Assessing the impact of mixing assumptions on the ...*Correspondence to: Fabrizio Fenicia, Centre de Recherche Public— Gabriel Lippmann, Department Environment and Agro-Biotechnologies,

HYDROLOGICAL PROCESSESHydrol. Process. (2010)Published online in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.7595

Assessing the impact of mixing assumptions on the estimationof streamwater mean residence time

Fabrizio Fenicia,1* Sebastian Wrede,1,2 Dmitri Kavetski,3 Laurent Pfister,1 Lucien Hoffmann,1

Hubert H. G. Savenije2 and Jeffrey J. McDonnell4

1 Centre de Recherche Public—Gabriel Lippmann, Department Environment and Agro-Biotechnologies, L-4422 Belvaux, Grand-Duchy ofLuxembourg

2 Water Resources Section, Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, NL-2600 GA Delft, TheNetherlands

3 Environmental Engineering, University of Newcastle, Callaghan, NSW 2308, Australia4 Institute for Water and Watersheds, Department of Forest Engineering, Resources and Management, Oregon State University, 015 Peavy Hall,

Corvallis, OR 97331-5706, USA

Abstract:

Catchment streamwater mean residence time (Tmr) is an important descriptor of hydrological systems, reflecting their storageand flow pathway properties. Tmr is typically inferred from the composition of stable water isotopes (oxygen-18 and deuterium)in observed rainfall and discharge. Currently, lumped parameter models based on convolution and sinewave functions areusually used for tracer simulation. These traditional models are based on simplistic assumptions that are often known to beunrealistic, in particular, steady flow conditions, linearity, complete mixing and others. However, the effect of these assumptionson Tmr estimation is seldom evaluated. In this article, we build a conceptual model that overcomes several assumptions madein traditional mixing models. Using data from the experimental Maimai catchment (New Zealand), we compare a complete-mixing (CM) model, where rainfall water is assumed to mix completely and instantaneously with the total catchment storage,with a partial-mixing (PM) model, where the tracer input is divided between an ‘active’ and a ‘dead’ storage compartment. Weshow that the inferred distribution of Tmr is strongly dependent on the treatment of mixing processes and flow pathways. TheCM model returns estimates of Tmr that are well identifiable and are in general agreement with previous studies of the Maimaicatchment. On the other hand, the PM model—motivated by a priori catchment insights—provides Tmr estimates that appearexceedingly large and highly uncertain. This suggests that water isotope composition measurements in rainfall and dischargealone may be insufficient for inferring Tmr . Given our model hypothesis, we also analysed the effect of different controls onTmr . It was found that Tmr is controlled primarily by the storage properties of the catchment, rather than by the speed ofstreamflow response. This provides guidance on the type of information necessary to improve Tmr estimation. Copyright 2010 John Wiley & Sons, Ltd.

KEY WORDS mean residence time; mixing models; partial-mixing; parameter inference

Received 31 August 2009; Accepted 30 November 2009

INTRODUCTION

In recent years, catchment streamwater mean residencetime Tmr [T], also known as ‘transit time’ or ‘waterage’, is increasingly used as a compact descriptor ofcatchment behaviour (McGuire et al., 2005; Hrachowitzet al., 2009). Tmr describes hydrological responses acrossa wide range of spatial and temporal scales, comple-menting traditional hydrometric streamflow information(McDonnell et al., 2007; Soulsby and Tetzlaff, 2008).In addition, it has been recently used as a constraintfor model structural development (Seibert and McDon-nell, 2002; Uhlenbrook and Leibundgut, 2002), reductionof parameter uncertainty (Vache and McDonnell, 2006b)and multi-criteria model calibration (Vache and McDon-nell, 2006a; Dunn et al., 2007).

* Correspondence to: Fabrizio Fenicia, Centre de Recherche Public—Gabriel Lippmann, Department Environment and Agro-Biotechnologies,L-4422 Belvaux, Grand-Duchy of Luxembourg.E-mail: [email protected]

Tmr is commonly inferred from measurements of stablewater isotope (oxygen-18 and deuterium) compositions inrainfall and discharge. Traditional estimation techniquesuse lumped parameter models, e.g. based on convolutionsor sinewave functions for tracer simulation (McGuireand McDonnell, 2006). These simple approaches requireseveral assumptions that are almost always violated inreal-world catchment systems (McGuire and McDonnell,2006), in particular: (i) steady flow conditions (constantrainfall and discharge), (ii) linear tracer input–outputrelations and (iii) instantaneous mixing of tracers withinthe entire catchment storage (Zuber and Maloszewski,2000). These assumptions are not supported by pro-cess understanding. For example, water fluxes withina catchment are generally unsteady and often followpreferential pathways between different storage com-partments such as groundwater, surface streams, etc.(e.g. Beven and Germann, 1982). In addition, runoffsystems are generally threshold-like and their responseto storm rainfall is highly nonlinear (e.g. Tromp van

Copyright 2010 John Wiley & Sons, Ltd.

