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**Hydrology and Earth System Sciences**
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**Technical note**
16 Dec 2019

**Technical note** | 16 Dec 2019

Technical note: Uncertainty in multi-source partitioning using large tracer data sets

^{1}Department of Geography and Water and Global Change Observatory, University of Costa Rica, San José, 2060, Costa Rica^{2}Institute for Applied Sustainability Research (iSUR), Quito, 170503, Ecuador^{3}Department of Mathematics, Universidad San Francisco de Quito, Quito, 170901, Ecuador^{4}Northern Rivers Institute, University of Aberdeen, Aberdeen, AB24 3UF, UK

^{1}Department of Geography and Water and Global Change Observatory, University of Costa Rica, San José, 2060, Costa Rica^{2}Institute for Applied Sustainability Research (iSUR), Quito, 170503, Ecuador^{3}Department of Mathematics, Universidad San Francisco de Quito, Quito, 170901, Ecuador^{4}Northern Rivers Institute, University of Aberdeen, Aberdeen, AB24 3UF, UK

**Correspondence**: Alicia Correa (alicia.correa@ucr.ac.cr)

**Correspondence**: Alicia Correa (alicia.correa@ucr.ac.cr)

Abstract

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The availability of large tracer data sets opened up the opportunity to investigate multiple source contributions to a mixture. However, the source contributions may be uncertain and, apart from Bayesian approaches, to date there are only solid methods to estimate such uncertainties for two and three sources. We introduce an alternative uncertainty estimation method for four sources based on multiple tracers as input data. Taylor series approximation is used to solve the set of linear mass balance equations. We illustrate the method to compute individual uncertainties in the calculation of source contributions to a mixture, with an example from hydrology, using a 14-tracer set from water sources and streamflow from a tropical, high-elevation catchment. Moreover, this method has the potential to be generalized to any number of tracers across a range of disciplines.

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Correa, A., Ochoa-Tocachi, D., and Birkel, C.: Technical note: Uncertainty in multi-source partitioning using large tracer data sets, Hydrol. Earth Syst. Sci., 23, 5059–5068, https://doi.org/10.5194/hess-23-5059-2019, 2019.

1 Introduction

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Tracer applications have dramatically increased over recent years across a wide range of disciplines (West et al., 2010). Applications in hydrology (Hooper, 2003; James and Roulet, 2006; Kirchner and Neal, 2013), ecology (Phillips and Gregg, 2003; Semmens et al., 2009b), anthropology (Ehleringer et al., 2008), conservation biology (Bicknell et al., 2014), nutrition (Magaña-Gallegos et al., 2018), environmental and ecosystem science (Bartov et al., 2013; Granek et al., 2009), and erosion and sediment transportation (Davies et al., 2018) have been the most prominent. Such widespread use of tracers was mainly facilitated by the availability of analytical techniques that provide highly sensitive, rapid multi-element analysis at a lower cost (Falkner et al., 1995). For example, the use of inductively coupled plasma mass spectrometry (ICP-MS) as one of the most relevant analytical techniques for elemental analysis (Helaluddin et al., 2016) led to the availability and use of large tracers sets (elements) in hydrological studies (Barthold et al., 2017; Belli et al., 2017; Correa et al., 2017; Kirchner and Neal, 2013; Mimba et al., 2017). Trace elements together with water stable isotopes (cavity ringdown laser absorption spectroscopy paved the way: Berman et al., 2009; Lis et al., 2008) as well as physical–chemical water parameters (e.g. electrical conductivity and pH) are now often used to improve understanding of hydro-geochemical cycles, flow pathways and runoff generation in hydrology. Furthermore, mixing models based on tracer mass balance equations are widely applied to identify the dominant sources of a mixture and their contribution dynamics.

In hydrological mixing models, the composition of the stream is assumed to
be an integrated mixture of signatures of different sources (Christophersen et al., 1990). The
proportional contributions of *n*+1 sources to the stream can be uniquely
determined using *n* different tracers (Christophersen
and Hooper, 1992). Bayesian methods have been developed to identify
multiple (>3) sources and compute their contributions to a
mixture in a two-dimensional mixing space (Parnell et al., 2010;
Stock et al., 2018). In this case, a unique solution is not feasible and a
higher uncertainty is attributed to the model (Phillips and
Gregg, 2001, 2003). On the other hand, end-member mixing analysis (EMMA) (Hooper, 2003) was developed to use multiple tracers as
input and, therefore, allows for a multi-dimensional space that potentially
increases the number of identifiable sources (Barthold
et al., 2011; Inamdar et al., 2013; Liu et al., 2004). In addition, the use
of multiple tracers can avoid bias and subjectivity in the input
information. Therefore, EMMA provides a robust and complete
conceptualization of catchment functioning and source interactions during
runoff generation (Iwasaki et al., 2015).
However, despite its benefits, the EMMA approach lacks a formal methodology
to assess the uncertainty of multiple end-members (Delsman et al., 2013) and their individual
uncertainties in the calculation of source contributions to a stream.

