## 3 Results
### Retrieval of LAI and Soil Moisture by Linear Regression  
The intrinsic dimensionality estimation as described in the previous section, suggest that most of the information content of the original spectral dataset can be well described by a set of 28 SI's (Table 1). We therefore restrict any further analysis to this reduced data set. In a first step, the simple linear dependencies between LAI and soil WC and single SI's are investigated. Standard linear regression was applied to extract the single SI with the maximum explanatory potential. Calculations were done for three variants: (1) using both crop types in one dataset and (2, 3) treating each crop species spring barley (2) and sugar beet (3) separately. Table 4 lists the coefficients of determination (R\({}^{2}\)) when predicting LAI and WC from a single SI for the three variants (1-3), as described above. Also given for each of the variants are the SI's (according to Table 1) for which the maximum R\({}^{2}\) is achieved. The results in Table 4 present a clear pattern: First, the R\({}^{2}\) values are relatively low in general with predictions being generally better for WC compared to LAI (maximum R\({}^{2}\) of 0.60 _vs._ 0.42) indicating that a single spectral index will not be sufficient to describe plant/soil variables over the range of conditions presented here by the individual nutrients and water treatments. Second, simple linear \begin{table}
\begin{tabular}{l c c} \hline \multicolumn{3}{c}{**Intrinsic dimensionality \(d\) of the datasets**} \\ \hline Methods/Dataset &
\begin{tabular}{c} all spectral bands \\ (\(N=2\),151) \\ \end{tabular} & spectral indices (\(N=28\)) \\ \hline _CorrDim_ & 1.03 & 1.58 \\ \hline _NNDim_ & 0.18 & 0.28 \\ \hline _MaxLike_ & 8.08 & 4.82 \\ \hline _PackNum_ & 0.00 & 0.00 \\ \hline _GMST_ & 5.96 & 3.69 \\ \hline _PCA*_ & 4.00 & 3.00 \\ \hline \end{tabular}  
* PCA has been included for comparison. \(d\) was derived by the number of PC with a minimum value of 0.25% for explaining total variance.  
\end{table} 
Table 3: Estimated intrinsic dimensionalities (d) of the complete spectral dataset including all spectral bands and of the dataset given by the pre-selected spectral indices. (CorrDim–correlation dimension estimator, NNDim–nearest neighbor dimension estimator, MaxLike–maximum likelihood estimator, PackNum–packing number estimator, GMST–geodesic minimum spanning tree estimator, PCA–principal component analysis). regression shows a better performance for both LAI and WC when applied to individual crop types rather than treating crops in a common application. In practice, this would mean that knowing crop species beforehand will significantly improve such retrieval, but will make the application of land use classification schemes necessary beforehand, thereby introducing potential misclassification error. The results from simple linear regressions (Table 4) as well as intrinsic dimensionality analysis of the spectral dataset suggest that multiple explanatory variables (3-4) should be able to more adequately explain the dynamic dependencies of WC, LAI and the vegetation spectral responses (SI's). In a next step, we therefore tested multiple linear regression using three explanatory SI's--one from each functional group (see Table 1)--leading to a scenario of 560 possible combinations of three-parameter explanatory inputs. Calculations were done in a similar way using three variants (1) for the entire dataset incorporating both crops together and for each species separately: Spring Barley (2) and Sugar Beet (3). Results are shown in Table 4 (right), demonstrating a significant improvement in the retrieval of WC and LAI from a combination of SI's for both the calculated R\({}^{2}\)-values as well as _rmse_.