A robust Bayesian Multi-Machine learning ensemble framework for probabilistic groundwater level forecasting

Accurate prediction of groundwater levels is crucial for effective water resource management in agricultural regions, where groundwater is a vital resource. In these areas, the complex nonlinear relationships between groundwater storage, agricultural water demand, climate, and surface water delivery pose challenges for traditional physically-based models. As an alternative, data-driven machine learning methods are often adopted as surrogate to capture such intricate relationships due to their high computational efficiency. Inevitably, using only one machine learning model is prone to underestimate prediction uncertainty and subjected to poor accuracy. This study proposes a robust Bayesian multi-machine learning ensemble framework for probabilistic groundwater level forecasting. The framework incorporates a comprehensive set of input factors, including autocorrelation, meteorological, hydrological, and human activity variables, to capture the lag effects and driving mechanisms of groundwater depth variations. Five rule-based constraints are applied to identify the optimal combination of predictive factors and their time lags. Five machine learning models are employed, and their hyperparameters are optimized using Bayesian optimization algorithm. The ensemble of models is constructed using the stacking algorithm, leveraging the strengths of different modeling approaches. The uncertainty analysis within the framework involves three key components: fitting and selecting marginal distributions, constructing joint distributions using Copula functions, and calculating Bayesian posterior distributions. This enables a quantitative assessment of the uncertainties associated with groundwater depth predictions. A case study in the YingGuo region of the Huaihe River Basin, China, demonstrates the effectiveness of the proposed framework. The ensemble predictions outperform individual models, and the probabilistic forecasts provide reliable confidence intervals. The flexibility of the uncertainty analysis framework in describing complex groundwater depth variables is also highlighted. The proposed approach offers a robust framework for analyzing the uncertainties in groundwater depth predictions, serving as a valuable tool for comprehensive groundwater resource assessment and effective risk management strategies in agricultural regions.