Tracer-based hydrograph separation techniques

2.4.3.1 Hydrograph separation based on the mass balance concept

The classical hydrograph separation technique is extensively described in KENDALL and

MCDONNELL (1998) and HOEG et al. (2000).

Hydrograph separation analyses are based on simple steady state mass balance equations of water and tracer fluxes in a catchment. Commonly a two-component mixing model (PINDER

and JONES, 1969) is used to quantify two different discharge components based on their tracer

concentrations and discharge volumes.

Theoretically, this concept allows the determination of n runoff components based on the observation of n-1 tracers and solving the following n linear mixing equations (HOEG et al.,

2000):

whereas QT is the total runoff, Q1, Q2, …, Qn are the different runoff components and c1ti, c2ti, …, cnti are the concentrations of one observed tracer ti. Because c1ti, c2ti, …, cnti represent the extreme possible concentrations for cT they are known as end-members (HOOPER et al., 1990). The application of this separation technique is restricted by several boundary conditions (SKLASH and FARVOLDEN, 1979):

ƒ The tracer concentrations of the different runoff components are significantly different.

ƒ Each input tracer concentration is constant in space and time or its variation is known.

ƒ The tracer behaves conservatively.

ƒ Contributions of an additional component are negligible, or their composition is identical to that of another component.

n T Q Q Q Q = ₁+ ₂ +...+ n t n t t T t T Q c Q c Q c Q c i = i + i +...+ i 2 2 1 1

Especially the demand for the tracer concentrations being constant in space and time or their variations being known is often insufficiently met in reality. In that way, introduced uncertainties have to be taken into account and quantified.

Hydrograph separation based on two-component mixing models was first applied in the 1970’s and used to determine the proportion of event and pre-event water contributing to storm runoff (SKLASH and FARVOLDEN, 1979; RODHE, 1981; OBRADOVIC and SKLASH et

al., 1986; STICHLER, 1987; WELS et al., 1991a). Thereby event water refers to the water that

is added to a catchment’s surface as rainfall or snowmelt during a storm event while pre-event water was held in the catchment prior to, and has been discharged into the stream channel during a storm event (BUTTLE, 1998).

Pre-event water was found to dominate largely the streamflow generation regardless of catchment scale, physical properties and climatic conditions (GENEREUX and HOOPER, 1998;

MARC et al., 2001). This implies that catchments store water for considerable periods of time

but then release it promptly during storm events (KIRCHNER, 2003).

It was soon understood that two components could not satisfactorily account for the variation of isotopic and chemical composition in the stream during stormflow. In more sophisticated studies three-component models where used to divide pre-event water further into soil water and groundwater (HOOPER et al., 1990; MCDONNELL et al., 1991; OGUNKOYA and JENKINS,

1993; HOEG et al., 2000) addressing rather the geographic than the time source of storm

runoff components. The concept was further extended by MEROT et al. (1995), LEE and

KROTHE (2001), UHLENBROOK and HOEG (2003) by refining the division of discharge

components applying four- and five-component hydrograph separation.

2.4.3.2 End-member mixing analysis (EMMA)

End-member mixing analysis (EMMA) is an analytical approach developed by CHRISTOPHERSEN et al. (1990) and HOOPER et al. (1990) that is closely connected to and

often combined with the classical hydrograph separation technique. It allows to model stream water chemistry as a mixture of representative end-members and to separate the hydrograph using multiple tracers simultaneously.

CHRISTOPHERSEN et al. (1990) observed that the chemical species in stream water that are

closely correlated with flow, are the same ones that exhibit marked differences in concentrations across soil horizons. They concluded that the extremes among the solutions, the so-called end-members, must mix in proportions such that their combined chemistry equates the observed stream water chemistry.

The restrictive assumptions for tracers mentioned before in the scope of hydrograph separation (2.4.3.1) are also valid within the framework of end-member mixing analysis. The end-member contributions are estimated by solving a constrained, overdetermined set of linear equations applying a least-square procedure.

End-member mixing analysis have been enhanced by introducing multivariate data analysis techniques – in particular principal component analysis (PCA) – to indicate the approximate rank of the mixture in question and thus to estimate the minimum number of end-members needed to describe the observed data (CHRISTOPHERSEN and HOOPER, 1992). The application

of principal component analysis enables to study the structure of variance within the data and results in a more efficient coordinate system describing the observed stream water chemistry. The detailed EMMA procedure is outlined by CHRISTOPHERSEN and HOOPER (1992) and

summarized in BROWN et al. (1999).

