Improving probabilistic prediction of daily streamflow by identifying Pareto optimal approaches for modeling heteroscedastic residual errors. Issue 3 (20th March 2017) - McInerney, David; Thyer, Mark; Kavetski, Dmitri; Lerat, Julien; Kuczera, George
Abstract: Reliable and precise probabilistic prediction of daily catchment‐scale streamflow requires statistical characterization of residual errors of hydrological models. This study focuses on approaches for representing error heteroscedasticity with respect to simulated streamflow, i.e., the pattern of larger errors in higher streamflow predictions. We evaluate eight common residual error schemes, including standard and weighted least squares, the Box‐Cox transformation (with fixed and calibrated power parameter λ ) and the log‐sinh transformation. Case studies include 17 perennial and 6 ephemeral catchments in Australia and the United States, and two lumped hydrological models. Performance is quantified using predictive reliability, precision, and volumetric bias metrics. We find the choice of heteroscedastic error modeling approach significantly impacts on predictive performance, though no single scheme simultaneously optimizes all performance metrics. The set of Pareto optimal schemes, reflecting performance trade‐offs, comprises Box‐Cox schemes with λ of 0.2 and 0.5, and the log scheme ( λ  = 0, perennial catchments only). These schemes significantly outperform even the average‐performing remaining schemes (e.g., across ephemeral catchments, median precision tightens from 105% to 40% of observed streamflow, and median biases decrease from 25% to 4%). Theoretical interpretations of empirical results highlight the importance of capturing the skew/kurtosis of raw residuals and reproducing zero flows. Paradoxically, calibration of λ is often counterproductive: in perennial catchments, it tends to overfit low flows at the expense of abysmal precision in high flows. The log‐sinh transformation is dominated by the simpler Pareto optimal schemes listed above. Recommendations for researchers and practitioners seeking robust residual error schemes for practical work are provided. Plain Language Summary: Predicting streamflow and water availability is a major scientific and engineering challenge, with global socioeconomic significance. Quantifying the uncertainty in streamflow predictions is a key component of risk‐based design and management of water systems. It enables decision‐makers to assess the likelihood that their investments will produce the desired outcome (e.g., reduced flood risk, increased environmental flows). Streamflow predictions at the catchment scale are often highly uncertain due to factors such as observation errors in the data and incomplete understanding of catchment physics. This study advances the field of catchment‐scale hydrological modeling by identifying the best‐performing error modeling schemes (from eight common approaches) that provide the most reliable, precise and unbiased streamflow predictions. These best‐performing schemes provide substantially tighter and more reliable predictions than other schemes under consideration, with some of the most pronounced improvements relating to streamflow prediction in ephemeral catchments. These findings provide hydrologists with robust modeling tools for quantifying predictive uncertainty in research and operational applications. Key Points: Choice of heteroscedastic error modeling approach significantly impacts on predictive reliability, precision, and bias, over 46 case studies Pareto optimal performance (out of eight residual error schemes) provided by Box‐Cox transform with power parameter λ fixed between 0 and 0.5 Empirically identified limitations of individual residual error schemes are interpreted using theoretical and synthetic analysis