## 2 Materials and Methods
### Land Surface Reflectance and Vegetation Index Data  
We used the surface reflectance data from MODIS (MOD09A1), Landsat (7 ETM+ and 8 OLI/TIRS), and Sentinel-2/A-2/B, which are accessible on the Google Earth Engine (GEE) platform [82]. For each flux tower site, we selected one MODIS pixel, one Landsat pixel, and one Sentinel-2 pixel centered on the tower coordinates.  
The MOD09A1 Collection 6 product [83] provides surface reflectance at a 500 m spatial resolution and 8-day temporal resolution. We employed the Google Earth Engine (GEE) platform [82] to calculate the enhanced vegetation index (EVI) [84] and land surface water index (LSWI) [85] using the surface reflectance data (see Equations (1) and (2)) and assessed the quality of individual observations via the quality band.  
\[\text{EVI}=2.5\times(\text{NIR}-\text{Red})/(\text{NIR}+6\times\text{Red}- 7.5\times\text{Blue}+1) \tag{1}\]  
\[\text{LSWI}=(\text{NIR}-\text{SWIR})/(\text{NIR}+\text{SWIR}) \tag{2}\]  
For Landsat 7 ETM+ and Landsat 8 OLI sensors, atmospherically corrected surface reflectance data at a 30 m spatial resolution were used to calculate the EVI and LSWI. We assessed the individual observations' data quality using cloud, shadow, water, and snow masks [86]. The blue band was used for an additional quality check by detecting the observations with cloudy and water pixels [87].  
Sentinel-2 provides optical data at high spatial resolutions (10 m, 20 m, 60 m) and a 10-day temporal resolution [88]. S2-A (launched June 2015) and S2-B (launched March 2017) have 13 spectral bands, including visible and NIR bands at a 10 m spatial resolution and red-edge and SWIR bands at a 20 m spatial resolution [89]. We used the 10 m S2 orthorectified atmospherically corrected surface reflectance available on GEE. We assessed the quality of individual observations using the cloud bit mask, cirrus bit mask, and blue band [87].  
For each eddy flux tower site, the time-series Landsat and Sentinel-2 observations were combined to form one time series of collated data. Both systems provide a 12-bit radiometric resolution with similar reflective wavelengths [88; 90; 91] and information sensed over the same areas. The similarities between the Landsat and Sentinel-2 spectral resolutions facilitates the combined use of their datasets in several different ways, including data fusion, as reported in previous studies [92]. Landsat and Sentinel-2 were combined without resampling the Landsat 30 m data into 10 m data, since the pixel chosen was within the tower's footprint and there were no changes in the crop management landcover typewithin the 30 m Landsat pixel. In our analysis, we integrated various vegetation indices derived from different optical datasets and, in cases where there were observations with overlapping dates, we selected a single observation for each date. This selection was based on the data quality (maximum NDVI), as in past studies [90; 92; 93]. Ultimately, the gaps in the combined Landsat/Sentinel time-series data were filled in, but the data were not subjected to smoothing. This decision was possible as there was at least one good-quality observation per week, and we ran the model at weekly intervals.  
We analyzed the time-series data of the EVI, LSWI, and GPP at the sugarcane sites to identify the start, end, and length of the sugarcane growing season (SOS, EOS, GSL). This information was determined using the VI data with the thresholds of EVI \(>\) 0.1 and LSWI \(>\) 0 for the SOS and EVI \(<\) 0.1 and LSWI \(<\) 0 for the EOS. The VI-based GSL was compared with a GPP-based GSL that used SOS \(>\) 1 g C m\({}^{-2}\) day\({}^{-1}\) and EOS \(<\) 1 g C m\({}^{-2}\) day\({}^{-1}\) over three consecutive 8-day periods as thresholds.