To our knowledge, the uncertainty estimation of source contributions to streams is based on Gaussian error propagation (Genereux, 1998) and was so far only calculated using one or two tracers simultaneously (MixSIAR: Parnell et al., 2010; Phillips and Gregg, 2001; Semmens et al., 2009a). Alternatively, when the number of sources is higher, the uncertainty is usually based on the sum of analytical errors, elevation effects and the spatial variability of end-member concentrations (Uhlenbrook and Hoeg, 2003). Hence, we propose an alternative methodology based on the first-order Taylor series approximation to estimate the uncertainty of individual end-members or sources (e.g. precipitation, soil water, groundwater) to a mixture (e.g. streamflow).

We illustrate this application using a multi-tracer data set from Correa et al. (2019b), in a three-dimensional space defined by a principal component analysis (PCA). In Correa et al. (2019b), the authors computed the uncertainties but without disclosing any details in the calculation and methodology used. The main objective of this technical note is, therefore, to explicitly describe the mathematical development that allows the calculation of partial derivatives, degrees of freedom and confidence interval limits for each source fraction contribution and, moreover, to provide the code and several examples for their calculation and reproducibility.

2 Uncertainty estimation method development

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In this section, the uncertainty estimation method presented in Phillips and Gregg (2001) is expanded for four source contributions to the mixture.

Let 𝒞 represent the following set of sources: *A*, *B*, *C* and *D*, and mixture *M*,
$\mathcal{C}=\left\{A,B,C,D,M\right\}$. In the following equations, *x*∈𝒞, $y\in \left\{\stackrel{\mathrm{\u203e}}{\mathit{\delta}},\stackrel{\mathrm{\u203e}}{\mathit{\lambda}},\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}\right\}$ and $z\in \left\{A,M,C\right\}$. *x*, *y* and *z* are variables that
belong to the following sets: *x* to the set of *A*, *B*, *C*, *D* and mixture *M*; *y* to the set of
medians of every projected source and mixture in each principal component
$\stackrel{\mathrm{\u203e}}{\mathit{\delta}}\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}$ respectively of the sub index
used; and *z* to the set of *A*, *M* and *C*. Furthermore,
${f}_{A},{f}_{B},{f}_{C}$ and *f*_{D} represent the contribution fraction of
sources *A*, *B*, *C* and *D* respectively to the mixture *M*.

The data required for this analysis are the median and standard deviations
(*σ*) of each of the sources (*A*, *B*, *C* and *D*) and the mixture *M*,
projected and expressed in the coordinates of the three-dimensional PCA
space. As well as this, the sample size (*n*) of each source is required. Details of
the application are presented in Sect. 3.2.

If the system is composed of Eq. (1)

$$\begin{array}{}\text{(1)}& \left\{\begin{array}{ccccccccc}{\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{A}{f}_{A}& +& {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{B}{f}_{B}& +& {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{C}{f}_{C}& +& {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{D}{f}_{D}& =& {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{M}\\ {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{A}{f}_{A}& +& {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{B}{f}_{B}& +& {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{C}{f}_{C}& +& {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{D}{f}_{D}& =& {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{M}\\ {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{A}{f}_{A}& +& {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{B}{f}_{B}& +& {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{C}{f}_{C}& +& {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{D}{f}_{D}& =& {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{M}\\ {f}_{A}& +& {f}_{B}& +& {f}_{C}& +& {f}_{D}& =& \mathrm{1}\end{array}\right.\end{array}$$

and has solution^{1} for *f*_{A}*f*_{B}*f*_{C}*f*_{D}>0, the
contribution fractions take the following form:

