## 4 Discussion  
The results presented in this study outline the importance of reducing the uncertainty in the relationship between satellite measurements and crop states variables caused by site and growth stage specific factors, in particular to use using remote sensing to map the G \(\times\) E \(\times\) M factors affecting crop growth. The importance of reducing the uncertainty is well illustrated by Figure 5, which shows that the set of allowable G \(\times\) M parameters in terms of consistency with the MODIS LAI retrievals (as measured by the RMSE) is strongly a function of the regression coefficients chosen. Based on the \"one place, one time, one equation\" concept [27], the appropriate regression coefficients for each time and place are ultimately different; therefore, auxiliary information is needed to select the appropriate regression coefficient column for each site and time to retrieve G \(\times\) M in Figure 5. This variability of the regression coefficients is best seen on the LAIGROUND dataset with high-resolution LANDSAT measurements in both the coefficients themselves and the large confidence \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Site** & **N** & \
\begin{tabular}{c} **RMSE Trained with** \\ **Actual Data** \\ \end{tabular} &
\begin{tabular}{c} **RMSE Trained with** \\ **Modeled Data** \\ \end{tabular} \\ \hline Beltsville & 26 & 0.84 & 0.97 \\ CEFLES2 & 26 & 0.77 & 0.87 \\ California & 59 & 1.40 & 1.39 \\ Italy & 35 & 1.39 & 1.26 \\ Missouri & 10 & 0.62 & 0.78 \\ NAFE06 & 14 & 0.51 & 0.47 \\ SEN3 EXP2009 & 10 & 0.87 & 0.79 \\ SMEX02-IA & 21 & 1.20 & 1.32 \\ SPARC & 45 & 1.87 & 1.83 \\
**All except Mead, Nebraska** & 267 & 1.30 & 1.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of LAI retrieval performance on all sites except Mead, Nebraska in LAGROUND dataset trained with actual and HM-modeled Mead, Nebraska LAI values. Only sites with \(\geq\)10 points listed site-by-site; all points included in last row. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Site** & **N** & \
\begin{tabular}{c} **RMSE Trained with** \\ **Actual Data** \\ \end{tabular} &
\begin{tabular}{c} **RMSE Trained with** \\ **Modeled Data** \\ \end{tabular} \\ \hline DE-Kli & 4 & 0.20 & 0.20 \\ FR-Gri & 1 & 0.20 & 0.10 \\ FR-Lam & 16 & 0.21 & 0.29 \\ IT-BCi & 32 & 0.19 & 0.35 \\ US-ARM & 1 & 0.22 & 0.37 \\ US-Bi2 & 12 & 0.26 & 0.30 \\ US-Ro1 & 27 & 0.13 & 0.28 \\
**All except Mead, Nebraska** & 93 & 0.19 & 0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of LUE\({}_{\text{C}\text{mopy}}\) retrieval performance on all sites except Mead, Nebraska in FLUX dataset trained with actual and HM-modeled Mead, Nebraska LUE\({}_{\text{C}\text{mopy}}\) values. intervals in Table 3, from which the range of the regression coefficients in Figure 5 was constructed. Less variability is seen on the FLUX dataset in Table 4 because this dataset has fewer points, smaller diversity in sites (points from Mead, Nebraska make up more than half the dataset), and is not designed to test the spatial variability of LAI in nearby plots in the same way as the LAIGROUND dataset; as a result, the LAIGROUND results in Table 3 are more appropriate for analyzing the variability between sites. Analyzed in conjunction with Figure 5, the regression coefficient variability in Table 3 makes it very difficult to use remote sensing for mapping \(\mathrm{G}\times\mathrm{E}\times\mathrm{M}\). This is because, as illustrated in Figures 4 and 5, the retrieval of \(\mathrm{G}\times\mathrm{E}\times\mathrm{M}\) is difficult due to equifinality (i.e., \"multiple combinations of parameters leading to similar simulation accuracy\") [13] and, especially when the observations are uncertain remote sensing retrievals, is ill-posed.