3.1 The Mixing Model Concept

The stable isotope composition of an element in a chemical compound dissolved in water (i.e., a solute) reflects its origin/source and, to some extent, the stable isotope fractionation processes that may have modified its initial signature during its transport and retention in solution. Failure to account for stable isotope fractionation processes may thus negatively affect the final conclusions of partitioning contributions from various sources of pollution.

A solute analysed in a water sample often originates from more than one source and therefore represents a mixture. This mixture is characterised by a mean stable isotope composition that is proportional to the relative contributions from all sources and the original stable isotope signatures of those sources. If the stable isotope signatures of individual sources are known and if they differ significantly from each other, then the stable isotope composition of the mixture can be used to estimate the relative contribution from each source. The calculation algorithm for this estimation is known as a stable isotope mass balance, and it can be applied to various elements in various solutes analysed in a single sample. However, an unambiguous solution of mass balance equations requires that the number of tracers used be not lower than one less than the number of sources under consideration (e.g., if three sources are contributing to a mixture, we need a minimum of two tracers; if four sources are considered, we need a minimum of three tracers, etc.). If fewer than the required number of tracers is available, these tracers can still be used for mass balance calculations, but an accurate calculation of the exact contributions will not be possible. Instead, possible ranges of contributions from each source can be obtained. The concept of mixing models is illustrated in the following three examples, which show different scenarios according to the different numbers of tracers available.

3.2 Calculation Examples

3.2.1 Example 1—Two-Source Mixing Using One Tracer

This is a simplified example of calculations for a two-source mixing model using only one stable isotope tracer (fractionation is not considered). The stable sulfur isotope composition, δ(34S), has been analysed in a water sample collected from a river at a site below a large area of agricultural farmland where high sulphate concentrations were detected. Two sources of sulphate were considered to be major contributors to the elevated sulfate concentrations in the river (SO42− = 800 mg/L): (1) oxidation of pyrite (FeS2) that naturally occurred in the metamorphic rocks common in the study area and (2) potash fertiliser (potassium sulphate K2SO4), applied in large quantities by farmers.

Input values

δ(34S)Pyr = 1.5 ± 0.5 ‰; the mean signature of pyrite samples collected from several rock outcrops.

δ(34S)Fer = 6.2 ± 0.6 ‰; the signature analysed in fertiliser samples collected from a few bags purchased by the farmers.

Mixture values

δ(34S)Spl = 5.2 ‰ the signature of sulfates in the river water sample.

Mass balance

The following set of simultaneous equations can be used for mass balance calculations of the relative contributions of fertiliser and pyrite to elevated  sulfate  concentrations in the river:

$$\delta \left( {^{34} S} \right)_{{{\text{Spl}}}} = x_{{{\text{Pyr}}}} \times \delta \left( {^{34} S} \right)_{{{\text{Pyr}}}} + x_{{{\text{Fer}}}} \times \delta \left( {^{34} S} \right)_{{{\text{Fer}}}}$$
(3.1)
$$1 = x_{{{\text{Pyr}}}} + x_{{{\text{Fer}}}} ,$$
(3.2)

where xPyr and xFer are the fractions of pollution (SO42−) from pyrite and fertiliser, respectively. Since only two sources are considered, the sum of xPyr and xFer is equal to 1 (by fraction) or 100 %; therefore, these equations can be easily solved, as follows:

$$x_{{{\text{Fer}}}} = {1} - x_{{{\text{Pyr}}}}$$
(3.3)

and, by substitution

$$\delta \left( {^{34} S} \right)_{{{\text{Spl}}}} = x_{{{\text{Pyr}}}} \times \delta \left( {^{34} S} \right)_{{{\text{Pyr}}}} + \left( {1 - x_{{{\text{Pyr}}}} } \right) \times \delta \left( {^{34} S} \right)_{{{\text{Fer}}}}$$
(3.4)

and this is finally rearranged to

$$x_{{{\text{Pyr}}}} = \frac{{\delta (^{34} S)_{{{\text{Spl}}}} - \delta (^{34} )S_{{{\text{Fer}}}} }}{{\delta (^{34} S)_{{{\text{Pyr}}}} - \delta (^{34} )S_{{{\text{Fer}}}} }}.$$
(3.5)

