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**Hydrology and Earth System Sciences**
An interactive open-access journal of the European Geosciences Union

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**Technical note**
14 Apr 2020

**Technical note** | 14 Apr 2020

Technical note: A two-sided affine power scaling relationship to represent the concentration–discharge relationship

^{1}Université Paris-Saclay, INRAE, HYCAR Research Unit, 92761 Antony, France^{2}Sorbonne Université, CNRS, EPHE, UMR Metis 7619, Paris, France

^{1}Université Paris-Saclay, INRAE, HYCAR Research Unit, 92761 Antony, France^{2}Sorbonne Université, CNRS, EPHE, UMR Metis 7619, Paris, France

**Correspondence**: Vazken Andréassian (vazken.andreassian@inrae.fr)

**Correspondence**: Vazken Andréassian (vazken.andreassian@inrae.fr)

Abstract

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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.

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Tunqui Neira, J. M., Andréassian, V., Tallec, G., and Mouchel, J.-M.: Technical note: A two-sided affine power scaling relationship to represent the concentration–discharge relationship, Hydrol. Earth Syst. Sci., 24, 1823–1830, https://doi.org/10.5194/hess-24-1823-2020, 2020.

1 Introduction

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The relationship between solute concentrations and river discharge (from now
on “*C*–*Q* relationship”) is an age-old topic in hydrology (see among
others Durum, 1953; Hem, 1948; Lenz and Sawyer, 1944). It would be impossible
to list here all the articles that have addressed this subject, and we refer
our readers to the most recent reviews (e.g., Bieroza et al., 2018; Botter
et al., 2019; Moatar et al., 2017) for an updated view of the ongoing
research on *C*–*Q* relationships.

Many complex models have been proposed to represent *C*–*Q* relationships, from
the tracer mass balance (e.g., Minaudo et al., 2019) to the
multiple regression methods (e.g., Hirsch et al.,
2010). Nonetheless, for the past 50 years the simple mathematical formalism
known as “power law” has enjoyed lasting popularity among hydrologists and
hydrochemists (see, e.g., Edwards, 1973; Gunnerson, 1967; Hall, 1970, 1971).
Over the years, however, some shortcomings of this relationship have become
apparent: recently, Minaudo et al. (2019) mentioned that,
“fitting a single linear regression on *C*–*Q* plots is sometimes questionable
due to large dispersion in *C*–*Q* plots (even log transformed)”. Also,
Moatar et al. (2017) present an extensive typology of shapes (in
log–log space) for the French national water quality database, which shows
that the power law must be modified to represent the *C*–*Q* relationship for
dissolved components as well as for particulate-bound elements.

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
*C*–*Q* relationship of some solutes as a natural extension of the power law.

2 Test dataset

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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 [Na^{+}], sulfate
[S-${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$], and chloride [Cl^{−}] – as well as electrical
conductivity (EC). This dataset was collected from June 2015 to March 2018,
averaging 20 700 measurement points.

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.

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.

3 Mathematical formulations

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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 (*C*) and discharge (*Q*)
(Eq. 1).

$$\begin{array}{}\text{(1)}& C=a{Q}^{b}\end{array}$$

From a numerical point of view, the relationship presented in
Eq. (1) is generally adjusted by first transforming
the dependent (*C*) and independent (*Q*) variables using a logarithmic
transformation and then adjusting a linear model
(Eq. 2).

$$\begin{array}{}\text{(2)}& \mathrm{ln}\left(C\right)=\mathrm{ln}\left(a\right)+b\mathrm{ln}\left(Q\right)\end{array}$$

Graphically, this is equivalent to plotting concentration and discharge in a
log–log space, where parameters *a* and *b* can be identified either
graphically or numerically, under the assumptions of linear regression.

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 *C*–*Q* relationship can now be analyzed on a wider span (Kirchner et al., 2004). Figure 1 shows an
example from our own high-frequency dataset: the 17 500 data points (which
correspond to the calibration period of Table 1)
represent half-hourly measurements collected over a 2-year period, during
which the catchment was exposed to a variety of high- and low-flow events,
thus providing a great opportunity for exploring the shape of the *C*–*Q*
relationship. This being said, we do not wish to imply that a similar
behavior could not been identified in medium- and low-frequency datasets,
which remain essential tools with which to analyze and understand long-term
hydrochemical processes (e.g., Godsey et al., 2009; Moatar et
al., 2017).

Figure 1 illustrates the inadequateness of the power
law for this dataset: the *C*-*Q* relationship evolves from a well-defined
concave shape on the left to a slightly convex shape on the right in the
log–log space. From the point of view of a modeler wishing to adjust a
linear model, one has gone beyond the straight shape that was aimed at. Note
that this is true for our dataset, and that it does not need to always be
the case: the log–log space can be well adapted in some situations (see
examples in the paper by Moatar et al., 2017).

