## 2 Materials and Methods
### The Link between Carbon Fluxes and Biomass  
In this study, BEPS uses daily LAI and hourly meteorological data as inputs and outputs GPP and NPP for each hour (g C m\({}^{-2}\) h\({}^{-1}\)); the hourly GPP and NPP are then binned into daily scale in units of g C m\({}^{-2}\) day\({}^{-1}\). Using GPP as an example, the accumulative GPP in unit of g C m\({}^{-2}\) on a specific day is defined as the sum of daily GPP from day 1 to day n:  
\[GPP_{A}=\sum_{i=1}^{n}GPP_{i} \tag{1}\]  
where \(GPP_{A}\) is also referred to as GPP for simplicity, hereafter; \(GPP_{i}\) is the daily total GPP for day \(i\); day 1 is defined as the first day when LAI is larger than 0.  
As summarized in the Introduction, there is justification to link crop GPP to crop biomass. Given an accumulated crop GPP (g C m\({}^{-2}\)) simulated by the BEPS over a given period of time in the growing season, the accumulated crop biomass (\(B\), in units of g m\({}^{-2}\)) in the same period is estimated as:  
\[B=GPP\cdot c_{1}\cdot c_{2} \tag{2}\]  
where \(c_{1}\) is the carbon use efficiency (CUE, the ratio of NPP to GPP) and ranges from 0 to 1 for different species [54]; CUE is often set to 0.5 as an average for global ecosystems [55]; \(c_{2}\) is the biomass to carbon ratio. As summarized by [PERSON] [23], the C content of biomass is almost always found to be between 45 and 50%; therefore, \(c_{2}\) is close to 2.0.  
As CUE increases with a decrease in temperature [56; 57], CUE in high latitudes is above 0.5. Since \(c_{2}\) is slightly larger than 2.0, the product of \(c_{1}\) and \(c_{2}\) is expected to be larger than 1.0 in high latitudes. Given the small variations in \(c_{1}\) and \(c_{2}\) in a specific area, it is desirable to directly link _GPP_ to biomass by a single coefficient \(C\) for practical applications:  
\[B=GPP\text{-}C \tag{3}\]  
Equation (3) assumes a linear relationship between GPP and biomass. The correlation coefficient between GPP and AGB for a few crop types will be examined since only AGB data are available in this study.  
Based on Equations (2) and (3), crop yield (\(Y\), in unit of \(\text{Kg}\) Ha\({}^{-1}\)) can also be estimated from the end-of-season crop _GPP_, although it is not the focus of this study:  
\[Y=GPP\text{-}c_{1}\text{-}c_{2}\text{-}c_{3}\text{-}c_{4} \tag{4}\]  
where \(c_{3}\) is the harvest index (HI), the ratio of harvested grain mass to total aboveground dry biomass; and \(c_{4}\) is 10, the factor to convert the unit from g m\({}^{-2}\) to kg to Ha\({}^{-1}\). As reported by [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] [30], the AGB explains more variations in crop yield than the peak LAI values.  
Similarly, we also compare crop biomass to accumulated LAI or NPP in a period in order to test their correlations. Accumulated LAI is defined as the sum of daily LAI from the date when LAI is larger than zero to a specific date in the growing season; the daily LAI is interpolated from remote sensed derived LAI. Similar to GPP, NPP and accumulated LAI also contain information from a time series rather than a single measurement.  
To estimate _AGB_ from _NPP_:  
\[AGB=NPP\text{-}c_{2}\text{-}c_{5} \tag{5}\]  
where \(c_{5}\) is ratio of AGB to total biomass. \(c_{5}\) may vary with different crops and can be calibrated; in this study, \(c_{5}\) is set to as 80%.  
A full diagram of the data processing flow is illustrated in Figure 1.