F. FENICIA ET AL.

Meerveld and McDonnell, 2006a,b; Zehe and Sivapalan,2009).

Alternatively, Tmr may be estimated using more com-plex flow and transport models. These include concep-tual models (e.g. Seibert et al., 2003; Page et al., 2007;Fenicia et al., 2008), compartmental mixing-cell mod-els (e.g. Harrington et al., 1999; Vache and McDon-nell, 2006a; Sayama and McDonnell, 2009) and solutetransport models using the advection–dispersion equation(e.g. Konikow and Reilly, 1998). These approaches over-come most assumptions of black-box lumped parametermodels. However, they are seldom applied for Tmr esti-mation, arguably due to more complicated structure anddata requirements.

Despite a growing number of models, current under-standing of how different mixing models affect the esti-mation of Tmr remains limited. How strongly are theestimates of Tmr dependent on the underlying modelhypotheses? Are traditional methods practically reliableand useful, notwithstanding being based on unrealisticassumptions? Are isotopic tracer composition measure-ments in rainfall and discharge sufficient for estimatingTmr , or is additional process information about the catch-ment necessary? Insights into these questions are a pre-requisite for meaningful use of Tmr , both as a catchmentdescriptor and as a constraint during model evaluation.

This work advances the case for using conceptual mod-els to explore the effects of mixing assumptions on theestimation of Tmr . Conceptual models allow systematichypothesis-testing and improvement guided by experi-mental data (Fenicia et al., 2008). In addition, their com-putational speed allows a comprehensive treatment ofuncertainty using Monte Carlo methods. This facilitatesa better understanding of catchment behaviour and mayhelp overcome inconsistencies arising from the use ofblack-box tracer simulation models (Kirchner, 2006), forexample, the paradox that water released within minutesor hours of rainfall events is often months to years old(Kirchner, 2003).

In this article, we use simultaneous flow and tracer sim-ulation to explore how the conceptualization of mixingmechanisms affects the inferred distribution of Tmr , andhence the inferred flow and tracer dynamics of a catch-ment. In doing so, we also develop an improved mecha-nistic understanding of the catchment behaviour throughuncertainty estimation and hypothesis-testing. Using datafrom the well-studied Maimai catchment in New Zealand,we construct two distinct mixing models—one with com-plete mixing, where water mixes instantaneously withinthe whole catchment storage, and the other one with par-tial mixing, where water follows preferential pathwaysand is separated into ‘active’ and ‘dead’ storage com-partments. This active–dead storage distinction closelyresembles the complex, yet highly qualitative descrip-tions of McDonnell (1990) for the Maimai site, wherethe rationale for old water movement through macroporeswas based on unrequited storage within the soil profileand non-participatory portions of the hillslope material.

The two conceptual models have the same hydrologicalstructure and differ solely in the way water (and hencetracers) mix within catchment compartments. Both mod-els allow simultaneous flow and solute simulation, andtherefore represent an improvement with respect to tra-ditional steady-state-flow convolution-based models fortracer simulation (this is verified by direct comparison).

Our main focus is on the influence of mixing assump-tions on the estimates and associated uncertainties in Tmr .We also investigate the dominant controls on Tmr . Thisis an interesting open question, since currently there is apoor understanding of whether Tmr is mostly affected by‘speed of response’ aspects such as hydrograph recessioncharacteristics (e.g. Vitvar et al., 2002), or by ‘storage’properties such as the capacity of the unsaturated zone(e.g. Dunn et al., 2007).

STUDY SITE

The Maimai study area on the South Island of NewZealand was chosen for this study due to its long his-tory of intensive hydrological research (see McGlynnet al., 2002). The availability of comprehensive and high-resolution data sets was essential for a meaningful under-standing of underlying catchment processes and facil-itated model development, application and interpreta-tion. Importantly, the comparatively simple and spatiallyhomogeneous physical characteristics and hydrologicalresponses of the Maimai area catchments make them ideal‘benchmark basins’ for model evaluation and testing (e.g.Seibert and McDonnell, 2002; Vache and McDonnell,2006a; Dunn et al., 2007; Fenicia et al., 2008).

This study focuses on the small forested M8 catchment,characterized by steep short slopes and deeply incisedchannels that drain an area of 3Ð8 ha (for details seePearce et al., 1986). Its soils are generally shallow (about60 cm on average) and remain highly saturated over theentire year. These highly transmissive soils are underlainby cemented conglomerate bedrock that appears largelyimpermeable (Mosley, 1979). Its climate is humid, withan average precipitation of about 2600 mm/a and a highlyresponsive 1550 mm/a stream runoff with almost no sea-sonal variation.

