Log Transformations Improve Reliability of Hourly Streamflow Predictions

Hourly streamflow data are increasingly available, offering a pathway to improve the accuracy of hydrological predictions. However, translating this high-resolution data into reliable probabilistic forecasts remains a significant technical challenge. A new study published in Hydrology and Earth System Sciences evaluates simple statistical methods for quantifying uncertainty in hourly streamflow predictions. The research, led by Cristina Prieto and colleagues, investigates how specific transformations of flow values and models of error persistence can enhance the statistical reliability of these predictions. The study focuses on practical, accessible methods that hydrologists and practitioners can implement without extensive computational resources.

The primary objective of the research is to provide a characterization of predictive uncertainty that is both robust and easy to apply. The authors examine common approaches for representing the heteroscedasticity of streamflow errors, where the magnitude of errors varies with the flow level. They also investigate methods for representing the autocorrelation of these errors, which refers to the tendency for errors to persist from one hour to the next. By combining a Box-Cox transformation with autoregressive models, the team aims to create a probabilistic framework that accurately reflects the inherent variability of hourly streamflow data.

The study utilizes a case study approach involving seven catchments located in Spain, Switzerland, and the United States. These catchments were selected to cover a range of climatic conditions, from humid to semi-arid environments. The specific sites include Lasarte in Spain, Wängi in Switzerland, and five catchments in the USA: Smith River, Gila, French Broad, Baron Fork, and San Francisco. The researchers employed hourly time series of rainfall, streamflow, and potential evapotranspiration to calibrate and evaluate their models. To ensure the robustness of the findings, the study excluded catchments with ephemeral flow or snow-dominated hydrology, which introduce additional complexities not addressed in this specific analysis.

The deterministic component of the model used in this study is the GR4H, a conceptual hydrological model designed for hourly time scales. GR4H is an hourly variant of the well-known daily GR4J model, retaining the same overall structure but adjusting internal parameters to suit higher temporal resolution. The model includes four calibration parameters: the capacity of the production store, the groundwater exchange coefficient, the capacity of the routing store, and the time base of the unit hydrograph. By using a conceptual model, the researchers aimed to balance accuracy with parsimony, ensuring that the methods proposed are feasible for widespread use in operational contexts.

For the probabilistic component, the researchers tested four fixed values of the Box-Cox power parameter to handle error heteroscedasticity. These values were λ=0, equivalent to the logarithmic transformation; λ=0.2; λ=0.5, the square root transformation; and λ=1, which corresponds to no transformation. The offset parameter A was fixed at 10% of the mean flow in each catchment to avoid numerical issues near zero flow. Additionally, the team compared three autoregressive models—AR1, AR2, and AR3—to represent the autocorrelation of residual errors. They also evaluated the potential benefit of adding moving average terms, testing ARMA models with up to three moving average components.

The validation strategy employed a split-sample approach to ensure that the model performance was assessed on independent data. In each catchment, the data were divided into two time periods. The model was calibrated on one period and used to generate probabilistic predictions for the other. This process was reversed, and the predictions from both validation periods were concatenated to create a comprehensive evaluation dataset. A warmup period of two to three years was included before the calibration period to allow the model to reach a steady state. This rigorous validation design helps to mitigate the risk of overfitting and provides a realistic assessment of the model’s predictive capability.

The results indicate that the choice of streamflow transformation has a major impact on predictive performance. When considering all flows, the logarithmic transformation achieved the best statistical reliability of predictions. The improvement in reliability was statistically significant compared to the other transformations. Furthermore, the precision and volumetric bias of the logarithmic transformation were not statistically significantly worse than those achieved by the other Box-Cox transformations. This finding suggests that the log transformation effectively stabilizes the variance of the errors, preventing the model from being dominated by high-flow events while maintaining accurate predictions for lower flows.

