Deep learning for the probabilistic prediction of semi-continuous hydrological variables – An application to streamflow prediction across CONUS

• Deep learning models for probabilistic prediction of semi-continuous variables. • Models use long short-term memory networks and probabilistic output layers. • Rectified Gaussian and hurdle distributions used in probabilistic output layers • Models tested for daily streamflow prediction in 421 catchments across CONUS. • Models with hurdle distribution provide best performance, especially at low flows. This study introduces two new end-to-end probabilistic deep learning (DL) models for predicting semi-continuous hydrological variables with a physical lower bound of zero. Both models use long short-term memory networks (LSTMs) with probabilistic output layers to predict the parameters of a probability distribution that fit observed streamflow (target) and respect its physical lower bound of zero. The first model is based on the rectified Gaussian (RG) distribution, which is akin to a Gaussian distribution that converts negative values to zero. The second model is a hurdle model, which includes two parts. The first part predicts the probability of the target being non-zero (the hurdle) while the second part predicts the probability distribution over non-zero values. Both RG and hurdle models are benchmarked against a state-of-the-art deterministic LSTM and evaluated on a daily streamflow prediction case study across 421 catchments in the contiguous United States. The hurdle model improved upon the mean normalized root mean squared error of RG and deterministic models by 9%. RG and hurdle models outperformed the deterministic model in mean low flow accuracy by 11% and 24%, respectively. When compared against third-party benchmarks, the hurdle model outperformed a hybrid process-based DL model by over 35% in low flow root mean squared error and performed competitively with a probabilistic benchmark in terms of reliability, sharpness, and accuracy. Overall, the RG and hurdle models are promising probabilistic DL models for predicting semi-continuous hydrological variables. However, the hurdle model is recommended as a new probabilistic benchmark for daily streamflow prediction across CONUS due to its improved low flow accuracy, reliability, and sharpness.