## 2 Materials and Methods
### Inversion of LIDFa and \(V_{cmo}\) from In Situ Measurements  
#### 2.3.1 LIDFa Inversion from the Diurnal Canopy Reflectance Spectra  
LIDFa is a leaf inclination distribution factor that will vary with the growth of wheat. In this study, the LIDFa values across the growing season were retrieved from the measured reflectance spectra by inverting the RTMo module with the look-up table (LUT) method. According to the GSA of the SCOPE model for reflectance simulation in [29], the LIDFa has a significant influence on the reflectance spectra in the region at around 500-1300 nm. Moreover, other key driving variables that govern the simulated reflectance spectra from 650 nm to 750 nm (including LAI, Cab, and Cdm) are all accurately measured from in situ experiments. Therefore, the LIDFa can be retrieved from our measured reflectance spectra from 645 nm to 805 nm. However, at different measurement times during the day, different LIDFa values will be retrieved due to the bi-directional reflectance characteristics of canopy and random measurement errors. So, the only and optimal LIDFa value must be determined for one day (hereafter denoted as the daily LIDFa). In addition, the adjacent two days were regarded as one day for LIDFa retrieval. The schematic overview of the LIDFa inversion is shown in Figure 2.  
First, the LIDFa at each measuring time was retrieved using the least root mean squared error (RMSE) method. Using the measured diurnal meteorological variables and other required inputs, the RTMo module was run with an LUT of different LIDFa values at half-hour time intervals for each fieldwork day. According to the six common kinds of leaf inclination distribution defined in [50], the LIDFa range was set to \(-1\)-\(1\) with an interval of 0.05. Thus, for every half-hour, 41 canopy reflectance spectra under different LIDFa conditions could be collected from the SCOPE model outputs. Next, for the \(i\)-th LIDFa value of the \(n\)-th spectra measurement, we calculated the RMSE of simulated reflectance spectra \(R_{\text{sim}}\) to the measured reflectance spectra \(R_{\text{mea}}\), as shown in Equation (3)  
\[RMSE_{\text{R}}(i,n)=\frac{\sum\limits_{\lambda=645}^{805}\lvert R_{\text{sim} }(\lambda,i,n)-R_{\text{mea}}(\lambda,n)\rvert}{N_{\lambda}} \tag{3}\]  
in which \(\lambda\) represents the spectral wavelength and \(N_{\lambda}\) is the number of the spectral bands of the QE pro; \(n\) ranges from 1 to \(N\), and \(N\) is the total number of the spectra measurements during one fieldwork day; and \(i\) ranges from 1 to 41. To avoid the influence of SIF emission, the apparent reflectance spectra around the absorption bands were smoothed by spline interpolation. The LIDFa,  
Figure 2: Schematic overview of the LIDFa inversion procedure. which produces the least RMSE, was selected as the retrieved LIDFa at the \(n\)-th spectra measurement (LIDFa\({}_{n}\)). Thus, a vector [LIDFa\({}_{1}\dots\)LIDFa\({}_{n}\dots\)LIDFa\({}_{N}\)] can be calculated to represent the various LIDFa estimations at all the measuring times during one fieldwork day. Secondly, the daily LIDFa was final determined by a majority voting method, as follows:  
\[\text{daily LIDFa}=\text{mod}\left[ \begin{array}{cccc}\text{LIDFa}_{1}^{-}& \dots&\text{LIDFa}_{n}^{-}&\dots&\text{LIDFa}_{N}^{-}\\ \text{LIDFa}_{1}&\dots&\text{LIDFa}_{n}&\dots&\text{LIDFa}_{N}\\ \text{LIDFa}_{1}^{+}&\dots&\text{LIDFa}_{n}^{+}&\dots&\text{LIDFa}_{N}^{+} \end{array} \right] \tag{4}\]  
in which LIDFa\({}_{n}^{-}\) and LIDFa\({}_{n}^{+}\) were calculated as LIDFa\({}_{n}\) minus and plus the LIDFa step in the LUT (0.05), respectively, and the mod is an operator that calculates a matrix's mode. Some strategies were applied to decrease the uncertainties. First, we adopted the mode instead of the mean value to represent the daily LIDFa to avoid the influence of the abnormal LIDFa values retrieved from incorrect measurements.