Multi-Step Ahead Probabilistic Forecasting of Daily Streamflow Using Bayesian Deep Learning: A Multiple Case Study

In recent decades, natural calamities such as drought and flood have caused widespread economic and social damage. Climate change and rapid urbanization contribute to the occurrence of natural disasters. In addition, their destructive impact has been altered, posing significant challenges to the efficiency, equity, and sustainability of water resources allocation and management. Uncertainty estimation in hydrology is essential for water resources management. By quantifying the associated uncertainty of reliable hydrological forecasting, an efficient water resources management plan is obtained. Moreover, reliable forecasting provides significant future information to assist risk assessment. Currently, the majority of hydrological forecasts utilize deterministic approaches. Nevertheless, deterministic forecasting models cannot account for the intrinsic uncertainty of forecasted values. Using the Bayesian deep learning approach, this study developed a probabilistic forecasting model that covers the pertinent subproblem of univariate time series models for multi-step ahead daily streamflow forecasting to quantify epistemic and aleatory uncertainty. The new model implements Bayesian sampling in the Long short-term memory (LSTM) neural network by using variational inference to approximate the posterior distribution. The proposed method is verified with three case studies in the USA and three forecasting horizons. LSTM as a point forecasting neural network model and three probabilistic forecasting models, such as LSTM-BNN, BNN, and LSTM with Monte Carlo (MC) dropout (LSTM-MC), were applied for comparison with the proposed model. The results show that the proposed Bayesian long short-term memory (BLSTM) outperforms the other models in terms of forecasting reliability, sharpness, and overall performance. The results reveal that all probabilistic forecasting models outperformed the deterministic model with a lower RMSE value. Furthermore, the uncertainty estimation results show that BLSTM can handle data with higher variation and peak, particularly for long-term multi-step ahead streamflow forecasting, compared to other models.