# Retrieval of Biophysical Crop Variables from Multi-Angular Canopy Spectroscopy
## 2 Materials and Methods
### Step-Wise Inversion of the LUT  
Inverting a LUT means comparing measured spectra with all PROSAIL model results and selecting the parameters that led to the best performing LUT members. Different cost functions can be used to quantify the agreement between measurement and model result. Most authors use the Root Mean Squared Error cost function type (\(\mathrm{RMSE_{cft}}\)), defined as Equation (3):  
\[\mathrm{RMSE_{cft}}=\sqrt{\frac{1}{n}\cdot\sum_{i=1}^{n}(\mathrm{R_{measured} }(\lambda_{i})-\mathrm{R_{simulated}}(\lambda_{i}))^{2}} \tag{3}\]  
By squaring the distances before extracting the root, larger deviations gain more influence in this term. Consequently, the \(\mathrm{RMSE_{cft}}\) favors results for which both spectral signatures match rather closely for all wavelengths. An alternative cost function tested is the Nash-Sutcliffe-Efficiency (\(\mathrm{NSE_{cft}}\)[50]) as defined in Equation (4),  
\[\mathrm{NSE_{cft}}=1-\frac{\sum_{i=1}^{n}(\mathrm{R_{measured}}(\lambda_{i})- \mathrm{R_{simulated}}(\lambda_{i}))^{2}}{\sum_{i=1}^{n}\left(\mathrm{R_{measured }}(\lambda_{i})-\overline{\mathrm{R}}_{\mathrm{measured}}\right)^{2}} \tag{4}\]  
Weighing the squared sum of distances between measured and simulated reflectance against the squared sum of distances between measured reflectance and the average measured reflectance of the complete spectrum. The mathematically simplest approach is the mean absolute error (MAE) as defined in Equation (5):  
\[\mathrm{MAE}=\sum_{i=1}^{n}\lvert\mathrm{R_{measured}}(\lambda_{i})-\mathrm{R _{simulated}}(\lambda_{i})\rvert \tag{5}\]  
In all three cases, \(\mathrm{R_{measured}}(\lambda_{i})\) is the measured and \(\mathrm{R_{simulated}}(\lambda_{i})\) is the modelled reflectance at wavelength \(\lambda\) for the \(i\)th spectral sensor band, whereas \(n\) corresponds to the total number of bands used for the optimization. For each sampling date, the s-t-s-geometry must be known. Prior to application of the cost function, the correct sub-LUT must be selected. At first, by analysis of the observed spectrum, the corresponding LUT is inquired, depending on the expected canopy background. Senescent vegetation does not only show distinct absorption features by leaf pigments, but also significant features in the SWIR domain. A new index that has been optimized for the EnMAP spectral configuration, the \(\mathrm{NPVI_{EnMAP}}\) (Equation (6)) is introduced. \(\mathrm{NPVI_{EnMAP}}\) is used to classify the background of a pixel as either soil (type A) or senescent vegetation (type B) based on a simple threshold. \[\mathrm{NPVI_{EnMAP}}=\frac{\mathrm{R_{2218}}}{\mathrm{R_{671}}} \tag{6}\]  
If \(\mathrm{NPVI_{EnMAP}}<1.4\), the spectrum is classified as type B and classified as type A for all other cases. Based on the angular constellation for each pixel a decision is made, which of the remaining 252 sub-LUTs shall finally be used for the inversion. The ill-posedness can be mitigated by narrowing the parameter constraints for the generation of the LUT. In this case, the user needs to have access to a priori information about the expected data range. These constraints make the approach more empirical and thus inconsistent with the proposed generic conviction of the study. For this reason, the ill-posed problem was dealt with by considering more than just the one best performing LUT member and its according parameter configuration [3]. The final results vary with the number of considered best fits (n\({}_{\text{bf}}\)). A tradeoff between singular (ill-posed) and multiple (over-balanced) solutions needed to be found for an optimal retrieval setup (n\({}_{\text{bf}}\) \(\in\) {1, 20, 50, 100, 200, 500, 1000}). For n\({}_{\text{bf}}>\) 1, the median is used to get the final parameter value. Figure 2 illustrates the necessity to include an adequate amount of LUT-members for the variable retrieval. Parameter constraints for the creation of the LUT can be narrowed down to further increase inversion performance (e.g., [42]).