## 3 Methods
### Phenological Analysis
#### 3.2.3 Optimal Parameterisation of Phenological Thresholds  
Optimising the threshold in Equation (4) depends on the specific phenophase and the type of crop that is targeted. For example, a lower threshold that effectively identifies the emergence stage in winter wheat might overlook the heading stage of the same crop or could inaccurately detect the emergence phase of another crop. Therefore, it is crucial to calibrate and optimise these thresholds to accurately represent a particular phenological stage. This optimisation process ensures that the thresholds are well-suited to explain the targeted phenophase, such as emergence or heading, for a given crop type.  
To achieve this, we have developed a two-step framework that optimises these thresholds using ground-phenological observations. This framework incorporates a novel optimisation method called Optuna [53] to obtain the optimal threshold for capturing a crop-specific phenological stage. The use of Optuna during the present study was due to its status as a next-generation optimisation framework that leverages advanced techniques such as Bayesian optimisation, gradient-based optimisation, and evolutionary algorithms for hyperparameter optimisation, making it suitable to tackle a variety of optimisation problems quickly and efficiently. The initial step in the proposed optimisation framework reduced the search space by identifying a promising range for the optimal thresholds for both SOS and EOS, examining values in intervals of 0.1 from 0 to 1. Subsequently, the second step fine-tunes and optimises the thresholds within the narrowed range determined in the first step, using steps of 0.01 from the initial threshold point \(-0.1\) to the initial threshold point +0.1. This process is summarised in Figure 3b.  
As mentioned in Section 2.2, the phenological stages linked to each DWD station are regarded as representative of the crops situated within a 5 km radius of the station's location. In light of this and to guarantee the calibration of each threshold using ground phenology observations from DWD, we computed the temporal differences \(\Delta t\) between the observed DOY of DWD and each estimated DOY of the phenophases for the parcels encompassed within the 5 km radius, according to the methodology introduced by [54] and as in Equation (5).  
\[\Delta t=\Delta_{\text{obs}}-\Delta_{\text{est}} \tag{5}\]  
Subsequently, the median values of the temporal difference \(\Delta t\) were calculated from the differences for each field within the station buffers for all investigated years. The median \(\Delta t\) values were used as measures of the accuracy of estimated phenophases. To demonstrate the principle of this optimisation process, we show a graphical representation of it for one DWD station in Figure 3. This procedure is repeated for all DWD stations, and the optimal threshold is returned when the lowest median \(\Delta t\) value is obtained between the DOY derived from the satellite and the observed DOY at all available DWD stations. Furthermore, the optimal threshold is compared between the SOS and the EOS to determine which yields the lower error in terms of \(\Delta t\). This helps in judging which metric to use for each crop-specific phenophase.