Under these climatic and physiographic conditions,lateral subsurface flow over the impervious bedrock isthe dominant runoff mechanism in the catchment. Whileslow continuous down-slope drainage prevails duringlow-flow conditions, storm-flow events are characterizedby significant preferential flows along cracks and lateralroot networks, as well as along soil horizons and the soilbedrock interface (McDonnell, 1990). Thus, the M8’sTmr , estimated at 4 š 1 months using oxygen-18 data(Pearce et al., 1986), is one of the shortest observed innatural catchments.

This study uses hourly rainfall, discharge, potentialevaporation data, as well as deuterium compositionmeasurements in the rainfall and the catchment outletstream. The modelling period spans 1986–1987, while

Copyright 2010 John Wiley & Sons, Ltd. Hydrol. Process. (2010)DOI: 10.1002/hyp

IMPACT OF MIXING ASSUMPTIONS ON MEAN RESIDENCE TIME ESTIMATION

continuous isotopic rainfall time series and event-basedstream isotope data were available from Septemberto December 1987. Isotope data covered the entirerainfall period and selected discharge events. A detaileddescription of the isotope collection approach is given byMcDonnell et al. (1991).

METHODOLOGY

We compare two different model hypotheses: complete-mixing (CM) and partial-mixing (PM) model structures.The assumption of partial mixing is often perceived to bemore realistic by the experimentalist (e.g. Seibert et al.,2003), and, as discussed later, corresponds closer to thecurrent perception of water movement in the Maimaicatchment. Note that the two model structures havean identical water balance conceptualization and differsolely in the way water and isotopes are mixed internally.For completeness, we also evaluate the performance ofa traditional steady-state exponential model (EM) on thesame dataset (e.g. Zuber and Maloszewski, 2000).

Hydrological model

The water fluxes are described using a single reservoircharacterized by an active storage Sa. The storage–discharge relationship of the reservoir was estimatedfrom multiple recession segments using Master RecessionCurve (MRC) analysis (Figure 1(a and b)) (Lamb andBeven, 1997; Fenicia et al., 2006).

As seen in Figure 1(b), the dependence of discharge onstorage appears to be piecewise linear with a smooth tran-sition. This behaviour was represented using two linearsegments connected by a logistic smoother (Kavetski andKuczera, 2007, see Appendix for details). The methodyields a smooth overall functional relationship, with con-tinuous derivatives of all orders.

The water balance of the reservoir is

dSa

dtD P � Q � E �1�

where Sa is the active storage [L], P is precipitation [L/T],Q is discharge [L/T] and E is evaporation [L/T].

The storage–discharge relationship of the reservoirmodel is defined as

Q D G�Sajk1, k2, Sbq, mq� �2�

where Q is discharge and G is a smoothed piecewise-linear function (see Appendix). The parameters of therelationship are as follows: k1 and k2 are the slopes ofthe linear segments separated by a breakpoint at Sa DSbq (Figure 1(b)) and mq is a dimensionless smoothingparameter (less smoothing as mq ! 0). For clarity, weuse the vertical bar j to separate the dependence of G onthe time-dependent state variable Sa from its dependenceon parameters that are fixed within a given run.

Reservoir evaporation E is assumed to be proportionalto the potential evaporation Ep, with a scaling parameterEs [�]. We used a smooth evaporation function such thatE�Sa D 0� D 0 and E�S� reaches a constant value EpEs

as S increases:

E�SajSbe, me� D EpEs G�Saj1/Sbe, 0, Sbe, me� �3�

where Sbe and me are the breakpoint and smoothingparameters, respectively.

The water balance (Equation (1)) is integrated usingthe implicit Euler method with fixed time step anda Newton–Raphson solver (with the latter typicallyconverging to double precision in two to five iterations)(see Kavetski et al., 2006a for details).

The water balance model has four calibration parame-ters: k1 [1/T], k2 [1/T], Sbq [L] and Es [–]. The smooth-ing parameter mq D 2 was determined from Figure 1.The parameters Sbe and me in the evaporation func-tion (Equation (3)) were considered part of the numericaldescription and were fixed at Sbe D 3 mm and me D 0Ð5.

Mixing model

In most catchments, the dynamics of the tracerresponse is strongly damped with respect to the inputsignal (Kirchner, 2003). In a catchment model, thisbehaviour can be reproduced by introducing an additionalstorage that does not alter the hydrological behaviour(streamflow), but influences the way water and tracersmix within it. This approach was suggested by Barnesand Bonell (1996) and finds support in recent studies (e.g.

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Figure 1. Master recession curve ((a) continuous line) and storage–discharge relation inferred from it (b)


F. FENICIA ET AL.

Sa

SdPd

Pa

Ed

EdPd Φ(D)

Ea

Q

P E

Q

E=Ea+Ed

P=Pa+Pd

S

(a) (b)

Figure 2. Water balance for the CM and PM models (a) and flux exchanges for the PM model (b)

Vache and McDonnell, 2006a; Dunn et al., 2007; Son andSivapalan, 2007; Fenicia et al., 2008). This dead storagecan be interpreted as representing the unsaturated zone,or a groundwater system that is not directly connected tothe stream.