$$\begin{array}{}\text{(2)}& \begin{array}{rl}& {f}_{A}={\displaystyle \frac{\left({\mathrm{\Phi}}_{M}-{\mathrm{\Delta}}_{M}\right)\left({\mathrm{\Lambda}}_{C}-{\mathrm{\Delta}}_{C}\right)-\left({\mathrm{\Lambda}}_{M}-{\mathrm{\Delta}}_{M}\right)\left({\mathrm{\Phi}}_{C}-{\mathrm{\Delta}}_{C}\right)}{\left({\mathrm{\Phi}}_{A}-{\mathrm{\Delta}}_{A}\right)\left({\mathrm{\Lambda}}_{C}-{\mathrm{\Delta}}_{C}\right)-\left({\mathrm{\Lambda}}_{A}-{\mathrm{\Delta}}_{A}\right)\left({\mathrm{\Phi}}_{C}-{\mathrm{\Delta}}_{C}\right)}}\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}={\displaystyle \frac{\mathrm{Num}}{\mathrm{Den}}},\\ & {f}_{C}={\displaystyle \frac{\left({\mathrm{\Delta}}_{M}-{\mathrm{\Lambda}}_{M}\right)-\left({\mathrm{\Delta}}_{A}-{\mathrm{\Lambda}}_{A}\right){f}_{A}}{\left({\mathrm{\Delta}}_{C}-{\mathrm{\Lambda}}_{C}\right)}},\\ & {f}_{B}={\mathrm{\Delta}}_{M}-\left({\mathrm{\Delta}}_{C}{f}_{C}+{\mathrm{\Delta}}_{A}{f}_{A}\right),\\ & {f}_{D}=\mathrm{1}-\left({f}_{C}+{f}_{B}+{f}_{A}\right),\end{array}\end{array}$$

where

$$\begin{array}{}\text{(3)}& \begin{array}{rl}& {\mathrm{\Delta}}_{x}={\displaystyle \frac{{\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}-{\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{D}}{{\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{B}-{\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{D}}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\mathrm{\Lambda}}_{x}={\displaystyle \frac{{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}-{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{D}}{{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{B}-{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{D}}},\\ & {\mathrm{\Phi}}_{x}={\displaystyle \frac{{\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}-{\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{D}}{{\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{B}-{\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{D}}}.\end{array}\end{array}$$

The partial derivatives of Eq. (2) are given by

$$\begin{array}{rl}& {\displaystyle \frac{\partial {f}_{A}}{\partial {y}_{x}}}={\displaystyle \frac{\mathrm{1}}{{\mathrm{Den}}^{\mathrm{2}}}}\left[\right[\left({\mathrm{\Lambda}}_{C}-{\mathrm{\Delta}}_{C}\right)\left({\displaystyle \frac{\partial {\mathrm{\Phi}}_{M}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{M}}{\partial {y}_{x}}}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+\left({\mathrm{\Phi}}_{M}-{\mathrm{\Delta}}_{M}\right)\left({\displaystyle \frac{\partial {\mathrm{\Lambda}}_{C}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{C}}{\partial {y}_{x}}}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-\left({\mathrm{\Phi}}_{C}-{\mathrm{\Delta}}_{C}\right)\left({\displaystyle \frac{\partial {\mathrm{\Lambda}}_{M}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{M}}{\partial {y}_{x}}}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-\left({\mathrm{\Lambda}}_{M}-{\mathrm{\Delta}}_{M}\right)\left({\displaystyle \frac{\partial {\mathrm{\Phi}}_{C}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{C}}{\partial {y}_{x}}}\right)]\mathrm{Den}\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-[\left({\mathrm{\Lambda}}_{C}-{\mathrm{\Delta}}_{C}\right)\left({\displaystyle \frac{\partial {\mathrm{\Phi}}_{A}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{A}}{\partial {y}_{x}}}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+\left({\mathrm{\Phi}}_{A}-{\mathrm{\Delta}}_{A}\right)\left({\displaystyle \frac{\partial {\mathrm{\Lambda}}_{C}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{C}}{\partial {y}_{x}}}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-\left({\mathrm{\Phi}}_{C}-{\mathrm{\Delta}}_{C}\right)\left({\displaystyle \frac{\partial {\mathrm{\Lambda}}_{A}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{A}}{\partial {y}_{x}}}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-\left({\mathrm{\Lambda}}_{A}-{\mathrm{\Delta}}_{A}\right)\left({\displaystyle \frac{\partial {\mathrm{\Phi}}_{C}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{C}}{\partial {y}_{x}}}\right)\left]\mathrm{Num}\right],\end{array}$$

