José Manuel

Tunqui Neira

Gaëlle Tallec

Vazken Andréassian

Mouchel Jean-Marie

José Manuel Tunqui Neira

Jean-Marie Mouchel

A combined mixing model for high-frequency concentration-discharge relationships

Keywords: Concentration-discharge relationships, high-frequency measurements, mixing equations

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Concentration-discharge ( -) relationships are interesting for a variety of purposes and potential users and because they employ simple approaches to describe complex hydro-chemical interactions, hydrologists and geochemists have been using and exploring them for over 70 years (see [START_REF] Chanat | Consistency of patterns in concentration-discharge plots[END_REF][START_REF] Durum | Relationship of the mineral constituents in solution to stream flow, Saline River near Russell, Kansas[END_REF][START_REF] Hem | Fluctuations in concentration of dissolved solids of some southwestern streams[END_REF][START_REF] Johnson | A working model for variation in stream water chemistry at Hubbard-Brook-experimental-forest, new-hampshire[END_REF][START_REF] Kirchner | Quantifying new water fractions and transit time distributions using ensemble hydrograph separation: theory and benchmark tests[END_REF][START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF]. The established relations can be purely empirical, used to infill lacking concentration data and compute fluxes over long time periods, or to characterize the regime export of chemical components in a number of catchments and empirically derive euristic patterns on a regional and global scale (e.g. [START_REF] Bieroza | The concentrationdischarge slope as a tool for water quality management[END_REF][START_REF] Meybeck | Daily variability of river concentrations and fluxes: indicators based on the segmentation of the rating curve[END_REF][START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF]. In some cases, the relations have been derived from a more complex model, describing a physical representation of water circulation and composition, from various mixing sources, with the objective to derive relevant information on internal functionning [START_REF] Johnson | A working model for variation in stream water chemistry at Hubbard-Brook-experimental-forest, new-hampshire[END_REF].

To modelrelationships, two classical approaches exist:

• The temporalpatterns relationship (i.e. fit parameters of vs , [START_REF] Musolff | Emergent archetype patterns of coupled hydrologic and biogeochemical responses in catchments[END_REF] try to explain the processes controlling the mobilization and delivery of chemical elements into streams (i.e. export regimes): dilution, flow-enhancement, or no-variation at all (see e.g. [START_REF] Salmon | Hydrological controls on chemical export from an undisturbed old-growth Chilean forest[END_REF], as well as biogeochemical transformations in river networks [START_REF] Minaudo | Seasonal and event-based concentration-discharge relationships to identify catchment controls on nutrient export regimes[END_REF]. These models are, first and foremost, based on observations of concentrations relatively stable over time, as electro-conductivity (EC) or chlorides [START_REF] Durum | Relationship of the mineral constituents in solution to stream flow, Saline River near Russell, Kansas[END_REF][START_REF] Hem | Fluctuations in concentration of dissolved solids of some southwestern streams[END_REF]. Often, the main objective is to identify the groundwater contribution to streamflow [START_REF] Durum | Relationship of the mineral constituents in solution to stream flow, Saline River near Russell, Kansas[END_REF]. These models are based on measurable quantities (discharge and in-stream chemical concentrations) [START_REF] Durum | Relationship of the mineral constituents in solution to stream flow, Saline River near Russell, Kansas[END_REF][START_REF] Hem | Fluctuations in concentration of dissolved solids of some southwestern streams[END_REF][START_REF] Tunqui Neira | Technical note: A two-sided affine power scaling relationship to represent the concentration-discharge relationship[END_REF] (see example, Eq. (1) and Eq. ( 2) in Table 1).

• The more complex n-component models have in addition the aim of quantifying the sources of the chemical concentrations measured in the river [START_REF] Barthold | How many tracers do we need for end member mixing analysis (EMMA)? A sensitivity analysis[END_REF]. Introduced at the end of the 1960s [START_REF] Hubert | Aspects hydrologique, géochimique et sédimentologique de la crue exceptionelle de la Dranse du Chablais du 22 sept. 1968[END_REF][START_REF] Johnson | A working model for variation in stream water chemistry at Hubbard-Brook-experimental-forest, new-hampshire[END_REF][START_REF] Pinder | Determination of the ground-water component of peak discharge from the chemistry of total runoff[END_REF], they are based on the chemical contribution of hydrological sources (i.e. groundwater, runoff, precipitation), assuming each one can be characterized by a constant concentration and assuming the existence of a methodology to identify the relative contribution of each source to streamflow. This mixing approach is based on the mass balance equation (see its simplest expression, Eq. (3), Table 1). [START_REF] Hall | Dissolved solids-discharge relationships .1. Mixing models[END_REF] gives an exhaustive presentation of these models, their different expressions and assumptions. The mixing model was applied at different time scale from flood and event scale [START_REF] Hubert | Aspects hydrologique, géochimique et sédimentologique de la crue exceptionelle de la Dranse du Chablais du 22 sept. 1968[END_REF][START_REF] Pinder | Determination of the ground-water component of peak discharge from the chemistry of total runoff[END_REF] to annual scale [START_REF] Johnson | A working model for variation in stream water chemistry at Hubbard-Brook-experimental-forest, new-hampshire[END_REF] and for different purposes such as hydrograph separation [START_REF] Pinder | Determination of the ground-water component of peak discharge from the chemistry of total runoff[END_REF], understanding of flowpaths in catchment [START_REF] Johnson | A working model for variation in stream water chemistry at Hubbard-Brook-experimental-forest, new-hampshire[END_REF], or the dynamics of the components of storm flow [START_REF] Hubert | Aspects hydrologique, géochimique et sédimentologique de la crue exceptionelle de la Dranse du Chablais du 22 sept. 1968[END_REF]. Despite their simplicity, the power-law models can yield excellent fits for some ions, explaining up to 90 % of the variance of the concentrations. For this reason, they are still widely used today [START_REF] Barco | Linking hydrology and stream geochemistry in urban fringe watersheds[END_REF][START_REF] Godsey | Concentration-discharge relationships reflect chemostatic characteristics of US catchments[END_REF][START_REF] Moatar | Riverine fluxes of pollutants: Towards predictions of uncertainties by flux duration indicators[END_REF][START_REF] Probst | Solute and particulate transports by the upstream part of the Garonne river[END_REF]. However, several authors have criticized the power-law models, underlining that they lump different hydrochemical processes and dynamics, and do not allow for an explicit physical flow-path interpretation [START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF][START_REF] Rose | Concentration-discharge relationships describe solute and sediment mobilization, reaction, and transport at event and longer timescales[END_REF]. [START_REF] Godsey | Concentration-discharge relationships reflect chemostatic characteristics of US catchments[END_REF] compared differentmodels, and concluded that it is difficult to find simple generalizable models that accurately represent the typical shape of therelationship, which are internally consistent, and make plausible assumptions about catchment behavior.

The variability ofrelationships could be due to a large number of processes, varying in space, time and with the characteristics of the catchment. The exact mechanisms leading torelations remain an open question. Both natural and anthropogenic factors affect the biogeochemical response of streams, and, while the majority of solutes show identifiable behaviors in individual catchments, only a minority of behaviors can be generalized [START_REF] Botter | Anthropogenic and catchment characteristic signatures in the water quality of Swiss rivers: a quantitative assessment[END_REF]. Some recent studies attempt to differentiate two distinct behaviors: chemodynamic and chemostasic export.

Chemostasic export is defined as relatively small variations in concentrations compared to discharge [START_REF] Musolff | Catchment controls on solute export[END_REF][START_REF] Thompson | Relative dominance of hydrologic versus biogeochemical factors on solute export across impact gradients[END_REF]. Chemostasis could be associated with constant rate of chemical weathering [START_REF] Godsey | Concentration-discharge relationships reflect chemostatic characteristics of US catchments[END_REF], "a legacy storage" of anthropogenic nutriment which buffer the variability in concentrations [START_REF] Basu | Nutrient loads exported from managed catchments reveal emergent biogeochemical stationarity[END_REF][START_REF] Clow | Mechanisms for chemostatic behavior in catchments: Implications for CO2 consumption by mineral weathering[END_REF], or with a significant hydraulic residence time compared to weathering kinetics [START_REF] Ameli | Primary weathering rates, water transit times, and concentration-discharge relations: A theoretical analysis for the critical zone[END_REF][START_REF] Maher | The role of fluid residence time and topographic scales in determining chemical fluxes from landscapes[END_REF].