This study conceptualizes the dead storage Sd as anadditional storage below the threshold that producesrunoff (Barnes and Bonell, 1996) (Figure 2). We notethat this dead storage represents all non-participatorystorage compartments, including soil water, water storedin bedrock depressions, etc.

Complete mixing. The CM model assumes instanta-neous mixing between the active and dead compartments.It is represented by the tracer balance equation:

d�cS�

dtD cpP � cE � cQ �4�

where S D Sa C Sd is the total storage, c is the isotopictracer composition [M/M] in the reservoir and cp isthe isotopic tracer composition [M/M] in the rainfall P.Expanding the derivative and rearranging yields:

dc

dtD 1

Sa C Sd

(cpP � c

(Q C E C dSa

dt

))�5�

The simulation of tracer dynamics in the CM model addsthe additional calibration parameter Sd [L]. To avoidconfusion, note that Sa is a state variable, whereas Sd

is a calibrated parameter.

Partial mixing. The complete-mixing hypothesis is notrealistic in most natural catchments. Even in the Maimaicatchment, which is characterized by shallow soil andimpermeable bedrock, it has been shown experimentallythat water is poorly mixed during storm events (McDon-nell, 1990). This is due to a two-component flow systemof rapid macropore flow and slow matrix flow. McDon-nell (1990) noted that rainfall water infiltrates into macro-pores, quickly reaching the soil–bedrock interface. As aresult, free (new) water develops at soil–bedrock inter-face and backs-up into the soil matrix, where it mixeswith much larger (old) water storage.

This rationale suggests a partial-mixing hypothesis,with at least two storage compartments. Based on thesearguments, we developed a mixing model that hypothe-sizes:

1. instantaneous mixing within the active and dead stor-age compartments, and

2. partial mixing between the two compartments (e.g.Seibert et al., 2003; Page et al., 2007).

Similary to the CM model, the dead storage is rep-resentative of all water that is not directly connected tothe stream. This may include soil water, as well as waterstored in bedrock depressions. While this conceptualiza-tion remains quite simplistic, it services our primary aimof investigating how different water-mixing assumptionsinfluence the inferred distribution of Tmr .

The water fluxes are depicted in Figure 2. We dis-tribute the precipitation between the active and deadstorages using a dimensionless coefficient Fr , so thatPd D FrP and Pa D �1 � Fr�P. In the context of waterbalance, the precipitation Pa that enters the dead stor-age is released into the active storage. We also assumethat total evaporation is partitioned between the two com-partments according to their relative storage fraction, i.e.Ea/E D Sa/S.

Unlike the CM model, the PM model separatelyevolves isotopic tracer compositions in the active anddead storages and hence uses a total of three, ratherthan two, state variables. It conceptualizes the mixingbetween these compartments using an internal flux DD�cd � ca�Sa, where D is a dispersion parameter [1/T],and ca and cd are the isotopic tracer compositions [M/M]in the active and dead storages, respectively.

We hypothesize that the mixing flux depends onthe active storage Sa because, at least a priori, weexpect larger exchanges during wetter conditions—inparticular, during storm events. The linear relation is usedfor simplicity; nonlinear dependencies could be readilyincluded, if warranted by the data. Note that the waterbalance for the dead storage is 0, whereas the water



balance for the active storage is still represented byEquation (1).

The tracer balance equation for the active storage is:

d�caSa�

dtD cpPa C cdPd � caE � caQ C D�cd � ca�Sa

�6�Rearranging for dca/dt yields:

dca

dtD 1

Sa

(cpPa C cdPd � ca

(Q C E C dS

dt

)

C D�cd � ca�Sa

)�7�

Analogously, the tracer balance equation for the deadstorage is:

dcd

dtD 1

Sd

(Pdcp � Pdcd C �ca � cd�

Sd

Sd C SaE

C D�ca � cd�Sa

)�8�

The PM includes the additional calibration parametersFr [�], D [1/T] and Sd [L]. Again, note that Sa is a statevariable, whereas Sd is a calibrated parameter.

It could be argued that the PM model hypothesizes amore physically motivated mixing mechanism than theCM model. In particular, the coefficient Fr representsthe effects of preferential flows that explain the rapidresponse of this catchment to rainfall (Mosley, 1979),whereas the flux reflects the macropore–matrix inter-actions responsible for the slow mixing within eventsobserved by McDonnell (1990).

Numerical solution. The CM model (Equation (5))and the PM model [coupled Equations (7 and 8)] aresolved numerically using an explicit Runge–Kutta solverwith a scaled error tolerance of 10�4. For computationalsimplicity and efficiency, the water fluxes from the waterbalance model (Equation (1)) are distributed uniformlyover each timestep of the mixing model (numerical errorsarising from this linearization are minor).

Model evaluation

Bayesian inference. We use a standard Bayesianapproach to estimate the hydrological and mixing param-eters of the CM and PM models and compare the perfor-mance of these approaches.

