Quantifying Uncertainty in Tracer&hyphen ... - CiteSeerX

WATER RESOURCES RESEARCH,

VOL. 34, NO. 4, PAGES 915-919, APRIL 1998

Quantifying uncertainty in tracer-basedhydrograph separations David

Genereux

Departmentof Geology,SoutheastEnvironmentalResearchProgram,Florida InternationalUniversity,Miami

Abstract. A methodis presentedfor quantifyingthe uncertaintyin two- and threecomponenttracer-basedhydrographseparations.The methodrelatesthe uncertaintyin computedmixingfractionsto both the tracer concentrations usedto perform the

hydrograph separation andtheuncertainties in thoseconcentrations. A two-component exampleand a three-component exampleillustratethe applicationof the method.The three-componentexampleyieldsuncertaintyresultsvery similarto thosefrom a previously publishedMonte Carlo analysisand requireslesscomputation. This questionhasgenerallybeen addressedwith an informal sensitivityanalysisof (1) or (2), in whicha rangeof valuesfor Hydrogaph separation(quantitativelyapportioningstorm eachmixingfraction is computedusinga range of component runoff in a streamor river amongcontributionsfrom different tracer concentrationsselectedby inspectionof the data. For water sources)representsone of the earliestforms of water- example, Rodhe[1981]used•80 datato separate springrunshedhydrologicanalysis.Early approachesto hydrographsep- off into groundwaterand meltwater from snow,and calculated arationby graphicalmeans[e.g.,Linsleyet al., 1982]havebeen therangeof groundwater ,mixing fractions (about_+0.15)that augmentedin recentyearsby methodsbasedon chemicaland couldarisefroma rangeof _+0.5%0 in the •80 of groundwaisotopictracers.These methodsgenerallyinvolve separating ter, or _+1%0in the •80 of meltwater. No specific confidence stormhydrographsinto two or more "components"or "end- limit (e.g., 90% or 95% confidence)was attachedto the unmembers"basedon naturally occurringchemicalor isotopic certaintyrange.Neal et al. [1990] followedthe samegeneral differencesbetweenor amongthe waterscontributingto storm approachin estimatingthe uncertaintyin hydrographseparafloW.Tracersusedin this contextincludemajor ions and alka- tions at their study site in Wales. Acid-neutralizingcapacity linity (or acid-neutralizingcapacity)[e.g., Pinder and Jones, (ANC) wasusedto separatesoilwater from deepwater. Com1969;Hooperet al., 1990;Nealet al., 1990;Muiholland,1993; putedfractionsof deepwater in the streamdifferedby asmuch Eshleman,1993],•80 and/or2H [e.g.,Sklashet al., 1976; as0.35,dependingon the concentrations usedto representsoil Hooperand Shoemaker,1986;DeWalleet al., 1988;McDonnell water and deep water. et al., 1990],and silica[e.g.,Hooperand Shoemaker,1986;Wels McDonnellet al. [1991]followeda similarapproachin their et al., 1991].This paperproposesa meansof formallyquanti- analysisof a late Novemberrainstormon the smallM8 catchfying the uncertaintyin tracer-basedhydrographseparations mentin New Zealand.Deuterium(2H) datawere usedto (and other water-mixingproblems),throughapplicationof partitiona hydrograph intopreeventandeventwater(event generaluncertaintypropagationtechniquesusedin other sci- water being precipitation,preeventwater being water stored entific and engineeringproblems. on the catchment, in the subsurface,before the start of the stormof interest).Hydrographseparationcalculations were 1.

Introduction

2.

