## 3 Theoretical Background and Improvements
### Improvements to Estimate Crop Productivity
Given that corn and soybeans in the Midwestern US have not emerged or in very early growth stages by mid-May, when the natural vegetation has already accumulated significant leaf area [41], it is appropriate to approximate the LAI of vegetation other than crops during the growing season by the maximum LAI before mid-May:  
\[\mathrm{LAI}_{\mathrm{other}}=\mathrm{max}(\mathrm{LAI}_{\mathrm{total,\,DOY \,n}-89\,N_{0}}) \tag{6}\]  
where \(\mathrm{LAI}_{\mathrm{other}}\) is the estimated LAI of vegetation other than crops; and \(\mathrm{max}\) (\(\mathrm{LAI}_{\mathrm{total,\,DOY\,n}=89\cdot N_{0}}\)) is the maximum LAI value between April (DOY 89) and before mid-May (DOY N\({}_{0}\)), as obtained from MOD15. The DOY N\({}_{0}\) value was assumed to be 129 for corn and 137 for soybeans because corn is typically planted approximately one to two weeks earlier than soybeans. Similar to Equations (3) and (4), the maintenance respiration of other vegetation is estimated to be:  
\[\mathrm{Leaf\_MR}_{\mathrm{other}}=\mathrm{LAI}_{\mathrm{other}}\ /\ \mathrm{ SLA}\ \times\ \mathrm{leaf\_mr\_base}\ \times\ \mathrm{Q10\_mr}^{\ \mathrm{[(Tavg-20.0)\ /\ 10.0]}} \tag{7}\]  
\[\mathrm{Froot\_MR}_{\mathrm{other}}=\mathrm{LAI}_{\mathrm{other}}/\ \mathrm{ SLA}\ \times\ \mathrm{froot\_leaf\_ratio}\ \times\ \mathrm{froot\_mr\_base}\ \times\ \mathrm{Q10\_mr}^{\ \mathrm{[(Tavg-20.0)\ /\ 10.0]}} \tag{8}\]  
\[\mathrm{MR}_{\mathrm{other}}=\mathrm{Leaf\_MR}_{\mathrm{other}}+\mathrm{Froot\_ MR}_{\mathrm{other}} \tag{9}\]  
where \(\mathrm{MR}_{\mathrm{other}}\) is the maintenance respiration of other vegetation; \(\mathrm{Leaf\_MR}_{\mathrm{other}}\) and \(\mathrm{Froot\_MR}_{\mathrm{other}}\) are the maintenance respiration of the leaves and fine roots of other vegetation, respectively; \(\mathrm{Tavg}\) (\({}^{\circ}\)C) is the \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Description** & **Corn** & **Soybean** & **Units** \\ \hline \(\mathcal{E}\) & the radiation use efficiency & 3.35 & 1.44 & \(\mathrm{g\cdot MJ}^{-1}\) PAR \\ RS & the root: shoot ratio & 0.18 & 0.15 & dimensionless \\ HI & the harvest index & 0.53 & 0.42 & dimensionless \\ MC & the moisture content of the grain & 0.11 & 0.10 & dimensionless \\ \hline \end{tabular}
\end{table} 
Table 2: The parameters used in our study for estimating crop yields. average daily temperature, as estimated from the DAO datasets; and the values for the other parameters are the same as the agricultural biome in the MOD17 algorithm. Because the ratio of NPP to GPP, termed as the carbon use efficiency (CUE), is modeled as a constant in many ecosystem models [29, 31], the GPP contribution from other vegetation can then be estimated to be:  
\[\mathrm{GPP_{other,\;DOY\;n}}=\mathrm{MR_{other,\;DOY\;n}}\;/\;(1-\mathrm{CUE}) \tag{10}\]  
where \(\mathrm{GPP_{other,\;DOY\;n}}\) is the GPP of other vegetation types for a specific DOY n, \(\mathrm{MR_{other,\;DOY\;n}}\) is the maintenance respiration, and \(\mathrm{CUE}\) is the carbon use efficiency. Published studies indicate a CUE value from 0.3 to 0.5 for natural vegetation such as forests [54]. We approximated a carbon use efficiency of 0.4 in this study. Our estimation of each GPP and MR component throughout the growing season is further illustrated in Figure 5, which presents the example of Olmsted County, Minnesota. The model was applied at the pixel level, and estimates were aggregated to the county level. The proposed algorithm captured the subtle variation of the GPP and MR caused by climatic factors in the time series. The sudden drop of MODIS GPP in the middle of the growing season (DOY 169) was most likely due to cloud contamination rather than harvest or other cultural practices. Clouds and aerosols can largely influence crop GPP in a short period of time by reducing the incident PAR [55].