$$\begin{array}{}\text{(4)}& \begin{array}{rl}& {\displaystyle \frac{\partial {f}_{C}}{\partial {y}_{x}}}={\displaystyle \frac{\mathrm{1}}{{\left({\mathrm{\Delta}}_{C}-{\mathrm{\Lambda}}_{C}\right)}^{\mathrm{2}}}}\left[\right[\left({\displaystyle \frac{\partial {\mathrm{\Delta}}_{M}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Lambda}}_{M}}{\partial {y}_{x}}}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-\left({\displaystyle \frac{\partial {\mathrm{\Delta}}_{A}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Lambda}}_{A}}{\partial {y}_{x}}}\right){f}_{A}-\left({\mathrm{\Delta}}_{A}-{\mathrm{\Lambda}}_{A}\right){\displaystyle \frac{\partial {f}_{A}}{\partial {y}_{x}}}]\left({\mathrm{\Delta}}_{C}-{\mathrm{\Lambda}}_{C}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-\left({\displaystyle \frac{\partial {\mathrm{\Delta}}_{C}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Lambda}}_{C}}{\partial {y}_{x}}}\right)\left[\left({\mathrm{\Delta}}_{M}-{\mathrm{\Lambda}}_{M}\right)-\left({\mathrm{\Delta}}_{A}-{\mathrm{\Lambda}}_{A}\right){f}_{A}\right]],\\ & {\displaystyle \frac{\partial {f}_{B}}{\partial {y}_{x}}}={\displaystyle \frac{\partial {\mathrm{\Delta}}_{M}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{C}}{\partial {y}_{x}}}{f}_{C}-{\mathrm{\Delta}}_{C}{\displaystyle \frac{\partial {f}_{C}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{A}}{\partial {y}_{x}}}{f}_{A}\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\mathrm{\Delta}}_{A}{\displaystyle \frac{\partial {f}_{A}}{\partial {y}_{x}}},\\ & {\displaystyle \frac{\partial {f}_{D}}{\partial {y}_{x}}}=-{\displaystyle \frac{\partial {f}_{C}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {f}_{B}}{\partial {y}_{x}}}-{\displaystyle \frac{\partial {f}_{A}}{\partial {y}_{x}}}.\end{array}\end{array}$$

It is trivial that

$$\begin{array}{}\text{(5)}& \begin{array}{rl}& {\displaystyle \frac{\partial {\mathrm{\Delta}}_{z}}{\partial {w}_{x}}}=\mathrm{0},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}w\in \left\{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}},\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}\right\};\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\displaystyle \frac{\partial {\mathrm{\Lambda}}_{z}}{\partial {w}_{x}}}=\mathrm{0},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}w\in \left\{\stackrel{\mathrm{\u203e}}{\mathit{\delta}},\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}\right\};\\ & {\displaystyle \frac{\partial {\mathrm{\Phi}}_{z}}{\partial {w}_{x}}}=\mathrm{0},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}w\in \left\{\stackrel{\mathrm{\u203e}}{\mathit{\delta}},\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}\right\},\end{array}\end{array}$$

where

$$\begin{array}{}\text{(6)}& {\displaystyle \frac{\partial {\mathrm{\Delta}}_{z}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}}={\left({\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{B}-{\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{D}\right)}^{-\mathrm{1}}\left\{\begin{array}{ll}\mathrm{1}& z\in \left\{A,C,M\right\}\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}x=z\\ -{\mathrm{\Delta}}_{z}& z\ne B\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}x=B\\ {\mathrm{\Delta}}_{z}-\mathrm{1}& z\ne D\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}x=D\\ \mathrm{0}& \mathrm{otherwise}\end{array}\right.,\text{(7)}& {\displaystyle \frac{\partial {\mathrm{\Lambda}}_{z}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}}={\left({\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{B}-{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{D}\right)}^{-\mathrm{1}}\left\{\begin{array}{ll}\mathrm{1}& z\in \left\{A,C,M\right\}\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}x=z\\ -{\mathrm{\Lambda}}_{z}& z\ne B\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}x=B\\ {\mathrm{\Lambda}}_{z}-\mathrm{1}& z\ne D\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}x=D\\ \mathrm{0}& \mathrm{otherwise}\end{array}\right.\end{array}$$

and

$$\begin{array}{}\text{(8)}& {\displaystyle \frac{\partial {\mathrm{\Phi}}_{z}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}}={\left({\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{B}-{\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{D}\right)}^{-\mathrm{1}}\left\{\begin{array}{ll}\mathrm{1}& z\in \left\{A,C,M\right\}\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}x=z\\ -{\mathrm{\Phi}}_{z}& z\ne B\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}x=B\\ {\mathrm{\Phi}}_{z}-\mathrm{1}& z\ne D\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}x=D\\ \mathrm{0}& \mathrm{otherwise}\end{array}\right..\end{array}$$