Chemostasic export processes could represent the long-term trend of basin chemistry and can therefore be identified as an average of concentrations [START_REF] Musolff | Catchment controls on solute export[END_REF]. Chemodynamic patterns are characterized as high variations in concentrations compared to discharge caused by flushing, enrichment behavior, or dilution behavior [START_REF] Musolff | Catchment controls on solute export[END_REF]. It can be controlled: (i) by activation of solute sources heterogeneously distributed in space; (ii) by threshold-driven transport of constituents or (iii) high reactivity of constituents [START_REF] Jones | Nitrate transport and supply limitations quantified using high-frequency stream monitoring and turning point analysis[END_REF][START_REF] Musolff | Catchment controls on solute export[END_REF][START_REF] Vaughan | High-frequency dissolved organic carbon and nitrate measurements reveal differences in storm hysteresis and loading in relation to land cover and seasonality[END_REF][START_REF] Zhang | An improved method for interpretation of riverine concentration-discharge relationships indicates long-term shifts in reservoir sediment trapping[END_REF]. We could also note that [START_REF] Zhang | An improved method for interpretation of riverine concentration-discharge relationships indicates long-term shifts in reservoir sediment trapping[END_REF] emphasized that temporal patternsmodels (including the parameter of Eq. ( 1)) may vary over time or with season, and a single b coefficient for the entire period of record is over-simplification and potentially misleading. The other difficulty of a generalization is also the difference in behavior and source of the different ions. Beyond the patterns, changes can occur whether due to weathering, meteorological or anthropic processes, affecting one ion more than another [START_REF] Knapp | Concentration-discharge relationships vary among hydrological events, reflecting differences in event characteristics[END_REF][START_REF] Musolff | Emergent archetype patterns of coupled hydrologic and biogeochemical responses in catchments[END_REF][START_REF] Rose | Concentration-discharge relationships describe solute and sediment mobilization, reaction, and transport at event and longer timescales[END_REF][START_REF] Zhi | Distinct Source Water Chemistry Shapes Contrasting Concentration-Discharge Patterns[END_REF].

To deal with the non-linearity of therelationship, some authors have proposed to increase the number of hydrological components (e.g. [START_REF] Evans | Causes of concentration/discharge hysteresis and its potential as a tool for analysis of episode hydrochemistry[END_REF][START_REF] Probst | Nitrogen and Phosphorus exportation in the Garonne basin (France)[END_REF], make specific catchment calibration [START_REF] Godsey | Concentration-discharge relationships reflect chemostatic characteristics of US catchments[END_REF], consider flow classes, e.g. low and high flows [START_REF] Meybeck | Daily variability of river concentrations and fluxes: indicators based on the segmentation of the rating curve[END_REF][START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF] or integrate significant time steps [START_REF] Kirchner | Quantifying new water fractions and transit time distributions using ensemble hydrograph separation: theory and benchmark tests[END_REF]. To gain insight into the linkages between chemical and hydrologic processes that yield the observed concentration-discharge relationship, research has often focused on only one of the modeling solutions (i.e. on either power-law model or mixing model).

The temporal patterns (dilution, constant, enrichment) or the mass balance (end-members mixing)

approaches are operational and widely used to describerelationships. There are also other compound methods in the literature, based on solute production models [START_REF] Ibarra | Differential weathering of basaltic and granitic catchments from concentration-discharge relationships[END_REF][START_REF] Maher | The role of fluid residence time and topographic scales in determining chemical fluxes from landscapes[END_REF][START_REF] Maher | Hydrologic regulation of chemical weathering and the geologic carbon cycle[END_REF] or on end-members models [START_REF] Bao | Understanding watershed hydrogeochemistry: 1. Development of RT-Flux-PIHM[END_REF][START_REF] Bouchez | River Mixing in the Amazon as a Driver of Concentration-Discharge Relationships[END_REF][START_REF] Zhi | Distinct Source Water Chemistry Shapes Contrasting Concentration-Discharge Patterns[END_REF]. However, none offers a parsimonious and generalized approach as proposed in this paper. To date, no simple conceptual model still allows combining a dynamic of concentrations with a dynamic of water flow. The new model proposed in this paper is trying to solve this problem, by combining the two approaches (i.e., power-law and mixing model). The originality of our approach is that we do not proceed by assuming an end-member concentration value to perform the baseflow-quickflow separation, but we use instead the inverse approach, an a priori assumption of the baseflow-quickflow using a hydrological hydrograph separation method to infer the source concentration values. All the parameters of the model proposed depend on the temporal variation of the stream discharge. Using the high frequency series of concentrations and discharges, observed during two years on the ORACLE-Orgeval observatory (France), we have tested the performances of this new model at different hydrological time scales and on a full range of flow and concentration.

The first aim of this paper is to present the model and its validation on one long high-frequency timeseries. The second aim is to discuss of its performances with regards to the classical models. Note that this paper is methodological: it does not focus on the different hydro-chemical processes involved in the composition of the stream solutes or on others watershed functioning considerations.

The combined mixing model

The combined model tested here associates arelationship with a two-component mixing equation:

• Asrelationship, we use the two-sided affine power scaling relationship (2S-APS), which we recently proposed to fit therelationship, as a natural extension of the wellknown and widely-used power law relationship (see Tunqui Neira et al., 2020);

• As mixing equation, we use the simplest of the existing schemes, i.e. a two-component equation distinguishing "base flow" and "quick flow" (e.g [START_REF] Pinder | Determination of the ground-water component of peak discharge from the chemistry of total runoff[END_REF].

To combine the 2S-APS relationship with the two-component mixing equation (see Eq. ( 2) and Eq.

(3), Table 1), we propose to write the concentrations of each of components in Eq. (3) (i.e. and ) as a function of total discharge, using the same transformation:

= + = +
The total discharge (and not the discharge of each of the two hydraulic components) was used to seek simplification, not increasing too much the number of parameters. We will discuss later the value of this simplification. Then, the mixing equation becomes:

= + + +
Eq. ( 4)

The "combining" procedure merges the two most used forms of equations of the hydro-chemical literature (i.e. Eq. ( 2) and Eq. ( 3)). The advantage of this new model is to improve the mixing approach by introducing dynamic concentrations for the sources.Depending on the values taken by parameters and in Eq. ( 4), three schematic situations can be identified, briefly described below and illustrated by Figure 1.

2.1. Case 1: chemostatic components (bb = bq = 0)

In this case, and are constant and independent of river discharge. Thus, is only influenced by the discharge components dynamics ( and ) (see Case 1 in Figure 1). The case 1 represents the chemostatic export characterizing a catchment controlled by large legacy stores (see [START_REF] Musolff | Catchment controls on solute export[END_REF]. It corresponds to the assumption of the classical mixing equation approach, where the and values can assume for example the average of concentrations, measured respectively during the dry and wet seasons (i.e. Saraiva [START_REF] Saraiva Okello | Hydrograph separation using tracers and digital filters to quantify runoff components in a semi-arid mesoscale catchment[END_REF][START_REF] Stewart | Calibration of base flow separation methods with streamflow conductivity[END_REF][START_REF] Zhang | Baseflow separation in a small watershed in New Brunswick, Canada, using a recursive digital filter calibrated with the conductivity mass balance method[END_REF].

Case 2: single 2S-APS relationship (ab =aq = a and bb = bq = b)

This case reduces to = (Case 2 in Figure 1) and to the simplerelationship:

= +

The observed concentration is only a function of discharge.

Case 3: General case (a and b are different)

This is the general case of the transformed mixing equation (Eq. ( 4)). Unlike the classical mixing equation (chemostatic components), the and values are not constant but vary individually as a function of stream flow (see Case 3 in Figure 1). The general case allows accounting for the temporal variation of the chemical components. It allows connecting concentration with total discharge and the hydrograph separation components ( and , i.e. with = -, see Table 1).

Figure 1: The three hypotheticalscenarios due to the values taken by parameters and of Eq. ( 4): case 1, case 2 and case 3. The green hatching represents the probable yields by our model in thespace.

Material and methods

Study site and datasets

The combined model (Eq. ( 4)) was applied to the high-frequency hydro-chemical dataset measured at the Oracle-Orgeval Observatory [START_REF] Tallec | Observatoire Oracle[END_REF] by the River Lab [START_REF] Floury | The potamochemical symphony: new progress in the high-frequency acquisition of stream chemical data[END_REF]. The Oracle-Orgeval Observatory is a small catchment located 70 km east of Paris, France. It is subjected to a temperate and oceanic climate, with annual average temperature of 11 ± 1 °C and a mean annual rainfall of 674 ± 31 mm [START_REF] Tallec | Introduction[END_REF]. The average measured streamflow at the Avenelles outlet (sub-catchment of 46 km 2 and location of the River Lab on the Oracle-Orgeval Observatory) is about 0.2 m 3 /s (1962 -2017), with minimum flows in summer (< 0.1 m 3 /s) and floods up to 10 m 3 /s in winter and spring. With respect to geology, the catchment is underlain entirely by limestone rocks, with two aquifers: the shallower aquifer of the Brie limestone and the deeper Champigny limestone aquifer [START_REF] Mouhri | Designing a multi-scale sampling system of stream-aquifer interfaces in a sedimentary basin[END_REF]. Land use is mostly agricultural with few villages, and with intensive farming practices, mainly based on mineral nitrogen fertilization [START_REF] Garnier | Reconnecting crop and cattle farming to reduce nitrogen losses to river water of an intensive agricultural catchment (Seine basin, France): past, present and future[END_REF]. Nearly 60% of the surface of the catchment is drained with tile drains. Among all ions measured every 30 minutes by the River Lab laboratory, we used 3 ions (see Table 2) whose behavior exhibit notable differences in the Orgeval catchment: sulfate, nitrate and chloride.

