## 2 Methods
### Application of the NARX Network  
When using historical coarse spatial resolution data as a priori knowledge for the estimation of high spatial resolution LAI, we chose to use the NARX neural network to train time series data to extract the phenological characteristics. The NARX network is a nonlinear autoregressive network with exogenous inputs [39]. It is not only applicable to modeling and predicting nonlinear systems and time series, but is also able to achieve favorable outcomes for LAI estimation [37]. The general form of the NARX network can be expressed as follows:  
\[y(t)=f(y(t-1),\ y(t-2),\ \ldots,\ y(t-i),\ u(t),\ u(t-1),\ \ldots u(t-j)) \tag{1}\]  
In this expression, \(y\) refers to the time series output signal, \(u\) to the exogenous input time series, and \(i\) and \(j\) to the time-delay orders of input and output, respectively, which approximate the function \(f\). For feedback neural networks, the tapped delay line (TDL) plays the most important role. According to the TDL (Figure 2), the inputs of a neural network are divided into external inputs and delayed external inputs, and the outputs are split into present outputs and delayed outputs. It is assumed that the current predicted value of a time series depends on both the predicted value of the moment before the present moment and the exogenous time series input parameters for the current and previous moments. The existence of the TDL relates the output results to the present input parameters and is the result of interactions between the first \(i\) input parameter(s) and the first \(j\) output result(s); the estimation results will not fluctuate due to errors in the current input parameters.  
A typical NARX network consists of input, hidden, and output layers and the input and output TDLs. During network training, standardization using input data is required. Moreover, a sigmoidal transformation function should also be used as a transmission function for hidden neurons. To obtain the best result, the suggested number of hidden neurons should not be too large; otherwise, the overfitting phenomenon will occur. To increase training speed, the Levenberg-Marquardt backpropagation algorithm is used to train the neurons.  
The overall procedure for using time series to estimate LAI recursively is presented in Figure 2. When the model training data were constructed, the reflectance values in the red and near-infrared (NIR) bands, which are the most sensitive to vegetation growth, were selected as input data. The cosine  
Figure 2: Architecture of the NARX network configured with tapped delay lines (TDLs).  
of the solar zenith angle (SZA), the cosine of the viewing zenith angle (VZA), and the cosine of the relative azimuth (RA) were also taken as the input data. Meanwhile, \(u\) and \(y\), which are the input and output variables of the NARX network, respectively, can be expressed by a time series as follows:  
\[\left\{\begin{array}{l}u=\left[\begin{array}{ccc}Red_{n1}&Red_{n2}&\ldots& Red_{nm}\\ Nir_{n1}&Nir_{n2}&\ldots&Nir_{nm}\\ SZA_{n1}&SZA_{n2}&\ldots&SZA_{nm}\\ VZA_{n1}&VZA_{n2}&\ldots&VZA_{nm}\\ RA_{n1}&RA_{n2}&\ldots&RA_{nm}\\ \end{array}\right]\\ y=\left[LAI_{n1}&LAI_{n2}&\ldots&LAI_{nm}\right]\end{array}\right. \tag{2}\]  
During NARX training, MODIS NABR and MODIS LAI data were used initially to acquire a prior LAI background field at MODIS pixel resolution. For the purpose of trained neural network optimization, data were grouped into training, validation, and testing data at a ratio of 0.7:0.15:0.15. Considering the availability and flexibility of the model, the TDL was set to 0 for the input variables and 1 for the output variables. Equation (1) can be rewritten as follows:  
\[y(t)=f(y(t-1),\ u(t)) \tag{3}\]  
In the methods section, we discuss how to use the recursive update model for HR LAI estimation and address how to create a NARX network and configure the network parameters.