Using the input and mixture data given above in the example and Eq. 3.5, the following can be calculated:

$$x_{{{\text{Pyr}}}} = \frac{5.2 - 6.2}{{1.5 - 6.2}} = 0.21$$
(3.6)
$$x_{{{\text{Fer}}}} = 1 - 0.21 = 0.79.$$
(3.7)

This means that the rock weathering and resulting pyrite oxidation contributes only to 21% of the total sulfate concentration of 800 mg/L observed in the river water, whereas the majority of the sulphate (79 %) originates from fertiliser. This can be translated into loads respective to concentrations of 168 mg/L originating from pyrite and 632 mg/L from fertiliser. If the outflow from the catchment is known, then the total load of sulfate originating from agricultural pollution discharged downstream can be estimated (e.g., mg/h or kg/day) by multiplying the outflow (e.g., L/h) by the concentration (e.g., mg/L). However, one drawback that must be kept in mind is that these numbers reflect the hydrochemical conditions at the time of sampling. As a result, they may change seasonally due to variability in hydrological conditions, the timing of fertiliser applications and whether crops are being planted or harvested. If more frequent (e.g., monthly) and long-term monitoring of the hydrochemical and stable sulfur isotope composition is conducted, and if the total annual load of fertiliser in the catchment is known, the annual agricultural pollution loads and the efficiency of fertiliser utilisation by plants can be estimated (see, e.g., Szynkiewicz et al. 2015a, b).

Note that for a two-source mixing model, the δ-value of the mixture must fall between the δ-values of the two considered sources (Fig. 3.1). If the δ-value of the mixture is outside the range covered by the δ-values of two sources, then the following scenarios are likely: (1) the sources have been incorrectly identified; (2) a third additional source is contributing to the mixture; or (3) the original stable isotope compositions of the sources have been significantly modified by stable isotope fractionation or equilibration processes, so the δ-values used are not a true reflection of the source signatures.

Fig. 3.1
A graphical illustration depicts the two-source mixing model. The delta value of the mixing falls between the delta values of the two sources.

Two-source mixing model concept (data as for Example 1). The δ(34S) value of the mixture must lie between the δ(34S) values of the sources. The size of the plotted bar is inversely proportional to the contribution from each source; therefore, a larger distance between the signature of the source and the signature of the mixture implies a lower contribution to the mixture

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3.2.2 Example 2—Three-Source Mixing Using Two Tracers

More than two major sources often contribute simultaneously to water pollution. In these cases, more tracers need to be analysed in each sample to allow calculation of the exact contributions from individual sources (e.g., if three sources are considered, a minimum of two different isotope tracers need to be used). This simplified example considers a three-source mixing model using two stable isotope tracers (fractionation was not considered). The stable nitrogen and oxygen isotope compositions [δ(15N) and δ(18O)] have been analysed in a water sample collected from a river outflowing from an agricultural area where high nitrate concentrations were detected. Three major NO3− sources were considered: precipitation, fertiliser and manure.

Input values

δ(15N)Prec = 5.2 ‰ and δ(18O)Prec = 40.8 ‰; the mean signatures of NO3 analysed in precipitation.

δ(15N)Fer = −2.3 ‰ and δ(18O)Fer = 5.0 ‰; the mean signatures of NO3 analysed in fertiliser.

δ(15N)Man = 25.0 ‰ and δ(18O)Man = 5.4 ‰; the mean signatures of NO3 analysed in manure.

Mixture values

δ(15N)Spl = 16.9 ‰ and δ(18O)Spl = 8.1 ‰; the signatures of a water sample from the river.

Mass balance

Equations for mass balance calculations can be prepared in the same way as depicted in Example 1, assuming that the stable isotope compositions of both nitrogen and oxygen remain unchanged and fully identify the initial signatures of the sources.