The model parameters and response error variancesare inferred from the observed discharge and isotopictracer compositions, Q and Qc, respectively, using Bayesequation

p�q, �q, �cjQ, Qc� D p�Q, Qcjq, �q, �c�p�q, �q, �c� �9�

where p�q, �q, �cjQ, Qc� is the posterior distribution of themodel parameters q, the standard deviation of flow errors�q and the standard deviation of isotopic compositionerrors �c, while p�q, �q, �c� is the prior information onthese quantities and p�Q, Qcjq, �q, �c� is the likelihood

function. In Equation (9), the tildes indicate quantitiesthat are observed and hence subject to sampling andmeasurement uncertainties.

In the absence of any additional independent knowl-edge, we used uniform (bounded) priors for all modelparameters and Jeffreys’ uninformative prior for �q and�c (Kavetski et al., 2006b). Note that both �q and �c,which reflect the residual uncertainties in the dischargeand isotopic compositions arising from data and modelerrors, are inferred using Equation (9).

Likelihood function. The likelihood function used forthe inference is

p�Q, Qcjq, �q, �c� DNq∏iD1

N��i�q, Qi�j0, �2q � ð

Nc∏jD1

N��j�q, Qcj�j0, �2c �

�10�

where N�xjm, s2� is the probability density function ofa Gaussian deviate x with mean m and variance s2, �i

is the residual error (difference between observed andsimulated responses), whereas Nq and Nc are the numbersof discharge and isotopic tracer composition samples,respectively.

The least-squares likelihood function, i.e.Equation (10), assumes that the model residuals, whichlump the effects of data and model errors, are independentand identically distributed Gaussian. It also assumes thatthe model and observation errors in the two responses areuncorrelated.

While these statistical assumptions can and have bequestioned (e.g. see discussion in Kavetski et al., 2002),deriving a more complicated likelihood function liesoutside the scope of this study, which focuses on thephysical motivation for the PM model and its comparisonwith the traditional CM approach. In future work, we willreport a more general Bayesian approach for estimatingand accounting for the auto- and cross-correlations in theresiduals when jointly calibrating the hydrological andtracer models to streamflow and tracer observations.

Inference of mean residence time. The mean residencetime in a multi-compartmental system with unsteady flowand full tracer recovery can be obtained from the tracerbreakthrough curve. Given a pulse injection of tracer intothe system at time t0, and assuming that the tracer canleave the system either through discharge Q or throughevaporation E, the mean residence time can be definedas follows:

Tmr D

∫A

∫ 1

t0

�cqQ C ceE�t dt da∫A

∫ 1

t0

�cqQ C ceE� dt da�11�

where cq is the isotopic tracer composition in Q, ce isthe isotopic composition in E, t is time, da is an arealelement and A is the control surface (catchment area) ofthe system.


F. FENICIA ET AL.

Equation (11) differs from traditional formulations(e.g. Goode, 1996) because it accounts for unsteadystreamflow and internal fluxes. Since these fluxes origi-nate in different compartments (e.g. evaporation from theactive and dead storages in the PM model) and hence mayhave different isotopic compositions, the integral over thecatchment area is used.

Since Equation (11) implies that Tmr D f�q, cq�, itwas used to obtain the posterior distribution of Tmr ,p�Tmr jQ, Qc�, corresponding to the posterior distributionp�q, �q, �cjQ, Qc� given by Equation (9).

RESULTS

The models were run on an hourly time step over theentire year of 1987. The first 10 days of simulation weretreated as a warm-up period and discarded from thecomputation of the performance statistics. The posteriordistribution Equation (9) was sampled using the MarkovChain Monte Carlo (MCMC) strategy described by Thyeret al. (2009) with a total of 39 000 model runs and5 parallel chains. During the first 2000 samples, thejump distribution was tuned one parameter at a time.During the next 2000 samples, the jump distribution wastuned by scaling its entire covariance matrix. Followingthis, the jump distribution was fixed and 35 000 sampleswere collected. The first 25 000 samples were treatedas a warm-up period and only the final 10 000 sampleswere used to report the parameter distributions. In allinferences, the Gelman-Rubin convergence test was veryclose to unity, which suggests adequate convergence ofthe MCMC chains to the stationary distribution.

Since the tracer data contained long gaps betweenevents, we used eight separate events within the observa-tion period for calibration. Before each event, the isotopictracer composition of the model compartment(s) wasreinitialized to the value before (when available) or afterthe event. However, the performance of each parametersample was evaluated on all events simultaneously.

Tmr was calculated using Equation (11), running themodels for the two full years 1986–1987 and addinga constant isotopic tracer composition to all modelreservoirs on 1st March 1986. While injecting the traceron a different date may lead to slightly different estimatesof Tmr , seasonality effects at the Maimai site are minorbecause the rainfall regime is relatively stable (seeSection on Study Site).