Chloride mainly come from rain inputs during the wet season and to a smaller extent from fertilizers [START_REF] Floury | Chemical weathering and CO2 consumption rate in a multilayered-aquifer dominated watershed under intensive farming: The Orgeval Critical Zone Observatory, France[END_REF]. Sulfate comes from the chemical weathering of gypsum, which makes it highly variable, depending on the season and the leaching of concerned localized underground layers [START_REF] Floury | Chemical weathering and CO2 consumption rate in a multilayered-aquifer dominated watershed under intensive farming: The Orgeval Critical Zone Observatory, France[END_REF][START_REF] Mouchel | Caractérisation de la géochimie des interfaces nappe-rivière du bassin des Avenelles[END_REF]. Nitrate mostly come from agricultural activities and fertilizers inputs, with specific seasonal leaching rates [START_REF] Garnier | Reconnecting crop and cattle farming to reduce nitrogen losses to river water of an intensive agricultural catchment (Seine basin, France): past, present and future[END_REF]. Finally, we also used electro-conductivity reflecting the presence of all ions in stream water (EC, see Table 2). The main data set (flow rates and chemical concentrations) covers the period between June 2015 and March 2018 (Table 2), i.e. 20,700 measurements over 33 months. To perform a split-sample test, according to [START_REF] Klemeš | Dilettantism in hydrology: Transition or destiny?[END_REF], we divided the dataset into two periods. Table 2 presents a first period used for the model calibration and a second period for the model validation.

Hydrograph separation

To apply the combined mixing model (Eq. ( 4)), we have to separate the hydrograph in order to compute the values of base flow ( ) and quick flow ( ). To perform the baseflow-quickflow separation (because we do not know the source concentration values) we use an a priori assumption of the baseflow-quickflow separation to infer the source concentration values (and not the usual inverse approach).

In this study, we use the Recursive Digital Filter (RDF) hydrograph separation approach. The RDF approach, adapted in the late 1970s from the signal-processing theory, is widely applied for hydrograph separation. Indeed, RDF methods are computationally efficient, easily automated and applied to long continuous streamflow records [START_REF] Chapman | Comment on "Evaluation of automated techniques for base flow and recession analyses[END_REF][START_REF] Eckhardt | How to construct recursive digital filters for baseflow separation[END_REF]. Among all the RDFmethods existing in literature (Brodie et al., 2007 p.62), we used the well-known Lyne-Hollick method (LH-RDF method) [START_REF] Lyne | Stochastic time-variable rainfall-runoff modelling[END_REF][START_REF] Nathan | Evaluation of automated techniques for base flow and recession analyses[END_REF]. Base flow is considered here as a low-frequency signal, and surface runoff as a high-frequency signal. By filtering out the highfrequency signal, the low-frequency signal (i.e. base flow) can be revealed [START_REF] Longobardi | Assessing baseflow index vulnerability to variation in dry spell length for a range of catchment and climate properties[END_REF][START_REF] Nathan | Evaluation of automated techniques for base flow and recession analyses[END_REF].

The LH-RDF method is defined as follows:

= min + ! " # $ + %, ' Eq. (5)
where , , and ( are respectively, the baseflow, the total flow, the LH-RDF filter parameter and the time.

Iterative application of the filter allows smoothing data and nullifying phase distortion. We have used the forward-backward-forward application proposed by [START_REF] Nathan | Evaluation of automated techniques for base flow and recession analyses[END_REF]. The LH-RDF method is characterized by one parameter ( ) which defines the speed of convergence of the filter.

It is a common practice (e.g. [START_REF] Longobardi | Hydro-geo-chemical streamflow analysis as a support for digital hydrograph filtering in a small, rainfall dominated, sandstone watershed[END_REF][START_REF] Zhang | Evaluating relative merits of four baseflow separation methods in Eastern Australia[END_REF] to adapt the filter parameter ( ) to the hydrological recession time constant of the catchment (τ). Otherwise, either or

would have an unwanted behavior on a seasonal time scale (too slow convergence of or too fast decrease of during flood events). Another important reason for this adaptation, is that the default value of (0.925, proposed by [START_REF] Nathan | Evaluation of automated techniques for base flow and recession analyses[END_REF] applied in small catchments controlled by the regional scale factors such as slope and shape, has shown poor performance when computing [START_REF] Ladson | A standard approach to baseflow separation using the Lyne and Hollick filter[END_REF][START_REF] Zhang | Evaluating relative merits of four baseflow separation methods in Eastern Australia[END_REF].

During seasons without significant recharge, stream flow may recess exponentially and follows the form:

Q ) = . exp ./ 0 = . 1 Eq. ( 6
)
where 1 is the so-called recession constant of the catchment.

The recession constant 1 can be obtained using the master recession curve (MRC) approach [START_REF] Nathan | Evaluation of automated techniques for base flow and recession analyses[END_REF]. A linear regression (i.e. plot Q vs Q see Figure 2A) allows establishing the recession constant K, which also represents the parameter of LH-RDF method (Eq. ( 5)). This analysis was applied on daily flow data of the Avenelles station from January 1, 2000 to September, 2018 as follows: the daily stream flow data of several seasons were overlapped according to the day of the year, starting from the beginning of June, and assuming that the stream flow decreases under a continuous recession process for the period June-September. The 1 value obtained about 0.90 has been calculated for a daily discharge (see Figure 2A). However, for high frequency discharge measurements (i.e. time step of 30 min or 0.5 hours), this value must be transformed at the appropriate time step as proposed by [START_REF] Eckhardt | A comparison of baseflow indices, which were calculated with seven different baseflow separation methods[END_REF] :

1 2.34 = $1 5 % 2.3 #6 7 = 0.99
With the LH-RDF method (i.e., Eq. ( 5)) and the calibrated value of 1 (i.e. 0.99 for the Avenelles catchment), we computed the values of the base flow ( ) and the quick flow ( ) (see example of the results Figure 2B). We should not fail to mention here that the baseflow-quickflow methods have been widely criticized:

see e.g. [START_REF] Beven | Hydrograph separation?[END_REF] for a vibrant indictment against them and the arguments raised by [START_REF] Pelletier | Interactive comment on "Hydrograph separation: an impartial parametrization for an imperfect method[END_REF] in the discussion of their paper, for a careful use of them. We do not wish to enter this debate. We will not claim to have identified a precise physical pathway and in this paper, we will thus limit our ambition to the identification of what Pelletier and Andréassian (2020) have named the "not-too-delayed" and the "delayed-enough" flow components (baseflow and quickflow).

Calibration of :, ; <=> ? parameters

The combined mixing model (Eq. ( 4)) was applied with the baseflow calculated by the LH-RDF method. For each ion and the EC, we used the parameter @ previously determined by Tunqui Neira et al. ( 2020) on the same data set, but without flow separation.

The extremely large number of values in this high-frequency dataset may cause problems for a robust identification of ( , ) parameters, 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 discharge values are in the range [2.6 m 3 s -1 , 12.2 m 3 s -1 ], and they correspond to the lowest concentrations). To address this question, we successively tested a large number of ( , ) pairs from Eq. (4) (@ remaining fixed). Each pair yields a series of simulated concentrations ( ) that can be compared with the observed concentrations ( A ).

Among the many numerical criteria that could be used, we chose the bounded version of the Nash and Sutcliffe (1970) efficiency criterion BCDE [START_REF] Mathevet | A bounded version of the Nash-Sutcliffe criterion for better model assessment on large sets of basins[END_REF], which is commonly used in hydrological modeling. BCDE 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 lowest concentrations and thus to avoid the issue of under-representation of high-discharge/low-concentration measurement points. If other criteria exists to test the model performances, the BCDE FA has the advantage to privilege neither the load nor the concentration and integrates both in a single criterion. Table 3 presents the formula for these numerical criteria. We retained as optimal the pair of ( , ) that yielded the highest BCDE FA value (we explored in a systematic fashion the range [1-5] for and [-1.2-1.2] for ). 2006) transforms NSE into NSEB, which varies between -1 and 1 (its optimal value). The advantage of this rescaled version is to avoid the occurrence of large negative values (the original NSE criterion varies in the range [-∞, 1]).

BCD FA F = 1 - ∑ A - # ∑ A -A IIIIII #
Eq. ( 7)

BCDE FA F = BCD FA F 2 -BCD FA F
Eq. ( 8)

BCD KALM = 1 - ∑ A - # ∑ A -A IIIIIIII #
Eq. ( 9)

BCDE KALM = BCD KALM 2 -BCD KALM
Eq. ( 10)

BCDE FA = 1 2 BCDE FA F + BCDE KALM Eq. ( 11
)
For each ion and for EC, Table 4 presents the results obtained for the @, and parameters calibrated for the Avenelles sub-catchment and their corresponding optimal BCDE FA criterion.

Although the parameters of the combined mixing model are shown here, we will not discuss them in this paper. Indeed, for now, without generalized studies covering several catchments it is not possible to interpret them physically. Note that for these reasons we have chosen to present Table 4 in this section and not in the Results and discussion section.

Performances of the model

We evaluate the combined model performance in calibration and validation mode. Due to the temporal density of the dataset, we have tested the performances of the models in calibration mode on widely different discharge ranges: over the entire calibration period and for selected storm events (from June 2015 to July 2017, see Table 2, Chapter 3.1). In validation mode, the performances have only been assessed over the entire period (from August 2017 to March 2018, see Table 2,Chapter 3.1).

The X Y and the standardized root mean square error (YZ[CD) allow assessing respectively accuracy and precision of the combined mixing model. Table 5 presents the formula for these numerical criteria.