Previous Attempts to Quantify Uncertainty

doneusing•2H valuesforpreevent waterthatwere1%0higher and lower than the best estimate of the true value; resultant

Tracer-basedseparationof a hydrographinto contributions from two water sources(components) relieson two massbalance equations,one for water and one for the tracer. These equationsare usuallysolvedfor the fractionof streamflowdue to each component: C2-

Cs

estimates of the fractionof preeventwater(fp) spanned a range of about +_0.05at most pointson the hydrograph. Bazemoreet al. [1994] used a Monte Carlo approachto estimatethe uncertaintiesin three-component(event water, soilwater, and groundwater)hydrographseparations basedon

chlorideand •80 data from ShaverHollow, a forestedwater-

Q•

shedin Virginia. The water and solutemassbalanceequations were solved50,000 times for each stream sample,and 68% confidenceintervalswere obtainedfor each mixing fraction. Cs-C• Q2 Hooper et al. [1990] pursuedan uncertaintyanalysisfor their f2- C2-c• Qs (2) three-componentseparationof streamflowat Panola Mounwheref is the fraction of total streamflowdue to a component tain, Georgia, using a mathematicalstartingpoint similar to (f• + f2 = 1), Q is volumetricflow rate, C is tracerconcen- that presentedbelow. tration, and subscripts1, 2, and S refer to components1 and 2 and streamwater, respectively.A key questionis how uncertainty in measuredtracer concentrationsis propagatedinto f• 3. Uncertainty in Two-ComponentSeparations and f2. In the casewhere a parametery is calculatedas a functionof Copyright1998 by the American GeophysicalUnion. severalmeasurands x•, X2, ''', Xn (i.e., y = G(x•, X2, ''', x,•)), and the uncertainty in eachmeasurandis independentof Paper number98WR00010. 0043-1397/98/98WR-00010509.00 uncertaintyin the others,the uncertaintyin y is related to the

f• = C2-C• Qs

(1)

915

916

GENEREUX: UNCERTAINTY IN TRACER-BASEDHYDROGRAPH SEPARATIONS

of thetermsfor C• andC2 shows that uncertainty in eachof themeasurands bythefollowing[Peters ment).Inspection is more sensitiveto uncertaintyin the componentaccounting et al., 1974;Meyer,1975;Taylor,1982;Kline,1985]: for morethanhalf of the mixture(e.g.,preeventwaterin the caseof mostseparations of eventandpreeventwater).

! -- •(øY •-•lWx 1)2 q-(øY) •-•2Wx2 •xnWxn Wy 2q-...q-(øY) 2 (3)

whereW represents theuncertainty in thevariablespecified in 4. Uncertaintyin Three-ComponentSeparations the subscript. Equation(3) is a simplifiedform of a more Theapproach to uncertainty analysis represented by(3) may generalexpression that allowsfor nonzerocovariances be- alsobe appliedto three-component mixingproblems,for tweenmeasurands (i.e.,the casewheretheuncertainty in one whichtwo tracersare required.For a casewherecomponents measurand maynotbeindependent of thatin another).In this 1, 2, and3 mixto formstreamwater,andtheconcentrations of moregeneralcase[e.g.,Meyer,1975,pp.39-40; Taylor,1982, two differenttracersare symbolized byA andB, the relevant pp. 175-178],there are termsof the form (Oy/Ox•)(Oy/ massbalanceequationsare Ox2)Wxlx2 underthe squarerootsymbol on the right-hand sideof the equation(Wx•x2is computed fromthe covariance Os = O• + 02 + 03 (5) of measurands 1 and2). The uncertainty analysis of Hooperet As = f•A• + f2A2 + f3A3 (6) al. [1990]retainedsomeof thecovariance terms,to allowfor the possibility that the concentrations of differentsolutes Bs = f•B• + f2B2+ f3B3 (7) mightbe correlated withina component or end-member, but (5)-(7) maybe solvedfor the mixingfractions in not betweencomponents. However,thereis a lackof strong Equations evidence supporting thepractical importance of thismoregen- termsof the eightmeasuredtracerconcentrations: eralapproach; someavailable datasuggest thatthesecovari-

