## 3 Methods
### Biomass Estimation Model: Support Vector Machine Algorithm  
In this study, we used a SVM algorithm to perform the regression analysis and generalize the biomass estimation model. The basic idea of the SVM algorithm is to find an optimal non-linear mapping \(\phi\) and the corresponding parameters through cross-validation; we mapped the input space to the high dimensional feature space \(\aleph\) and found the linear regression relationship in this space.  
The SVM algorithm model can be described as follows:  
\[f\left(x\right)=\left\langle\,w,\varphi\left(x\right)\,\right\rangle+b,with\;w \in\aleph,b\in R \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Input MODIS \(t_{k}\)** & **Input Landsat \(t_{k}\)** & **Input MODIS \(t_{0}\)** & **Validation Landsat \(t_{0}\)** \\ \hline
**Scheme 1** & 09/06/2007-09/13/2007 & 09/08/2007 & 05/17/2007–05/24/2007 & 05/19/2007 \\  & DOY 249 & & DOY 137 & 05/19/2007 \\ \hline
**Scheme 2** & 09/22/2006–09/29/2006 & 09/21/2006 & 05/17/2007–05/24/2007 & 05/19/2007 \\  & DOY 265 & & DOY 137 & 05/19/2007 \\ \hline
**Scheme 3** & 08/29/2005-09/05/2005 & 09/02/2005 & 05/17/2007–05/24/2007 & 05/19/2007 \\  & DOY 241 & & DOY 137 & 05/19/2007 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Description of the input images used for the validation of the three schemes for data fusion.  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Statistics**} & **Formula 1 ** \\ \hline Mean value & \(\overline{p}=\frac{N}{i-1}\sum\limits_{j=1}^{M}\frac{p_{(i,j)}}{N\times M}\) \\ \hline Standard deviation & \(\sigma=\sqrt{\frac{1}{N\times M}\sum\limits_{i=1}^{N}\frac{1}{i-1}\left(p_{(i,j)}-\overline{p}\right)^{2}}\) \\ \hline Entropy & \(\mathrm{H}=-\sum\limits_{j=1}^{M}\sum\limits_{i=1}^{N}P_{(i)}lnP_{(i)}\) \\ \hline Average gradient & \(\overline{x}=\frac{1}{\left(M-1\right)\left(N-1\right)}\times\sum\limits_{i=1} ^{M-1}\sum\limits_{j=1}^{N-1}\sqrt{\frac{\left(p\left(i,j\right)-p\left(i+1,j \right)\right)^{2}+\left(p\left(i,j\right)-p\left(i,j+1\right)\right)^{2}}{2}}\) \\ \hline Mean absolute difference & \(\overline{D}=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{M}\frac{\left|p_{(i,j)}-q_{( i,j)}\right|}{N\times M}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Statistics for assessment of the quality of the synthetic images.  
where \(w\) and \(b\) are regressions coefficients and \(\varphi\left(x\right)\) is the mapping function. \(R\) is the space of the output pattern. The flow chart for the SVM algorithm is shown in Figure 3.  
Figure 3 is similar to a three layer neural network. The input vector for input layer nodes is \(x_{1},x_{2},x_{3}\cdots x_{n}\). The middle layer node for the kernel function is \(k\left(x_{i},x\right)\). The middle layer and output layer are connected by the weight vector, and the weight vector is \(W=\left(w_{1},w_{2},\ldots,w_{n}\right),\) which is essentially the coefficients of kernel function \(k\left(x_{i},x\right)\). The output is the estimated AGB \(f\left(x\right)\), where \(\lambda_{i},\ \lambda_{i}^{*},\) and \(b\) are coefficients and \(k\left(x_{i},x\right)\) is the kernel function. The basic kernel functions that can be used include the linear function, polynomial function, radial basis function (RBF), and sigmoid function. In this study, we decided to adopt the RBF to build the SVM-AGB model because the SVM model based on the RBF had highest accuracy; moreover, it is a recommended kernel function to acquire satisfactory results [55]. Two parameters are important for the RBF, namely, \(C\) and \(\gamma\). We used the recommended grid search method and cross-validation method [55] to find the optimal parameters for the RBF. In our study, the input parameter \(x\) was the MODIS NDVI or synthetic NDVI, while \(f\left(x\right)\) was the above ground biomass estimated by the MODIS NDVI or synthetic NDVI.