Improving probabilistic prediction of daily streamflow by identifying Pareto optimal approaches for modeling heteroscedastic residual errors

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.