$$\delta^{15} N_{{{\text{Spl}}}} = x_{{{\text{Prec}}}} \times \delta \left( {^{15} {\text{N}}} \right)_{{{\text{Prec}}}} + x_{{{\text{Fer}}}} \times \delta \left( {^{15} {\text{N}}} \right)_{{{\text{Fer}}}} + x_{{{\text{Man}}}} \times \delta \left( {^{15} {\text{N}}} \right)_{{{\text{Man}}}}$$
(3.8)
$$\delta \left( {^{18} {\text{O}}} \right)_{{{\text{Spl}}}} = x_{{{\text{Prec}}}} \times \delta \left( {^{18} {\text{O}}} \right)_{{{\text{Prec}}}} + x_{{{\text{Fer}}}} \times \delta \left( {^{18} {\text{O}}} \right)_{{{\text{Fer}}}} + x_{{{\text{Man}}}} \times \delta \left( {^{18} {\text{O}}} \right)_{{{\text{Man}}}}$$
(3.9)
$$1 = x_{{{\text{Prec}}}} + x_{{{\text{Fer}}}} + x_{{{\text{Man}}}}$$
(3.10)

where xPrec, xFer and xMan are the fractions of NO3 originating from precipitation, fertiliser and manure, respectively. Since the contributions are assumed to arise from only these three sources, the sum of xPrec + xFer + xMan is equal to 1 (by fraction) or 100 %. These simultaneous equations (Eqs. 3.8, 3.9 and 3.10) can be mathematically solved by substitutions.

The solution of this mass balance model, using the input and mixing data for Example 2, is as follows:

$$\begin{aligned} & x_{{\text{Prec }}} = 0.08 \\ & x_{{\text{Fer }}} = 0.24 \\ & x_{{\text{Man }}} = 0.68 \\ \end{aligned}$$

In Example 2, the majority of the NO3− pollution detected in the river originates from manure, at 68 %, and fertilisers, at 24 %, whereas precipitation delivers only 8 %.

Note that for a three-source mixing model, the δ-value of the mixture must lie within the triangle (or polygon if more sources are considered) restricted by the δ-values of the three considered sources. The triangle shows the range of all possible δ-values which can originate from mixing of the considered sources proportionally to their contributions (Fig. 3.2).

Fig. 3.2
A graphical illustration depicts the three-source mixing model. It includes a triangle with three sides labeled, manure, fertilizers, and precipitation. The ranges of manure at 68%, fertilizers at 24%, and precipitation at 8%.

Three-source mixing model concept (data as for Example 2). The signature of the mixture, in proportion to the contributed fractions, must fall between the signatures of the three sources; therefore, it will lie within the yellow mixing triangle. For simplification, fractionation was not considered in this example

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3.2.3 Example 3—Three-Source Mixing Using One Tracer Only

Often, more than two major sources contribute to water pollution, but an inadequate number of tracers is available (e.g., if three sources are considered, but only one isotope tracer is available). In this scenario, the exact contribution from each source cannot be calculated. Instead, ranges of contributions may be calculated that can provide information about possible maximum and minimum contributions. Example 3 presents the same scenario with the same three sources discussed in Example 2, but only one tracer, δ(15N) has been used for the calculations.

Input values

δ(15N)Prec = 5.2 ‰; a mean signature of NO3 analysed in precipitation.

δ(15N)Fer = −2.3 ‰; a mean signature of NO3 analysed in fertiliser.

δ(15N)Man = 25.0 ‰; a mean signature of NO3 analysed in stored manure.

Mixture values

δ(15N)Spl = 16.9 ‰; the signature of a water sample from the river.

Mass balance

The mass balance equations for this scenario cannot be mathematically solved because three unknowns cannot be calculated using only two equations. Instead, the ranges and the combination of values for solving these equations can be obtained using, for example, IsoSource software (Phillips and Gregg 2003) (See input data in Appendix 2).

$$\delta \left( {^{15} {\text{N}}} \right)_{{{\text{Spl}}}} = x_{{{\text{Prec}}}} \times \delta \left( {^{15} {\text{N}}} \right)_{{{\text{Prec}}}} + x_{{{\text{Fer}}}} \times \delta \left( {^{15} {\text{N}}} \right)_{{{\text{Fer}}}} + x_{{{\text{Man}}}} \times \delta \left( {^{15} {\text{N}}} \right)_{{{\text{Man}}}}$$
(3.11)
$$1 = x_{{{\text{Prec}}}} + x_{{{\text{Fer}}}} + x_{{{\text{Man}}}}$$
(3.12)

where xPrec, xFer and xMan are fractions of NO3 originating from precipitation, fertiliser and manure, respectively. Since the contributions from three sources are considered, the sum of xPrec + xFer + xMan is equal to 1 (by fraction) or 100 %.