For example, for *f*_{A} we have

$$\begin{array}{rl}& {\displaystyle \frac{\partial {f}_{A}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}}={\displaystyle \frac{\mathrm{1}}{{\mathrm{Den}}^{\mathrm{2}}}}\left[\right[{\displaystyle \frac{\partial {\mathrm{\Delta}}_{M}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}}\left({\mathrm{\Phi}}_{C}-{\mathrm{\Lambda}}_{C}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{C}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}}\left({\mathrm{\Phi}}_{M}-{\mathrm{\Lambda}}_{M}\right)]\mathrm{Den}-[{\displaystyle \frac{\partial {\mathrm{\Delta}}_{A}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}}\left({\mathrm{\Phi}}_{C}-{\mathrm{\Lambda}}_{C}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\displaystyle \frac{\partial {\mathrm{\Delta}}_{C}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}}\left({\mathrm{\Phi}}_{A}-{\mathrm{\Lambda}}_{A}\right)\left]\mathrm{Num}\right],\\ & {\displaystyle \frac{\partial {f}_{A}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}}={\displaystyle \frac{\mathrm{1}}{{\mathrm{Den}}^{\mathrm{2}}}}\left[\right[{\displaystyle \frac{\partial {\mathrm{\Lambda}}_{C}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}}\left({\mathrm{\Phi}}_{M}-{\mathrm{\Delta}}_{M}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\displaystyle \frac{\partial {\mathrm{\Lambda}}_{M}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}}\left({\mathrm{\Phi}}_{C}-{\mathrm{\Delta}}_{C}\right)]\mathrm{Den}-[{\displaystyle \frac{\partial {\mathrm{\Lambda}}_{C}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}}\left({\mathrm{\Phi}}_{A}-{\mathrm{\Delta}}_{A}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\displaystyle \frac{\partial {\mathrm{\Lambda}}_{A}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}}\left({\mathrm{\Phi}}_{C}-{\mathrm{\Delta}}_{C}\right)\left]\mathrm{Num}\right],\end{array}$$

$$\begin{array}{}\text{(9)}& \begin{array}{rl}& {\displaystyle \frac{\partial {f}_{A}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}}={\displaystyle \frac{\mathrm{1}}{{\mathrm{Den}}^{\mathrm{2}}}}\left[\right[{\displaystyle \frac{\partial {\mathrm{\Phi}}_{M}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}}\left({\mathrm{\Lambda}}_{C}-{\mathrm{\Delta}}_{C}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\displaystyle \frac{\partial {\mathrm{\Phi}}_{C}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}}\left({\mathrm{\Lambda}}_{M}-{\mathrm{\Delta}}_{M}\right)]\mathrm{Den}-[{\displaystyle \frac{\partial {\mathrm{\Phi}}_{A}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}}\left({\mathrm{\Lambda}}_{C}-{\mathrm{\Delta}}_{C}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\displaystyle \frac{\partial {\mathrm{\Phi}}_{C}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}}\left({\mathrm{\Lambda}}_{A}-{\mathrm{\Delta}}_{A}\right)\left]\mathrm{Num}\right].\end{array}\end{array}$$

Using Eq. (9), the first-order Taylor series approximation (Taylor, 1982) for the variance (*σ*^{2}) of *f*_{A}
evaluated at the mean can be calculated by

$$\begin{array}{}\text{(10)}& \begin{array}{rl}& {\mathit{\sigma}}_{{f}_{A}}^{\mathrm{2}}=\sum _{x}{\left({\displaystyle \frac{\partial {f}_{A}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}^{\mathrm{2}}+\sum _{x}{\left({\displaystyle \frac{\partial {f}_{A}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}^{\mathrm{2}}\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+\sum _{x}{\left({\displaystyle \frac{\partial {f}_{A}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}^{\mathrm{2}}=\sum _{y}\sum _{x}{\left({\displaystyle \frac{\partial {f}_{A}}{\partial {y}_{x}}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}.\end{array}\end{array}$$

To calculate *γ*_{A} (the Satterthwaite, 1946,
approximation for the degrees of freedom), we define ${f}_{A{y}_{x}}={c}_{A}{\left(\frac{\partial {f}_{A}}{\partial {y}_{x}}\right)}^{\mathrm{2}}$, where *c*_{A} is a scale
constant that relates ${f}_{A{y}_{x}}$ with the respective derivative. It means
that *f*_{A} with respect to *y*_{x} can be a scalar multiple of the
derivative value.

