# Regional Crop Gross Primary Productivity and Yield Estimation Using Fused Landsat-MODIS Data
## 2 Materials and Methods
### Satellite Based Crop GPP and Yield Modeling
#### 2.2.2 Cropland GPP Estimation  
GPP for the seven major crop types in MT during 2008 to 2015 was calculated using an LUE framework similar to the MODIS MOD17 model [14]:  
\[\text{GPP}=LUEmax\times f(VPD)\times f(T)\times fPAR\times SWrad\times 0.45 \tag{1}\]  
\[f(T)= \begin{cases}0&T\leq T_{min}\\ \frac{T-T_{min}}{T_{max}-T_{min}}&T_{min}<T<T_{max}\\ 1&T\geq T_{max}\end{cases}  \tag{2}\]  
\[f(VPD)= \begin{cases}0&VPD\geq VPD_{max}\\ \frac{VPD_{max}-VPD}{VPD_{max}-VPD_{min}}&VPD_{min}<VPD<VPD_{max}\\ 1&VPD\leq VPD_{min}\end{cases}  \tag{3}\]  
where \(LUEmax\) is a prescribed maximum light use efficiency (g C M)\({}^{-1}\)), which is specific for different biome types (Table 2); photosynthetically-active radiation (_PAR_) is estimated as 45% of shortwave radiation (_SWrad_\(\times\) 0.45; [48]); _VPD_ is the mean daily vapor pressure deficit (Pa); T is the minimum daily surface air temperature (\({}^{\circ}\)C); _VPD\({}_{max}\), VPD\({}_{min}\), T\({}_{max}\) and \(T_{min}\) represent respective maximum and minimum _VPD_ and temperature (\(T\)) levels for plant photosynthesis, which vary for different plant types (Table 2); _f(VPD)_ and _f(T)_ are dimensionless (0 to 1) scalar linear ramp functions ranging between optimal (1) and fully constrained (0) levels under unfavorable atmospheric moisture deficit (_VPD_) and minimum daily temperature conditions; the product of _f(VPD)_ and _f(T)_ describes the daily LUE and GPP reduction from potential (_LUEmax_) conditions due to environmental stress. The LUE model environmental response functions are prescribed for different biome types within a global Biome Properties Look-Up Table (BPLUT) that distinguishes two major cropland functional types for cereal and broadleaf growth forms (Table 2). For estimating GPP, the seven major MT crop types were grouped into each of these two general plant functional types as either cereal (maize, barley, durum wheat, spring wheat, winter wheat) or broadleaf (peas, alfalfa) BPLUT categories. The daily minimum air temperature (T) and shortwave radiation (SWrad) inputs to the LUE model were obtained from a Gridded Surface Meteorological dataset (Gridmet, [49]), which provides relatively high spatial resolution (\(\sim\)4 km) daily surface meteorological data from 1979 to 2017 over the continental United States. The daily _VPD_ inputs were derived using relative humidity and air temperature data from Gridmet. The daily surface meteorology for each 30-m pixel used for GPP estimation was subsampled from the overlying 4-km resolution Gridmet cell. The BPLUT parameters (_VPD\({}_{max}\), VPD\({}_{min}\), T\({}_{min}\)_ and _T\({}_{max}\)_) for the cereal and broadleaf crop types were obtained from the Soil Moisture Active Passive (SMAP) L4C model GPP product [16]. The L4C GPP product uses a similar LUE model framework and MODIS vegetation observations as the MOD17 GPP product, but the L4C BPLUT parameters are calibrated using historical tower eddy covariance CO\({}_{2}\) flux observations from a global tower carbon-flux network (FLUXNET) that includes multiple cropland tower sites. The fraction of photosynthetically-active radiation (_fPAR_) absorbed by crops was calculated from the 30-m 8-day fused NDVI as [1,50]:  
\[fPAR=\frac{NDVI-NDVI_{min}}{NDVI_{max}-NDVI_{min}}*(fPAR_{max}-fPAR_{min}) \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Crop Type** & _LUEmax_ (g C M)\({}^{-1}\)) & _VPD\({}_{max}\)_ (Pa)** & _VPD\({}_{min}\)_ (Pa)** & _T\({}_{max}\)_ (\({}^{\circ}\)C)** & _T\({}_{min}\)_ (\({}^{\circ}\)C)** \\ \hline Cereal crop & 2.55 & 6940 & 1 & 45.85 & \(-\)23.15 \\ Broadleaf crop & 2.5 & 7000 & 1500 & 27.85 & \(-\)2.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: LUE model Biome Properties Look-Up Table (BPLUT) used for the MT cropland GPP calculations. where _fPAR\({}_{max}\)_ and _fPAR\({}_{min}\)_, representing the maximum and minimum _fPAR_ values of vegetation in the domain during the growing season, were assigned respective values of 0.95 and 0.01, following [1] and [50].