## 3 Methods
### The STARFM Algorithm for NDVI Image Fusion  
In this study, we obtained higher temporal and spatial resolution NDVI data by fusing MODIS and Landsat images with the use of the STARFM [14]. Previous studies have demonstrated the applicability of using STARFM for generating synthetic NDVI [18; 22; 29; 54]. Among these studies, [PERSON] _et al._[18] and [PERSON] _et al._[54] further verified that use of the NDVI from Landsat and MODIS images as the input for STARFM can produce higher precision compared with blending reflectance data. Therefore, we used Landsat NDVI and MODIS NDVI directly as the input data for generating the synthetic NDVI.  
Through establishing a linear relationship between MODIS and TM, STARFM can generate synthetic images based on an image pair of MODIS and TM NDVI data and the MODIS NDVI on the prediction date. Supposing that land cover types and system errors do not change over time, we can predict the NDVI at Landsat's spatial resolution for date (\(t_{o}\)) through using MODIS observations for date (\(t_{o}\)) and one or several pairs of Landsat and MODIS images acquired at the same date (\(t_{k}\)) [14]:  
\[L\left(x_{i},y_{j},t_{o}\right)=M\left(x_{i},y_{j},t_{o}\right)+L\left(x_{i},y _{j},t_{k}\right)-M\left(x_{i},y_{j},t_{k}\right) \tag{4}\]  
where \(L\left(x_{i},y_{j},t_{o}\right)\) is the predicted pixel \(\left(x_{i},y_{j}\right)\) at Landsat resolution for date \(t_{o}\), \(M\left(x_{i},y_{j},t_{o}\right)\) is the MODIS pixel \(\left(x_{i},y_{j}\right)\) for date \(t_{0}\), \(L\left(x_{i},y_{j},t_{k}\right)\) is the Landsat pixel \(\left(x_{i},y_{j}\right)\) for date \(t_{k}\), and \(M\left(x_{i},y_{j},t_{k}\right)\) is the MODIS pixel \(\left(x_{i},y_{j}\right)\) for date \(t_{k}\).  
However, in actual cases, the relationships between the NDVI of MODIS and TM are much more complicated. To solve this problem, STARFM uses a moving window (the size of the window was 750 m in our study) to acquire reflectance information for the neighboring pixels (750 m \(\times\) 750 m), and then, it weights those pixels with a combination of spectral, temporal, and spatial differences among the MODIS and TM images as follows [14]:  
\[L\left(x_{w/2},y_{w/2},t_{o}\right)=\sum_{i=1}^{w}\sum_{j=1}^{w}\sum_{k=1}^{ n}W_{ijk}\times\left(M\left(x_{i},y_{j},t_{o}\right)+L\left(x_{i},y_{j},t_{k} \right)-M\left(x_{i},y_{j},t_{k}\right)\right) \tag{5}\]  
where \(L\left(x_{w/2},y_{w/2},t_{o}\right)\)is the Landsat pixel \(x_{w/2},y_{w/2}\) for date (\(t_{o}\)), which is decided by the accumulation of adjacent pixels \(\left(x_{i},y_{j}\right)\) multiplied by the different weights \(W_{ijk}\). The time interval between \(\mathrm{t_{0}}\) and \(\mathrm{t_{k}}\), the NDVI value difference between Landsat and MODIS, and the relative distance among surrounding neighboring points are altogether used to determine the weights \(W_{\mathrm{ijk}}\).  
Because of the lack of available TM images with little cloud contamination during the growing season, TM images with good quality were not available for some years. To solve this problem, we used TM and MODIS images pairs (\(L\left(x_{i},y_{j},t_{k}\right)\) and \(M\left(x_{i},y_{j},t_{k}\right)\)) from the adjacent years (even as much as a 2-yr difference) as the input image pairs.  
To examine the quality of the fusion results, we tested the accuracy of the fused images by using TM as the base image with the following three schemes: date \(t_{0}\) and date \(t_{k}\) from the same year; date \(t_{0}\) and date \(t_{k}\) from adjacent years; and date \(t_{0}\) and date \(t_{k}\) with 2-yr intervals. The input and validated images are shown in Table 2.  
We chose five statistical indices (_i.e._, mean value, standard deviation, entropy, average gradient, and mean absolute difference) to assess the quality of the predicted synthetic NDVI products (Table 3).