## 2 Materials and Methods
### Methods
#### 2.3.3 Satellite Retrieval and Crop Growth Model Sensitivity Analysis  
Third, we analyze the sensitivity of the crop growth model to its G \(\times\) M inputs and analyze how uncertainty in the satellite retrieval of LAI propagates to the uncertainty in estimation of its G \(\times\) M inputs. Specifically, we perform new Hybrid-Maize simulations based on the inputs used in [90], varying the planting density, the planting date, and the seed's growing degree days to maturity from their actual values, and observe the error in the modeled LAI with respect to the measured LAI for the modified simulations. As the emergence date is directly input into the simulations in [90], a preliminary set of Hybrid-Maize simulations is used to determine the appropriate planting date in Hybrid-Maize for the observed emergence date and then this planting date is varied in the sensitivity analysis. This method of determining the planting date to be varied is used in place of the actual planting date to remove the uncertainty caused by modeling the planting to emergence time (as in [90]). Comparison of the modeled LAI is performed with both the actual measured ground-truth LAI and the measured LAI retrieved from the MODIS measurements. To visualize the effect of the uncertainty in the regression coefficients, the error is shown for a range of regression coefficients determined from the confidence intervals obtained by bootstrapping in the previous subsection. Specifically, the slope of the regression is linearly varied from its minimum lower bound to its maximum upper bound while the intercept of the regression is simultaneously varied from its maximum upper bound to its minimum lower bound. As a large value for the intercept compensates for a lower value in the slope and vice versa, this method generates a realistic space within which to analyze the variation of the regression coefficients. 3.4 Evaluation of Uncertainty of LAI and LUE\({}_{\text{Canopy}}\) Retrievals Due to Site and Growth Stage Specific Factors with Temporal Analysis  
Fourth, we assess the uncertainty of LAI and LUE\({}_{\text{Canopy}}\) retrievals with temporal analysis due to site and growth stage specific factors. Due to the \"one place, one time, one equation\" concept [27], different regression equations should be used to retrieve the LAI and LUE\({}_{\text{Canopy}}\) at different sites and growth stages (different times). Furthermore, data from different years may also appear to require different regression equations because the interannual difference in weather and agromanagement is very significant [13] and can cause large differences in secondary factors. Therefore, different years can also be considered different sites for the purposes of this analysis. In order to separate uncertainty caused by site and growth stage specific factors from other types of uncertainty, we use temporal analysis and focus on the retrieval of the temporal change in LAI and LUE\({}_{\text{Canopy}}\). Errors caused by site and growth stage specific factors should be strongly positively correlated at the same place and nearby times; as a result, errors should partially cancel out when retrieving the temporal change as opposed to the actual values themselves. Thus, in order to assess the extent of the uncertainty caused by site and growth stage specific factors, the retrieval error of the change in LAI and \(\text{LUE}_{\text{Canopy}}\) is compared to the theoretical error of the change in LAI and \(\text{LUE}_{\text{Canopy}}\) assuming temporal independence of error. To perform the temporal uncertainty analysis for LAI, we use the LAIGROUND dataset as the baseline retrieval and apply the LANDSAT-trained leave-one-site-out regression coefficients from Equation (4) to the MODIS MCD43A4 BDRF-adjusted daily surface reflectance time series to obtain retrievals of LAI with daily resolution. The NIR band is designated as Band 2 on MODIS (0.84-0.88 um), while the Red band is designated as Band 1 on MODIS (0.62-0.67 um).