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.

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, C=A,B,C,D,M. In the following equations, x∈C, y∈δ‾,λ‾,ϕ‾ and z∈A,M,C. 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 δ‾λ‾ϕ‾ respectively of the sub index used; and z to the set of A, M and C. Furthermore, fA,fB,fC and fD 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) 1δ‾AfA+δ‾BfB+δ‾CfC+δ‾DfD=δ‾Mλ‾AfA+λ‾BfB+λ‾CfC+λ‾DfD=λ‾Mϕ‾AfA+ϕ‾BfB+ϕ‾CfC+ϕ‾DfD=ϕ‾MfA+fB+fC+fD=1 and has solution

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 ∑xfx=1.

The partial derivatives of Eq. (2) are given by ∂fA∂yx=1Den2[[ΛC-ΔC∂ΦM∂yx-∂ΔM∂yx+ΦM-ΔM∂ΛC∂yx-∂ΔC∂yx-ΦC-ΔC∂ΛM∂yx-∂ΔM∂yx-ΛM-ΔM∂ΦC∂yx-∂ΔC∂yx]Den-[ΛC-ΔC∂ΦA∂yx-∂ΔA∂yx+ΦA-ΔA∂ΛC∂yx-∂ΔC∂yx-ΦC-ΔC∂ΛA∂yx-∂ΔA∂yx-ΛA-ΔA∂ΦC∂yx-∂ΔC∂yx]Num],

4∂fC∂yx=1ΔC-ΛC2[[∂ΔM∂yx-∂ΛM∂yx-∂ΔA∂yx-∂ΛA∂yxfA-ΔA-ΛA∂fA∂yx]ΔC-ΛC-∂ΔC∂yx-∂ΛC∂yxΔM-ΛM-ΔA-ΛAfA],∂fB∂yx=∂ΔM∂yx-∂ΔC∂yxfC-ΔC∂fC∂yx-∂ΔA∂yxfA-ΔA∂fA∂yx,∂fD∂yx=-∂fC∂yx-∂fB∂yx-∂fA∂yx.

It is trivial that 5∂Δz∂wx=0,w∈λ‾,ϕ‾;∂Λz∂wx=0,w∈δ‾,ϕ‾;∂Φz∂wx=0,w∈δ‾,λ‾, where 6∂Δz∂δ‾x=δ‾B-δ‾D-11z∈A,C,Mandx=z-Δzz≠Bandx=BΔz-1z≠Dandx=D0otherwise,7∂Λz∂λ‾x=λ‾B-λ‾D-11z∈A,C,Mandx=z-Λzz≠Bandx=BΛz-1z≠Dandx=D0otherwise and 8∂Φz∂ϕ‾x=ϕ‾B-ϕ‾D-11z∈A,C,Mandx=z-Φzz≠Bandx=BΦz-1z≠Dandx=D0otherwise.

For example, for fA we have ∂fA∂δ‾x=1Den2[[∂ΔM∂δ‾xΦC-ΛC-∂ΔC∂δ‾xΦM-ΛM]Den-[∂ΔA∂δ‾xΦC-ΛC-∂ΔC∂δ‾xΦA-ΛA]Num],∂fA∂λ‾x=1Den2[[∂ΛC∂λ‾xΦM-ΔM-∂ΛM∂λ‾xΦC-ΔC]Den-[∂ΛC∂λ‾xΦA-ΔA-∂ΛA∂λ‾xΦC-ΔC]Num],

9∂fA∂ϕ‾x=1Den2[[∂ΦM∂ϕ‾xΛC-ΔC-∂ΦC∂ϕ‾xΛM-ΔM]Den-[∂ΦA∂ϕ‾xΛC-ΔC-∂ΦC∂ϕ‾xΛA-ΔA]Num].

Using Eq. (9), the first-order Taylor series approximation (Taylor, 1982) for the variance (σ2) of fA evaluated at the mean can be calculated by 10σfA2=∑x∂fA∂δ‾x2σδ‾x2+∑x∂fA∂λ‾x2σλ‾x2+∑x∂fA∂ϕ‾x2σϕ‾x2=∑y∑x∂fA∂yx2σyx2.

To calculate γA (the Satterthwaite, 1946, approximation for the degrees of freedom), we define fAyx=cA∂fA∂yx2, where cA is a scale constant that relates fAyx with the respective derivative. It means that fA with respect to yx can be a scalar multiple of the derivative value.

In this case, we get 11γA=∑y∑xfAyxσyx22∑y∑xfAyxσyx22nyx-1.

Note that whatever the value of cA is, Eq. (11) leads to γA=∑y∑x∂fA∂yx2σyx22∑y∑x∂fA∂yx2σyx22nyx-1, and if we set fAyx∗=∂fA∂yx2 then the numerator of the last equation can be replaced by σfA22. In other words, we can use Eq. (10) and the derivatives Eq. (9) to estimate the value of γA, resulting in fAyx=cAfAyx∗. Of course, it is required that cA is constant with respect to yx. Then, 12γA=σfA22∑y∑x∂fA∂yx2σyx22nyx-1.

Let w∈C∖A. The first-order Taylor series approximation for the variance of fw can be calculated by (as above Eq. 10) 13σfw2=∑x∂fw∂δ‾x2σδ‾x2+∑x∂fw∂λ‾x2σλ‾x2+∑x∂fw∂ϕ‾x2σϕ‾x2=∑y∑x∂fw∂yx2σyx2.

If we construct γw as γA, we get γw=∑y∑xfwyx∗σyx22∑y∑xfwyx∗σyx22nyx-1, where fwyx=cwfwyx∗ and fwyx∗=∂fw∂yx2 with cw constant with respect to yx, then we finally get 14γw=σfw22∑y∑x∂fw∂yx2σyx22nyx-1.

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: 15fw±t0.05γwσfw, where t0.05,γ is the Student's t for α=0.05 (two-tailed) (Walpole et al., 2017) and γ degrees of freedom related with σfw.

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.