 0.04 \\
1988-05-29 & 67.08 & 0.03 & 32.86 & 0.01 & 0.02 \\
1989-06-17 & 67.88 & 0.00 & 0.00 & 32.11 & 0.01 \\
1991-05-22 & 67.60 & 0.00 & 0.00 & 32.36 & 0.03 \\
1993-05-27 & 67.66 & 10.85 & 0.00 & 21.48 & 0.00 \\
1994-05-30 & 68.71 & 0.01 & 0.00 & 31.25 & 0.03 \\
1996-06-04 & 64.96 & 0.01 & 0.00 & 35.03 & 0.00 \\
1997-05-22 & 64.03 & 0.02 & 0.00 & 35.95 & 0.00 \\
2000-05-30 & 67.21 & 0.02 & 32.76 & 0.01 & 0.00 \\
2002-05-28 & 67.35 & 30.31 & 0.00 & 0.00 & 2.34 \\
2004-06-10 & 66.00 & 0.02 & 0.00 & 33.96 & 0.01 \\
2005-06-13 & 67.19 & 9.19 & 0.00 & 23.57 & 0.04 \\ \hline \end{tabular}
\end{table}
Table 2: Target dates used for Vegetation Dynamics 1.0 and pixel share of selected observations relative to the target dates (\(\Delta\)D).  
performed to minimize the model error \(E_{\text{b}}\)[83], while both enforcing non-negativity of fractions \(F_{i}\)[84], and honoring the sum-to-one constraint  
\[\sum_{i=1}^{m}F_{i}=1. \tag{2}\]  
For congruency with [PERSON] et al. [42], we selected the same reference endmembers from field measurements (Table 3), i.e., rock (limestone with lichen), soil (Luvisol), photosynthetically active vegetation (PV, _Quercus coccifera_), and spectrometric shade for scaling the fractions for physical interpretability (Equation (3)). The four-endmember model was carefully selected considering the inherent dimensionality of Landsat TM data [42]. The limitation to four endmembers leads to non-photosynthetic vegetation (NPV) falling into the soil fraction component. This does not affect our comparison, given that the interpretation focuses on the trends in photosynthetic active vegetation here. The derived PV fraction was used for all further analyses. For simplicity, we will use \(f\) without subscript to refer to the shade-normalized PV fraction for the remainder of the manuscript.  
\[f=F_{PV}/(1-F_{Shade}). \tag{3}\]