This technical note deals with the mathematical representation of concentration–discharge relationships. We propose a two-sided affine power scaling relationship (2S-APS) as an alternative to the classic one-sided power scaling relationship (commonly known as “power law”). We also discuss the identification of the parameters of the proposed relationship, using an appropriate numerical criterion. The application of 2S-APS to the high-frequency chemical time series of the Orgeval-ORACLE observatory is presented here (in calibration and validation mode): it yields better results for several solutes and for electrical conductivity in comparison with the power law relationship.

The relationship between solute concentrations and river discharge (from now
on “

Many complex models have been proposed to represent

This technical note presents a two-sided affine power scaling relationship
(named “2S-APS”) that can be seen as a generalization of the power law.
And although we do not wish to claim that it can be universally applicable,
we argue here that it allows for a better description and modeling of the

We used the half-hourly (every 30 min) hydrochemical dataset collected by
the in situ River Lab laboratory at the Orgeval-ORACLE observatory
(Floury et al., 2017; Tallec et al., 2015). A
short description of the study site is given in Appendix A1. We used
dissolved concentrations of three ions – sodium [

As our main objective in this note is to compare the performance of two relationships (the new 2S-APS and the classic power law), we divided our dataset into two parts to perform a split-sample test (Klemeš, 1986): we used June 2015 to July 2017 for calibration (of both relationships), and August 2017 to March 2018 for validation. Table 1 presents the main characteristics of both periods.

Summary of high-frequency dissolved concentrations and electrical conductivity (EC; average, minimum, maximum values and coefficient of variation) from the River Lab at the Orgeval-ORACLE observatory, divided into two groups: June 2015 to July 2017 (calibration period) and August 2017 to March 2018 (validation period).

Table 1 shows a slight difference in the coefficient of variation (CV), which represents the dispersion of data with respect to their average value between the calibration and the validation period: this is due to the number of data used, which is much larger in the case of the calibration period.

For over 50 years, a one-sided power scaling relationship
(commonly known as power law) has been used to represent and model the
relationship between solute concentration (

In many cases, the power law appears visually adequate (and conceptually
simple), which explains its lasting popularity. With the advent of
high-frequency measuring devices in recent years, the size of the datasets
has exploded, and the

Concentration–discharge relationship observed at the
Orgeval-ORACLE observatory (measurements from the River Lab) for chloride
ions [

Figure 1 illustrates the inadequateness of the power
law for this dataset: the

As a progressive alternative to the one-sided power scaling relationship
(power law), we propose to use a two-sided affine power scaling (2S-APS)
relationship as shown in Eq. (3) (Box and Cox,
1964; Howarth and Earle, 1979).

Thus, for large values of

Evolution of the shape of the concentration–discharge scatterplot
for chloride ion with two-sided affine power scaling (2S-APS) and an
increasing value of parameter

Because the hydro-biogeochemical processes that control the transport and
reaction of ions are different, different ionic species may have a

Coefficient of determination (

The results given in Table 2 show the better quality
of the fit obtained with the optimal value of

The extremely large number of values in this high-frequency dataset may
cause problems for a robust identification over the full range of discharges
using a simple linear regression. Indeed, the largest discharge values are
in small numbers (in our dataset only 1 % of discharges are in the range
[2.6, 12.2 m

To address this question, we successively tested a large number of (

Numerical criteria used for optimization (

We retained as optimal the pair of (

In Appendix A2, we show that our proposed methodology for the identification
of parameters

The optimal values of

Summary of values

The five NSE criteria (defined in Table 3) used to identify the parameters of the 2S-APS relationship have also been computed for the power law relationship. The results are given in Table 5: the values obtained for the 2S-APS relationship are always higher than those calculated for the power law relationship.

NSE criteria computed for the three ions and EC.

Also for comparing the two relationships, we used the RMSE criterion. The
results are shown in Table 6; they illustrate (for
our catchment) the better performance (i.e., lower RMSE value) of the proposed
2S-APS relationship for the three ions (sodium, sulfate, and chloride) over
the power law relationship. For EC, there is a slight advantage over the
power law. A test of the equality of variance (

Summary of values of RMSE criterion calculated for the three ions and EC.

