## 3 Model Calibration and Optimization  
### Model Calibration and Uncertainty Quantification  
The genetic coefficients of the CSM-CROPGRO-Cotton model were calibrated through a hybrid approach combining the Generalized Likelihood Uncertainty Estimation (GLUE) framework [37] with gradient-based optimization, ensuring robust parameter identifiability under non-linear model constraints. Leveraging phenological and yield data from 2023 field trials (cultivar _Zhongmian 113_), the calibration process systematically addressed equifinality challenges inherent in crop models. Post-calibration validation utilized independent 2024 experimental data, with model performance evaluated across four statistical metrics. The statistical metrics used for model evaluation, including ARE (Average Relative Error), RMSE (Root Mean Square Error), \(r^{2}\) (Coefficient of Determination), and \(d\) (Willmott's Index of Agreement), were chosen based on their ability to comprehensively assess model performance. RMSE and ARE are suitable for the validation of agricultural models [38]. ARE provides a percentage-based measure of the average relative error between simulated and observed values, which is particularly useful for understanding the magnitude of errors relative to observed values. RMSE quantifies the standard deviation of the residuals, indicating the spread of the errors. A lower RMSE value indicates better model performance. The coefficient of determination (\(r^{2}\)) measures the proportion of the variance in the observed data that is explained by the model, with values closer to one indicating better fit. [PERSON]'s Index of Agreement (\(d\)) is a normalized measure that accounts for both the variance and the bias in the model predictions, with values closer to one indicating better agreement between simulated and observed data. These metrics together provide a comprehensive assessment of model accuracy and reliability:  
\[\text{ARE}=\frac{100}{n}\sum_{i=1}^{n}\frac{|s_{i}-o_{i}|}{|o_{i}|}\quad(\%) \tag{4}\]  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(s_{i}-o_{i})^{2}}\quad(\text{kg} \,\text{ha}^{-1}) \tag{5}\]\[r^{2}=\left(\frac{\sum_{i=1}^{n}(o_{i}-\bar{o})(s_{i}-\bar{s})}{\sqrt{\sum_{i=1}^{ n}(o_{i}-\bar{o})^{2}\sum_{i=1}^{n}(s_{i}-\bar{s})^{2}}}\right)^{2} \tag{6}\]  
\[d=1-\frac{\sum_{i=1}^{n}(s_{i}-o_{i})^{2}}{\sum_{i=1}^{n}(|s_{i}-\bar{o}|+|o_{i}- \bar{o}|)^{2}} \tag{7}\]  
where \(n\) denotes observational replicates, \(s_{i}\) and \(o_{i}\) represent simulated and observed above-ground biomass (kg ha\({}^{-1}\)), respectively, with \(\bar{o}\) as the observational mean. The normalized RMSE (nRMSE = RMSE/(o_{\max}-o_{\min})\)) facilitates cross-scale performance benchmarking.  
Model fidelity was deemed acceptable under established thresholds: \(r^{2}>0.65\), \(d>0.80\), and nRMSE \(<15\%\) for agricultural system models [38].  
Figures 3 and 4 illustrate the comparison between the inferred results of our model and the actual results for seed cotton and lint cotton yields. From the model evaluation, we observe a good fit between the model and the real data. This indicates that our model can be utilized for subsequent simulation experiments and can serve to optimize fertilization practices in actual cotton cultivation operations.