In this case, we get

$$\begin{array}{}\text{(11)}& {\mathit{\gamma}}_{A}={\displaystyle \frac{{\left(\sum _{y}\sum _{x}{f}_{A{y}_{x}}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{\sum _{y}\sum _{x}\frac{{\left({f}_{A{y}_{x}}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{{n}_{{y}_{x}}-\mathrm{1}}}}.\end{array}$$

Note that whatever the value of *c*_{A} is, Eq. (11) leads to

$${\mathit{\gamma}}_{A}={\displaystyle \frac{{\left(\sum _{y}\sum _{x}{\left(\frac{\partial {f}_{A}}{\partial {y}_{x}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{\sum _{y}\sum _{x}\frac{{\left({\left(\frac{\partial {f}_{A}}{\partial {y}_{x}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{{n}_{{y}_{x}}-\mathrm{1}}}},$$

and if we set ${f}_{A{y}_{x}}^{\ast}={\left(\frac{\partial {f}_{A}}{\partial {y}_{x}}\right)}^{\mathrm{2}}$ then the numerator of the last equation can be replaced
by ${\left({\mathit{\sigma}}_{{f}_{A}}^{\mathrm{2}}\right)}^{\mathrm{2}}$. In other words, we can use Eq. (10) and the derivatives Eq. (9) to estimate the value of *γ*_{A},
resulting in ${f}_{A{y}_{x}}={c}_{A}{f}_{A{y}_{x}}^{\ast}$. Of course, it is required
that *c*_{A} is constant with respect to *y*_{x}. Then,

$$\begin{array}{}\text{(12)}& {\mathit{\gamma}}_{A}={\displaystyle \frac{{\left({\mathit{\sigma}}_{{f}_{A}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{\sum _{y}\sum _{x}\frac{{\left({\left(\frac{\partial {f}_{A}}{\partial {y}_{x}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{{n}_{{y}_{x}}-\mathrm{1}}}}.\end{array}$$

Let $w\in \mathcal{C}\setminus \left\{A\right\}$. The first-order Taylor
series approximation for the variance of *f*_{w} can be calculated by (as
above Eq. 10)

$$\begin{array}{}\text{(13)}& \begin{array}{rl}& {\mathit{\sigma}}_{{f}_{w}}^{\mathrm{2}}=\sum _{x}{\left({\displaystyle \frac{\partial {f}_{w}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{\stackrel{\mathrm{\u203e}}{\mathit{\delta}}}_{x}}^{\mathrm{2}}+\sum _{x}{\left({\displaystyle \frac{\partial {f}_{w}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{x}}^{\mathrm{2}}\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+\sum _{x}{\left({\displaystyle \frac{\partial {f}_{w}}{\partial {\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{\stackrel{\mathrm{\u203e}}{\mathit{\varphi}}}_{x}}^{\mathrm{2}}=\sum _{y}\sum _{x}{\left({\displaystyle \frac{\partial {f}_{w}}{\partial {y}_{x}}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}.\end{array}\end{array}$$

If we construct *γ*_{w} as *γ*_{A}, we get

$${\mathit{\gamma}}_{w}={\displaystyle \frac{{\left(\sum _{y}\sum _{x}{f}_{w{y}_{x}}^{\ast}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{\sum _{y}\sum _{x}\frac{{\left({f}_{w{y}_{x}}^{\ast}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{{n}_{{y}_{x}}-\mathrm{1}}}},$$

where ${f}_{{w}_{{y}_{x}}}={c}_{w}{f}_{{w}_{{y}_{x}}}^{\ast}$ and ${f}_{{w}_{{y}_{x}}}^{\ast}={\left(\frac{\partial {f}_{w}}{\partial {y}_{x}}\right)}^{\mathrm{2}}$ with *c*_{w}
constant with respect to *y*_{x}, then we finally get

$$\begin{array}{}\text{(14)}& {\mathit{\gamma}}_{w}={\displaystyle \frac{{\left({\mathit{\sigma}}_{{f}_{w}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{\sum _{y}\sum _{x}\frac{{\left({\left(\frac{\partial {f}_{w}}{\partial {y}_{x}}\right)}^{\mathrm{2}}{\mathit{\sigma}}_{{y}_{x}}^{\mathrm{2}}\right)}^{\mathrm{2}}}{{n}_{{y}_{x}}-\mathrm{1}}}}.\end{array}$$

The upper and lower confidence interval limits for each end-member fraction can be calculated using partial derivatives and the 95 % confidence intervals constructed as follows:

$$\begin{array}{}\text{(15)}& {f}_{w}\pm {t}_{\mathrm{0.05}{\mathit{\gamma}}_{w}}{\mathit{\sigma}}_{fw},\end{array}$$

where *t*_{0.05,γ} is the Student's *t* for *α*=0.05
(two-tailed) (Walpole et al., 2017) and *γ* degrees of
freedom related with *σ*_{fw}.

