## 3 Results
### Retrieval of LAI and Soil Moisture by Linear Regression
Interestingly, in four of the six applications, the SI's that produced the best fit within the univariate regression analysis was not among the best combination of SI's in the multiple regression approach, indicating that some of the individual SI's might be better able to represented the combined plant response to water and nutrient stress, but a combination of each multiple SI's are more selective \begin{table}
\begin{tabular}{c c c c c} \hline  & & \multicolumn{2}{c}{**Univariate Linear**} & \multicolumn{2}{c}{**Multivariate Linear Regression**} \\  & & \multicolumn{2}{c}{**Regression**} & \multicolumn{2}{c}{**(3 variables)**} \\  & & R\({}^{2}\) & SI & R\({}^{2}\) & SI \\  & & (rmse) & & (rmse) & \\ \cline{3-5}  & & 0.43 & MSI & 0.49 & RG, PSRI, MSI \\  & & (8.45) & & (8.01) & mNDVI\_705, PRI, NWI\_1 \\
**WC** & & 0.57 & NWI\_2 & 0.65 & RG, PSRI, MSI \\
**[Vol.\%]** & only barley & (7.14) & NWI\_2 & 0.65 & ONDVI, PRI, NWI\_1 \\  & only sugar & 0.60 & NDVI\_a & 0.65 & RG, PSRI, MSI \\  & & (7.32) & & (6.82) & RG, PSRI, MSI \\  & both crops & 0.20 & mSR\_705 & 0.57 & ONDVI, PRI, NDWI\_2130 \\  & & (1.40) & & (1.03) & ONDVI, PRI, NDWI\_2130 \\
**LAI** & only barley & 0.42 & GNDVI & 0.67 & ONDVI, PRI, NDWI\_2130 \\
**[m\({}^{2}\)m\({}^{-2}\)]** & only sugar & (1.40) & & (1.06) & ONDVI, PRI, NDWI\_2130 \\  & & 0.27 & MSR & 0.33 & ONDVI, PSRI, NWI\_2 \\  & only sugar & (0.55) & & (0.53) & ONDVI, PRI, NWI\_2 \\ \hline \end{tabular}
\end{table} 
Table 4: Coefficients of determination (_R\({}^{2}\)_) and root mean square error (rmse, in brackets) of simple univariate and multiple (three-fold) linear regressions between LAI/WC and pre-selected SI’s (see Table 1). Values listed are the maximum _R\({}^{2}\)_ values and the corresponding root mean square error (_rmse_) when varying through all 28 indices for univariate linear regression and through all combinations of SI’s (one from each of the three functional groups, see Table 1) for multiple linear regression. This analysis has been performed using data for “both crops” (Spring Barley and Sugar Beet) together as well as separately (“only barley” and “only sugar”). The columns “SI” refers to the spectral index (see Table 1) achieving the best fit. to a certain aspect of plant stress response and perform better. In particular the retrieval of LAI (for both crops) has gained drastically from inclusion of additional SI's confirming findings of e.g., [19] that LAI is not only influences by leaf chlorophyll concentration but also by other biochemical and crop/leaf structural factors that are possibly reflected in other SI's. While the results from dimensionality analysis suggested the use of 3-4 independent variable SI's best representing the spectral information, we have investigated the consideration of further SI's into the modeling procedure as shown in Figure 5. While considerable improvements in the prediction of LAI is achieved by an increase to three variables, only small increases in the model performance are observed by including additional SI's as explanatory variables. For the WC retrieval, already one SI achieves relatively high R\({}^{2}\)--values that only slightly improved by adding further SI's. The overall performance of the multiple-(3-SI)-regression-retrieval approach with R\({}^{2}\)-values of 0.65 for WC (both crops), and 0.67 and 0.33 for LAI (barley and sugar beet, respectively) compare very much to the upper limit of results that have been summarized by [19] showing a range of 0.05-0.66 for the retrieval of LAI. However, it needs to be considered, that our experiments have been performed under local, partly idealized glasshouse conditions, with direct control on illumination conditions and without the disturbance of atmospheric effects.