# Estimation of Time-Series Forest Leaf Area Index (LAI) Based on Sentinel-2 and MODIS
## 2 Materials and Methods
### Data and Preprocessing
#### 2.3.1 Construction of Remote Sensing Model for LAI  
Referring to the previous research results [70; 71; 72], the NDVI Normalized Difference Vegetation Index (NDVI), Simple Ration Index (SR), Difference Vegetation Index (DVI) and Atmospherically Resistant Vegetation Index (ARVI) were selected as the input parameters of the model. The specific calculation formulae are shown in Table 3.  
In this study, the estimation of the LAI was carried out in the following three ways: (1) constructing linear and nonlinear estimation models using a single vegetation index, i.e., establishing linear, exponential, logarithmic, multiplicative power and quadratic polynomial regression models for each vegetation index, separately, with the LAI; (2) constructing multivariate linear estimation models using multiple vegetation indices; (3) establishing principal component regression analysis models for computational analysis.  
To compare the differences in the LAI estimation by the different vegetation indices and different estimation models, the measured LAI values in the study area were divided into 110 training samples and 29 test samples. The vegetation indices were substituted into the regression equation calculated from the training set to obtain the estimated value of \(LAI_{mod}\), and then the coefficient of determination \(R^{2}\) and root mean square error _RMSE_ of the regression equation were calculated for the calculated result \(LAI_{mod}\) and its corresponding true value of \(LAI_{real}\). The degree of correlation of the equation fit is judged according to \(R^{2}\), which has a value of between 0 and 1. If \(R^{2}\) is closer to 1, it indicates that the reference value of the fitted equation is higher. The calculation formula of \(R^{2}\) and _RMSE_ are as follows.  
For linear regressions:  
\[R^{2}=1-\sum_{i=1}^{n}(LAI_{\text{mod}i}-LAI_{\text{real}i})^{2}/\sum_{i=1}^{n }\bigl{(}LAI_{\text{real}i}-LAI_{\text{realavg}}\bigr{)}^{2} \tag{1}\]  
For non-linear regressions:  
\[R^{2}_{adjust}=1-\Bigl{(}1-R^{2}\Bigr{)}(n-1)/(n-k-1) \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Sample Plot** & **Coordinates** \\ \hline I & \(117.3185^{\circ}\) N, \(42.40883^{\circ}\) E \\ II & \(117.2736^{\circ}\) N, \(42.44807^{\circ}\) E \\ III & \(117.2676^{\circ}\) N, \(42.44339^{\circ}\) E \\ IV & \(117.2674^{\circ}\) N, \(42.44396^{\circ}\) E \\ \hline \hline \end{tabular}
\end{table}
Table 2: Coordinates of the Time-series LAI sample plots.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Vegetation Index** & **Formulation** \\ \hline NDVI & (NIR \(-\) R)/(NIR \(+\) R) \\ SR & NIR/R \\ DVI & NIR \(-\) R \\ ARVI & (NIR \(-\) RB)/(NIR \(+\) RB), RB \(=\) NIR \(-\) \(\gamma\)(B \(-\) R) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Expressions for Vegetation Indices.  
\[RMSE=\sqrt{\sum_{i=1}^{n}(LAI_{\text{modi}}-LAI_{\text{reali}})^{2}/n} \tag{3}\]  
Here, \(n\) is the number of samples, \(k\) is the number of independent variables. If \(R^{2}\) is closer to 1 and \(RMSE\) is closer to 0, the estimation result of the model is closer to the real value.