To analyze the C-Q relationship, these models use a statistic regression linear and non-linear. The one-component models using linear regression describe the C-Q relationship from a power-law equation. A log-log transformation allows linearizing the C-Q relationship. The one-component models using non-linear regression describe the C-Q relationship from a hyperbolic model, without transformation. For the two types of one-component model, the regression parameters or the indices derived from them, describe the different C-Q relationships and consequently different associated hydro-chemical processes.
To alleviate the problems mainly due to the linear regression approach of the log-log transformation (i.e. sensitivity to outliers, overfitting), Johnson et al. (1969) developed the hyperbolic model. The hyperbolic model uses as the power-law model only one component to describe the C-Q relationship (i.e. the total discharge Q). This model was developed by Johnson et al. (1969) from the mass balance equation form: 𝐶0.𝑉0+ 𝐶1.𝑉1= 𝐶(𝑉0+ 𝑉1) Eq. (4).
Where the concentration of the solute before dilution is 𝐶0, 𝑉0 is the volume before dilution, 𝐶1 is the diluent concentration, 𝑉1 is the volume of dilution and 𝐶 is the concentration from the mixed solution.
With the Eq. (4), Johnson et al. (1969) stated the following hypotheses:
(i) when discharge (𝑄) approaches zero, 𝑉1 also approaches zero, thus (𝑉0+ 𝑉1)→ 𝑉0.
(ii) at any given time if 𝑉1> 𝑉0, 𝑉1 is directly proportional to discharge (𝑉1=𝛾𝑄, where 𝛾 is the residence time for water moving through the soil without chemical influences).
(iii) every volume before dilution (𝑉0) has a characteristic initial concentration (𝐶0) for any given natural ion. Under these hypotheses, Eq. (4) can be rewritten as: 𝐶0.𝑉0+ 𝐶1.𝛾𝑄= 𝐶(𝑉0+ 𝛾𝑄).
To approximate the physical processes involved, n-component models or mixing equations, still based on mass balance equation, were developed (Dewalle et al., 1988; Pinder and Jones, 1969; Sklash et al., 1976; Stewart et al., 2007; Uhlenbrook and Hoeg, 2003). These models assume that the chemical concentrations of the stream water are the result of the mixing of various chemical components inherent in the different flow components. In this model, the composition or chemical signature of water is constant and different from each other components (Pinder and Jones, 1969). The mixing models or n-component models, may include several flow and chemical components (e.g. Dewalle et al., 1988; Miller et al., 2017; Uhlenbrook and Hoeg, 2003). The simplest, with two-component was initially applied by La Sala Jr. (1967): “The water in a stream usually is a mixture of surface runoff and groundwater. The chemical quality of the stream water is therefore, determined both the chemical qualities of overland runoff and groundwater and by the proportion of each”. Following this simple definition, the equation of the 2-components mixing model could be written: 𝐶 = 𝐶𝑏𝑄𝑏𝑄+ 𝐶𝑞𝑄𝑞𝑄 Eq. (8).
Where 𝐶 is the concentration observed in the stream, 𝑄 is the total observed discharge, 𝑄𝑏 is the groundwater discharge assimilated to the baseflow, 𝑄𝑞 is the runoff discharge assimilated to the quickflow, 𝐶𝑏 is the characteristic concentration of groundwater and 𝐶𝑞 of runoff.
The baseflow represents the longer-slow discharge derived from the groundwater (Hall, 1968; Nathan and McMahon, 1990). The quickflow represents the direct response to a rainfall event including overland flow (runoff) and direct rainfall onto the stream surface (direct precipitation) (Linsley Jr et al., 1958; Pinder and Jones, 1969).
Mixing models are used by geochemist to study hydrochemistry processes and estimate the end-member sources of the stream water quality (e.g Evans and Davies, 1998; Genereux et al., 1993; Probst, 1985; Uhlenbrook and Hoeg, 2003). They are also used by hydrologists and hydrogeologists to separate quickflow from baseflow (e.g. Nathan and McMahon, 1990; Zhang et al., 2017; Zhang et al., 2013). Dependent on discipline (geochemistry, hydrology or hydrogeology) and purpose, lateral movement in the soil profile (interflow) is integrated to the baseflow (e.g. Buttle, 1994; Klaus and McDonnell, 2013; Sklash et al., 1976) or the quickflow (e.g. Brodie et al., 2007; Chapman and Maxwell, 1996; Nathan and McMahon, 1990). Baseflow and quickflow could be also assimilated to the “old water” and “new water” used by geochemists who work with isotopes (Bansah and Ali, 2017; Klaus and McDonnell, 2013). Whatever the discipline and the components number, the key points of the mixing models are the hydrograph separation and /or the quantification of sources contribution.