Table 5: Numerical criteria used for model performance comparison (Cobs -observed concentration, Csim -simulated concentration, ( -the time step, B -the number of observed concentration)

X Y % = 100 * ∑ -A ∑ A
Eq. ( 13)

YZ[CD % = 100 * ^1 B ∑ A -FLK # A
Eq. ( 14)

Results and discussion

Performance of the combined model over the entire period

Table 6 presents the evaluation of the combined mixing model, in calibration and validation mode over the entire period. Whatever the mode (calibration or validation) and whatever the solute species considered, the optimal YZ[CD and X Y values are obtained for the general case (Case 3, see Table 6). The results also show that a single 2S-APS relationship (Case 2) explains better the variations of the stream water concentrations than a mass balance equation with constant concentration components (Case 1) (see YZ[CD and X Y values from case 1 to case 2, Table 6). This means that -within the adopted modelling framework and in as much as the hydrograph separation can be considered as hydrologically relevant -the concentrations of the flow components cannot be considered constant across time (i.e., the behavior of the Avenelles catchment is nonchemostatic). The stream water quality of the Avenelles sub-catchment appears strongly influenced by discharge processes.

The most evident improvement of the case 3 model is observed for chloride and EC (see YZ[CD of 5 % with a negative X Y less than 0.5% in calibration mode, Table 6). Less importantly, sulfate YZ[CD is also much improved when variable concentrations are introduced in the quick and slow components (see YZ[CD around 10%, Case 3, in calibration and validation mode Table 6). In calibration mode, the sulfate X Y is comparable to that obtained for chloride and EC (less than 0.5%, see Table 6).

According to [START_REF] Floury | Chemical weathering and CO2 consumption rate in a multilayered-aquifer dominated watershed under intensive farming: The Orgeval Critical Zone Observatory, France[END_REF], chloride would come from the Brie aquifer with mainly external input from rainfall during the wet season. According to [START_REF] Mouchel | Caractérisation de la géochimie des interfaces nappe-rivière du bassin des Avenelles[END_REF], the variability of sulfate concentrations is related to the spatial heterogeneity of gypsum lenses and to the temporary variability of the water table heights. The spatial and temporal variability of sulfate and chloride concentrations is due to several chemo-dynamical processes, well modeled by the general case (case 3).

The lowest values of the general case are obtained for nitrate (see YZ[CD about 27.6% and a X Y about -11.2%, case 3, in calibration mode, and both > 100 % in validation mode, Table 6). Nitrate present the largest seasonal variability, strongly related to farming practices [START_REF] Garnier | Reconnecting crop and cattle farming to reduce nitrogen losses to river water of an intensive agricultural catchment (Seine basin, France): past, present and future[END_REF] and chemo-dynamic processes especially in the hyporheic-zone [START_REF] Floury | Chemical weathering and CO2 consumption rate in a multilayered-aquifer dominated watershed under intensive farming: The Orgeval Critical Zone Observatory, France[END_REF]. This larger variability, either environmental or anthropogenic, cannot be simulated by our model, whatever the case.

Figure 3 illustrates the simulations of the models, in the first period of calibration, in regard with the observed time series. As theoretically expected (see Figure 1), the simulated concentrations are more or less dispersed, from linear case 2 to very dispersed case 1 (see Figure 3). The case 3, as shown previously by the YZ[CD and the X Y (see Table 6), is the one closest to the observed concentrations. For the nitrate, while cases 1 and 2 seem to overestimate the concentrations, case 3 has the advantage, despite its poor performance, of covering all the nitrate concentration values.

A significant feature, common to almost all ions (except nitrate and sulfate) and cases of the model, is the lower concentration, compared to (see black lines vs pink lines, Figure 3). For the nitrate ion, concentration becomes a more important contributor in the three cases, until becoming preponderant in the case 3 (see nitrate, pink line Figure 3). For sulfate, is the main contributor to the stream concentration for discharges greater than ~ 0.5 m 3 .s -1 (see sulfate, pink lines Figure 3).

This result confirm studies carried out at the Orgeval-ORACLE observatory [START_REF] Billy | Factors controlling nitrate concentrations in surface waters of an artificially drained agricultural watershed[END_REF][START_REF] Floury | Chemical weathering and CO2 consumption rate in a multilayered-aquifer dominated watershed under intensive farming: The Orgeval Critical Zone Observatory, France[END_REF][START_REF] Garnier | Reconnecting crop and cattle farming to reduce nitrogen losses to river water of an intensive agricultural catchment (Seine basin, France): past, present and future[END_REF][START_REF] Mouchel | Caractérisation de la géochimie des interfaces nappe-rivière du bassin des Avenelles[END_REF][START_REF] Mouhri | Designing a multi-scale sampling system of stream-aquifer interfaces in a sedimentary basin[END_REF]. Indeed, according to the results, the principal contributor to the stream concentration in the Avenelles catchment would be first groundwater (i.e. ) for chloride and EC. During the wet season (i.e. for the discharge beyond 0.5 m 3 .s -1 ) for the sulfate and whatever the season for nitrate, the principal contributor is the quickflow concentrations (i.e. ). However, the combined mixing model does not seem to be able to consider the quickflow concentrations in a good way; i.e. integrating either the variations of the subsurface hydrological compartment or the variations of concentrations linked to this compartment.

Figure 3: Simulated concentration ( , in blue) obtained of each case of our combined method plotted in thespace for the three ions and EC in the calibration period. In addition, we have plotted the computed values for (black line) and (pink line) for each case studied. Note that in case 2 the lines of and overlap each other. Grey points represent the observed concentrations.

Performances of the combined model for selected storm events

We selected two representative flood events of short duration, at the end of fall (November 2015, low groundwater level, low soil moisture and no contribution of tile drains, see Figure 4A) and during winter (February 2016, high groundwater level, high soil moisture and tile drains contribution, see Figure 4B). A longer wet season flood event of March 2016, well covered during 15 days by the concentration data set, is also added (see Figure 4C). Figure 4 presents observed data and the three cases of simulations with the combined mixing model as a function of time for the three events. For the three selected storm events, Table 7 presents the YZ[CD and X Y values obtained in a calibration mode with the three cases of the combined model.

At the event scale, except for nitrate during the dry season, we obtain the same results than with the entire period: the case 3 shows the optimal YZ[CD and X Y values for all ions and EC (see YZ[CD and X Y from case 1 to case 3, Table 7). The potential advantage of the general case of the combined mixing model (case 3) is to decouple rising and decreasing flow periods with changing / and / ratios (see hydrograph separation, Figure 4). The recursive filter is efficient in separating the base flow from the quick flow. It simulates a quick response of the quick flow at the beginning of flood periods, and a progressive increase of the base flow during the second part of flood periods. Accordingly, the two components model improves the simulation of complex concentration changes during flood events.

We notice that the best results are obtained for the longest flood period: the performances of the case 3 are improved for the long event compared to short events (see YZ[CD and X Y from short event to long event during wet period, Table 7). As shown in Figure 4C, because of the longer duration of the flood, the recursive filter simulates important variations of the / ratio during the event: during most of peak flow period, most of the discharge is due to the quick flow component, while the quick flow component turns back to zero after day 10. As expected from the wider range of / ratio, the model simulates more difference between concentrations during the rising part of the event than the descending part (Figure 4C).

We also observe that during short events, the simulated concentrations are slightly shifted compared to the observed concentration (see example for EC, Figure 4). This comes from the limit of our hydrograph separation filter in a flood with a longer duration, the bias due to the time lag is less significant and a wider distribution of the / ratio, as simulated by the RDF method, is expected (see Figure 4).

During the wet and dry hydrological seasons, almost all ions behavior (except for nitrates) and EC show a dilution pattern with water of lower concentration followed by a step of increasing concentration (see dotted black line A , Figure 4). A common pattern with three stages can be observed in most storm events.

During the first stage we observed a slight increase in concentration during the initial increase in the discharge; then, in the second stage, which is generally short in time, the concentrations strongly decrease while the discharges quickly increase. In the third stage, both concentrations increase again while discharges continue to decrease and once more reach a lower flow and higher concentration (see Figure 4). To discuss the performances obtained for each case of our combined model, we use the stages outlied above.

The first stage would correspond to a pre-event pattern during which the concentrations come mostly from the groundwater pool [START_REF] Evans | Causes of concentration/discharge hysteresis and its potential as a tool for analysis of episode hydrochemistry[END_REF][START_REF] Rose | Concentration-discharge relationships describe solute and sediment mobilization, reaction, and transport at event and longer timescales[END_REF]. The following stages correspond to the emergence of soil waters and/or runoff mixing in varying proportions during the event [START_REF] Evans | Causes of concentration/discharge hysteresis and its potential as a tool for analysis of episode hydrochemistry[END_REF][START_REF] Rose | Concentration-discharge relationships describe solute and sediment mobilization, reaction, and transport at event and longer timescales[END_REF] (see Figure 4). The contribution of these pools (and the dilution that accompanies it) persists for some time after the peak flow, but in increasingly weaker proportions compared with those of the groundwater. Note that in the dry season, for all ions and EC, the soft dilution stage (first stage) is interrupted by rapid and abrupt dilution (Figure 4).

This last dilution would correspond to a larger proportion of rain water; it is indistinguishable during the wet season, because rain water is mixed with water from tile drains during this period of the year [START_REF] Billy | Factors controlling nitrate concentrations in surface waters of an artificially drained agricultural watershed[END_REF].

