## 2 Methods
### Recursive Update Model  
The algorithm proceeds through several stages shown in Figure 1, processing the images from one one one-year time series at a time. The neural network is trained with coarse-resolution data, and a set of weights is obtained. To renew the weights, high spatial resolution LAI and coarse resolution data are simultaneously provided to the neural network, and the neural network is retrained with the new training set in each iteration. The recursive update process of the neural network is as follows:  
Step 1: Create the NARX network for background information.  
We use the MODIS NBAR and LAI time series of 2008-2010 (i.e., D(1) in Figure 1) as input for the neural network and to train the NARX neural network to estimate the LAI for each pixel. The neural network is trained on a reflectance database comprising radiative transfer model simulations. The relationship between the MODIS LAI and the reflectance time series is modeled (i.e., M(1) in Figure 1) and represented by the parameters of the neural network, which are taken as the time series background for further processing.  
Step 2: HR LAI estimation  
The HR time series surface reflectance in 2011 is used as input for the neural network (M(1)) to estimate HR LAI. Note that in this step, each HR pixel is taken as a subpixel of the related MODIS pixel. The background information of each MODIS pixel can be used as a priori knowledge of approximately 272 (500*500/30*30) HR pixels. In this step, although the retrieved HR LAI has a 30 m spatial resolution, it also contains substantial MODIS pixel information.  
Step 3: Retraining of NARX networks.  
High spatial resolution information is added to the training data, and the neural network is retrained using the same structure. In this step, the MODIS reflectance data of 2008-2010 and the HR reflectance data of 2011 are set as new inputs for the neural network, and the MODIS LAI of 2008-2010 and HR LAI of 2011 are set as the corresponding outputs to retrain the NARX neural network for each HR pixel. The new training dataset is called D(2). After creating a new training set, the neural network is retrained with D(2). The new set of NARX network weights is found (M(2)). This process changes the initial weights and biases of the network and may produce an improved network after retraining. Through updating of the neural network, the new coefficients contain information from both MODIS and the 30 m resolution datasets. Consequently, using the MODIS (500 m) and HR (30 m) data as the background, the LAI time series with HR pixels (30 m) is estimated.  
Step 4: Recursive model construction.  
Figure 1: A flowchart for the Nonlinear Auto-Regressive with Exogenous Inputs (NARX) network-based recursive update model to estimate Leaf area index (LAI). D and M in the plots stand for training data and the NARX network, respectively. (1), (2),... (i) represent the training step number. The green dotted box represents step 1. The red dotted box represents step 2. The blue dotted box represents step 3.  
Steps 2 and 3 are repeated for a number of iterations until all HR datasets are involved in the training. The effect of HR data on the LAI results gradually increases during the estimation process. In these cases, the recursive procedures provide model improvement.  
By conducting recursive training, the proportion of HR information in the training data was increased, and the result can be considered HR LAI data. With the gradual increase in HR spatial detail, the retrieved results achieved the full complement of HR pixels. Using the proposed method, the neural network feedback is continually updated. As the HR images in the training data increase, a more accurate estimate of the 30 m resolution LAI can be obtained.