Parameter distributions

CM model. The Nash–Sutcliffe performance (NS) ofthe CM model with respect to flows is quite high, 0Ð85.The NS of the tracer output is much lower, i.e. 0Ð37.However, it stressed that: (1) the tracer signal is stronglydamped, and therefore the average of the observation isalready a relatively good representation of the data and(2) the sampling uncertainty is relatively high in relationto the dynamics. Reassuringly, the inferred values of �c

are close to the sampling uncertainty in the deuterium

composition measurements, which is commonly between1‰ and 2‰ (Pearce et al., 1986). This suggests thatit would be difficult to expect much higher modelperformances.

The parameter distributions of the CM model areshown in Figure 3. All model parameters appear to bewell identifiable and display little correlation (apart froma mild correlation between Sbq and k2). Figure 3 alsoshows the estimated distribution of Tmr and its corre-lation with model parameters. Note that Tmr itself wasnot calibrated—it is a derived quantity computed fromthe samples. It can be seen to be very strongly depen-dent on Sd, with minimal statistical dependence on othermodel parameters. Given the good identifiability of Sd

for the CM model, the marginal distribution of Tmr isitself well identifiable, with likely values in the range of60–120 days. This result agrees with previous studies ofthe Maimai catchment, which estimated Tmr as approxi-mately 4 months (Pearce et al., 1986). Interestingly, Tmr

depends primarily on the dead storage Sd and displaysvery little correlation with other model parameters. Wewill comment on this result in the Discussion.

Traditional steady-state model. To provide a compari-son with current approaches for Tmr estimation, we alsoevaluated a traditional EM (e.g. Zuber and Maloszewski,2000). The EM is essentially a single-parameter linearreservoir that convolves the isotopic tracer compositionthrough an exponential function. The EM was calibratedand evaluated on the selected events using the sameapproach as for the CM and PM models. The EM pro-vided a very poor fit to the tracer data, with NS D 0Ð05.This low value is likely due to the steady-state hypothesisunderlying the traditional EM model, which completelyneglects the rainfall-runoff dynamics. This is unsurprisingand simply confirms that traditional ‘steady-state’ tracermodels are unsuitable for high temporal resolution sim-ulations (such as hourly in this case).

PM model. The PM model performs slightly better thanthe CM model, raising the NS efficiency of the tracer fitfrom 0Ð37 to 0Ð45. This difference can be attributed tothe PM model having three parameters (Sd, Fr and D)controlling the tracer response, whereas the CM modelhas only one parameter (S).

The hydrological parameters inferred from the PMmodel have almost identical distributions as for the CMmodel (hence not shown). This is a direct consequenceof the CM and PM models sharing the same hydrologicalconceptualization.

The distributions of the mixing model parameters areshown in Figure 4. D displays large variability, whereasFr appears well identified. The upper bound of Sd ispoorly identifiable (or at least not within the prior upperbound of 4000 mm allowed during calibration). Due toa strong correlation between Tmr and Sd (same as in theCM model), the poor identifiability of Sd translates intopoor identifiability of Tmr . The lower bound of Tmr is



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F. FENICIA ET AL.

102 103

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Figure 4. Distribution and correlation between parameters of the CM model (hydrology-related parameters not shown). The Tmr distribution derivedfrom parameter samples is also shown

Figure 5. Cross-cuts of the NS index calculated for deuterium composition data alone

close to that estimated with the CM model, but the upperbound is much higher.

The strong correlation between D, Fr and Sd wasinvestigated using cross-sections of the NS measure forthe isotope data fit (Figure 5). In this analysis, parameterswere varied, two at a time, while keeping all othersfixed at near-optimal values identified during the modelcalibration.

Figure 5(a) shows that when the exchange flux associ-ated with D is small, the damping effect can be largelyexplained by a fixed partitioning of rainfall betweenactive and dead storages (Fr). When D is large, Fr

becomes less identifiable, since the damping can beexplained either by a dispersion flux, or by partitioningof rainfall, or by a combination of both. There is insuffi-cient information to identify the process solely from thegiven data and model conceptualization.

Figure 5(b) demonstrates a negative correlation bet-ween Sd and D. This implies that, keeping all other

parameters fixed, the same damping of tracer signalcan be achieved either by a reduction of the (mixing)dispersion flux, or by increasing the dead storage, or bya combination of both.

Finally, Figure 5(c) shows that Sd tends to be poorlyidentifiable for large values of Fr . This occurs becauseSd behaves like a large reservoir that receives a cer-tain fraction of the precipitation and releases it stronglydamped. Therefore, Sd provides a constant ‘baseline’contribution to the isotope signal, whereas Sa providesthe variable time-dependent contribution. These com-ponents are weighted by the parameter Fr to fit thedegree of damping in the observed data. This for-mulation of the system does not require Sd to havean upper bound. Also, note that Sd does not partici-pate in the water balance (except for the minor effectthrough Ed) and hence is not directly informed byobserved streamflow. The combination of these fac-tors may explain why, in the absence of additional



12/10/87 14/10/87 16/10/87 18/10/87

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Figure 6. Simulated and observed specific discharge (a) and deuterium compositions (b) for a selected event. The size of the bubbles in (b) isproportional to the observed rainfall

information or constraints, Tmr is pushed against its priorupper bound (Figure 4).

