# Near-Surface and High-Resolution Satellite Time Series for Detecting Crop Phenology
## 2 Materials and Methods
### Methodology
#### 2.2.2 Crop Phenological Modeling  
For both PhenoCam and PlanetScope imagery, the pre-processed time series was fitted using the Beck double logistic model [64]. The double logistic model had been widely utilized to model the vegetation phenological dynamics, as the parameters of the model  
Figure 3: The influence of dormant season vegetation on the PhenoCam GCC time series curves. (**a**) The dormant season vegetation is before the growing season; (**b**) The dormant season vegetation is after the growing season. The GCC time series was pre-processed with off-season fluctuations replaced by the median dormant season values.  
could be linked with the characteristic phenological process [24]. The model could also capture the relatively short growing season of crops without overestimating the growing season duration [19]. Built on the double logistic model, the [PERSON] model was employed in this study to model the pre-processed seasonal time series as a function of time using six unique parameters (Equation (2)).  
\[VI(t)=wVI+(mVI-wVI)\times\left(\frac{1}{1+e^{(-a_{1}\times(t-a_{2}))}}+\frac{1}{1 +e^{(b_{1}\times(t-b_{2}))}}-1\right) \tag{2}\]  
Here, \(VI(t)\) denotes the modeled value of a vegetation index (e.g., GCC or NDVI) at time \(t\). \(wVI\) is the baseline value of the vegetation index during the dormant season and is determined through the imagery pre-processing (Section 2.2.1). \(mVI\) is the maximum value of the vegetation index throughout the growing season. \(a_{2}\) is the inflection point in the curve upward (or greening) direction and \(a_{1}\) controls the corresponding rate of curve increase at \(a_{2}\). Similarly, \(b_{2}\) is the inflection point in the curve downward (or senescing) direction and \(b_{1}\) denotes the corresponding rate of curve decrease at \(b_{2}\). The parameters in the Beck model were estimated using the iterative non-linear least squares.  
The Beck double logistic model maintained two unique properties to improve the phenological modeling process. First, the Beck model replaced the dormant season vegetation index values lower than a pre-defined dormant value with that pre-defined one to reduce the impact of snow, ice, cloud, etc. In this study, the pre-defined dormant value was the median dormant season value of a vegetation index identified in image pre-processing (Section 2.2.1). Second, the Beck model incorporated a weighting factor that placed lighter weights on the values overestimated by the fitted model to diminish the influence of adverse weather conditions (e.g., sub-optimal illumination conditions) negatively biasing the observed values.  
We also tested two other generalized double logistic models (e.g., Elmore and Klosterman models) with additional parameters considered for tracking the crop seasonal phenological dynamics [35; 65]. The Elmore double logistic model could account for the declining trend of vegetation greenness in the summer months, while the Klosterman double logistic model took into account the varying changing rates near the asymptotes of the logistic functions as well as the changes of observation values during both dormant and summer periods. Despite the more generalized functions, these two models had degraded performance in our study sites compared to the [PERSON] double logistic model, partly due to the lack of summer green-down process in certain agricultural sites or the increasingly large number of parameters to be optimized. Thus, the [PERSON] double logistic model was employed in this study as the functional representation of phenological time series for retrieving the crop phenological transition dates.