## 4 Discussion
Figure 4 does a good job of showing the ill-posedness of the \(\mathrm{G}\times\mathrm{M}\) retrieval even when using ground-truth LAI measurements; interestingly, due to the availability of the entire time series when using MODIS measurements in Figure 5, some combinations of \(\mathrm{G}\times\mathrm{M}\) identified as probable in Figure 4 are not probable in Figure 5 for any combination of regression coefficients. This is an illustration of the importance of the number of measurements [13; 100] needed to perform \(\mathrm{G}\times\mathrm{E}\times\mathrm{M}\) retrievals and the frequent, low-cost observations provided by satellites may be one of the most promising technologies to achieve that goal [101]. Although the uncertainty caused by site and growth stage specific secondary factors is well-known [27; 28; 29; 50; 51], it is difficult to isolate it from other sources. One approach to understand it [29; 50; 51] is to include variables connected to the secondary factors that cause it in the regression methodology. Unfortunately, this approach requires that the secondary factors causing the uncertainty are known and recorded or measured prior to the analysis being conducted. As a result, these studies can miss some of the factors causing the issue and underestimate its extent. Another approach is to train a global relationship between the satellite variables and crop state variables, ignoring the secondary factors [27]. In this case [27], the issue is seen from the variability of the regression coefficients, as in our analysis in Tables 3 and 4, as well as indirectly from the variability in the leave-one-site-out RMSE error. However, this method cannot be used to exclude other sources of uncertainty from the retrieval, such as random error and the mixed-pixel effect [102]. In this study, temporal analysis is used to avoid these alternate sources [27; 29; 50; 51] of uncertainty in determining the portion caused by site and growth stage specific secondary factors. The results in Table 5 show that the modeled \(\Delta\)LAI is retrieved from the MODIS measurements with significantly better performance in terms of both \(|\mathrm{r}|\) and RMSE as compared to the theoretical values assuming temporal independence of error, indicating significant site and time correlation of error. These results are also reproduced with actual \(\Delta\)LUE\({}_{\mathrm{Canopy}}\) measurements across multiple CO\({}_{2}\) flux tower sites in Figures 8 and 9. Both the results with modeled \(\Delta\)LAI and actual \(\Delta\)LUE\({}_{\mathrm{Canopy}}\) indicate a significant portion of the error can be removed by either predicting the secondary factors [29; 50; 51] or developing better methods to remove their influence, such as identifying vegetation indices less sensitive to the secondary factors [32; 45; 46]. The difference between the actual and theoretical \(|\mathrm{r}|\) and RMSE for both \(\Delta\)LAI and \(\Delta\)LUE\({}_{\mathrm{Canopy}}\) provide an indication of the possible reduction in uncertainty by addressing the issue with secondary factors. The change in performance with respect to the value of \(\Delta\) is driven by two factors:  
* As \(\Delta\) increases, the correlation between the error in the retrieved LAI or LUE\({}_{\mathrm{Canopy}}\) at t\({}_{2}\) relative to t\({}_{1}\) decreases because the measurements are more likely to be in different growth stages. * As \(\Delta\) increases, the magnitude of the retrieved \(\Delta\)LAI or \(\Delta\)LUE\({}_{\mathrm{Canopy}}\) increases relative to the remaining error which is not cancelled when calculating the change in the retrieved variables from the variables themselves, i.e., \(\mathrm{e}[\mathrm{t}_{2}]-\mathrm{e}[\mathrm{t}_{1}]\). As a result of these opposing error-influencing forces, a single value for the improvement in the performance that could be obtained by reducing the influence of the secondary factors cannot be reported; however, as seen from Table 5 and Figures 8 and 9 the improvement can be quite dramatic.