## 4 Data Processing
### Modeling Phenology
Of those four models, the logistic function has been used most frequently in phenology (and other ecological) studies (in [21; 34; 88]). A smaller number of investigations explored the suitability of the hyperbolic tangent [100; 101; 102], probably because of its similarity with the logistic function. Rather fewer researchers seem to have considered the Gaussian [103; 104] or the sine functions. This paper aimed in part to explore which of those formulations might be more appropriate or more advantageous for this purpose. These four mathematical models are described in detail in Appendix B. The shape of each elementary S-shaped function is determined by four independently adjustable parameters, typically to specify the (Julian) date and value of the start and of the end of the simulated fluctuation. Imposing a continuity condition between the two functions of each pair reduces the number of parameters to be adjusted to seven. The process of fitting a double S-shaped model to the data reduces to an inversion procedure, where the goal is to optimize a goodness of fit criterion: it is typically implemented as an iterative algorithm to minimize a figure of merit function, such as the root mean square difference between the model and the observed values (\(\chi^{2}\)). When this condition is realized, the values of the model parameters are deemed to optimally describe the seasonal evolution of the simulated geophysical variable, and the model, together with these newly established parameters, can be used to generate a complete time series, at whatever frequency is required. Data points can be assigned weights during this inversion process, to allow those with smaller uncertainties to have more influence on the outcome than points with larger uncertainties. The inverse of the estimated standard deviation provided by the JRC-TIP algorithm for each FAPAR value was used as the weight in this investigation. The iterative inversion process is terminated when the value of \(\chi^{2}\) changes by less than \(10^{-3}\) between successive iterations, i.e., when there is little benefit to be gained by searching for another minimum in the current neighborhood. This approach requires at least four FAPAR values within each of the growing and decay phases, and more than seven values over the entire vegetative season to ensure that the number of measurements exceeds the number of model parameters to be determined [105]. The average number of valid FAPAR observations across all pixel locations in the FAPAR record is 288, which, over a period of 18 years, yields around 16 observations per season. However, those are very unevenly distributed, with many more data points during the dry season (under clear skies) than during the rainy season, as was shown in Section 4.2.1. In fact, the process of inverting the models against those data can fail or lead to unrealistic results whenever there are fewer than four observations during the growing phase, which is always the most cloudy. In addition to systematically fitting each of these double S-shaped models to each vegetative season, a combined approach was also explored, whereby the best model for each individual season is selected, to increase the number of successful model inversions and decrease the average \(\chi^{2}\) statistic. Once a parametric double S-shaped model has been successfully fitted to the remote sensing measurements, classical techniques based on an inspection of the first and second derivative of this model can be used to determine the start, end, and length of the growing season (in [22; 106], among many others).