Though the EMMA approach itself has been less applied compared to the vast literature covering the use of classical hydrograph separation, clearly its advantages consist in:

ƒ its compatibility with classical hydrograph separation,

ƒ the generation of testable hypotheses that focus future field efforts,

ƒ the identification of geographical source areas,

ƒ the simultaneous application of multiple tracers and

ƒ the overdetermination of the algebraic solution

(CHRISTOPHERSEN et al., 1990; HOOPER et al., 1990; CHRISTOPHERSEN and Hooper, 1992;

MULHOLLAND, 1993; OGUNKOYA and JENKINS, 1993; BAZEMORE et al., 1994, ELSENBEER

et al., 1995, BROWN et al., 1999; BURNS et al., 2001).

Common end-members identified have been soil water, groundwater, organic horizon water, hillslope subsurface water, riparian zone water or event (rain) water to name just a few.

Temporal variations of end-member chemistry have been taken into account by OGUNKOYA

and JENKINS (1993) who tested “fixed”, “time-invariant” and “temporally varying” end-

member concentrations. Depending on the size of variation, BURNS et al. (2001) allowed end-

member chemistry to vary over time (large variations) or to be represented by their median value (small variations).

2.4.3.3 Uncertainty analysis

The validity of conclusions drawn from EMMA-based hydrograph separation is largely dependent on the conducted uncertainty analysis.

The success of EMMA-based hydrograph separation and the determination of end-member proportions are based on the adequate chemical differentiation of these source waters (JOERIN

et al., 2002). Given that, at the catchment scale, spatial and temporal variability in end- member composition is usually unknown or difficult to characterize (HOEG et al., 2000) and

that the determination of tracer concentrations can be subject to sampling and analytical errors, it is evident that the hydrograph separation approach inherits large uncertainties. In addition to parametric and natural uncertainties the strong simplistic model hypotheses that mixing models refer to (JOERIN et al., 2002) introduce significant uncertainties into the

obtained results. While the former sources of error can be summarized as statistical uncertainty, the latter is referred to as model uncertainty.

Among other, UHLENBROOK and HOEG (2003) conclude that results of hydrograph

separations should not be taken as exact numbers and are of mostly qualitative nature unless combined with additional field data. The following paragraphs offer a brief introduction to uncertainty analysis in conjunction with EMMA-based hydrograph separation.

Focusing on parametric variability only, early research (RODHE, 1981; NEAL et al., 1990;

MCDONNELL et al., 1991) addresses uncertainty in terms of sensitivity analysis where tracer

concentrations change within the range of observed data and requantification of the models results in an array of end-member proportions. HOOPER et al. (1990) introduced first-order

Taylor series expansion into sensitivity analysis and where thus the first to invoke a formalized statistical approach.

Concentrating on both, the tracer concentrations used to perform the hydrograph separation and the uncertainties in the tracer concentrations itself, GENEREUX (1998) uses Gaussian

error propagation to determine uncertainties in the computed mixing fractions of two- and three-component hydrograph separation. This technique was frequently applied (BROWN et

al., 1999; HOEG et al., 2000; BURNS et al., 2001), extended to an even higher number of end-

members and refined concerning the addressed uncertainty sources (UHLENBROOK and

HOEG, 2003).

BAZEMORE et al. (1994) incorporated both the effects of spatial variability of the end-member

concentrations as well as the laboratory analytical error into uncertainty analysis and were the first to apply the, to some extent advanced, Monte Carlo approach. This computer-based method simulates probability distributions of possible end-member contributions for each collected stream sample, a technique applied and adapted by DURAND and TORRES (1996),

RICE and HORNBERGER (1998) among others.

JOERIN et al. (2002) investigated for the first time both the statistical uncertainty of mixing

models due to chemical variability inside components applying a less restricted Monte Carlo method, and the model uncertainty by comparison of alternative hypotheses.

Recently SOULSBY et al. (2003) used a Bayesian model (BREWER et al., 2002) to estimate

uncertainty. Therein, end-members are assumed to arise from bivariate normal distributions whose mean vectors and co-variance matrices can be estimated. Markov Chain-Monte Carlo methods are used to model the average and 95 percentile upper and lower bounds of end- members during storm events. This method, its implicit assumptions and shortcomings are discussed in detailed by BEVEN (2004) who concludes that subjectivity, for example in the

choice of model structures or input and boundary condition errors, will remain a significant part of uncertainty estimation for the foreseeable future.