## 3 Results
### Retrieval of LAI and Soil Moisture by Non-Linear Regression Trees  
In addition to the multiple-linear regression approach, we have investigated the possible improvement and benefit of using non-linear techniques in the retrieval algorithm. A large variety of different techniques have been developed over the past decades, many of them emerging from the area of artificial intelligence, data mining and machine learning. Among those neuronal networks [54],  
Figure 5: R\({}^{2}\) improvement within multiple linear regression analysis due to increase of explanatory variables for LAI (continuous line) and theta (discontinuous line) for both crops species together (top), (2) for barley (middle), and (3) for sugar beet (bottom). fuzzy rules based approaches [55], support vector machines [56], random trees [57] and classification and regression trees (CART) [58] are the most popular. The CART method was chosen here as it represents a simple, comprehensible, easily to implement and widely used method. It is a nonparametric modeling approach that can describe the responses of a dependent from a set of independent variables. In our application, CART recursively partition the set of SI's to find increasingly homogeneous subsets based on SI's splitting criteria using variance minimizing algorithms. The (dependent) LAI and WC data are partitioned into a series of descending left and right child nodes derived from parent nodes. Once the partitioning has ceased, the child nodes are designated as terminal nodes. For CART calibration, a cost function is defined as the product of the estimated probability of a node times the cost of a node, that is the average squared error over the observations in that node, summed over all terminal nodes [58]. The \"optimal\" size of the tree has been extracted using 10-fold cross-validation and choosing the smallest tree that is within one standard error of the minimum-cost sub-tree. As CART, as well as all the other non-linear techniques mentioned above, show the ability to \"overfit\" the available data, we have compared multiple-linear regression (as presented in Section 3.3) and CART in a cross-validation framework using a leave-one-out strategy. Results shown in Table 5 using the combination of SI's achieving the best results in the multiple-linear-regression analysis presented before (Table 4) clearly demonstrate the difficulties of using non-linear techniques. Using the standard settings of the CART tree building procedure, only the retrieval of LAI for both crops shows superior results to multiple-linear regression, while \"fine-tuning\" of individual settings in the tree-building has been shown to improve the validation results for one of the dependent variable and crop combinations, the decrease at the same time for the others. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Multiple Linear Regression**} & \multicolumn{1}{c}{**CART**} \\  & & \(\text{R}^{2}\) & \(\text{R}^{2}\) \\  & & (rmse) & (rmse) \\ \hline \multirow{4}{*}{**WC**} & both crops & 0.48 & 0.42 \\  & & (8.08) & (8.59) \\  & only barley & 0.63 & 0.43 \\  & & (6.59) & (8.25) \\  & only sugar & 0.64 & 0.61 \\  & & (6.94) & (7.17) \\  & both crops & 0.55 & 0.59 \\  & & (1.05) & (1.01) \\
**LAI** & only barley & 0.65 & 0.58 \\
**[m\({}^{2}\)m\({}^{-2}\)]** & only sugar & (1.08) & (1.20) \\  & only sugar & 0.30 & 0.18 \\  & & (0.54) & (0.59) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Coefficients of determination (R\({}^{2}\)) and root mean square errors (_rmse_) for multiple linear regression and CART between LAI and WC using the best combinations of pre-selected SI’s from the analysis in Section 3.3 within a cross-validation (leave-one-out) framework. Results are representing cross-validation results for both crops (Spring Barley and Sugar Beet) together and separately for each one.