## 3 Methodology  
### Machine Learning Methods  
The relationship between remote sensing data and target variables may often be nonlinear, making machine learning algorithms particularly suitable for addressing such complexities. These algorithms excel at handling nonlinear, heteroscedastic problems and extracting valuable insights from vast amounts of remote sensing data [44; 45]. In our  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Site Name** & **Observation Period** & **Longitude** & **Latitude** & **Elevation** & **Surface Types** \\ \hline Daxing (DXC) & 2008–2010 & 116.43\({}^{\circ}\) & 39.62\({}^{\circ}\) & 20 m & Maize/wheat \\ Guantao (GTC) & 2008–2010 & 115.13\({}^{\circ}\) & 36.52\({}^{\circ}\) & 30 m & Maize/wheat \\ Huaiali (HL) & 2016–2017 & 115.79\({}^{\circ}\) & 40.35\({}^{\circ}\) & 480 m & Maize/wheat \\ Luancheng (LC) & 2007–2018 & 114.41\({}^{\circ}\) & 37.53\({}^{\circ}\) & 50 m & Maize/wheat \\ Yucheng (YC) & 2003–2010 & 116.60\({}^{\circ}\) & 36.95\({}^{\circ}\) & 28 m & Maize/wheat \\ Xinxiang (XX) & 2019–2020 & 114.25\({}^{\circ}\) & 35.22\({}^{\circ}\) & 74 m & Maize/wheat \\ \hline \hline \end{tabular}
\end{table}
Table 2: The details of flux tower sites.  
Figure 2: The histograms of observed (a) ET and (b) NEE.  
study, we employed two machine learning algorithms--Random Forest Regression (RFR), and Backpropagation Neural Network (BPNN)--for fitting and estimating water (ET) and carbon (NEE) fluxes. These algorithms have been extensively utilized in establishing remote sensing models, including crop yield estimation [46], soil moisture estimation [4], leaf area index estimation [47], plant biomass estimation [48], and soil organic matter content estimation [49]. Additionally, we used Multiple Linear Regression (MLR) as a traditional fitting algorithm for comparison with these machine learning methods.  
(1) Random forest regression (RFR)  
RFR is an ensemble learning method primarily used for regression problems. It is based on the ensemble of decision trees and improves the accuracy and robustness of predictions by constructing multiple decision trees and averaging or majority voting on their prediction results. The following are the main steps and characteristics of the Random Forest Regression algorithm:  
* Random Sampling--Using the bootstrap sampling method to randomly extract multiple sample sets from the original training data set, with each sample set having the same size as the original data set;
* Decision Tree Construction--For each sample set, construct a decision tree. During the tree construction process, introduce randomness to increase the diversity of the trees, such as randomly selecting features for splitting;
* Each decision tree independently predicts new data;
* Ensemble Prediction--Averaging the prediction results of all decision trees to obtain the final prediction result.
* Backpropagation Neural Network (BPNN)  
The BP Neural Network regression algorithm is a multi-layer feedforward network trained using the error backpropagation algorithm, widely applied to various regression prediction problems. The main steps of the BP neural network regression algorithm include:  
* Network Initialization--Randomly set the weights and biases between neurons in each layer;
* Forward Propagation of Input Signals--Propagate the input signals forward through the network and calculate the output of neurons in each layer;
* Error Calculation--Calculate the error based on the network's output and the desired output;
* Error Backpropagation--Propagate the error signal backward and adjust the weights and biases of neurons in each layer according to the gradient descent method or other optimization algorithms;
* Iterative Training--Repeat steps \(b\) to \(d\) until the maximum number of iterations.  
In this study, we employed Python and SkLear to program the selected regression algorithms. The parameters of the two machine learning algorithms are presented in Table 3.