The solution of this mass balance model, using the input and mixing data for Example 3, is as follows:

$$\begin{aligned} & X_{{\text{Prec }}} = 0\;{\text{to}}\;0.41\quad ({\text{mean}}\;0.20) \\ & x_{{\text{Fer }}} = 0\;{\text{to}}\;0.30\quad ({\text{mean}}\;0.15) \\ & x_{{\text{Man }}} = 0.59\;{\text{to}}\;0.70\quad ({\text{mean}}\;0.65). \\ \end{aligned}$$

These calculations do not provide exact values; however, the obtained information can still be very useful as it confirms that the major source of nitrate in the polluted river is manure and that its contribution is not lower than 59 % and not higher than 70 % with mean 65 %. The remaining contribution, between 15 and 20 %, originates from fertiliser and precipitation, which can vary from 0 to 41 % and 0 to 30 %, respectively.

The mixing models all assume good mixing of water and pollution, but this is not always the case. Inhomogeneity may introduce additional uncertainty into mass balance calculations. In addition, none of the examples presented considered stable isotope fractionations. A fractionation may occur during chemical reactions, physical phase changes, or biological processes, particularly if these processes are not completed and not all substrates are exhausted. The effects of stable isotope fractionations can be corrected, but this will require more advanced modelling and obtaining specific fractionation factors for each stable isotope, chemical compound and physicochemical condition at the time of the reaction (e.g., Lewicki et al. 2022). All results, calculations and interpretations reflect the conditions at the time of sampling, and these can vary over seasons and can be particularly impacted by changes in the hydrological conditions (Dogramaci and Skrzypek 2015; Szynkiewicz et al. 2015a; Dogramaci et al. 2017).

3.3 Overview of Available Software for Mass Balance and Fractionation Models

Simple mixing models that use the number of tracers lower by one than the number of sources (t = s−1) can be solved algebraically and do not require advanced software. The solution can be derived from simultaneous equations, as presented in examples 1 and 2 above, and the final equation is then solved manually or implemented in a calculation spreadsheet. These options will provide an exact mathematical solution, and the uncertainty of the final calculation will be associated with the laboratory analyses and accurate determination of the signatures of sources and mixtures, but not with the calculation formula that provides the exact solution. The algebraic solution is not applicable in cases where fewer tracers are available (t ≤ s−2). That situation usually requires the calculation of a range of possible combinations of values solving equations, or alternatively, a probability calculation can be attempted. A few models have been published in recent years using different calculation methods, although most of these models have been developed for assessing dietary contributions in food web studies (e.g., Phillips et al. 2014). Although they were intended for ecological studies, they can easily be adopted in pollution studies. Here, the most popular models are presented: IsoSource, MixSIAR and FRAME.

3.3.1 IsoSource—Algebraic Solution for Complex Models

The IsoSource mixing model was developed to allow the calculation of the proportional contributions from a large number of sources using an algebraic algorithm. The software (v 1.3.3) works on Microsoft Windows systems (including Windows 10) and accepts input values for up to five tracers/isotopes and for a maximum of ten sources. All possible scenarios are calculated at user-defined increments and tolerances. The results are returned as a table with fractions of the relative contributions that solve the mixing model equations. Minimum, maximum and mean values are calculated and allow the determination of the possible range of contributions from each source. The software allows the calculation of fractions for one mixture at a time and does not assess uncertainty. The results are recorded as separate report files and need to be combined manually by the user. The system is easy to use, and all inputs are entered manually into a provided window. IsoSource does not have the option of uploading input files.

Website: https://www.epa.gov/eco-research/stable-isotope-mixing-models-estimating-source-proportions

Primary reference: Phillips and Gregg (2003)

Alternatives: LP_Tracer (Bugalho et al. 2008); Moore–Penrose pseudoinverse (Hall-Aspland et al. 2005); SOURCE/STEP (Lubetkin and Simenstad 2004).

3.3.2 MixSIAR—Bayesian

MixSIAR is an R package that uses Bayesian statistical methods for (1) determining the proportional contribution of different sources to a mixture and (2) assessing dispersion in the isotope space (Stock et al. 2018). This package, like its earlier version, SIAR, and like IsoSource, was developed for stable isotope studies of food webs. The first functionality (1) can be easily adopted for any mixing model, whereas the second functionality (2), although it is used to assess the distribution of stable isotope results, is specific for ecological studies and not directly applicable to pollution studies. Bayesian statistical distributions are used for both the characterisation of source contributions and uncertainties. The package is a ready-to-use product and could be used in R-Studio, but elementary knowledge of R coding is required. The model accepts input files, and batches of data can be processed.