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).

$$\begin{array}{}\text{(3)}& {C}^{{\scriptscriptstyle \frac{\mathrm{1}}{n}}}=a+b{Q}^{{\scriptscriptstyle \frac{\mathrm{1}}{n}}}\end{array}$$

From a numerical point of view, the relationship presented in
Eq. (3) is equivalent to first transforming
the dependent (*C*) and independent (*Q*) variables using a so-called
Box–Cox transformation (Box and Cox, 1964), and then adjusting a
linear model. In comparison with the logarithmic transformation, the
additional degree of freedom offered by *n* allows for a range of
transformations, from the untransformed variable (*n*=1) to the logarithmic
transformation (*n*→∞). This “progressive” property was
underlined long ago by Box and Cox (1964): when *n* takes high values,
Eq. (3) converges toward the one-sided power
scaling relationship (power law) (Eq. 1). The
reason is simple:

$${C}^{{\scriptscriptstyle \frac{\mathrm{1}}{n}}}={e}^{{\scriptscriptstyle \frac{\mathrm{1}}{n}}\mathrm{ln}C}\approx \mathrm{1}+{\displaystyle \frac{\mathrm{1}}{n}}\mathrm{ln}C$$

when *n* is large.

Thus, for large values of *n*, Eq. (3) can be written
as

$$\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{n}}\mathrm{ln}C\approx a+b+{\displaystyle \frac{b}{n}}\mathrm{ln}Q.$$

That is equivalent to

$$\mathrm{ln}C\approx A+b\mathrm{ln}Q\phantom{\rule{1em}{0ex}}(\text{with}A=n(a+b-\mathrm{1}\left)\right).$$

The progressive behavior and the convergence toward the log–log space are clearly evident in Fig. 2.

Because the hydro-biogeochemical processes that control the transport and
reaction of ions are different, different ionic species may have a *C*–*Q*
relationship of distinct shape (Moatar et al., 2017). In
Fig. 3, we show the behavior of three ions and the
EC from the same catchment and the same dataset (all four from the
Orgeval-ORACLE observatory) with different transformations (*n*=1, 3, 5 and
logarithmic transformation). The optimal shape was chosen numerically: we
transformed our data series of *C* and *Q* using different values of *n*
(i.e., ${C}^{\ast}={C}^{\mathrm{1}/n}$ and ${Q}^{\ast}={Q}^{\mathrm{1}/n}$) and logarithmic
transformation (i.e., ${C}^{\ast \ast}=\mathrm{log}\left(C\right)$ and ${Q}^{\ast \ast}=\mathrm{log}\left(Q\right)$). With these transformed values, we performed
a linear regression and computed parameter *a* and *b* and the coefficient
of determination (*R*^{2}) (see Table 2). The *n*
considered as optimal has the highest *R*^{2} value (see
Table 2). However, we could also have followed the
advice of Box et al. (2016, p. 331) and done it visually
(Fig. 3).

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

4 Numerical identification of the parameters for the 2S-APS relationship

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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^{3} s^{−1}], and they correspond to the
lowest concentrations; see Fig. 1).

To address this question, we successively tested a large number of (*a*,*b*)
pairs from Eq. (3) (*n* remaining fixed at the
optimal value given in Table 2). Each pair yields a
series of simulated concentrations (*C*_{sim}) that can be compared with the
observed concentrations (*C*_{obs}). Among the many numerical criteria that
could be used, we chose the bounded version of the Nash and Sutcliffe (1970) efficiency criterion NSEB (Mathevet et al., 2006), which is commonly
used in hydrological modeling. NSEB can be computed on concentrations or on
discharge-weighted concentrations (which corresponds to the load). We chose
the average of both, because we found that it allows more weight to be given
to the extremely low concentrations and thus to avoid the issue of
under-representation of high-discharge/low-concentration measurement points.
Table 3 presents the formula for these numerical
criteria.

We retained as optimal the pair of (*a*,*b*) that yielded the highest
NSEB_{comb} value (we explored in a systematic fashion the range [1–5]
for *a* and [−1.2–1.2] for *b*).

In Appendix A2, we show that our proposed methodology for the identification
of parameters *a*, *b* and *n*, based on the NSEB_{comb}
criterion, is effective also from the point of view of the predictive
confidence interval.

5 Results

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The optimal values of *a* and *b* corresponding to the simulation of each
ion and EC with the highestNSEB_{comb} criterion and the *n* value
identified in Fig. 3 and
Table 2 are presented in Table 4.

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.

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 (*F* test) was performed between the
RMSE obtained for the two relationships: because of the very large number of
points in our dataset, all differences were highly significant (*p* value <0.001).

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.

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.

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.

6 Conclusion

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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.

Appendix A

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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 ([Mg^{2+}], [K^{+}],
[Ca^{2+}], [Na^{+}], [Sr^{2+}], [F^{−}], [${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$]
[${\mathrm{NO}}_{\mathrm{3}}^{-}$], [Cl^{−}], [${\mathrm{PO}}_{\mathrm{4}}^{\mathrm{3}-}$]). In addition, a set of
physico-chemical probes is deployed to measure pH, conductivity, dissolved
O_{2}, dissolved organic carbon (DOC), turbidity, and temperature. The
discharge is measured continuously via a gauging station located at the
River Lab site.

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.

Data availability

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Data availability.

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

Author contributions

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Author contributions.

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.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

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.

Financial support

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Financial support.

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

Review statement

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Review statement.

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

References

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Short summary

This paper 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, based on high-frequency chemical time series of the Orgeval-ORACLE observatory.

This paper deals with the mathematical representation of concentration–discharge relationships....

Hydrology and Earth System Sciences

An interactive open-access journal of the European Geosciences Union