Hydrograph separation methods
Hydrograph separation is the oldest age topic in hydrology (Hall, 1968). It is a deconstructive method of streamflow as a two-component or multi-component process (Mei and Anagnostou, 2015). The most commonly used scheme is the two-component scenario that considers streamflow consisting of direct flow (i.e. quick surface or subsurface flow) and baseflow (i.e. flow that comes from groundwater storage or other delayed source) (Hall, 1968; Tallaksen, 1995). As shown in Figure 10, classical hydrograph separation is defined in terms of the delay or lag times of the components, without implication of origin (Hall, 1968).
The tracers-based methods are based on the use of isotopes and geochemical tracers to perform the hydrograph separation. This type of method dates back to the late 1960s (Hubert et al., 1969; La Sala Jr., 1967; Pinder and Jones, 1969). It is perceived as a transcendental step in the hydrograph separation. Since unlike the graphical methods based on the arbitrary interpretation of the flood hydrograph (and widely used until then), tracers based approach are measureable, objective and based on components of the water itself (Klaus and McDonnell, 2013). These methods assume that stream water results from the mixture of several sources, and that each source has a constant and unique chemical composition (Pinder and Jones, 1969).
The first studies were performed at the level of a storm event. The hydrograph separation was made into two components: “old water”, which represents the flow from groundwater, and “new water” that generally represented by precipitation and inflow (Buttle, 1994; Klaus and McDonnell, 2013). These terms continue to be used to this day (e.g. Bansah and Ali, 2017).
The tracers based approach is founded on five assumptions (according to Buttle, 1994; Klaus and McDonnell, 2013):
i. the isotopic/tracer content of the event and the pre-event water are significantly different.
ii. the event water maintains a constant isotopic/tracer signature in space and time.
iii. the isotopic/tracer signature of the pre-event water is constant in space and time.
iv. contributions from the vadose zone is negligible, or the isotopic/tracer signature of the soil water is similar to that of groundwater.
v. surface storage contributes minimally to streamflow. The tracer based methods can divided into two groups: (i) isotopic tracers and (ii) geochemical tracers.
This approach is generally used for a flood event. Two natural isotopes can be used: oxygen-18 (18O), first applied by Sklash and Farvolden (1979) and deuterium (2H), used for the first time by Sklash et al. (1996). Since they are part of the water molecule, these isotopes of the water molecule are ideal conservative tracers. Added naturally during precipitation events and once free from evaporative exposure, they are only subject to changes due to mixing end-members (Buttle, 1994; Klaus and McDonnell, 2013). Even if weekly or even daily isotope measurements are now becoming available for many catchments (Kirchner, 2019), isotopic analyses can be very laborious and expensive, especially for long term study (Lott and Stewart, 2016; Stewart et al., 2007). Moreover, the conservative character of these istopic tracers can be discussed which increases the uncertainties of the method (Kirchner, 2019). Klaus and McDonnell (2013) illustrated the differences in estimated source fractions due to the effect of evaporated soil water and its differential impact on 2H and 18O signatures and underline the need of a careful assessment of potential end-members (Figure 15).
Geochemical tracer methods was first performed by Pinder and Jones (1969) with different solutes and Kunkle (1965) with electro-conductivity (EC). Still on the assumption of the mass balance, the concentrations of the chemical constituents are related to actual physical processes and flowpaths in the basin that generate the different flow components (Stewart et al., 2007).
There are two principal geochemical tracer methods. Either the concentration of each chemical components are measured (e.g. Genereux et al., 1993; Uhlenbrook and Hoeg, 2003) or it is assumed that the chemical component of the baseflow is constant and can be defined as the maximum concentration calculated in the dry period. The chemical component of the quickflow is also considered constant and can be defined as the minimum concentration during high flows (e.g. Stewart et al., 2007; Zhang et al., 2013).
Unlike isotopes, this approach is easier applied to long term studies, especially with EC, that can be inexpensively measured concurrently with stream flow measurements (Lott and Stewart, 2016). However, the assumption of a constant and uniform signature for every component is often fulfilled within short intervals (e.g. within an event) (Gonzales et al., 2009). According to Sherson et al. (2015) the uncertainty of hydrograph separations could be due to laboratory analytical errors, the spatial and temporal variability of component chemical compositions or the non-conservation of the tracer.
