## 2 Materials and Methods
### Methods
#### 2.3.3 Satellite Retrieval and Crop Growth Model Sensitivity Analysis
The training of the LAI retrieval algorithm is performed on the LAIGROUND dataset with LANDSAT measurements for two reasons:  
* Using the LAIGROUND dataset with LANDSAT imagery better allows for the use of exact point measurements in fields and is thus less likely to be subject to uncertainty in training due to the inhomogeneity of LAI in the field, which can be significant [95]. * Training on high-resolution LANDSAT imagery as opposed to moderate-resolution MODIS imagery is preferable due to the significance of the mixed-pixel effect and neighboring pixels of other land types (including other crops) [95; 96]. In addition, a scaling effect correction algorithm is not used to correct for the uncertainty in applying a regression trained on LANDSAT data to MODIS data as these algorithms generally require _a priori_ information on the subpixel contents of the moderate resolution MODIS pixels [95; 96] which is not readily available. For this reason, training on MODIS pixels would likely not provide a benefit with respect to the uncertainty as it is likely that the bias caused by LAI inhomogeneity and the mixed pixel effect varies strongly from site to site [95; 96]. With these daily LAI retrievals from MODIS measurements, we calculated the change in LAI as  
\[\Delta LAI[t]=LAI[t+\Delta-1]-LAI[t-\Delta+1], \tag{7}\]  
where \(\Delta\) is in days and termed the \(\Delta\) window. The MODIS-retrieved \(\Delta\)LAI is compared to the crop growth model predicted \(\Delta\)LAI using the correlation coefficient absolute value (\(|\text{r}|\)) and RMSE. These metrics are compared to the theoretical \(|\text{r}|\) and RMSE if the error of retrieved LAI \([t+\Delta-1]\) and LAI \([t-\Delta+1]\) were independent with a RMSE equivalent to the leave-one-site-out RMSE calculated in Section 2.3.2. In this case, the theoretical RMSE and \(|\text{r}|\) can be calculated as  
\[RMSE(\Delta LAI[t])_{\textit{Thoror}}=RMSE(LAI[t+\Delta-1]-LAI[t-\Delta+1])= \sqrt{2}RMSE(LAI[t]), \tag{8}\]  
\[\left|\text{r}\left(\Delta LAI[t]\right)_{\textit{Thoror}}\right|=\left| \frac{cov(\Delta LAI_{\textit{actual}}+e_{\Delta LAI},\Delta LAI_{\textit{ actual}})}{\sqrt{var(\Delta LAI_{\textit{actual}}+e_{\Delta LAI})var(\Delta LAI_{ \textit{actual}})}}\right|=\left|\frac{1}{\sqrt{1+\left[\frac{\sqrt{2}RMSE( LAI[t])}{\sigma(\Delta LAI_{\textit{actual}})}\right]^{2}}}\right|, \tag{9}\]  
The uncertainty analysis for \(\text{LUE}_{\text{Canopy}}\) is complicated by the presence of high frequency components that need to be smoothed by Equation (3) in order to fully understand the temporal resolution of the retrieval. As the baseline retrieval methods with LANDSAT cannot account for the effects of the temporal smoothing because LANDSAT does not make daily measurements, the baseline retrieval must be retrained with MODIS measurements. Thus, leave-one-site-out regression is used to determine the regression coefficients in  
\[\overline{\textit{LUE}_{\textit{Canopy}}}=p\overline{\textit{EVI2}}+q, \tag{10}\]  
where \(\overline{\textit{EVI2}}\) is the moving average of EVI2 defined as  
\[\overline{\textit{EVI2}}[t]=\frac{1}{2N-1}\sum\
olimits_{i=-N+1}^{N-1}EVI2[t+i], \tag{11}\]With these leave-one-site-out regression coefficients, a baseline RMSE for the retrieval of \(\overline{LUE_{\text{Canopy}}}\) can be identified. In addition, as we have the benefit of a daily time series of MODIS measurements, \(\overline{\Delta LUE_{\text{Canopy}}}\) (defined in the same way as \(\Delta\text{LUE_{\text{Canopy}}}\) in Equation (2) can be determined by training a direct regression  
\[\overline{\Delta LUE_{\text{Canopy}}}=r\Big{(}\overline{EV2}[t+\Delta-1]- \overline{EV2}[t-\Delta+1]\Big{)}+s, \tag{12}\]  
in place of using Equation (10). The regression coefficients in Equation (12) are determined by leave-one-site-out cross-validation and the performance is compared to the theoretical \(|\text{r}|\) and RMSE performance defined in Equations (8) and (9) (with \(\text{LUE_{\text{Canopy}}}\) substituted for LAI).