## 2 Materials and Methods
### Methods  
In this subsection, we discuss the methods we use to evaluate the influence of site and growth stage specific secondary factors on the relationship between crop state variables and satellite measurments and the retrievability of \(\text{G}\times\text{M}\) factors from satellite data. We focus on LAI and GPP in this study because these variables are some of the most commonly retrieved from remote sensing [92]. GPP also serves as a good complement to LAI because, unlike LAI, it is measured on a daily time scale at CO\({}_{2}\) eddy-covariance tower stations. Thus, it can be used to provide validation of the temporal analysis performed on crop growth model simulations of LAI. In addition, it should be noted that, as in [67], the methods in this paper can be applied to crop growth model simulated variables whose time series are more difficult to measure than LAI and GPP, providing a basis to analyze performance over a wide range of crop state variables.  
As daily GPP strongly depends on the daily SRAD, studies analyzing satellite-derived GPP must account for the strong temporal variability of SRAD when performing retrievals; this is because the variability in SRAD can mask the much smaller variability component in GPP caused by changes in the leaves, plants, and canopy structure [93]. A common technique to do so is correlating the product  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Source(s)** & \multicolumn{2}{c}{**Sites**} & \multicolumn{2}{c}{**Variables**} \\ \hline  & & **Name** & **Latitude** & **Longitude** & **Name** & **Years** \\ \hline  & & US-Ne1 [35] & 41.17 & \(-\)96.48 & & 2001–2009 \\  & & US-Ne2 [35] & 41.16 & \(-\)96.47 & & 2001–2009, odd years \\  & [76] & US-Ne3 [35] & 41.18 & \(-\)96.44 & GPP & 2001–2009, odd years \\ Flux Tower & [76] & US-Ro1 [77] & 44.71 & \(-\)93.09 & SRAD & 2005, 2009, 2011, 2013 \\ Data & & US-Bi2 [78] & 38.11 & \(-\)121.54 & Ground-truth & 2017–2018 \\ (Dataset & & US-ARM [79] & 36.61 & \(-\)97.49 & LAI & 2008 \\ \cline{2-6} FLUX) & & DE-Ki [80] & 50.89 & 13.52 & Planting & 2007, 2012 \\  & & FR-Gri [81] & 48.84 & 1.95 & Date & 2008, 2011 \\ GHG Europe & & FR-Lam [82] & 43.5 & 1.24 & Harvest Date & 2006, 2008, 2010 \\  & & IT-BCI [83] & 40.52 & 14.96 & & 2004–2009 \\  & & NL-Lan [84] & 51.95 & 4.90 & & 2005 \\ \hline  & & Beltsville & 39.02 & \(-\)76.85 & & 1998 \\  & & CEFILES2 [85] & 44.37–44.46 & 0.19–0.41 & & 2007 \\  & & California [86] & 35.48–39.22 & \(-\)122.14—119.28 & & (N = 26) \\ LAI & & Italy (IT-BCI) [83] & 40.52 & 14.96 & & 2008–2009 \\ Validation & [27] & Mead (US-Ne1 to US-Ne3) [35] & 41.16 & \(-\)96.46 & Ground-truth & 2001–2012 \\ LAIGROUND) & & Missouri [87] & 39.22 & \(-\)92.12 & & (N = 10) \\  & & NAFE06 [88] & \(-\)35.08—34.65 & 145.87–146.3 & & 2006 \\  & & SN3 EXP2009 [85] & 39.02–39.08 & \(-\)2.13—2.08 & & (N = 14) \\  & & SMEX02-IA [89] & 41.76–42.67 & \(-\)93.73—93.28 & & 2002 \\  & & SPARC [85] & 39.03–39.15 & \(-\)2.18—1.88 & & (N = 45) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Ground-truth data sources.  
of the remote sensing measurement and SRAD with daily GPP, as opposed to the remote sensing measurement itself [93]. To achieve a result identical to [93], we analyze the canopy light use efficiency (\(\text{LUE}_{\text{Canopy}}\)) in place of the GPP, which we define as  
\[\textit{LUE}_{\text{Canopy}}=\frac{\textit{GPP}}{SRAD}, \tag{1}\]  
As the definitions of various light use efficiencies are not standardized in the literature, we need to clarify that \(\text{LUE}_{\text{Canopy}}\) is essentially equivalent to \(\text{LUE}_{\text{Inc}}\) in [94], except that incident photosynthetically active radiation (\(\text{PAR}_{\text{inc}}\)) is used in place of SRAD. In addition, we wish to note that for the purposes of this study, the criticism of \(\text{LUE}_{\text{Inc}}\) in [94] does not apply because our goal in calculating \(\text{LUE}_{\text{Canopy}}\) is simply to remove the influence of SRAD and not any plant-based process.