ancetermsareof minorsignificance. Forexample, 222Rn and

AsB2- AsB3+ A2B3- A2Bs+ A3Bs- A3B2

f•=A•B2 - A•B3 +A2B3 - A2B• +A3B• - A3B2

(8) Ca wereusedby Genereux et al. [1993]to carryout a threecomponent separation of streamflow atWalkerBranchWaterAsB3 - AsB• + A•Bs - A•B3 + A3B• - A3Bs shedin Tennessee; analysis of concentrations fromtwoof the f2 = A•B2 - A•B3 +A2B3 - A2B• +A3B• - A3B2 (9) threecomponents showedthat the two solutes werepoorly correlated witheachother(R2 -- 0.13 for soilgroundwater, AsB• - AsB2+ A•B2 - A•Bs + A2Bs- A2B• R 2 -- 0.0003 for bedrockgroundwater; the analysiswasnot f3: A•B2 - A•B3 +A2B3 - A2B• +A3B• - A3B2 carriedout on the third component,vadosezonewater,beof eachmixingfractionwith respectto each causethe 222Rnand Ca concentrationsfor that component Partialderivatives alongwiththeuncertainty in were derivedfrom differentsamples).Also,Bazemore et al. of theeighttracerconcentrations, [1994]foundnosignificant correlation between C1and•80 eachtracer concentration,would be usedas shownin (3) to concentrations in two of'the three components in their sepa- compute theuncertainty in themixingfractions, f•, f2, andf3. ration (the two soluteswere, however,correlatedin event water,thoughthiswasnot incorporated into the uncertainty analysis because of therelatively minorroleof eventwaterin 5.

Defining Uncertainty in Tracer

Concentrations the separations). One hassomechoicein the typeof uncertaintyto propagate Applicationof (4) (or its equivalentfor three-component

(i.e.,the definition of W), exceptinsofaras consistency requiresthat all the uncertainty values"... mustbe the same kindof quantity: eitherall average errors,all standard deviations,etc."[Meyer,1975,p. 39].Application of (3) to (1) gives

mixing)requires tracerconcentrations andan estimate of the uncertainty in each.Usingseparation of eventand preevent wateras an example,sampling of soilwaterandgroundwater can define the variability in the isotopiccompositionof

preevent water,Cp; spatial variability beforethe startof a

{[C2-C512[C5-C1] 2

stormis the main concern[e.g.,Sklashand Farvolden,1979; McDonnellet al., 1991].Likewise,incrementalsamplingof

+[(C2C1)

precipitation overthe courseof a stormcandefinethevariabilityin eventwaterisotopic composition, Ce;temporalvariabilityis generally the mainconcern[e.g.,McDonnell et al.,

Wf 1-- (•

el)2Wcl q- (C2- el)2Wc2

(4)

Application to (2) wouldgivethesameresult,sinceeachpar- 1990;PionkeandDeWalle,1992],thoughon largewatersheds couldpotentially be an issueaswell.Thusa tial derivativeof (2) is simply-1 timesthe correspondingspatialvariability meananda standarddeviation(o-)canbe computed for each partialderivativeof (1). toapply some formofweighting in Frominspection of (4) it is apparentwhya largedifference ofCpandCe.It ispossible of themeanandstandard deviation values(e.g., between the tracer concentrationof the two componentsis computation of themeanCp to subsurbeneficial; thelargerC2 - C•, thesmaller Wf•. Also,it canbe givingextraweightin computation seenthat the uncertaintyin f• is mostsensitive to uncertainty facewaterfrom an areaof thewatershedthoughtto contribute or givingextraweightin computation of in Cs,because themultipliers onWc•andWc2differfromthe moreto streamflow, the mean C e to event water falling during periods of higher multiplier onWcsbyfactors of (C2 - Cs)/ (C2 - C•) = fl

intensity). In a sense, weighting isimplicitin the and(Cs - C•)/(C2 - C•) = f2, respectively, factors whose precipitation sites,at leastfor Cp, andin practice absolutemagnitudes are lessthan 1 for mixturesof compo- choiceof measurement of the meanandstandard deviation of Cp maydenents1 and 2. This is fortunatebecauseCs is often the tracer estimates collection program(numberandlocation concentration with the smallestuncertainty(typicallyjust the pendon thesample theproperweighting functions to usein analyticaluncertainty of the streamconcentration measure- of sites).In general,