In all storm events (dry and wet), for all ions (except nitrate) and EC, case 3 best simulates the third stage of the storm event (where the mixing of two end members is most evident, see Figure 4). In the second stage of the wet season storm events (Figure 4B and C), we can observe that for the chloride and sulfate ions and EC, the 3 cases have an adequate performance (although none manages to simulate the maximum dilution point, see Figure 4). In the dry season storm event (Figure 4A), the second stage of chloride and sulfate ions and EC is better simulated by case 2, followed nearby case 3; case 1 has the lowest performance of the simulations in this stage. Finally in the first stage of each of the storm events for the chloride ion, sulfate ion and EC we note the same dynamics that occurred with the second stage for case 1 and 2: case 3 performs better A during the 2 storm events of the wet season, and case 2 performs better in the storm event of the dry season (see Figure 4A).

From this analysis we can deduce that in the dry season (Figure 4A), the first and second stages are almost completely dominated by the groundwater pool (both in terms of concentration and discharge), minimizing the other contributing pools (i.e. soil, runoff, represented by and ). For this reason case 2 (assuming total discharge ≈ groundwater) has the better performance in simulating the A of chloride and sulfate ions in addition to EC. However, in the wet season events (Figure 4B and C), and although in the first and second stage the groundwater pool is still the main contributor, the contribution of the other pools (i.e. soil, runoff, represented by and ) are much more notorious. Therefore, case 3 has better performances for chloride and sulfate ions and EC.

In the case of nitrates, we must acknowledge that the combined mixing model fails to fit the observed data whatever the event (see nitrates, Figure 4). Unlike other ions, nitrates are more concentrated in soils and drain water than in groundwater or rainwater. According to [START_REF] Garnier | Curative vs. preventive management of nitrogen transfers in rural areas: Lessons from the case of the Orgeval watershed (Seine River basin, France)[END_REF], in the Avenelles sub-catchment, sub-root nitrate concentrations average 22 mgN.L -1 , close to the average concentration observed in drains in the same area (26 mgN.L -1 ). Nitrate concentrations in the Brie aquifer are only around 13.2 mgN.L -1 , whereas in rainfall they are about 0.75 mgN.L -1 [START_REF] Mouchel | Caractérisation de la géochimie des interfaces nappe-rivière du bassin des Avenelles[END_REF][START_REF] Floury | Chemical weathering and CO2 consumption rate in a multilayered-aquifer dominated watershed under intensive farming: The Orgeval Critical Zone Observatory, France[END_REF]. For nitrate ions, a third component representing soil water would be needed as well as an appropriate parameterization of the seasonal nature of fertilizer applications. Several studies [START_REF] Burns | Monitoring the riverine pulse: Applying high-frequency nitrate data to advance integrative understanding of biogeochemical and hydrological processes[END_REF][START_REF] Sebestyen | Coupled hydrological and biogeochemical processes controlling variability of nitrogen species in streamflow during autumn in an upland forest[END_REF] have demonstrated that for nitrate ion, more than two different concentration pools (i.e. groundwater, soil water, etc.) are involved in rising and declining limbs of hydrograph.

Conclusions

Synthesis

The new concentration-discharge model presented in this paper associates a classical concentrationdischarge relationship with a classical two-component mixing equation. The originality of our approach lies in the fact that we do not proceed in the usual way by assuming a source concentration value to perform the baseflow-quickflow separation: we use the inverse approach, and use an a

priori assumption of the baseflow-quickflow separation to infer the source concentration values.

Our approach allows a better estimation of streamflow ionic concentration series for most ions at inter-annual scale, except for nitrate (which do not exhibit a clearrelationship), with improved X Y and YZ[CD criteria. This shows the advantage of coupling a time dynamic hydrological model with staticrelations for each of the flow components.

Limits of our approach

The first limit of our approach is apparent for the nitrate case: nitrate are known to be poorly described by concentration-discharge relationships, and our model does not allow improving much the issue. The two-member hydrograph separation tested here may be a limiting factor: for nitrate ions, the two components hydrograph separation ( ~ and ) seems to be inadequate, while for sulfate, chloride and EC, it seems sufficient. The base flow separation method used in this article carries naturally its share of uncertainties due to the arbitrary and speculative hypotheses used in its conception [START_REF] Beven | Hydrograph separation?[END_REF][START_REF] Brutsaert | Long-term groundwater storage trends estimated from streamflow records: Climatic perspective[END_REF][START_REF] Cheng | Automated selection of pure base flows from regular daily streamflow data: objective algorithm[END_REF] . The "simple" separation into two components cannot explain the complexity of the behavior of the nitrate ions. Indeed, several studies [START_REF] Miller | Estimating Discharge and Nonpoint Source Nitrate Loading to Streams From Three End-Member Pathways Using High-Frequency Water Quality Data[END_REF][START_REF] Probst | Nitrogen and Phosphorus exportation in the Garonne basin (France)[END_REF] have shown that the introduction of a third component (which would come to represent the soil pool component) is a prerequisite to represent the behavior of nitrate.

We also need to underline that we acknowledge that our approach remains conceptual, and we do not claim to have identified 'physically' the water masses that produce the quick and base flow.

Further perspectives

As the model does not simulate the decoupling of discharge and concentration patterns, due to a short time lag explained by hydraulic mechanisms; doing so, would be an interesting extension of this

work. An isotope study would be helpful to this aim.

Another priority would be to test the implementation of our model elsewhere, especially on highfrequency water quality stations [START_REF] Kirchner | The fine structure of water-quality dynamics: the (high-frequency) wave of the future[END_REF]. Different types of chemical signatures, reflecting different types of hydrological and hydro-chemical functions underlying the transfer processes, could be studied in detail, including different flow decomposition concepts and may be otherrelations possibly including specific seasonal features.

Another alternative would be to test this new methodology with other hydrograph separation methods than the RDF Lyne-Hollick to prove their validity (e.g. the methods developed by [START_REF] Eckhardt | How to construct recursive digital filters for baseflow separation[END_REF][START_REF] Pelletier | Hydrograph separation: an impartial parametrisation for an imperfect method[END_REF].

If high-frequency measurements become more and more available for science [START_REF] Kirchner | The fine structure of water-quality dynamics: the (high-frequency) wave of the future[END_REF], their implementation at large scale still difficult and low frequencies measurements (i.e. daily, weekly) remain the rule [START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF]. Thus, it remains important to continue methodological development through modelling which allow us, with a limited number of measurements and a good representativeness, to assess the quality of rivers in large scale, in the framework of low frequency monitoring. Ultimately, further developments of the combined mixing model should go in this direction.

  of the base flow (mgL -1 ) Cq : ion concentration of the quick flow (mgL -1 )

  Figure 2: A. Scatter plot of daily discharge Q against Q during recession periods, Avenelles station. Dashed black line: line through origin with slope 1 5 = 0.90. B. Example of hydrograph separation obtained with the LH-RDF method and the constant recession value 1 2.34 (i.e., 0.99), Avenelles station, with blue line as the baseflow and black line the total flow.

Figure 4 :

 4 Figure 4: For the three selected storm events (short flood events during the dry season (November 2015) (A) and wet season (February 2016) (B) and long flood event (March 2016) (C)), as a function of time (days): the flow and the hydrograph separation; the comparison for solutes and EC of observed (black line) and simulated concentrations from the Case 1 (orange line), the Case 2 (violet line) and the Case 3 (green line) of the combined model.

Table 1 :

 1 Classicalrelationships used in literature

	Approach	Equation	Number of parameters	Formula	Eq. n°
	Temporal -patterns	Power-law Two-sided power-law equation	2 3	= = + Q	Eq. (1) Eq. (2)
		mixing model (with at
	Mixing end-	least two sources
	member sources	which can be
		simplified with base
		flow and quick flow)

Table 2 :

 2 High-frequency measurements of chemical concentrations (average, minimum, maximum values and difference between quantiles 90 and 10 divided by the mean (CV)) from the River Lab at the Avenelles outlet, Oracle-Orgeval Observatory (June 2015 to July 2017 for calibration period and August 2017 to March 2018 for validation period).

	Solute	Unit	Mean	Calibration period (June 2015 to July 2017) Min Max	CV
	Sodium	mg.L -1	13	2	17	0.12
	Sulfate	mgS.L -1	19	2	32	0.19
	Chloride	mg.L -1	30	4	40	0.15
	EC	µS.cm -1	704	267	1015	0.11
			Validation period (August 2017 to March 2018)
	Sodium	mg.L -1	13	3	17	0.24
	Sulfate	mgS.L -1	18	3	26	0.27
	Chloride	mg.L -1	29	4	40	0.29
	EC	µS.cm -1	576	171	813	0.25

Table 3 :

 3 Numerical criteria used for optimization (Cobs -observed concentration, Csim -simulated concentration, Q -observed discharge). The[START_REF] Nash | River flow forecasting through conceptual models part I-A discussion of principles[END_REF] efficiency (NSE) criterion is well known and widely used in the field of hydrology. The rescaling proposed byMathevet et al. (

Table 4 :

 4 Values set for : in the combined model and (; ? ,? ? ) and (; P ,? P ) parameters for each case and solute. Values obtained for optimal BCDE FA criterion. Note that case 1 corresponds to chemostatic components, case 2 to the single 2S-APS relationship and case 3 to general case of the combined model. BCDE FA is optimum in 1.

