## 3 Theoretical Background and Improvements  
### A Brief Description of the MOD17 Algorithm  
Although the MOD17 algorithm has been described in detail in the MOD17 User's Guide [36], we briefly describe it here because the algorithm is fundamental to understanding our proposed method. The MOD17 product [37,48] estimates the gross primary production (GPP) and net photosynthesis (PSN) of each pixel every 8 days and estimates the GPP and NPP for each pixel annually. These estimates are produced based on three upstream datasets: the MODIS land-cover classification product (MOD12Q1) [4], MODIS LAI/FPAR product (MOD15) [42], and large-scale meteorological data from the NASA Data Assimilation Office (DAO). The coefficients for each biome type (decided based on MOD12Q1) are defined in a Biome Parameter Look-Up Table (BPLUT) from a general ecosystem model [49].  
The daily MODIS GPP is estimated as:  
\[\mathrm{GPP}=\mathrm{PAR}\times\mathrm{FPAR}\times\varepsilon_{g}\times f(\varepsilon) \tag{1}\]  
where PAR (\(\mathrm{MJ}\cdot\mathrm{day}^{-1}\)) is the daily incident photosynthetically active radiation, as estimated using the DAO data; FPAR (dimensionless) is the fraction of absorbed photosynthetically active radiation, as obtained from the upstream MOD15 product; \(\varepsilon_{g}\) (\(\mathrm{g}\cdot\mathrm{C}\cdot\mathrm{MJ}^{-1}\) PAR) is the radiation use efficiency (RUE) for GPP calculations when the environment is not limiting plant carbon uptake, as defined in the BPLUT for the agricultural biome; and \(f(\varepsilon)\) accounts for the influences of environmental stress, such as temperature and vapor pressure deficit, as calculated based on the DAO data and BPLUT.  
The concept of daily PSN considers the maintenance respiration of leaves and fine roots:  
\[\mathrm{PSN}=\mathrm{GPP}\;-\;\mathrm{Leaf}\_\mathrm{MR}\;-\;\mathrm{Froot} \_\mathrm{MR} \tag{2}\]  
where the daily maintenance respiration is estimated for the leaves (Leaf_MR) and fine roots (Froot_MR) separately:  
\[\mathrm{Leaf}\_\mathrm{MR}\;=\mathrm{LAI}\;/\;\mathrm{SLA}\;\times\;\mathrm{ leaf}\_\mathrm{mr}\_\mathrm{base}\;\times\;\mathrm{Q10}\_\mathrm{mr}^{\mathrm{[(Tavg -20.0)/10.0]}} \tag{3}\]  
\[\mathrm{Froot}\_\mathrm{MR}=\mathrm{LAI}\;/\;\mathrm{SLA}\;\times\;\mathrm{ froot}\_\mathrm{leaf}\_\mathrm{ratio}\;\times\;\mathrm{froot}\_\mathrm{mr}\_\mathrm{base}\;\times\;\mathrm{Q10}\_ \mathrm{mr}^{\mathrm{[(Tavg-20.0)/10.0]}} \tag{4}\]  
where LAI (\(\mathrm{m}^{2}\cdot\mathrm{leaf}\cdot\mathrm{m}^{-2}\) ground area) is the leaf area index and SLA (projected leaf area \(\mathrm{m}^{2}\cdot\mathrm{kg}^{-1}\cdot\mathrm{leaf}\) C) is the specific leaf area for a given pixel; leaf_mr_base and \(\mathrm{froot}\_\mathrm{mr}\_\mathrm{base}\) (\(\mathrm{kg}\cdot\mathrm{C}\cdot\mathrm{kg}\cdot\mathrm{C}^{-1}\cdot\mathrm{ day}^{-1}\cdot\mathrm{20}\) C) are the maintenance respiration of leaves and fine roots per unit mass at 20 C, respectively; \(\mathrm{froot}\_\mathrm{leaf}\_\mathrm{ratio}\) (unitless) is the ratio of the fine root to leaf mass; \(\mathrm{Q10}\_\mathrm{mr}\) (unitless) is an exponent shape parameter that controls respiration as a function of temperature; and Tavg (\({}^{\circ}\)C) is the average daily temperature. LAI is obtained from MOD15, Tavg is estimated from the DAO meteorological data, and other biome-specific coefficients are obtained from BPLUT.