GENEREUX:

UNCERTAINTY

IN TRACER-BASED

the face of heterogeneoussubsurfacetracer signalsor timevaryingrainfall signalsare not known,so simpleaveragesand standarddeviationsare computedand used [e.g.,Bazemoreet al., 1994]. In additionto preeventversuseventwater separations, there are many published examplesof what Sklash et al. [1976] termed "geographicsource"separations.These involveseparation of water reachingthe stream into contributionsfrom different geographicsourceareas,suchas soil versusbedrock, mineral soil versus organic soil, and saturated zone versus unsaturatedzone [Pinderand Jones,1969;Hooperet al., 1990; Genereuxand Hemond, 1990;Kleissenet al., 1990;Kobayashiet al., 1990; Neal et al., 1990; Robson and Neal, 1990; Hendershot

HYDROGRAPH

SEPARATIONS

917

Table 2. Tracer Data [Sklashet al., 1986] and Uncertainty Resultsfor Separationof a StreamSampleFrom the M8 catchment,September21, 1983 Preevent

Number of samples

Mean•i2H Standard

deviation

Event

6

Stream

5

1

Cp = -45.6 r e = -10.6 Cs = -31.9 5.60

3.87

na a

t (95%) t (70%) 95% uncertainty 70% uncertainty

2.571 1.156 14.40 6.47

2.776 1.190 10.74 4.61

naa 1.036a 2a 1

Uncertainty infp or

78

17

5

fe accountedfor, %

et al., 1992;Genereuxet al., 1993;Mulholland, 1993]. Because these separationsrely on tracer concentrationswith distinct spatial differencesthat persist over the period of interest (storm or season),both spatialvariabilityand temporalvariability in concentrationsare of interest,and shouldbe considered in samplingand in computingW valuesfor the compo-

confidence level)for •i2Hanalyses; the valuewasusedhereas the

typicallybasedon one sample(no replicates).Temporalvariability in Cs over the hydrographis not "uncertainty"but rather the hydrologicsignalof interest,and data analysisdoes not involve temporal averaging.Also, if stream samplesare collectedin the sameplacein a streamwell mixedhorizontally and vertically, there is no spatial variability to consider in definingCs (the measuredCs is understoodto apply at one point alongthe channel).In this casethe laboratoryprecision

samplescollectedbefore peak streamflow:Ce -- -10.6%o (o- = 3.87%0) [SMashet al., 1986, Figure 4]; the isotopic compositionand rainfall amountsfor the five sampleswere -8.9%0 (5 mm), -3.3%0 (2 mm), -6.2%0 (5 mm), -12.1%o (12 mm), and -12.3%o (15 mm). For preevent and event

Here,na denotes not available. Resultsarefp = 0.61, fe = 0.39, +0.28 (95% confidence),or _+0.13(70% confidence). aSklashet al. [1986] reported +_2as the analyticalprecision(95%

uncertainty(95% confidence)in Cs; the uncertaintyin Cs for 70% confidencewas estimatedby taking half of the value for 95% confidence(to give a number, _+1, that would be approximatelythe standard deviation)and multiplyingit by the asymptotict value for 70% nents. Separationcalculations with (1)-(2) or (8)-(10) are carried confidence(1.036). Thoughthere is only one streamsampleanalyzed for the separation,the uncertaintyin Cs for that sampleis basedon the out for individualstreamsamplescollectedover the courseof precisiondeterminedfrom analysisof numerousreplicatesof other a hydrograph;thus the Cs value usedin a givencalculationis water samples,as discussed in the text.

waters

the standard

deviations

of the tracer

concentrations

were multiplied by appropriate t values to give uncertainty of thesoluteanalyses provides a goodestimate of Wcs. While any set of consistentuncertainty(W) valuesmay be estimatesat two levelsof confidence:70% (approximatelyone propagatedusing(4), usingstandarddeviations(o-)multiplied standarddeviation)and 95% (Table 2). The uncertainty terms(i.e.,(Oy/Ox)2Wx 2) for preevent waby t valuesfrom the Student'st distribution(each t for the same confidencelevel, such as 95%) has the advantageof ter, event water, and stream water accounted for 78%, 17%, respectively, infp (Table2). providinga clear meaning(tied to a confidenceinterval) for and5% of thetotaluncertainty,

thecomputed uncertainty: f• _+Wf• wouldcorrespond to, for example,95% confidencelimits on f•.

