Revisiting the Power-Law Function as a Model for Concentration-Discharge Relationships
Abstract
Variation in the concentration (C) of dissolved constituents with discharge (Q) is a fundamental characteristic of watersheds. The C-Q relationship must be understood to calculate solute budgets and to explore the coupling of biogeochemical and hydrological processes. The power-law function is commonly used to describe C-Q relationships (C = aQb), originating from two papers by Francis Hall published in the 1970s. However, few have revisited the origins of the power law, particularly given the advent of high-frequency sensing and availability of long-term datasets. Thus, it is important to re-evaluate the usage and interpretation of power-law functions, as other nonlinear functions may be more suitable for fitting observed data.
Here, we evaluate seven alternative nonlinear C-Q functions originally defined by Hall that result from the dilution of a single mixing volume with a nonlinear discharge-storage (Q-S) relationship. Using conductivity and discharge data from weekly grab samples and high-frequency sensors in the Lamprey River (New Hampshire, USA), we quantify model performance for different sampling methods and intervals. We show that several distinct C-Q models are applicable, but the simple power-law relationship performs best for all sampling intervals. Model performance is sensitive to two central parameters: n, which is the power-law exponent describing the relationship between active storage and discharge (Q = kSn), and C0, which represents the concentration of inputs to the mixing volume. We found that for a power-law Q-S slope n > 1, dilution of a mixing volume can generate various C-Q power-law slopes of b > -1. We discuss key model assumptions and evaluate whether dilution of a non-linear reservoir is a realistic representation of watershed functioning. Our results confirm that a simple power-law equation is the most appropriate single, continuous function to describe C-Q relationships.- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2022
- Bibcode:
- 2022AGUFM.H52J0584L