	Solute	n	Case 1	ab 1.9	bb 0	aq 1.3	bq 0	QRST UVW? 0.46
	Sulfate	5	2	2.2	-0.55	2.2	-0.55	0.69
			3	2.3	-0.7	2	-0.4	0.73
			1	1.8	0	1.6	0	0.39
	Nitrate	5	2	1.8	-0.1	1.8	-0.1	0.41
			3	2.3	-1.2	2.1	-0.3	0.45
			1	3.3	0	1.6	0	0.52
	Chloride	3	2	3.7	-1	3.7	-1	0.83
			3	3.6	-0.7	3.3	-0.8	0.86
			1	3.8	0	3.1	0	0.61
	EC	5	2	4.2	-0.7	4.2	-0.7	0.77
			3	4.2	-0.7	3.9	-0.5	0.83

Table 6 :

 6 Values obtained for the X Y and the standardized RMSE (YZ[CD for each case and solute, for an application of the combined model over the entire period. Note that case 1 corresponds to chemostatic components, case 2 to the single 2S-APS relationship and case 3 to general case of the combined model.

	Solute Case Sulfate 1 2 3 Nitrate 1 2 3 Chloride 1 2 3 EC 1 2 3	Calibration mode ghiRS % ?j;g % 18.3 7.3 11.5 3.8 11.3 0.4 48.9 46.7 31.8 30.4 27.6 -11.2 13.8 -4.5 6.3 -0.7 5.3 -0.3 10.3 1.5 6.0 2.4 5.2 -0.5	Validation mode ghiRS % ?j;g % 24.1 11.7 14.8 -11.9 10.0 -6.7 317 315 267 266 144 119 20.0 -0.7 12.6 -3.3 12.1 -0.5 16.8 7.9 9.7 0.5 9.6 -2.7

Table 7 :

 7 Values obtained for standardized RMSE (YZ[CD) and X Y for each case and solute, for an application of the combined model on selected storm events. Note that case 1 corresponds to chemostatic components, case 2 to the single 2S-APS relationship and case 3 to general case of the combined model.

	Solute Sulfate Nitrate Chloride EC	Case 1 2 3 1 2 3 1 2 3 1 2 3	Short flood event, dry season Figure 4 A ghiRS % ?j;g % 16.2 -19.0 10.3 10.1 7.9 -3.0 13.1 10.1 20.3 18.2 21.2 16.5 28.4 -33.9 5.7 3.9 6.4 -5.8 13.2 -15.0 8.2 8.3 4.2 -2.5	Short flood event, wet season Figure 4 B ghiRS % ?j;g % 16.9 22.0 6.2 9.8 4.4 3.6 37.9 45.6 36.8 43.8 34.4 9.4 11.3 -8.9 5.7 -7.1 5.6 -6.2 8.7 8.6 7.1 8.2 4.2 1.6	Long flood event, wet season Figure 4 C ghiRS % ?j;g % 32.3 37.7 3.7 1.7 3.1 -0.5 58.8 64.4 36.2 41.3 40.2 -34.2 16.3 13.8 7.0 -6.0 4.9 1.3 16.3 16.9 5.3 1.4 3.9 -1.1

Acknowledgements

The first author acknowledges the Peruvian Scholarship Cienciactiva of CONCYTEC (grant no. 099-2016-FONDECYT-DE) for supporting his PhD study at INRAE and the Sorbonne University. The authors acknowledge the EQUIPEX CRITEX program (grant no. ANR-11-EQPX-0011) for the data availability.


MD Sarah Tebeka
email: sarah.tebeka@aphp.fr

M.D Pierre A Geoffroy

M.D Caroline Dubertret

M.D Yann Le Strat

Sadness and the continuum from well-being to depressive disorder: Findings from a representative US population sample

Keywords: Anxiety/Anxiety disorders, Bipolar disorder, Depression, Eating disorders, Epidemiology, Substance Use Disorders major depressive disorder, OR: odds ratio, PTSD: post-traumatic stress disorder

Objective: Sadness is a common symptom in the general population. We tested the hypothesis that sadness is an intermediate state on a continuum from well-being to major depressive disorder (MDD).

Methods: Using data from The National Epidemiologic Study of Alcohol and Related Conditions III (NESARC-III), a large and representative US population sample, we assessed the prevalence of sadness, its sociodemographic and clinical correlates, using three nonoverlapping groups: (i) non-depressed sad participants, (ii) non-sad non-depressed participants and (iii) depressed participants. We estimated sensitivity and specificity of sadness.

Results: Sadness was frequent in the general population 34.3%), and present in almost all participants with MDD (99.6%). Sad (N=4,593) and MDD participants (N=4,593) and 7,889 respectively) shared common sociodemographic characteristics. Compared to controls, sad and MDD participants presented more psychiatric disorders, including anxiety, substance use, psychotic, eating and personality disorders. Sadness was an intermediate state, sad individuals reporting more psychiatric disorders than controls, but less than participants with MDD. Sadness demonstrated a very high sensitivity (99.6 %), with a good specificity (83.8%) for MDD.

Limitations:

The NESARC assessed sadness over lifetime, which may involve memorization bias.

Conclusion:

Our study confirms the existence of a depressive continuum. Sadness is frequent in general population, and shares correlates with MDD. We have also shown a continuum where sadness is an intermediate state between well-being and psychiatric disorders. With high sensitivity and specificity, sadness appears as a clear MDD prodrome and at-risk state, and may be a symptom of a transdiagnostic distress process.

INTRODUCTION

The continuum from mental well-being to psychiatric disorders has been described for decades [START_REF] Shevlin | The Psychosis Continuum: Testing a Bifactor Model of Psychosis in a General Population Sample[END_REF][START_REF] Van Os | A systematic review and meta-analysis of the psychosis continuum: evidence for a psychosis proneness-persistence-impairment model of psychotic disorder[END_REF]. A better understanding of this continuum is of major interest in order to better disentangle the physiopathology, and to identify at-risk states. Numerous studies have particularly studied the place of psychotic symptoms within a continuum [START_REF] Ayuso-Mateos | From depressive symptoms to depressive disorders: the relevance of thresholds[END_REF][START_REF] Kingdon | Everybody gets stressed . . . it's just the way we react that differs[END_REF][START_REF] Lewinsohn | Clinical implications of "subthreshold" depressive symptoms[END_REF][START_REF] Shevlin | The Psychosis Continuum: Testing a Bifactor Model of Psychosis in a General Population Sample[END_REF][START_REF] Van Os | A systematic review and meta-analysis of the psychosis continuum: evidence for a psychosis proneness-persistence-impairment model of psychotic disorder[END_REF]. Linscott and van Os have proposed two conditions to confirm the psychotic continuum hypothesis : first, psychotic symptoms must exist and be more frequent in non-clinical population than characterized psychotic disorder; symptoms and characterized disorders should share same correlates and risk factors [START_REF] Van Os | A systematic review and meta-analysis of the psychosis continuum: evidence for a psychosis proneness-persistence-impairment model of psychotic disorder[END_REF].

Following this methodology, we recently tested the hypothesis that depressive symptoms also lies along a continuum [START_REF] Tebeka | A study in the general population about sadness to disentangle the continuum from well-being to depressive disorders[END_REF]. In a sample of 38,694 participants, we showed that depressive symptoms ranged from non-pathological sadness (reported by 29.8% of the sample), to major depressive disorder according to DSM-5 criteria (MDD, reported by 12.0% of the sample). In this previous work, sadness and MDD shared socio-demographic and clinical correlates. Moreover, sadness appeared to be an intermediate state between mental health and MDD. Participants with sadness and participants with MDD were more likely to report non-depressive psychiatric disorders than participants without sadness or MDD, participants with sadness constituting an intermediate group between MDD and controls. The results were consistent with a depressive continuum hypothesis. Finally, we also suggested that sadness could be used as a screener of "at risk" mental state, with a very high sensitivity for MDD (94.2%) and as expected, poor specificity (36.3%) [START_REF] Tebeka | A study in the general population about sadness to disentangle the continuum from well-being to depressive disorders[END_REF].

Although sadness is a "gate symptom" for MDD diagnosis, being one of the two cardinal criteria with anhedonia, it is relevant to assess the clinical value of this frequent symptom. The aims of the present study were threefold. First, we wanted to replicate and confirm the existence of a depressive continuum in a non-European sample. Second, we wanted to assess the strength of the associations between sadness and non-depressive psychiatric disorders. Thirdly, we wanted to test the sensitivity and specificity of sadness to screen Major Depressive Disorder (MDD).

METHODS

Sample

The National Epidemiologic Study of Alcohol and Related Conditions III (NESARC-III) is a nationally representative survey of U.S. civilian population. It included 36,309 participants, 18 years and older, in 2012-2013. Reponses rates was 60.1% in NESARC-III [START_REF] Grant | Source and Accuracy Statement: National Epidemiologic Survey on Alcohol and Related Conditions-III (NESARC-III)[END_REF]. Trained interviewers conducted face-to-face interviews. More information on NESARC-III sample are available.

The research protocol, including informed consent procedures, received ethical approval from the institutional review boards of Westat and the National Institutes of Health for NESARC-III.

Measures

Sociodemographic Measures

Age at interview was categorized into four age-bands (18-29 years, 30-44 years, 45-59 years, 60+ years). Educational level was classified into: (i) less than high school, (ii) high school graduate and (iii) some college or higher. Marital status was classified into (i) married, (ii) widowed/separated/divorced and (iii) never married. Personal income during a year was classified into: (i) $0-19,999, (ii) $20,000-34,999, (iii) $35,000-69,999, and (iv) $70,000 or greater. Race/ethnicity was categorized into: (i) White, (ii) Black, (iii) American Indian/Alaska Native, (iv) Asian/Native Hawaiian/Pacific Islander and (v) Hispanic/Latino. Region of residence was classified into: (i) Northeast, (ii) Midwest, (iii) South, and (iv) West.

