## 1 Introduction
Currently, many scientists have calculated global vegetation fPAR and published numerous products that are updated regularly. Most of these calculations are based on physical models, including MODIS _f_PAR and LAI products [20], with a temporal resolution of up to 4 days/500 m. The GLASS product [21] has a temporal resolution of 8 days/250 m, and Hi-GLASS couples physical models with machine learning algorithms and achieves a 30 m _f_PAR inversion using Landsat satellite data [22]. The same products include MISR [23], MERIS [24], SeaWiFS [25], and GEOV1 [26] and so on. These products have spatial resolutions ranging from 500 m to 50 km and temporal resolutions mostly ranging from 8 days to 1 month. Most products are calculated based on radiative transfer models or neural network algorithms [27]. However, there is still no globally scaled long-term time series product available. Scholars around the world have conducted significant research on vegetation productivity based on existing products, contributing to important studies at large temporal and spatial scales [4; 28]. In the North China Plain, the winter wheat planting to harvesting period is approximately from mid to late March to early June, spanning about 3 months. The cropping structure in this region is complex, with fragmented and heterogeneous wheat plots. Even with the use of 30 m spatial resolution _f_PAR products,the presence of mixed pixels remains possible. Ground-based measurement methods cannot provide comprehensive _f_PAR data and integrating physical models and machine learning methods into existing cloud computing platforms presents challenges. Existing products often require extensive computation prior to release, making real-time release within the plant growth cycle challenging. The vegetation index has been proven to be an effective parameter for vegetation monitoring, as it exhibits a more comprehensive ability to reflect the growth status of the preparation in comparison to the reflectance [29]. Therefore, the vegetation index method has become a relatively simple and feasible way to quickly estimate large amounts of real-time _f_PAR data. Many studies have shown a good linear or non-linear relationship between vegetation indices and _f_PAR, which are widely used in lighting efficiency models such as the GLOPEM model [30], the VPM model [31], and the EC-LUE model [11]. In these models, the relationship between vegetation indices and _f_PAR is defined as \(f\text{PAR}=a\times\text{VI}+b\)[32]. Other studies have also suggested that \(f\text{PAR}=a\times\text{e}^{\text{f}\text{VI}}\)[27]. Relevant studies on the calculation of _f_PAR using vegetation indices are listed in the Appendix A. In a previous study, _f_PAR estimation was conducted in Shandong Province in the North China Plain [33]. Spectral radiometers were used to simulate reflectance bands of Sentinel-2 data and calculate several vegetation indices for _f_PAR estimation. Ground experiments were conducted during the jointing, heading, and flowering stages of winter wheat. The experimental results suggested that the red edge band made a significant contribution to the _f_PAR calculation. However, the experiments did not take into account the important factors of different levels of water stress in wheat and the differences between varieties. It remains to be investigated whether the linear relationship between _f_PAR and vegetation indices is altered by water stress and variety differences. Therefore, this study mainly focuses on the following tasks: (1) finding vegetation indices suitable for calculating winter wheat _f_PAR at different growth stages, varieties, and water stress levels and establishing a correlation model between vegetation indices and _f_PAR; (2) evaluating the estimation accuracy of the model under different varieties, water stress levels and growth stages; and (3) assessing the reliability of the model at the satellite scale.