System response and predictive limits

Figure 6 shows the model performance (streamflowand tracer predictions) for the largest event in theobservation period. Figure 6(a) shows the hydrograph,whereas Figure 6(b) shows the deuterium composition inthe rainfall and in the discharge. The isotopic compositionin the discharge is strongly damped with respect tothe rainfall (note the different scales of the y-axis inFigure 6(b)).

Figure 6(b) shows the 5–95% predictive intervalsincluding parametric and residual uncertainty of isotopictracer compositions simulated by the two models. Thesebounds are essentially the same for the CM and PMmodels. Hence, while the two models provide verysimilar fits to the streamflow and tracer data, they yieldvery different estimates of Tmr .

DISCUSSION

While the CM model provides a well-defined estimateof Tmr , the PM model provides a much more uncertainestimate. In addition, the PM hypothesis leads to Tmr

estimates that are much larger than those correspondingto the CM hypothesis. However, both models providevery similar predictions and uncertainty estimates of theisotope signal. This suggests that the estimation of Tmr

and its posterior uncertainty can be very sensitive to theassumptions underlying the hypothesized mixing model.

The interpretation of internal mixing processes basedon input–output conservative tracer data is ambiguousbecause different model components may have similareffects on the output (see Renard et al. (2010), for arigorous discussion of model identifiability). Our find-ings are consistent with those of Page et al. (2007), who

attempted to simulate observed stream chloride concen-trations at Plynlimon (Wales). Based on a similar modelsetup (mobile–immobile storage mixing), they deter-mined that parameter equifinality prevented the identi-fication of effective catchment mixing volumes and coef-ficients. Since the way water mixes within the catchmentis likely to have a strong effect on the age of water, esti-mates of Tmr calibrated solely to input–output data—andespecially under CM assumptions—should be interpretedand used with considerable care.

Even at the Maimai site, which appears to be a rel-atively well-mixed catchment compared to the range ofsystems encountered elsewhere, complete mixing is con-sidered unrealistic by the experimentalist. In particular,McDonnell (1990) has proposed a rationale for old watermovement through macropores based on unrequited stor-age within the soil profile and non-participatory portionsof the hillslope material. As a result, Tmr can be expectedto be larger than estimated assuming complete mixing.

Generally, the uncertainty in Tmr estimates could bereduced using additional independent insights into thefunctioning of the catchment. In particular, a better qual-itative understanding of storage and flow characteristicsand pathways can improve the selection of mixing mod-els and constrain the uncertainty in model parameters.Obtaining and using this type of knowledge will requirea close exchange between modeler and experimentalist(Seibert and McDonnell, 2002). For example, previousstudies have shown that the Maimai catchment is charac-terized by shallow soil and impermeable bedrock (Pearceet al., 1986). This may a priori limit the depth of thedead storage and constrain the estimates of Tmr .

Another important result is that, in this analysis, Tmr

appears independent from the hydrology-related param-eters, but instead is controlled by the total storage ofthe catchment that participates in the mixing process.This result contradicts some previous studies, whichproposed estimating Tmr from hydrograph recessions


F. FENICIA ET AL.

(Wolock et al., 1997; Vitvar et al., 2002). Referring toFigure 1(b), in the absence of significant dead storage,Tmr is largely determined by recession constants. How-ever, as the dead storage becomes dominant with respectto the active storage, it becomes the main control on Tmr .

In contrast, other studies (e.g. Vache and McDonnell,2006a; Dunn et al., 2007; Son and Sivapalan, 2007;Fenicia et al., 2008) found a strong dependency of Tmr

(or of the isotope response in general) on unsaturatedand other storage. These studies support our conclusionthat Tmr is correlated much stronger to storage properties,rather than to measures of streamflow response speed.

Finally, the traditional convolution-based mixingmodel based on steady-state flow assumptions providedan unacceptably poor representation of the tracer signal(see Section on Parameter Distributions). While perhapsproviding some limited utility for long-term averageddata (e.g. monthly), such models ignore essential catch-ment dynamics and therefore cannot yield useful insightsinto the short-term mixing mechanisms operating in arealistic catchment.

While this study focused solely on the Maimai catch-ment and therefore the generality of its conclusions needsfurther investigation, we argue that physically motivatedmodels for flow and tracer simulation, such as the con-ceptual mixing models investigated in this study, aremore useful than black-box approaches. In particular,they can provide more integrated and physically realis-tic approaches for understanding and predicting pollutanttransfer in watershed systems, and are more likely toexplain apparent hydrological paradoxes, such as rainfallforcings resulting in rapid stream responses with waterthat is months to years old (Kirchner, 2003).