Sex, urbanicity and US born was collected.

Sadness

Sadness was assessed lifetime by a single question: "In your entire life, have you ever had a time when you felt sad, blue, depressed, or down nearly every day for at least 2 weeks?".

Assessment of psychiatric disorders

The AUDADIS-IV was used to assess psychiatric diagnoses according to DSM-IV criteria [START_REF] Chatterji | Reliability of the alcohol and drug modules of the Alcohol Use Disorder and Associated Disabilities Interview Schedule--Alcohol/Drug-Revised (AUDADIS-ADR): an international comparison[END_REF]. We reported lifetime prevalence for MDD (excluding bipolar disorder), bipolar disorder (BD), any anxiety disorders (including panic disorder, social phobia, and generalized anxiety disorder), Post-Traumatic Stress Disorder (PTSD), eating disorders (including anorexia nervosa, bulimia nervosa and binge eating disorder) and personality disorders (including schizotypal, borderline and antisocial personality disorders), substance use disorders (SUD, including tobacco, alcohol and cannabis), and psychotic disorder [START_REF] Bourgin | Prevalence and correlates of psychotic-like experiences in the general population[END_REF][START_REF] Tebeka | Parental Divorce or Death During Childhood and Adolescence and Its Association With Mental Health[END_REF]. The AUDADIS-IV used a single question to screen suicide attempts.

Statistical analyses

In order to make comparisons easier, we used the same statistics to our previous work [START_REF] Tebeka | A study in the general population about sadness to disentangle the continuum from well-being to depressive disorders[END_REF].

We divided the sample into three independent groups as follows: 1) participants without neither sadness nor MDD (controls), 2) participants with sadness but without MDD and 3) participants with MDD, with or without sadness (MDD).

We compared the prevalence of sadness and the sociodemographic characteristics in the three groups, to valid the continuum criterion proposed by van Os and Linscott [START_REF] Linscott | An updated and conservative systematic review and meta-analysis of epidemiological evidence on psychotic experiences in children and adults: on the pathway from proneness to persistence to dimensional expression across mental disorders[END_REF]. We also compared the prevalence of psychiatric disorders in the three groups.

Weighted estimates and standard errors (SEs) were computed using SUDAAN,version 11.01 (Research Triangle Park,NC). This software implements a Taylor linearization to adjust SEs of estimates for complex survey sampling design effects including clustering data.

Multivariate logistic regression models were estimated with simultaneous entry of socio-demographics covariates. Adjusted odds ratios (ORs), 95% confidence intervals (CIs) are presented to reflect association strength and significance.

Finally, the sensitivity and specificity of sadness for MDD risk were estimated. We also calculate the sensitivity and specificity for the other co-occurring psychiatric disorders: anxiety, substance use, personality, psychotic and eating disorders.

RESULTS

Prevalence rate of sadness (Table 1).

The prevalence of sadness was 34.29% in our sample (12,449 of 36,309 participants interviewed). Sadness without MDD concerned 12.65% of our sample.

The prevalence of MDD was 21.73%. In the MDD subgroup, almost all participants reported sadness (99.54%).

Sociodemographic characteristics (Table 1).

Participants with sadness and those with MDD were more likely than controls to be women (46.44% of women in controls, 57.46% in sad people, in 65.12% in MDD), nonmarried (60.27% of women in controls, 54.36% in sad people, in 52.47% in MDD) and divorced (11.54% of women in controls, 17.33% in sad people, in 18.85% in MDD).

Participants with sadness and those with MDD were also more likely than controls to have lower personal incomes (13.70% of high household incomes in controls, 10.91% in sad people, in 9.98% in MDD) and to be born in U.S. (81.56% of women in controls, 86.29% in sad people, in 90.25% in MDD sample). They were no difference on urbanicity and region of residence between the three groups.

-Table 1 about here -

Clinical characteristics of the sample.

The prevalence of suicide attempt was 1.34% in controls, 9.11% in sad people and 14.32% in MDD. Participants with sadness and those with MDD presented respectively 28.08% and 43.37% anxiety disorders, 49.50% and 57.90% SUD and 4.17% and 3.75% psychotic disorder. Participants with sadness and those with MDD presented respectively 2.67% and 4.32% eating disorders, and 20.68% and 27.44% personality disorders (Table 2).

All psychiatric disorders (previous suicide attempts, anxiety disorders, SUD, psychotic disorder, eating disorders and personality disorders) were significantly associated with both sadness and MDD status, in comparison to controls ( 2).

Validity of sadness as a marker of depression

Sadness demonstrated a very high sensitivity (SE = 99.6 %), with a good specificity (SP = 83.8%) for MDD.

Sadness had a lower sensitivity for other disorders (35.1% for anxiety disorders, 20.7% for substance use disorders, 42.0% for personality disorders, 37.1% for psychotic disorders and 42.6% for eating disorders). The specificity of sadness was also high for all psychiatric disorders assessed: 86.6% for anxiety disorders, 86.7% for substance use disorders, 86.1% for personality disorders, 84.2% for psychotic disorders and 84.1% for eating disorders (Table S1).

DISCUSSION

In an independent sample representative of the general US population, we replicated our previous findings on the continuous nature of sadness. Sadness was frequent both in the general population and in participants with MDD. Its prevalence was 34.3% in the whole sample, ranging from a low 12.6% of non-MDD participants to a high 99% in participants with MDD. We also confirmed that participants with sadness without MDD and those with MDD shared the same sociodemographic correlates: woman gender, never-married and separated marital status, low-and medium-income level, and being US born.

In the same way, as in our previous study in French population sample, we confirmed a positive association between sadness and various psychiatric disorders. In this study, we had a broader assessment of psychiatric disorders, as cannabis use disorder, eating disorders and personality disorders. They were all associated with sadness. Of note, these associations remained significant after adjustments for sociodemographic variables. The rates of psychiatric disorders in participants with sadness without MDD were higher than controls, but lower than that observed in participants with MDD. Thus, sadness appears as an intermediate state between mental well-being and MDD, validating the symptom severity continuum hypothesis [START_REF] Ayuso-Mateos | From depressive symptoms to depressive disorders: the relevance of thresholds[END_REF].

The prevalence of sadness was comparable in this US sample to the prevalence found in our previous European study [START_REF] Tebeka | A study in the general population about sadness to disentangle the continuum from well-being to depressive disorders[END_REF]. Moreover, the sociodemographic correlates of sadness found in this study, in particular women gender, education level, marital status and incomes, were similar to those found in previous studies concerning depressive symptoms [START_REF] Moreno-Küstner | Suicidality in primary care patients who present with sadness and anhedonia: a prospective European study[END_REF][START_REF] Schomerus | Continuum beliefs and stigmatizing attitudes towards persons with schizophrenia, depression and alcohol dependence[END_REF].

At first glance, migration status exhibit discrepant results between US and French sample. Being born in the U.S was associated with sadness in the sample, while an inverse association between migratory status (first, second and third generations separated or pooled) and sadness was found in the French sample. Immigration in Europe is usually associated with poorer health conditions, specially psychiatric disorders [START_REF] Bourque | A meta-analysis of the risk for psychotic disorders among first-and second-generation immigrants[END_REF][START_REF] Mindlis | Mood disorders in first-and second-generation immigrants: systematic review and meta-analysis[END_REF][START_REF] Rolland | Alcohol Use Disorders and Immigration up to the Third Generation in France: Findings from a 39,617-Subject Survey in the General Population[END_REF]. However, U.S studies describe an "immigrant paradox'": immigrants have better health outcomes than non-migrants. This has been particularly described for many psychiatric disorder, specially mood, anxiety and psychotic disorders [START_REF] Oh | Immigration and psychotic experiences in the United States: Another example of the epidemiological paradox?[END_REF][START_REF] Salas-Wright | Immigrants and mental disorders in the united states: New evidence on the healthy migrant hypothesis[END_REF], 2014).

With sensitivity about 99%, sadness is a relevant symptom to screen MDD. Sadness is the most common symptom for any given depressive episode. Over a life course in which a patient may have multiple episodes of depression, the probability of sadness during at least one episode is very high. Whereas we found a good specificity for sadness in MDD (83.8%), sadness is also a symptom significantly associated with other psychiatric disorders (anxiety, post-traumatic stress, eating, and personality disorders). Thus, the specificity of sadness was also high for other psychiatric disorders (anxiety, substance use, personality, psychotic and eating disorders). These results suggest that sadness is not only a prodrome and atrisk state of MDD but may be a symptom of a transdiagnostic distress process. This hypothesis needs to be confirmed in prospective studies and might be the marker of a broader process of psychological distress, as well as a good screening marker of at-risk state and mental suffering Some limits of this work should be kept in mind. On the one hand, a memory bias can affect the retrospective assessment of sadness over the entire life. Moreover, we did not compare bipolar disorders in our MDD groups because of the definition of this group, which exclude BD. In addition, the design of a cross-sectional study narrows the scope of our conclusions.