^{∗} Coordinates of end-members and stream (mixture) medians in
three-dimensional

space (*U*1, *U*2 and *U*3). *n* represents the sample size.

3 Application

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This methodology was tested using data from a high-elevation (3500–3900 m a.s.l.) tropical catchment (7.53 km^{2}) located in southern Ecuador
(3^{∘}4^{′}38^{′′} S, 79^{∘}15^{′}30^{′′} W). The mean
annual precipitation for this study site is 1300 mm (Padrón et al., 2015), the
mean annual discharge is 860 mm yr^{−1}. The catchment is of a volcanic
origin and dominated by volcanic Histosol (24 %) and Andosol (72 %)
soils (Quichimbo et al., 2012), both with a high percentage of
organic matter content (450 and 310 g kg, respectively) (Quichimbo
et al., 2012) and large water-holding capacities (Buytaert et al., 2006). Histosols
are primarily located at the valleys and covered by cushion plants, while
Andosol soils are predominant on the hillslopes under a cover of tussock
grass. Nearly saturated conditions of the riparian zone are observed
year-round, and a spring is located in the north-western part of the
catchment. Streamwater samples from five nested streams were collected weekly
from March 2013 to April 2014 (*n*=257) and at a higher frequency during
experimental campaigns. Additionally, bi-weekly water samples from four
potential end-members, rainfall (RF), soil water from Andosols (AN) and
Histosols (HS), and spring water (SW) (*n*∼30, for each
end-member) were collected. A multi-tracer (14 tracers) data set of
conservative tracers was obtained from each water sample (Na, Mg, Al, Si, K,
Ca, Rb, Sr, Ba, Ce, V, Y, Nd) at the Institute for Landscape Ecology and
Resource Management of the Justus Liebig University using an ICP-MS (Agilent
7500ce, Agilent Technologies) and the electrical conductivity (EC) was
measured in situ. More detailed information on the study site and data set
can be found in Correa
et al. (2017, 2019b).

Using the classic EMMA approach (Christophersen and
Hooper, 1992), data from end-members SW, HS, AN, RF and stream *M* were
projected into a three-dimensional space (Correa et al., 2019b) and
presented in Fig. 1. The resulting median and standard deviation of
end-members and stream coordinates are shown in Table 1. Furthermore, Fig. 2 shows the distribution of projected samples from individual end-members in
the PCA coordinates.

The example (3) considers outliers included at the positive extreme of the
data set of each source and (4) outliers included at the negative extreme. ^{∗} Coordinates of end-members and stream (mixture) medians in three-dimensional
space (*U*1, *U*2 and *U*3). *n* represents the sample size.

The example (5) considers 3 times the standard deviation of the original data
set and (6) 5 times the standard deviation of the original data set. ^{∗} Coordinates of end-members and stream (mixture) medians in three-dimensional
space (*U*1, *U*2 and *U*3). *n* represents the sample size.

The uncertainty range of each of the four end-member contributions to the
stream was determined using the above developed Eq. (15) based on the
first-order Taylor series approximation (Eq. 14) (MATLAB code in Correa et al., 2019a). The *f*_{w} gives the proportion of *w*
in *M* and ${\mathit{\sigma}}_{{f}_{w}}^{\mathrm{2}}$ gives the variances of *w*. The upper uncertainty
limit was computed as ${f}_{w}+{t}_{\mathrm{0.05},\mathit{\gamma}}{\mathit{\sigma}}_{fw}$ and the lower
limit as ${f}_{w}-{t}_{\mathrm{0.05},\mathit{\gamma}}{\mathit{\sigma}}_{fw}$. This procedure was applied
to all end-members. The resulting uncertainty estimates for each source
end-member are shown in Table 5.

Note that the set of sources *A*, *B*, *C* and *D* used for the development of the
equations are represented here by SW, HS, AN and RF in this specific order.
*U*1, *U*2 and *U*3 represent the principal components PC1, PC2 and PC3,
respectively.

From the above-mentioned data set, we have generated six examples to assess the sensitivity of the uncertainty calculation to the source sample size, the artificial inclusion of outliers (upper and lower extremes) and the increased standard deviations of the source data sets.

The example (1) was computed considering the initial 50 % and (2) the remaining 50 % of the sample sets.

The example (3) was computed after including outliers at the positive extreme of the data set and (4) including outliers at the negative extreme.

The example (5) was computed considering 3 times the standard deviation of the original data set and (6) 5 times the standard deviation of the original data set.

The first example considers 50 % of the samples (collected in the first half of the monitoring period) from each source. The median, standard deviation and sample size are input data (Table 2) to calculate the uncertainty bands (Table 6).

