# A Multimodal Data-Driven Framework for Enhanced Wheat Carbon Flux Monitoring
## 2 Materials and Methods
### Multimodal Carbon Flux Estimation Model
The sequential arrangement of both modules enables mutual reinforcement of feature refinement at both the channel and spatial levels. Following feature refinement, the maps are then passed to a GRU layer. GRUs are well-suited for sequential data due to their simple structure, rapid convergence, and effectiveness in modeling short- and long-term dependencies, making them ideal for smaller datasets or real-time applications [22]. Through its gating mechanism, the GRU captures both short-term and long-term dependencies in carbon flux dynamics, enabling accurate simulations of crop carbon flux parameters (GPP estimates) across different growth stages. The single-timestep features extracted by the CNN-attention framework are processed through the GRU to estimate stage-specific carbon flux values. Given the input data with varying temporal resolutions, the GRU extracts temporal features separately for meteorological data and remote sensing indices based on distinct growth stages of winter wheat. Specifically, for each defined growth stage, the GRU independently models meteorological data and remote sensing indices (such as NDVI, EVI, and LAI, typically with 4-day to 8-day intervals). This strategy allows the GRU to effectively capture short-term variability in meteorological conditions and more gradual, stage-specific vegetation dynamics reflected by remote sensing indices, resulting in robust and meaningful feature representations aligned closely with the physiological patterns of wheat growth. As a result, the MCFEW model leverages both spatial features (extracted by CNNs) and temporal dependencies (captured by the GRU), utilizing complementary information from multimodal data sources to significantly improve the accuracy and interpretability of carbon flux estimations. To clearly assess the model's generalization capability and avoid overfitting, the dataset was split into independent training and testing subsets according to an 8:2 ratio, ensuring no overlap between the subsets. Specifically, 80% of the samples were randomly selected for training the model, and the remaining 20% were reserved exclusively for independent validation. This split accounted explicitly for both spatial and temporal variability, meaning that the validation set contains distinct spatial locations and time periods not used during training. The Mean Squared Error (MSE) loss function was chosen for training the model because it strongly penalizes larger prediction errors, making it highly sensitive to significant deviations between predicted and observed values. This property is particularly beneficial for carbon flux modeling, where large errors could severely impact the reliability of ecological assessments and management decisions. Additionally, MSE is smooth and differentiable, which makes it suitable for efficient optimization via gradient-based methods commonly used in deep learning models. Mathematically, the MSE is defined as  
\[\text{MSE}=\frac{1}{m}{\sum_{i=1}^{m}}(y_{i}-\hat{y}_{i})^{2} \tag{2}\]  
where \(y_{i}\) represents the observed value, \(\hat{y}_{i}\) is the predicted value by the model, and \(m\) denotes the total number of observations.