Website: http://brianstock.github.io/MixSIAR/

Primary reference: Stock et al. (2018)

Alternatives: MixSIR (Moore and Semmens 2008; Ward et al. 2010); SIAR (Parnell et al. 2010); IsotopeR (Hopkins and Ferguson 2012) and FRUITS (Fernandes et al. 2014), isoWater (Bowen et al. 2018).

3.3.3 FRAME—Bayesian, Markov Chain Monte Carlo

FRAME (isotope FRactionation And Mixing Evaluation) is a newly developed software that works on the Windows platform. The FRAME mathematical algorithm uses the Markov chain Monte Carlo model to estimate the contribution of individual sources and processes, as well as the probability distributions of the calculated results. It has a user-friendly graphical interface that can simultaneously determine mixing proportions and the progression of the fractionating process. The fractionation can be defined by the user for specific requirements of the particular isotope system (e.g., an open or closed system). The model can integrate up to three isotopic signatures for each compound used to identify sources contributing to the mixture. This model has been specifically developed for the assessment of mixing and emissions in agriculture pollution studies; therefore, it can be used for any mixing model for pollution studies. The software is easy to use, and datasets can be uploaded using the provided spreadsheet templates.

Website: https://malewick.github.io/frame/

Primary reference: Lewicki et al. (2022).

Alternatives: isoWater (Bowen et al. 2018) (https://cran.r-project.org/web/packages/isoWater/index.html)

3.3.4 Other Models and Calculation Spreadsheets

A few other calculation spreadsheets and models can be applicable to agro-contaminant studies (e.g., Hydrocalculator, EasyIsoCalculator). Many studies require estimations of different processes, including evaporative losses in different parts of a catchment, retention times, or evaporation over inflow ratio in lakes or dam reservoirs. Evaporative loss also informs about the evapo-concentration of solutes and can confirm evaporation as a process driving high concentrations of solutes. These calculations are rather too complex to solve manually and can be easily preformed using Hydrocalculator (Skrzypek et al. 2015).

Tracer studies that use compounds with artificially elevated concentrations of heavier isotopes are used to track pollution dispersal or chemical uptake by biological organisms. In these studies, stable isotope results are reported using different expressions (e.g., fraction, ratio or delta). The EasyIsoCalculator spreadsheet allows fast and effortless recalculation between the main expressions (Skrzypek and Dunn 2020a, b).

3.3.5 Hydrocalculator—Estimating Evaporative Losses

The Hydrocalculator software (v 1.3) works on the Microsoft Windows system and allows the calculation of evaporative losses from a water body using the stable hydrogen and oxygen isotope composition of the water. The mathematical algorithm is based on a modified version of the Craig–Gordon model. The user can choose between a steady-state and non-steady-state model and select one of three options for estimating the stable isotope composition of atmospheric moisture. The software has an easy-to-use graphical interface. The data can be entered manually in calculator mode or uploaded as a template file for batch conversion. Outputs are recorded in a spreadsheet file (CSV, comma-separated values).

Website: http://hydrocalculator.gskrzypek.com

Primary reference: Skrzypek et al. (2015).

Alternatives: FRAME (Lewicki et al. 2022), isoWater (Bowen et al. 2018).

3.3.6 EasyIsoCalculator—Recalculating Between Fractions, Delta Values and ppm

EasyIsoCalculator is a calculation spreadsheet that allows recalculation of the main expressions of isotope compositions (isotope ratio, fraction and delta) for five light elements: hydrogen, carbon, nitrogen, oxygen and sulphur. The spreadsheet allows the selection of one of many absolute ratios defining zero points on isotope delta scales. The spreadsheet has an easy-to-use interface with dropdown menus, and its open code allows the addition of new values. Up to fifty results can be calculated at once using the copy-and-paste option.

Website: http://easyisocalculator.gskrzypek.com/

Primary reference: Skrzypek and Dunn (2020a, b)

Alternatives: n/a.