## 5 Discussion  
In this research, MODIS, NBAR, and LAI time series from 2008 to 2010 were used as background information to calculate the initial parameters of the NARX neural network. Then, HR observation data of 2011 were input into the trained neural network to estimate HR LAI data for 2011, and both MODIS and HR data were used to update the parameters of the NARX neural network, which was  
Figure 10: Time series comparison between the mean of the LAI at Huailai and the corresponding LAI of the MODIS product: (**a**) 2013, (**b**) 2014. The error bars represent the standard deviation from the MODIS LAI product (black) and the retrieved LAI (red). Figure 9: Spatiotemporal LAI maps of the Huailai area in 2013 and 2014 (100 \(\times\) 100 pixels, 30 m spatial resolution) predicted by the NARX model. then used as the background to estimate the 2012 LAI. This procedure is recursively repeated year by year until both MODIS background data and all HR data were involved in the training. The empirical model has high precision in specific regions, but the model seems to lack generic capacity and may not be applicable to other areas, and radiative transfer models are mostly under-determined and ill-posed. The number of unknowns is typically much larger than the number of independent observations [55]. For the estimation of surface parameters with obvious annual cycle variation characteristics, such as LAI, the variation characteristics can be extracted from a long-term low-resolution dataset. It can be used as supplementary material for HR observation data and constrained inversion for LAI. At the same time, our recursive update method effectively solves the difference between low-resolution background data and high-resolution sub-pixels, which can effectively improve the estimation accuracy of HR LAI. Compared to the empirical and radiative transfer models, meanwhile, our model combining low-resolution images and high-resolution images with machine learning algorithms. The recursive update model only requires multiyear low-resolution background data, sequential variation in the data from low-resolution LAI products were used as a reference for HR LAI estimation, and HR LAI retrieval can be conducted on this basis. Due to the characteristics of the NARX neural network, the model has a strong capacity for fault tolerance to the input data, the noise in the HR surface reflectance data can be effectively suppressed. The estimation results are more consistent with the phenological changes of maize LAI. In this paper, estimating the HR LAI using the NARX model can be considered a semi-physical model, because the background data is obtained by inversion using the radiative transfer model. Compared with the downscaling and data fusion method, the method in this paper has obvious differences. Coarse resolution data need to be used only in the early stages of the estimation process to provide background information support, and all subsequent estimations are performed using high spatial resolution data. The general neural network algorithm requires pure pixels for training in the estimation process, but our method does not require pure pixels. During the training process, the background information is extracted based on the growth information of each pixel. The recursive updating model is insensitive to the outliers in the observed data. This will not cause errors in the estimation results due to fluctuations in a certain period of time in the pixel. The comparison of the model estimation results with the ground measurements demonstrates that the two results show relatively good consistency. Compared with the results without BRDF correction, the accuracy of the estimation results is significantly improved after the BRDF correction. Previous research has shown that MODIS BRDF spectral parameters can be used to reduce the BRDF effects of high spatial resolution data [56; 57; 58].