In terms of error autocorrelation, the study found that higher-order autoregressive models generally provided a better representation of the residual error structure. The AR2 model captured autocorrelation much better than the AR1 model, and in some catchments, the AR3 model performed even better than AR2. The addition of moving average terms to the AR3 model did not provide a significant improvement in the partial autocorrelation of the innovations. This suggests that a simple AR3 model is sufficient to capture the short-term memory of the errors in hourly streamflow predictions. The findings are consistent with the idea that error persistence is a key factor in the reliability of probabilistic forecasts.

The study also explored the performance of the models when aggregated from hourly to daily and monthly time scales. This aspect is crucial for practical applications where users may require predictions at different temporal resolutions. The results showed that aggregating from hourly to daily scales had little effect on performance, with reliability and precision remaining relatively stable. However, further aggregation to the monthly scale resulted in a degradation of reliability in most catchments. This loss of reliability at the monthly scale is likely due to deficiencies in the treatment of longer-scale error autocorrelation and the reduced number of data points available for evaluation.

For the top 5% of flows, which are often of primary interest in flood forecasting, the performance rankings became more variable across catchments. The square root transformation (BC05) achieved the best-median reliability for these high flows, while the no-transformation model (SLS) had the best-median precision and volumetric bias. However, the differences in reliability and precision were not statistically significant. This indicates that the transformations recommended for the full range of flows are not significantly disadvantaged when applied to high flows. The BC05 transformation was particularly attractive for high flows due to its lower volumetric bias, offering a useful alternative for applications focused on peak flow estimation.

The authors recommend the logarithmic transformation as the primary choice for hourly predictions when reliability is the highest priority. If precision is the main concern, the logarithmic transformation is still recommended, with the BC02 transformation as a viable alternative. For applications where low volumetric bias is critical, the Log or BC02 transformations are advised. These recommendations are broadly consistent with earlier work on daily streamflow predictions, suggesting that the principles of uncertainty quantification can be transferred across time scales with minor adjustments. The simplicity of these methods makes them attractive for operational use, where computational efficiency and ease of implementation are paramount.

The study acknowledges several limitations, including the relatively small number of catchments compared to similar studies at daily scales. The findings are therefore conditional on the specific hydroclimatology of the seven catchments studied. The authors suggest that future research should expand to a wider range of catchments, including those with snow-dominated or ephemeral hydrology. Additionally, the study focuses on statistical performance metrics such as reliability, precision, and bias, which provide a general assessment of model performance. However, specific applications may require different performance criteria, such as the frequency of false alarms in flood forecasting or the accuracy of low-flow predictions for ecological health.

Despite these limitations, the study provides valuable practical guidance for hydrologists seeking to implement probabilistic models for hourly streamflow predictions. The findings demonstrate that simple Box-Cox transformations and autoregressive models can significantly improve the reliability of uncertainty estimates. By offering a clear framework for selecting residual error models, the research facilitates the uptake of probabilistic predictions in water resources management. As hourly data become more widely available, these methods will play an increasingly important role in advancing our understanding of catchment processes and improving the accuracy of operational forecasts.

Subject of Research: Probabilistic uncertainty quantification for hourly streamflow predictions using Box-Cox transformations and autoregressive models.

Article Title: Simple Box-Cox probabilistic models for hourly streamflow predictions

Article References: Prieto, C., Kavetski, D., Fenicia, F., Kirchner, J. W., McInerney, D., Thyer, M., & Álvarez, C. (2026). Simple Box-Cox probabilistic models for hourly streamflow predictions. Hydrology and Earth System Sciences, 30(21), 6907-6925. https://doi.org/10.5194/hess-30-6907-2026

Image Credits: AI Generated

DOI: 10.5194/hess-30-6907-2026

Keywords: hydrology, streamflow prediction, probabilistic modeling, Box-Cox transformation, uncertainty quantification, autoregressive models, water resources, hydrological modeling, Simple, Box-Cox, probabilistic, models