This replication study based on a representative of the U.S. population makes our first results more robust. It confirms the importance of considering the sadness symptom as an atrisk state. This simple assessment could be considered for early intervention to identify at-risk state, with a good specificity for MDD. These results consolidate the idea of a depressive continuum ranging from normal sadness to severe MDD. Further studies should assess if other symptoms, including anhedonia, also fit into a depressive continuum.

DECLARATION OF INTEREST

Authors have no conflicts of interest related to this work

Table 2 ).

 2

	Compared with participants with MDD, those with sadness were significantly less
	likely to be diagnosed with various psychiatric disorder, including previous suicide attempts
	(OR=1.55; 95% IC [1.31-1.83]), anxiety disorders (OR=1.84; 95% IC [1.69-2.01]), SUD

TABLE 1 .

 1 Sociodemographic characteristics in individuals with sadness without major depressive disorder (MDD), individuals with MDD and control subjects (no sadness, no MDD) -odd ratio (OR) adjusted for all other sociodemographic characteristics

		Control	Participants	Participants	Participants with	Participants with	Participants with MDD
		subjects	with sadness	with MDD	sadness without MDD	sadness without MDD	vs
		(no sadness, no	without MDD			vs		vs	controls
		MDD)				controls	Participants with
								MDD
		N= 23,827	N= 4,593	N= 7,889
		% (SE)	% (SE)	% (SE)	OR	95% CI	OR	95% CI	OR	95% CI
	Age
	18-29y	22.47 (0.45)	18.49 (0.68)	21.10 (0.59)	Ref		Ref		Ref
	30-44y	25.51 (0.35)	24.33 (0.85)	27.21 (0.64)	1.33	1.17-1.50	0.99	0.86-1.13	1.32	1.19-1.46
	45-64y	32.84 (0.37)	39.05 (0.87)	39.30 (0.64)	1.59	1.40-1.80	0.84	0.73-0.97	1.37	1.25-1.50
	65y+	19.18 (0.48)	18.13 (0.69)	12.39 (0.47)	1.11	0.96 -1.30	0.54	0.46-0.65	0.61	0.53-0.70
	Sex
	Men	53.56 (0.37)	42.54 (0.83)	34.83 (0.67)	Ref		Ref
	Women	46.44 (0.37)	57.46 (0.83)	65.17 (0.67)	1.42	1.32-1.54	1.41	1.28-1.55	2.03	1.89-2.17
	Education level
	Less than high	13.54 (0.46)	12.76 (0.73)	11.56 (0.53)	Ref		Ref		Ref
	school
	High school	26.10 (0.56)	26.63 (0.83)	24.32 (0.78)	1.07	0.93-1.23	0.91	0.78-1.07	0.96	0.85-1.08
	Some college or	60.36 (0.78)	60.61 (1.07)	64.12 (0.97)	1.13	1.00-1.29	0.99	0.85-1.16	1.10	0.99-1.21
	higher
	Marital status
	Married/ Common-	60.27 (0.57)	54.36 (1.01)	52.47 (0.75)	Ref		Ref		Ref
	law
	Single / never	22.71 (0.49)	21.22 (0.83)	5.99 (0.33)	1.34	1.14-1.57	1.09	0.91-1.30	1.46	1.25-1.71
	married
	Separated /	11.54 (0.25)	17.33 (0.62)	18.85 (0.45)	1.53	1.38-1.68	1.16	1.04-1.30	1.75	1.63-1.88
	Divorced
	Widower	5.49 (0.19)	7.08 (0.41)	22.69 (0.64)	1.13	1.01-1.26	1.08	0.94-1.24	1.21	1.10-1.33
	Personal incomes
	0 -19,999 $	41.98 (0.59)	48.32 (1.07)	49.13 (0.80)	Ref		Ref		Ref
	20,000 -34,999$	21.07 (0.35)	21.18 (0.77)	19.01 (0.52)	0.85	0.77-0.94	0.90	0.80-1.02	0.77	0.70-0.84

62 Abbreviations: SE; Standard Error; OR, odds ratio; CI, confidence interval; MDD, Major Depressive DisorderTABLE 2 .

 2 Prevalence and odds ratio (OR) of psychiatric disorders comparing individuals with sadness without depressive disorders (MDD) to those with MDD, and control subjects adjusted for sociodemographic characteristics

Abbreviations: SE; Standard Error; OR, odds ratio; CI, confidence interval; PTSD, post-traumatic stress disorder; MDD, Major Depressive Disorder

		Control	Participants	Participants	Participants with	Participants with	Participants with
		subjects	with sadness	with MDD	sadness without MDD	sadness without		MDD
		(no sadness,	without			vs		MDD		vs
		no MDD)	MDD			controls		vs		controls
							Participants with
								MDD
		N= 23,827	N= 4,593	N= 7,889
		% (SE)	% (SE)	% (SE)	OR	95% CI	OR	95% CI	OR	95% CI
	Suicide attempt	1.34 (0.09)	9.11 (0.58)	14.32 (0.54)	6.45	5.38-7.73	1.55	1.31-1.83	10.26	8.73-12.04
	Any Substance Use disorder	37.02 (0.58)	49.50 (0.96)	57.90 (0.75)	1.75	1.61-1.90	1.33	1.22-1.46	2.36	2.20-2.53
	Alcohol use disorder	24.38 (0.51)	32.76 (0.86)	41.15 (0.85)	1.63	1.49-1.80	1.37	1.23-1.53	2.27	2.10-2.45
	Tobacco use disorder	23.00 (0.50)	34.07 (0.87)	38.93 (0.82)	1.71	1.57-1.85	1.18	1.07-1.29	2.05	1.90-2.21
	Cannabis use disorder	4.35 (0.21)	7.88 (0.47)	11.11 (0.55)	1.96	1.69-2.27	1.43	1.21-1.68	2.83	2.46-3.25
	Bipolar disorder	1.25 (0.10)	16.16 (0.62)	0 (0)	15.88	13.32-18.93	-	--	-	--
	Any anxiety disorder	10.11 (0.29)	28.08 (0.67)	43.37 (0.77)	3.14	2.87-3.45	1.84	1.69-2.01	5.81	5.31-6.36
	Panic disorder	1.98 (0.13)	8.74 (0.45)	12.97 (0.54)	4.06	3.36-4.90	1.39	1.20-1.62	5.69	4.82-6.71
	Social anxiety disorder	1.61 (0.12)	5.69 (0.36)	8.79 (0.42)	3.37	2.80-4.06	1.51	1.29-1.76	5.05	4.31-5.91
	Generalized anxiety disorder	2.66 (0.13)	11.92 (0.53)	20.38 (0.57)	4.45	3.85-5.15	1.82	1.61-2.06	8.14	7.06-9.40
	Post-traumatic stress disorder	2.03 (0.12)	11.22 (0.55)	15.52 (0.62)	5.40	4.

64-6.29 1.33 1.16-1.53 7.50 6.54-8.61

	Any eating disorder	0.69 (0.06)	2.67 (0.29)	4.32 (0.24)	3.44	2.68-4.41	1.42	1.11-1.81	4.84	3.96-5.93
	Anorexia nervosa	0.37 (0.05)	1.27 (0.24)	1.86 (0.18)	2.93	1.87-4.58	1.18	0.78-1.77	3.36	2.60-4.35
	Bulimia nervosa	0.07 (0.02)	0.31 (0.09)	0.71 (0.14)	4.08	1.71-9.76	1.90	0.92-3.93	6.21	3.20-12.06
	Binge eating disorder	0.26 (0.04)	1.32 (0.18)	1.95 (0.17)	4.55	3.18-6.52	1.37	0.96-1.96	6.65	4.78-9.24
	Any personality disorder	5.55 (0.22)	20.68 (0.79)	27.44 (0.72)	4.55	4.03-5.14	1.43	1.29-1.59	6.67	6.00-7.42
	Antisocial	1.41 (0.09)	4.61 (0.36)	4.88 (0.33)	3.53	2.90-4.30	1.07	0.87-1.32	4.00	3.34-4.78
	Borderline	3.93 (0.17)	17.87 (0.74)	24.21 (0.65)	5.29	4.67-5.98	1.43	1.28-1.60	7.74	6.95-8.63
	Schizotypal	1.61 (0.11)	9.00 (0.49)	10.05 (0.49)	5.82	4.91-6.89	1.11	0.94-1.31	6.87	5.75-8.20
	Psychotic disorder	1.28 (0.15)	4.17 (0.42)	3.75 (0.31)	3.08	2.29-4.13	0.89	0.69-1.15	2.84	2.27-3.55

TABLE S1 . Sensitivity and specificity of sadness for psychiatric disorders.

 S1

		Sensitivity Specificity
	Major depressive disorder	0.996	0.84
	Any anxiety disorder	0.35	0.866
	Any substance use disorder	0.21	0.87
	Any eating disorder	0.42	0.24
	Psychotic disorder	0.37	0.84
	Any personality disorder	0.42	0.86

The authors are grateful to the National Institute on Alcohol Abuse and Alcoholism for providing the necessary data for this research. This manuscript was prepared using a limited access dataset obtained from the NIAAA and does not reflect the opinions or view of NIAAA or the U.S. government.

Financial Support

No funding was secured for this study.

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

DATA AVAILABILITY

This manuscript was prepared using a limited access data setobtained from the NIAAA and does not reflect the opinions or viewsof the NIAAA or the U.S. Government. Research data are not shared.