The second example considers the remaining 50 % of the samples and was similarly executed (Table 2).

In the third example, outliers were artificially included at the upper positive end of data sets for each source at each coordinate, respectively. The outliers consisted of twice the maximum positive value of the observed data (Table 3).

Using the same criteria, the negative extremes were included in the fourth example (Table 3).

Sources affected by dispersed data clouds were taken into account by an increase in the standard deviation. We considered two cases, the first, in example five, increasing 3 times the value of the standard deviation of the initial data set (Table 4) and finally, increasing the standard deviation 5 times for the sixth example (Table 4).

The results of this analysis are presented in Tables 6–8. In examples 1 and 2 the sample size reduction from 24 to 12 and 13 samples respectively (Table 6) had a minimal effect (less than 3 %) on the calculation of the uncertainty ranges compared to the original complete set (Table 1). The fractions of source contributions did not experience changes. The inclusion of outliers affected the values of the medians at levels of the second decimal (Table 3) concerning the median of the initial data (Table 2). However, the standard deviations increased in a range of 1.2 to 2.5 times the original value for AN and HS, and more for RF (2.5 to 10.5) and drastically more for SW (4 to 20 times wider). These variations were reflected in the widening (1 % to 12 %) of uncertainty bands for all existing cases (Table 7) in comparison with those calculated from the original data set (Table 5). Furthermore, the widening of the standard deviations to 3 and 5 times their initial values resulted in an increase in the range of uncertainty between 2 % and 22 % for the first case and between 5 % and 37 % for the second case. For the latter, the minimum limit of the uncertainty range was reached in all the reported cases. The large number of samples used in these exercises was reflected in high degrees of freedom.

4 Summary and remarks

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Our methodology was developed to calculate the contribution of sources to the mixture and its associated uncertainty (based on multiple tracer sets) and was shown to be effective in real application cases. The application of the method reflected that the calculations of the uncertainty ranges of multiple source contributions to a mixture do not experience significant changes with sample size reduction or inclusion of outliers. Rather, it shows marginally different results by incorporating standard deviations from widely dispersed data.

The methodology, based on Phillips and Gregg (2001) combined with EMMA applications (Hooper, 2003) presents high potential for use as an alternative method to the simple sum of analytical errors (Uhlenbrook and Hoeg, 2003) or the Bayesian approach (Parnell et al., 2010; Stock et al., 2018). We provide a tool to close the gap in studies of mixing processes when a larger number of source contributions (>3) and related uncertainty estimates are needed for a more complete conceptualization (Iwasaki et al., 2015).

The MATLAB code provided and the illustrative examples facilitate the understanding of the methodology and promote future scientific applications. We are confident that the use of this methodology will help the scientific community that is increasingly using large tracer sets in its research to obtain results supported by a sound uncertainty analysis.

Code and data availability

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Code and data availability.

A MATLAB code to calculate the fractions of end-member contribution to the mixture and their associated uncertainties is freely available at https://zenodo.org/record/3518700 (last access: 5 November 2019; Correa et al., 2019a), as well as input data (used in this study) as an example for the code run and an instruction note.

Author contributions

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Author contributions.

AC and CB conceptualized the methodology. AC was responsible for the data collection and analysis. DOT and AC programmed and evaluated the MATLAB code with collected data. AC wrote the paper with contributions from all co-authors.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

Alicia Correa and Christian Birkel would like to acknowledge support by a UCR postdoctoral fellowship awarded to Alicia Correa, UCREA, and the Water and Global Change Observatory at the Department of Geography, UCR. The authors thank the Central Research Office (DIUC) of the Universidad de Cuenca for making available part of the tracer data sets. We are especially grateful for the constructive comments that were provided by the referees, which greatly improved the quality of the technical note.

Review statement

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Review statement.

This paper was edited by Markus Hrachowitz and reviewed by two anonymous referees.

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The system has a solution if the vector of
mixture *M* is on the polyhedron generated by the vectors of sources *A*, *B*, *C* and *D* such that $\sum _{x}{f}_{x}=\mathrm{1}$.

Short summary

The applications and availability of large tracer data sets have vastly increased in recent years leading to research into the contributions of multiple sources to a mixture. We introduce a method based on Taylor series approximation to estimate the uncertainties of such sources' contributions. The method is illustrated with examples of hydrology (14 tracers) and a MATLAB code is provided for reproducibility. This method can be generalized to any number of tracers across a range of disciplines.

The applications and availability of large tracer data sets have vastly increased in recent...

Hydrology and Earth System Sciences

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