Revisiting the Origins of the Power‐Law Analysis for the Assessment of Concentration‐Discharge Relationships.

Concentration‐discharge (C‐Q) relationships are frequently used to understand the controls on material export from watersheds. These analyses often use a log‐log power‐law function (C = aQb) to determine the relationship between C and Q. Use of the power‐law in C‐Q analyses dates to two seminal papers by Francis Hall (1970, https://doi.org/10.1029/WR006i003p00845) and Francis Hall (1971, https://doi.org/10.1029/WR007i003p00591), where he compared six increasingly complex hydrological models, concluding the power‐law had the greatest explanatory power. Hall's analyses and conclusions, however, were based on a limited data set, with assumptions regarding water volume and storage, and used simple model selection criteria. While the power‐law is applied widely, it has not been rigorously tested and evaluated in over 50 years. We reexamined Hall's original models across time scales using 8 years of high‐frequency and weekly specific conductance data and evaluated model performance using more sophisticated model selection criteria. While we found the power‐law analysis remains one of the best performing models, other models performed equally as well including the log‐linear functional form. Model performance was similar at the sub‐daily to weekly scale but varied with sampling method. More complex models performed poorly relative to simpler models and tended to underpredict concentration at flow extremes due to constraints in fitting model parameters to the observed data. While we conclude, based on the data analyzed here, that the power‐law remains a suitable model for C‐Q analyses, opportunities exist to refine and differentiate among C‐Q models based on underlying assumptions of data distribution, recession analyses, and for applying models to reactive solutes. Plain Language Summary: Understanding how solute concentrations respond to changes in river flow remains a fundamental challenge in water resources science. Evaluating the relationship between solute concentrations and river flow (or discharge) can provide insight into how watersheds are structured and how they function. Concentration‐discharge (C‐Q) relationships can be used to estimate the rate of material export from terrestrial landscapes to surface waters. Many C‐Q analyses use a power‐law analysis and model to determine the response of C to variation in Q; yet this model and its embedded assumptions have not been rigorously tested following the development of water quality sensors that provide high‐frequency data. Here we revisit eight mathematical models originally developed by Hall (1970, https://doi.org/10.1029/WR006i003p00845) and Hall (1971, https://doi.org/10.1029/WR007i003p00591) that were initially evaluated with only 36 data points. We reexamine Hall's models using eight years of 15‐min specific conductivity data and find that while the power‐law model still is one of the best models to use in the evaluation of C‐Q relationships. And overall, simpler models outperform more complex models. We discuss many of the assumptions, such as constant load, that underpin C‐Q analyses to demonstrate that future studies could further parameterize C‐Q analyses for more insight on the mechanisms driving solute‐discharge relationships. Key Points: We re‐evaluate equations proposed by Francis Hall to assess concentration‐discharge (C‐Q) relationships using newly available long‐term and high‐frequency data setsAcross time steps we find that log‐log and log‐linear models perform equally well to describe C‐Q relationshipsParametrization of storage‐discharge relationships via recession analyses provides additional insight to C‐Q relationships [ABSTRACT FROM AUTHOR]