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.

State updating and calibration period selection to improve dynamic monthly streamflow forecasts for an environmental flow management application

“…However, the reliability of all model configurations in this catchment is already relatively high without state updating. All other metrics (sharpness, bias, CRPS and NSE) show improvements from state updating in catchment C2, suggesting potential trade-offs in performance, similar to that found by Crochemore et al (2016) and McInerney et al (2017). This slight reduction in reliability is not considered to have a significant detrimental impact on the PD produced for this practical application.…”

“…λ = 0.5 was used, as this setting was shown to produce good predictive performance (especially in terms of sharpness and bias) in ephemeral catchments by McInerney et al (2017). The offset is set as A = 1 × 10 −5 mm month −1 .…”

“…3.2, the GR4J model is calibrated at the daily scale to observed streamflow using the standard least squares likelihood function, because it better captures the high daily flows, important for estimating the monthly volumes. The post-processing error model for the monthly volumes is designed to capture the predictive uncertainty in these monthly volumes, in particular the heteroscedasticity and skew of the residuals (McInerney et al, 2017;Refsgaard, 1997). These choices of residual error models at the daily and monthly timescales contribute to the study objectives of reliable forecasts at the monthly timescale (see another example in Lerat et al, 2015).…”

“…Importantly, the case study catchments in this work are ephemeral and dry, with low runoff ratios. These types of catchments are known to be challenging to model (McInerney et al, 2017;Ye et al, 1997). For example, the models predict a streamflow response in 2002 and 2005 in Fig.…”

Multi-Source Uncertainty Analysis in Simulating Floodplain Inundation under Climate Change

“…Further, we used the 5th and 95th percentiles over each simulation and over each time step as the synthetic minimum and maximum water levels for the corresponding climate projection. The approach for the predictive uncertainty is based on heteroscedastic error modeling that represents the aggregated effects of data and model structural errors of the best model run found [48]. We applied the Box-Cox [49] transformations with a fixed transformation value of λ = 0.5 to get a robust statistical model for the residuals of discharge simulations.…”