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.

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

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

The data required for this analysis are the median and standard deviations
(

If the system is composed of Eq. (1)

The system has a solution if the vector of
mixture

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

It is trivial that

For example, for

Using Eq. (9), the first-order Taylor series approximation (Taylor, 1982) for the variance (

To calculate

In this case, we get

Note that whatever the value of

Let

If we construct

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:

Three-dimensional mixing space generated using stream data, where
the median of end-members are projected.

Median and standard deviation (SD) of end-members and stream projected in three-dimensional space for the study period 2013–2014.

Boxplots of end-members projected in the three-dimensional mixing
space for the study period 2013–2014. The

Median and standard deviation (SD) of end-members and stream projected in three-dimensional space considering 50 % of the data sets.

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

This methodology was tested using data from a high-elevation (3500–3900 m a.s.l.) tropical catchment (7.53 km

Using the classic EMMA approach (Christophersen and
Hooper, 1992), data from end-members SW, HS, AN, RF and stream

Median and standard deviation (SD) of end-members and stream projected in three-dimensional space including artificial outliers.

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.

Median and enlarged standard deviation (SD) of end-members and stream projected in three-dimensional space.

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.

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

Note that the set of sources

Uncertainty of individual end-member contributions to the stream and Satterthwaite (1946) approximation for the degrees of freedom calculated for the study period 2013–2014.

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.

Uncertainty of individual end-member contributions to the stream and Satterthwaite (1946) approximation for the degrees of freedom computed considering 50 % of the data sets.

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

Uncertainty of individual end-member contributions to the stream and Satterthwaite (1946) approximation for the degrees of freedom computed after including artificial outliers.

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

Uncertainty of individual end-member contributions to the stream and Satterthwaite (1946) approximation for the degrees of freedom computed with enlarged standard deviations.

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.

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 (

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.

A MATLAB code to calculate the fractions of end-member contribution to the
mixture and their associated uncertainties is freely available at

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.

The authors declare that they have no conflict of interest.

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.

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