## 2 Materials and Methods
### Methods
#### 2.5.2 Time-Series Phenological Modelling  
Following the phenological pre-processing of the time-series curve, the next step involves modeling the seasonal smoothed phenological curve to elucidate its developmental trajectory and capture rapid changes during the growing season. This modeling phase is depicted in Figure 2, transitioning from the smoothed dashed lines to the solid yellow or red lines. For this modeling task, the study opted for double logistic functions, a type of curve-fitting-based phenological model, owing to their widespread utilization  
Figure 2: The variations observed in four distinct phenological characterization methods employed in this study. The dots represent the original crop measurements, while the dashed lines illustrate the fitted lines following the pre-processing step. Solid lines represent the phenological modeled curves, with winter wheat in yellow and maize in red. Vertical dashed lines in light blue indicate the auto-recognized rotation points between winter wheat and maize. (**a**) Displays transitions extracted using the threshold-based (TR) method for winter wheat (yellow) and the derivative-based (DB) method for maize (red). (**b**) Showcases transitions extracted using the curvature-based (CU) method for winter wheat (yellow) and the Gu (GU) method for maize (red). in phenological monitoring and superior performance compared to other smoothing algorithms [31]. Furthermore, double logistic functions exhibit better modeling performance for relatively short and rapid-growing seasons, which are common in agricultural crops, when contrasted with alternative algorithms like Fourier analysis and asymmetric Gaussian functions [72]. In this study, two variants of double logistic functions, namely the Beck and Gu models, were employed, alongside the smoothing spline function as a simpler alternative for comparison. The [PERSON] model, which establishes a minimal baseline for the growing-season curve, has been shown to mitigate the influence of spurious measurements during the fitting process [72]. Beck model characterizes the temporal variations of the growing season using six parameters:  
\[f(t)=a_{base}+(a_{max}-a_{base})\times\left(\frac{1}{1+e^{(-m_{1}\times(t-m_{ 2}))}}+\frac{1}{1+e^{(n_{1}\times(t-n_{2}))}}-1\right) \tag{2}\]  
In the equation, \(t\) represents DOY, and \(f(t)\) represents the fitted value at \(t\). Additionally, \(a_{base}\) signifies the minimum baseline or off-season value, while \(a_{max}\) represents the maximum value during the growing season. Double logistic functions posit that two piecewise logistic functions of time, one for the upward and the other for the downward trend, can effectively depict the phenological development of the monitored plant [31]. Here, \(m_{2}\) and \(n_{2}\) denote the inflection points corresponding to the onset and conclusion of the ascending and descending portions of the pre-processed curve, while \(m_{1}\) and \(n_{1}\) denote the rates of increase and decrease at \(m_{2}\) and \(n_{2}\), respectively. The Gu model, in contrast, offers greater generality compared to the Beck model, encompassing additional parameters for accommodating diverse phenological curve fitting requirements [73]. These supplementary parameters enhance the model's robustness and adaptability in tracking phenological trajectories. However, the broader set of parameters may pose challenges in phenological modeling when working with lower-quality original data. The function for the Gu model is as follows:  
\[f(t)=a_{0}+\frac{a_{1}}{\left(1+e^{-\frac{1-m_{2}}{m_{1}}}\right)^{m_{4}}}- \frac{a_{2}}{\left(1+e^{-\frac{1-m_{2}}{m_{1}}}\right)^{n_{4}}} \tag{3}\]  
In the provided equation, \(t\) represents the DOY, and \(f(t)\) signifies the fitted value at \(t\). The parameters \(a_{0}\), \(a_{1}\), \(a_{2}\), \(m_{1}\), \(m_{2}\), \(m_{4}\), \(n_{1}\), \(n_{2}\) and \(n_{4}\) are the empirical parameters specific to the Gu model. These parameters are going to be literately set to fit the smoothed time-series data.