Figure 4 illustrates the comparison of the quality of simulation over the entire calibration dataset between the power law and 2S-APS relationships. In general, the two-sided affine power scaling relationship yields better simulated concentrations than the classic power law relationship for the two ions (according to the results of Table 6). This is particularly evident over the low concentrations (see Fig. 4). This better performance is more apparent in the case of sodium and chloride ions.

Comparison of simulated concentrations with observed
concentrations for

For the validation mode, we applied the above-calibrated relationships to a different time period (August 2017 to March 2018). We used as in Table 5 the five NSE criteria (see Table 3) to compare the performance between the two relationships studied. The results are given in Table 7. As in the calibration period, the values obtained for the 2S-APS relationship are higher than those calculated for the power law.

NSE criteria computed for the three ions and EC.

Also, as in the calibration mode, we computed the RMSE criterion. The results are shown in Table 8. The RMSE criterion illustrates (for our catchment) the better performance of the proposed 2S-APS relationship over the power law relationship for all the solutes. Unlike the calibration case, the quality of the simulation of EC using the 2S-APS relationship has a much better performance than the one simulated by the power law relationship.

Summary of values of RMSE criterion calculated for the three ions and EC with the validation dataset.

In this technical note, we tested and validated a three-parameter relationship (2S-APS) as an alternative to the classic two-parameter one-sided power scaling relationship (commonly known as “power law”), to represent the concentration–discharge relationship. We also proposed a way to calibrate the 2S-APS relationship.

Our results (in calibration and validation mode) show that the 2S-APS relationship can be a valid alternative to the power law: in our dataset, the concentrations simulated for sodium, sulfate, and chloride and the EC are significantly better in validation mode, with a reduction in RMSE ranging between 15 % and 26 %.

Naturally, because the data used for this study come from a single catchment, wider tests will be necessary to judge of the generality of our results.

In June 2015, the “River Lab” was deployed on the bank of the Avenelles
River (within the limits of the Orgeval-ORACLE observatory, see
Fig. A1) to measure the concentration of all major
dissolved species at high frequency (Floury et al.,
2017). The River Lab's concept is to “permanently” install a series of
laboratory instruments in the field in a confined bungalow next to the
river. River Lab performs a complete analysis every 30 min using two
Dionex^{®} ICS-2100 ionic chromatography (IC) systems by
continuous sampling and filtration of stream water. River Lab measures the
concentration of all major dissolved species ([

All the technical qualities, calibration of the equipment, comparison with laboratory measurements, degree of accuracy, etc. have been well described in a publication by Floury et al. (2017).

We have computed the predictive confidence interval, a well-known methodology used in linear regression (Jonnston, 1972, pp. 154–155; see also the discussion in Andréassian et al., 2007), to verify whether the 2S-APS relationship and the associated parameter identification methodology increase or decrease the uncertainty with respect to the power law relationship (linear regression with log transformation). We show two intervals: 50 % and 95 %. The results are given in Fig. A2: clearly, the predictive interval (blue surface for a 50 % predictive confidence interval, red for 95 %) is much narrower for the 2S-APS relationship than for the power law relationship. This can only reinforce our preference for the 2S-APS relationship.

Location of the River Lab (red dot) on the Avenelles River, Orgeval-ORACLE observatory.

Predictive confidence interval computed for the 2S-APS relationship and the power law for the three ions and the EC relationship. In blue the 50 % and in red the 95 % predictive confidence intervals.

Data will be available in a dedicated database website after a contract accepted on behalf of all institutes.

GT was in charge of the development and construction of the database. VA and JMTN conceptualized the methodology. JMTN performed the methodology on the catchment dataset. JMM was in charge of the statistical analysis of the proposed methodology. All authors framed the study and contributed to the interpretation of the results and to the writing of the paper.

The authors declare that they have no conflict of interest.

The first author acknowledges the Peruvian Scholarship Cienciactiva of CONCYTEC for supporting his PhD study at Irstea and Sorbonne University. The authors acknowledge the EQUIPEX CRITEX program (grant no. ANR-11-EQPX-0011) for the data availability. We thank François Bourgin for his kind review.

This research has been supported by the Peruvian Scholarship Cienciactiva of CONCYTEC (grant no. 099-2016-FONDECYT-DE).

This paper was edited by Roger Moussa and reviewed by Renata Romanowicz and one anonymous referee.