Linear and nonlinear ensemble deep learning models for karst spring discharge forecasting

Forecasting karst spring discharge is crucial for groundwater resource management in karst aquifers. These aquifers, with their inherent heterogeneity and complexity influenced by a myriad of environmental factors and hydrological processes, often show nonlinear and nonstationary behaviors. This study introduces novel multivariate multi-step ensemble models, both linear and nonlinear, to forecast karst spring discharge based on predictions from long short-term memory (LSTM), gated recurrent units (GRU) and one-dimensional convolutional neural network (1D-CNN). Linear and nonlinear ensemble learners including simple average method (SAM), linear regression (LR) and support vector regression (SVR) with Bayesian optimization (BO) are used to aggregate obtained results from base models and produce the final ensemble results. These proposed models, named as Ensem-SAM, Ensem-LR and Ensem-SVR-BO, are implemented for daily spring discharge forecasting across various lead times and time steps at Barton Springs, Texas, USA. Notably, these ensemble models outperform the individual base models in prediction accuracy and consistency. The results demonstrate that the ensemble framework can effectively leverage the strengths of diverse deep learning models and complement their limitations, thereby gaining strong generalization capabilities and robust performance, especially with diversiform and nonlinear data. The most marked disparities between individual base models and ensemble models arise with a short time step or a long lead time. Among all, Ensem-SVR-BO exhibits the best generalization capability and delivers accurate and robust prediction results even when one or all base models stuck in local optima.