## 3 Theoretical Background and Improvements
### Improvements to Estimate Crop Productivity  
Based on the MOD17 algorithm, we attempted to improve the crop yield estimation in the Midwestern US using the following techniques: (1) crop-type specific masks were derived from fine-resolution NASS CDL maps to identify relatively pure MODIS pixels; (2) a specific RUE wasapplied for each crop type; and (3) contributions to primary production from other vegetation types were deducted when possible. First, the MOD17 algorithm relies on the MODIS land cover product (MOD12), a classification scheme that does not differentiate crop types. Several studies have indicated that the use of the MODIS land cover product is likely to overestimate the agriculture areas relative to those reported by the NASS [18]. For this reason, we analyzed relatively pure corn or soybean pixels based on spatially disaggregated CDL maps (Figure 2) rather than applying a general cropland mask based on the MODIS land cover product. By counting corn pixels in the CDL maps, we observed that the estimated corn areas agreed well with the harvested corn areas reported by the NASS in 2011 (Figure 4A). The soybean areas were slightly overestimated, especially for the top soybean-producing counties (Figure 4B). It is appropriate to assume that pixels with more than 75% corn or soybean cover are relatively pure endmembers [41, 50]. The satellite data (resampled to 500 m) were spatially averaged based on those representative pixels to derive the mean values for each county every 8 days. Second, we applied crop-type specific RUE values for corn and soybeans (Table 2) instead of a biome-wide value in the MOD17 algorithm to estimate the yields. The MOD17 algorithm prescribes a universal RUE value for all crops, regardless of their types [36, 49]. However, the field measurements that quantify RUE values based on the relationship between biomass accumulation and cumulative radiation interception suggest that the carbon sequestration abilities vary widely across crop types [34, 51]. [PERSON] and [PERSON] [52] reviewed several studies and reported that the seasonal corn RUE ranges from 2.6 to 3.4 g\(\cdot\)MJ\({}^{-1}\) and soybean RUE ranges from 1.32 to 2.52 g\(\cdot\)MJ\({}^{-1}\). To account for the RUE differences, we applied 3.35 g\(\cdot\)MJ\({}^{-1}\) for corn and 1.44 g\(\cdot\)MJ\({}^{-1}\) for soybeans as obtained from recent field  
Figure 4: Comparisons between crop areas calculated from CDL maps and reported by NASS in 2011 for counties in the Midwestern US. (**A**) Corn area estimates; (**B**) Soybean area estimates. measurements in Iowa [53]. Although some studies observed that the RUE values varied across time [38,52], the RUE values applied here were derived as seasonal mean values in the experiments. Because the respiration costs of annual crop plants were deducted from the field measurements of accumulated biomass, the use of these RUE values would result in crop productivities that can be linked to crop yields directly. Finally, our analysis attempted to address the productivity contributed by vegetation other than crops because pixels are commonly mixed at the subpixel level of MODIS. The conceptual partition of mixed pixels could avoid the biased estimation of crop yields. In other words, the GPP for each pixel was partitioned into two components in our algorithm:  
\[\mathrm{GPP}_{\mathrm{total,\,DOY\,n}}=\mathrm{GPP}_{\mathrm{crop,\,DOY\,n}}+ \mathrm{GPP}_{\mathrm{other,\,DOY\,n}} \tag{5}\]  
where \(\mathrm{GPP}_{\mathrm{total,\,DOY\,n}}\) is the GPP value for a specific Day Of Year (DOY) n; \(\mathrm{GPP}_{\mathrm{crop,\,DOY\,n}}\) is the contribution from crops (_i.e._, corn or soybeans) for the same time period, and \(\mathrm{GPP}_{\mathrm{other,\,DOY\,n}}\) is the contribution from other vegetation types. Separating each component in mixed pixels is challenging in remote sensing models because dynamic information of subpixel land-cover composition is often unavailable. Here, we propose an algorithm to approximate the \(\mathrm{GPP}_{\mathrm{other,\,DOY\,n}}\) component based on plant LAI.