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

Abstract

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.

Linking Dynamic Water Storage and Subsurface Geochemical Structure Using High‐Frequency Concentration‐Discharge Records

“…A standard metric for C‐Q relationships is based on a power law, with fit parameters specific to each dissolved species (e.g., Bouchez et al., 2017; Clow & Mast, 2010; Godsey et al., 2009; Ibarra et al., 2016; Knapp et al., 2022; Li et al., 2021; Moquet et al., 2016; Musolff et al., 2015; Rose et al., 2018; Stewart et al., 2022; Wymore et al., 2023): $$$C=\alpha {Q}^{b}$$$ where C is the concentration of the considered species and Q the discharge of the river. When the b‐ value is 0, the species is considered perfectly chemostatic (Godsey et al., 2009), and its concentration is essentially independent of Q .…”

“…For Na + , Ca 2+ , Mg 2+ , and SO 4 2− over the five flood events (Figure 5), the ascending limb of the hydrograph is always more chemostatic than the recession and the loops are therefore systematically clockwise. By contrast, the NO 3 − C-Q relationships do not present any systematic pattern over the five floods (Figure S14 A standard metric for C-Q relationships is based on a power law, with fit parameters specific to each dissolved species (e.g., Bouchez et al, 2017;Clow & Mast, 2010;Godsey et al, 2009;Ibarra et al, 2016;Knapp et al, 2022;Li et al, 2021;Moquet et al, 2016;Musolff et al, 2015;Rose et al, 2018;Stewart et al, 2022;Wymore et al, 2023):…”

Establishing fluvial silicon regimes and their stability across the Northern Hemisphere

“…We evaluated the CQ relationship at each site using log Clog Q slope (Eq. 4; Wymore et al 2023). We calculated CQ slope from the raw concentration and discharge data.…”

Hydrologic and Landscape Controls on Rock Weathering Along a Glacial Gradient in South Central Alaska, USA

“…The general form is presented below in Equation 6 (Godsey et al., 2009; Hall, 1970; Moon et al., 2014), $$$C={aq}^{b}$$$ where C is concentration (mass solute/volume water), q is runoff (area normalized discharge, in length/time) and a and b are fitting parameters. The fitting parameter b is utilized to describe the behavior of solutes, where a value of −1 is pure dilution (mass of solutes remains the same while volume of water increases), a value of 0 is chemostasis (concentration remains the same regardless of discharge) and a value > 0 indicates increased concentration of solutes with higher discharge (Godsey et al., 2009 see also Bouchez et al., 2017; Ibarra et al., 2017; Torres et al., 2015; Wymore et al., 2017, 2023). The relationships highlighted by the b variable reflect the landscape and hydrologic drivers of solute generation and can be utilized to infer chemical constraints on rock weathering (Clow & Mast, 2010; Ibarra et al., 2017; Torres et al., 2015; Von Blanckenburg et al., 2015).…”