## 2 Materials and Methods
### Statistical Analysis
#### 2.4.2 Partial Least Squares Regression (PLS-R) Modeling  
Partial least squares regression (PLS-R) is recently becoming popular in remote sensing spectroscopy to predict plant traits [15; 53; 55; 56; 57; 58; 59; 61; 62; 78]. PLS-R is a machine learning technique, typically used to predict an outcome (y) from a set of predictors (x\({}_{1}\), x\({}_{2}\),..., x\({}_{\mathrm{p}}\)). In addition, PLS-R can handle highly correlated predictors. In a nutshell, PLS-R decomposes the predictor matrix (X) into a set of loadings and scores with the objective of maximizing co-variance between the scores and y. This is repeated for a given number of latent variables (LV) as the number of loadings and scores necessary to explain sufficient variance in y. The optimal number of LV can be found by employing cross validation [79; 80]. We modeled leaf A, leaf g\({}_{\mathrm{s}}\), and leaf T\({}_{\mathrm{r}}\) using VNIR reflectance, T\({}_{\mathrm{L,Rad}}\) and h\({}_{\mathrm{c}}\) as predictors. In our study, although h\({}_{\mathrm{c}}\) was manually measured, it was defined as a predictor as it can be estimated by photogrammetry techniques with UAVs [52] or from LiDAR sensors. In total, the number of samples totaled 144 for soybean and 126 for maize (Table 1). To assess the synergies of using only hyperspectral or adding other types of remote sensing information, we conducted PLS-R modeling multiple times: (i) with only hyperspectral data (Hyper), (ii) adding T\({}_{\mathrm{L,Rad}}\), and (iii) including h\({}_{\mathrm{c}}\), for both crops together, for only soybean and only maize data. PLS-R was implemented using the Python package Scikit-learn [81; 82]. PLS-R Preprocessing  
Growth and spatial effects are often considered a challenge when using remote sensing and physiology measurements [20]. To remove undesirable obstructive systematic variation, we carried out pre-processing of the spectral data. Standard normal variate (SNV) [83] was applied performing mean and variance normalization for each observation using the equation  
\[\overline{x}_{i,j}\ =\ \frac{x_{i,j}-\mu_{i}}{\sigma_{i}},\ i\ =\ 1\ \ldots\ N,\ j\ =\ 1\ \ldots\ K \tag{1}\]  
where \(N\) accounts for the number of samples and \(K\) for the number of optical hyperspectral wavelengths. \(\mu_{i}\) and \(\sigma_{i}\) are the mean and standard deviation of spectral data in each sample, respectively. \(x_{i,j}\) indicates the value of the \(i\)'th number observation in the \(j\)'th predictor and \(\overline{x}_{i,j}\) is the corrected predictor after using SNV with zero mean and unit variance. Standardization of the predictor data was subsequently performed (Equation (2)) as suggested by [PERSON] and [PERSON] (1986) [80] and also applied in other remote sensing studies to predict crop function [15; 53]. \[\overline{x}_{i,j}\ =\ \ \frac{\overline{x}_{i,j}-\ \mu_{j}}{\sigma_{j}} \tag{2}\]  
where \(\mu_{j}\) and \(\sigma_{j}\) are the mean and standard deviation of samples in each spectral wavelength, respectively. \(\overline{x}_{i,j}\) indicates the corrected predictor after standardization. Similarly, standardization of response data was applied as  
\[\overline{y}_{i}\ =\ \ \frac{y_{i}-\ \mu}{\sigma} \tag{3}\]  
where \(\mu\) and \(\sigma\) are the mean and standard deviation of samples for the selected response variable, respectively. \(\overline{y}_{i}\) indicates the corrected response after standardization. PLS-R Selection and Assessment  
The dataset was randomly split into calibration (70%) and validation (30%) sets [55; 62]. Using the same dataset, we applied PLS-R over 500 bootstrapped iterations to evaluate the uncertainty of PLS-R models. Bootstrapping is a technique that generates random samples of calibration and validation datasets. Bootstrapping has been found to be the most suitable strategy to validate predictability, ensuring reproducibility and generalizability of the final model [84].