Furthermore, it is clearly preferable to use physi-cally motivated models, rather than simplistic or black-box approaches, to predict quantities such as Tmr ornew-water/old-water fractions when evaluating the per-formance of more complex flow and transport models(Vache and McDonnell, 2006a; Son and Sivapalan, 2007).It is stressed that, when the model is poorly identifi-able, quantities inferred from data, such as Tmr or new-water/old-water fractions, can be much more uncertainthan the data itself.

CONCLUSION

This study investigated the impact of different watermixing assumptions on the estimation of streamwatermean residence time Tmr . While mean residence timeis increasingly used for catchment characterization andmodel evaluation, in practice, its estimation is often per-formed using models and underlying hypotheses that aredemonstrably unsupported by physical and/or statisticalevidence.

In a case study based on the experimental Maimaicatchment in New Zealand, we compared the meanresidence time Tmr estimated using a complete-mixing(CM) versus a partial-mixing (PM) conceptual model.

The latter was motivated by a priori physical insights intothe Maimai catchment and allowed for water exchangebetween an active and a dead storage compartmentthat does not contribute to the overall catchment waterbalance. The hydrological and mixing parameters ofboth models were inferred statistically. Both the PMand CM models account for transient dynamics duringstorm events and fit the tracer observations much betterthan convolution-based models based on steady-statehydrology.

While the CM and PM models provided similar fits tothe stream tracer signal, with NS indices of 0Ð37 versus0Ð45, respectively (and with predictive bounds of similarwidth), their Tmr estimates varied considerably. Thisemphasizes that model evaluation and comparison basedsolely on output series may be highly unreliable, and thatdetailed evaluation of internal states and associated modelcharacteristics can detect critical differences in modelbehaviour and interpretation.

Although CM-based Tmr estimates are well identified,those derived from the PM model were considerablylarger and more uncertain. This occurs because of themathematical behaviour of the PM conceptualization andformulation. Hence, although the PM model agrees betterwith experimental perceptions of the mixing mechanismsin the Maimai catchment, its poor identifiability suggeststhat further independent understanding of mixing pro-cesses is needed to constrain the model. It also followsthat Tmr estimates derived without a careful indepen-dent testing of underlying mixing assumptions may beunreliable. Importantly, this highlights that improving thephysical realism of a model does not necessarily improveits statistical identifiability, and indeed the contrary maybe quite common.

Model analysis also suggested that Tmr is controlledprimarily by storage properties, rather than by speed-of-response characteristics. This finding implies thatestimating Tmr from recession constants is unreliable,and supports previous work linking Tmr to unsaturatedand other storage.

Further analysis is warranted for other catchmentsto investigate the generality of our conclusions. Nev-ertheless, we argue that conceptual models providedeeper insights into catchment dynamics than traditionalconvolution-based mixing models requiring steady-stateassumptions, linearity, complete mixing, etc. Conceptualmodels are more flexible and, especially when combinedwith additional data and insights, can be used within sys-tematic hypothesis-testing and improvement frameworksto improve our understanding of catchment dynamics andthe interaction between different catchment processes.

ACKNOWLEDGEMENT

This work was supported by the National Research Fundof Luxembourg (Grants FNR/08/AM2c/50 and TR-PHDBFR07-047-Investigating scale dependency of runoffgeneration).



APPENDIX: LOGISTIC SMOOTHER FORPIECEWISE-LINEAR SEGMENTS

The logistic smoother used in this work is defined asfollows (Kavetski and Kuczera, 2007):

�xjm� D m�x C ln[1 C exp��x�]� �A.1�

where x is the independent variable and m is a shapeparameter.

Next, define F as

F�xjk1, k2, xb, m� D

yb C k1�x � xb�, if xm < ln�ε�yb C k2�x � xb�, if xm > � ln�ε�yb C k1�x � xb�C�k2 � k1��xmjm�, otherwise

�A.2�where k1 is the slope of the first linear segment, k2

the slope of the second linear segment, yb D k1xb, xm D�x � xb�/m and ε is the machine precision (e.g. 10�16 fordouble precision computation).

To ensure F�x D 0� D 0, we define

G�xjk1, k2, xb, m� D F�xjk1, k2, xb, m�

� F�0jk1, k2, xb, m� �A.3�

Both F and G are infinitely differentiable, which is anecessary (and usually sufficient) condition for yieldingsmooth models with smooth objective functions. Thesmoothness of the model and its objective function isvery beneficial in both calibration and prediction: itallows using fast derivative-based optimization methods,simplifies sampling, yields more stable and well-behavedmodel predictions, etc. (Kavetski and Kuczera, 2007).

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Assessing the impact of mixing assumptions on the ...*Correspondence to: Fabrizio Fenicia, Centre de Recherche Public— Gabriel Lippmann, Department Environment and Agro-Biotechnologies,