Classic one-sided power scaling relationship (power law)
Since at least 50 years ago, 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. (13)). 𝐶=𝑎𝑄𝑏 q. (13).
From a numerical point of view, the relationship presented in Eq. (13) is generally adjusted by first transforming the dependent (𝐶) and independent (𝑄) variables using a logarithmic transformation, and then adjusting a linear model (Eq. (14)). ln (𝐶)=ln(𝑎)+𝑏.ln(𝑄) Eq. (14).
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.
Factor analysis (FA) or positive matrix factorization (PMF)
The term, ‘factor analysis’ (FA), is ambiguous. FA means principal component analysis (PCA): singular value decomposition (SVD), selection of dimension, and rotations. Statisticians often remark that this is not FA at all according to their definition of FA. In statistics, FA means investigation of correlations of random variables. This leads to a non-linear computation, which cannot be done with SVD. To avoid the ambiguous term ‘FA’, Paatero and Tapper (1994) called the statistician FA, ‘positive matrix factorization’ or PMF.
Although highly used in the air pollution domain to perform end-members apportionment of particulate matter (see detailed review of existing methods in Popoola et al., 2018), the PMF method is recent for the domain of hydrochemistry (Capozzi et al., 2018; Haji Gholizadeh et al., 2016; Zanotti et al., 2019). The PMF method utilizes statistical techniques to reduce the data to meaningful terms for identifying the chemical end-members and to estimate the end-member contributions.
The form of the PMF model, most widely used to analyze the chemical network data, is the bilinear model: two matrices, G and F, leading to a reproduction of the dataset variability as a linear combination of a set of constant factors profile and their contribution to each sample (Reff et al., 2007).
The main advantages of this method compared to PCA are that: (i) it takes into account the analytical uncertainties often associated with measurements of environmental samples and (ii) forces all of the values in the solution profiles and contributions to be positive, which can lead to a more realistic representation (Reff et al., 2007; Zanotti et al., 2019).
According to Zanotti et al. (2019), the main limitations of this method, which would lead to its poor development in the field of catchment hydrochemistry, are that PMF requires only data expressed as concentration while some typical measurement of water samples have different units (e.g. pH, Electrical Conductivity, Oxidation Reduction Potential, isotopes analysis, age tracers). Thus, the dataset cannot be directly fed into a PMF model. Moreover, a proper monitoring network, specifically designed to capture the variability of the hydrochemical system, is crucial to obtain a complete representation of the system with a PMF analysis. Under some cases, where a single end-member is present, a multivariate statistical analysis such as PMF might not be appropriate (Zanotti et al., 2019).
Table of contents :
2 Scientific questions of this thesis
3 Structure of the thesis
Part I A brief review of concentration-discharge (C-Q) relationships
2 Non-univocal concentration-discharge (C-Q) relationship (hysteresis)
3 Concentration-discharge (C-Q) relationship models
3.1 One-component models
3.1.1 Power-law model
3.1.2 Hyperbolic model
3.2 N-component models
3.2.1 Mixing models
4 Hydrograph separation methods
4.1 Non tracers-based method
4.1.1 Recession curves analysis methods
4.1.2 Filtering methods
4.2 Tracers-based methods
4.2.1 Isotopic tracers
4.2.2 Geochemical tracers
5 Quantification of the end-members
5.1 Eigenvector analysis
5.1.1 Principal Component Analysis (PCA)
5.1.2 Factor analysis (FA) or positive matrix factorization (PMF)
5.2 Classification analysis
5.2.1 Cluster analysis (CA)
5.2.2 Discriminant analysis (DA)
5.3 Mixed analysis
5.3.1 End member mixing analysis (
5.3.2 MIX method
6 Study site – The Oracle-Orgeval observatory
6.1 Characteristics of Oracle-Orgeval observatory
6.1.1 Location and brief history
6.1.2 Topography and Climate
6.1.3 Geology and hydrogeology
6.1.5 Land use and hydro-agricultural infrastructures
6.2 Hydrological measurements of Oracle-Orgeval observatory
6.3 Chemical measurements of Oracle-Orgeval observatory
6.3.1 Long-term monitoring
6.3.2 River Lab. and high-frequency measurements
6.3.3 Typology of high-frequency measurements of ions concentrations during flow events
6.4 Hydrological and chemical behavior of the catchment
