## 3 Methods
### Construction of Vegetation Index Curves of Crops  
Vegetation indices are widely used for the qualitative assessment of vegetation coverage and growth status and have become important indicators for crop monitoring. In this study, the average values of NDVI, EVI, GCVI, LSWI, NDWI, SAVI, RVI, NDBI, and GRVI during the growing season were selected for classifying the major crop types in the study area. The calculation formulas for each vegetation index are provided in Table 2.  
Each crop type has a specific planting pattern and growth curve characteristics, which can be identified using remote sensing time-series imagery [25; 52]. To address cloud contamination and irregular observation frequency [53], we employed temporal image compositing at 15-day intervals. Spectral indices derived from fused Landsat 8/9 and Sentinel-2 data were aggregated by calculating the mean value of all valid observations within each 15-day window, generating continuous time-series data. The Savitzky-Golay filter, proven effective for time-series reconstruction in complex terrain [11; 13], was applied to smooth phenological curves. We employed linear least squares to fit a low-order (second-order) polynomial within a moving window (size = 9), effectively eliminating noise while preserving phenological trends [54].  
The vegetation index time series highlighted distinct growth patterns and key phenological stages corresponding to specific image acquisition dates (Figures 4 and 5). We identified June 15 to August 20 as the optimal classification window (Figure 5a), coinciding with peak developmental stages, when intercrop spectral differences become most pronounced. Based on this time window, we further selected vegetation index statistical features with strong separability (Figure 5b) to reduce the number of input features, thereby improving the computational efficiency and learning capability of the classifier. Meanwhile, to complement optical data limitations, SAR image composites were generated at three-month intervals across four phenologically distinct periods: January-March (dormant season), April-June (emergence phase), July-September (peak growth), and October-December (senescence stage). This multi-temporal SAR synthesis captured structural variations during critical growth transitions, particularly useful for cloud-prone regions.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Indicators** & **Formula** & **Reference** \\ \hline Normalized Difference Vegetation Index (NDVI) & \(\frac{NIR-Red}{NIR+Red}\) & [46] \\ Land Surface Water Index (LSWI) & \(\frac{NIR-SWIR1}{NIR+SWIR1}\) & [13] \\ Enhanced Vegetation Index (EVI) & \(\frac{2.5(NIR-Red)}{NIR+Red-7.5 Blue+1}\) & [47] \\ Ratio Vegetation Index (RVI) & \(\frac{NIR}{Red}\) & [48] \\ Green Chlorophyll Vegetation Index (GCVI) & \(\frac{NIR}{Green}-1\) & [49] \\ Soil-Adjusted Vegetation Index (SAVI) & \(\frac{NIR-Red}{NIR+Red+0.5}\times 1.5\) & [17] \\ Normalized Difference Built-up Index (NDBI) & \(\frac{SWIR1-NIR}{SWIR+NIR}\) & [50] \\ Green Ratio Vegetation Index (GRVI) & \(\frac{Green-Red}{Green-Red}\) & [51] \\ Normalized Difference Water Index (NDWI) & \(\frac{Green-NIR}{Green+NIR}\) & [52] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calculation of each remote sensing indicator.