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

Abstract

AbstractConcentration‐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.