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<p>As water scarcity and global warming can affect agricultural irrigation, predicting crop water requirements is essential to improving irrigation water-use efficiency. Existing studies have focused on the complex structure of evapotranspiration, which is driven by several meteorological factors, to estimate reference evapotranspiration (ET<sub>0</sub>) using machine-learning techniques. However, due to the data-intensive requirements and the intricate structures and frameworks of machine-learning models, most of these studies are unsuitable in case of limited data availability, resulting in a lack of meteorological data for ET<sub>0</sub> estimation. In this study, a novel Hybrid-Distance Whale Optimization Algorithm (MWOA) is proposed for fine-tuning the super learner ensemble model. The proposed MWOA enhances the traditional WOA update rules by integrating a cosine-distance term to adjust the distance vector during exploration and exploitation, and by adaptively regulating step sizes based on the angular dissimilarity between candidate solutions and the best-so-far position. This directionally informed scaling enhances search diversification while ensuring stable convergence inside the hyperparameters space. In addition, permutation feature importance is employed for feature reduction, enabling the model to prioritize the most influential meteorological variables and reduce computational complexity. The proposed SL architecture integrates support vector regression and multi-layer perceptron as base learners, while a cross-validation-driven strategy further mitigates overfitting and improves generalization. Together, these advancements yield a high-performance, interpretable ET<sub>0</sub> prediction framework. In addition to the standalone SL, the measurements were compared with SL combined with particle swarm optimization and whale optimization algorithm. The results reveal that MWOA-SL can significantly enhance prediction accuracy, especially when limited meteorological data are available. Specifically, it can improve the mean squared error in a range from 14.42 to 88.36%, the mean absolute error in a range from 5.73 to 61.49%, and the mean absolute percentage error in a range from 5.83 to 66.04% when compared to the standalone SL. This explicitly demonstrates that MWOA-SL can significantly improve ET<sub>0</sub> prediction under constrained meteorological data.</p>

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Hybrid-distance whale optimization for super learner ensemble modeling of reference evapotranspiration

  • 期刊论文

  • 发表日期：

    2026年7月10日

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  • Mohammed Saied Aly,
  • Jian-Jun Han,
  • Yan Xiao,
  • Jie Liu

摘要

As water scarcity and global warming can affect agricultural irrigation, predicting crop water requirements is essential to improving irrigation water-use efficiency. Existing studies have focused on the complex structure of evapotranspiration, which is driven by several meteorological factors, to estimate reference evapotranspiration (ET0) using machine-learning techniques. However, due to the data-intensive requirements and the intricate structures and frameworks of machine-learning models, most of these studies are unsuitable in case of limited data availability, resulting in a lack of meteorological data for ET0 estimation. In this study, a novel Hybrid-Distance Whale Optimization Algorithm (MWOA) is proposed for fine-tuning the super learner ensemble model. The proposed MWOA enhances the traditional WOA update rules by integrating a cosine-distance term to adjust the distance vector during exploration and exploitation, and by adaptively regulating step sizes based on the angular dissimilarity between candidate solutions and the best-so-far position. This directionally informed scaling enhances search diversification while ensuring stable convergence inside the hyperparameters space. In addition, permutation feature importance is employed for feature reduction, enabling the model to prioritize the most influential meteorological variables and reduce computational complexity. The proposed SL architecture integrates support vector regression and multi-layer perceptron as base learners, while a cross-validation-driven strategy further mitigates overfitting and improves generalization. Together, these advancements yield a high-performance, interpretable ET0 prediction framework. In addition to the standalone SL, the measurements were compared with SL combined with particle swarm optimization and whale optimization algorithm. The results reveal that MWOA-SL can significantly enhance prediction accuracy, especially when limited meteorological data are available. Specifically, it can improve the mean squared error in a range from 14.42 to 88.36%, the mean absolute error in a range from 5.73 to 61.49%, and the mean absolute percentage error in a range from 5.83 to 66.04% when compared to the standalone SL. This explicitly demonstrates that MWOA-SL can significantly improve ET0 prediction under constrained meteorological data.