Part II Revisiting the concentration-discharge (C-Q) relationships
Chapter 1: Revisiting the one-component models – power law model
Technical Note: A two-sided affine power scaling relationship to represent the concentration–discharge relationship
2 Tested dataset
3 Mathematical formulations
3.1 Classic one-sided power scaling relationship (power law)
3.2 Limits of the power law
3.3 A two-sided affine power scaling relationship as a progressive alternative to the power law
4 Numerical identification of the parameters for the 2S-APS relationship
5.1 Results in calibration mode
5.2 Results in validation mode
7 Appendix 1 – Description of the River Lab
8 Appendix 2 – Graphical representation of the numerical identification of parameters
Chapter 2: Revisiting the n-component models – mixing model Hydrograph separation issue using high-frequency chemical measurements
1.1 Hydrograph separation: an age-old issue
1.2 A variety of solutions proposed for hydrograph separation
1.3 Joining the strengths of hydrological and chemical approaches for hydrograph separation
1.4 Using high-frequency chemical measurements to revisit the hydrograph separation issue
2.1 Study site and data set
2.2 Hydrograph separation methods
2.2.1 Lyne – Hollick method (LH)
2.2.2 Eckhardt method (ECK)
2.2.3 Hydrological recession time constant of the catchment
2.2.4 MB method
2.3 Comparison of the two RDF methods through an hydrological and a chemical calibration
2.3.1 The hydrological recession time constant (τ) from hydrological MRC approach
2.3.2 Sensibility of the RDF methods to the hydrological recession time constant of the catchment (τ)
2.3.3 The hydrological recession time constant of the catchment (τ) from MB approach
2.4 Comparison between the components 𝑪𝒃𝒋 and 𝑪𝒒𝒋 obtained from the optimal hydrograph separation and the field chemical dataset.
3 Results and Discussion
3.1 Hydrological approach to calculate the hydrological recession time constant of the catchment (τ)
3.2 Baseflow exploration from RDF methods with hydrological recession time constant (τ) values
3.3 MB approach to find the optimal hydrological recession time constant
3.4 Comparison between the components 𝑪𝒃𝒋 and 𝑪𝒒𝒋 obtained from the optimal hydrograph separation and the field chemical dataset.
Chapter 3: Combining the one- and n-component models Combining concentration-discharge relationships with mixing models
2 Procedure for combining mixing models and C-Q relationships
2.1 Case 1: chemostatic components ( bb = bq = 0)
2.2 Case 2: single 2S-APS relationship (ab =aq = a and bb = bq = b)
2.3 Case 3: General case (a and b are different)
3 Application of the combining model
3.1 Study site and datasets
4 Results and discussion
4.1 Identification of the parameters and overall performance of the models
4.2 Performances for selected storm events
Chapter 4: Identification and quantication of the end-members
Identification of potential end members and their apportionment from downstream high – frequency chemical data
1.1 Formulation of the problem and main resolution techniques
1.2 Identification and contribution of end members methods
1.2.1 Tracer mass balance methods (TMB)
1.2.2 Multivariate statistical methods (MS)
1.2.3 Principal Component Analysis (PCA) and End member mixing analysis (EMMA) .
1.2.4 Positive Matrix Factorization method (PMF)
1.2.5 MIX method
1.3 Scope of this paper
2 Material and method
2.1 Study site
2.2 Data set processing
2.3.1 Optimization function
2.3.2 Sensitivity analysis of end members
3 Results and discussion
3.1 Relation between the criterion VR and A/B ratio
3.2 Average monthly concentrations of the potential end-members and their respective apportionment
3.3 Sensitivity analysis of end members
3.4 Potential end-members to identify observed evolution in a synthetic manner .
3.5 Potential end-members versus pre-identified possible end-members
5.1 Cluster analysis for evolution of end-members using a variable VR (0.02<VR<0.10)
5.2 Appendix-2: Summary for the month of November 2015 of: a) the initial position of the concentration with respect to the triangle of end-members. b) Position of the concentrations after their projection in the triangle. c)
Part III Conclusions and Perspectives
1.1 Main achievements of the thesis
1.1.1 The affine power scaling relationship
1.1.2 Calibration of the Hydrograph separation
1.1.3 Combining of affine relationship and mixing model
1.1.4 Identification and quantification of potential end-members
1.2 The develop of a parsimonious model