6.

Two-Component Example Deuterium data from Sklashet al. [1986] were usedto esti-

Whilethestandard deviation of Cpwasabout1.4timeslarger thanthatof Ce (5.6versus 3.9),theuncertainty fromCpwas 78/17 = 4.6 timeslarger,becauseOfp/OCp> Ofp/OCe;as notedin section3, the greatersensitivity offp to Cp arisesin this casebecausepreeventwater accountedfor more than half of the mixture (i.e., the streamwater).

mateeventandpreeventwatermixingfractions(fe andfp) and their uncertaintiesfor peak stormflow duringthe Septem-

ber 21, 1983,stormon the M8 catchment.Data on the 2H

7. Uncertainty Plot: Separation of Event and Preevent Water With •80

content of preeventwater were availablefrom six subsurface Rewriting(4) in the followingform emphasizesthe imporsites(Table 1). The isotopiccompositionof eventwater was estimated as the depth-weightedmean of five precipitation tanceoffp (andfe), andof Ce - Cp (thedifference between the isotopiccompositionof event and preevent waters), in

Table 1. Preevent2H Data for the September 21, 1983, Storm on the M8

Site

Catchment

152H,%o

Source a

SL5S Site A Pit 5

-43.1 -42.3 -45.5

Table 5 Table 5 Table 5

Seep SL5D SLBD

-44.0 -56.7 -41.8

Table 5 Figure 7 Figure 10

Mean

of six sites

-45.6

aTable or figure of Sklashet al. [1986] from which the value was taken.

controlling theuncertainty infp orfe:

fP Wcp] -lt-[(We WfP-'{[(CeCp) -3-(re- Cp) WCs

(11)

Attempts to rederive a similar equationpresentedby Rodhe [1987,p. 204, equation9.3] and reconcileit with (11) suggest that Rodhe's equationis missinga term and that the correct form of the expressionis given by (11). Figure 1 showsfive curvesgeneratedwith (11) usinguncertaintyvalues(95% confidence)whichseemreasonablebasedon the literaturereport-

918

GENEREUX:

UNCERTAINTY

IN TRACER-BASED

8. 1.0

0.3

0.8

0.6

'

0.4

I

!

0.2

0.3

/

from ShaverHollow, Virginia,in order to illustratethe application of the method,to comparethe resultsof the methodpresentedherewith the earlierMonte Carlo analysis of Bazemoreet al. [1994],andto demonstrate theconstruction of an "uncertainty budget"that can be used to quantifythe different sourcesof

uncertainty. TheC1and•80 datafortheJune1992storm(Table

AC=3 •L AC=4• ac=6

, 0.0

• 0.2

,

• 0.4

,

3) were taken from Bazemoreet al. [1994] (data on the three components from their Table 1 on page52, data on the stream watersampleat peakflowon the hydrograph from theirFigure4 on page57). Thisis the samesamplediscussed in somedetailby Bazemore et al. [1994].For consistency with the notationin (6)-

0.1

•

AC=10 0.0

Three-Component Example

carried outusing chloride and•80 datafora stream watersample

0.2

0.2

SEPARATIONS

The methodof uncertaintyanalysisdescribed in section4 was

0

I

HYDROGRAPH

••

• 0.6

,

• 0.8

,

(10),A andB areusedto designate •80 andC1concentrations, andthesubscripts 1, 2, 3, andS referto even•water,soilwater,

0.0 1.0

groundwater,and streamwater,respectively. Tracer

Figure1. Uncertainty infp andfe asa functionoffp andfe for hydrographseparationof event and preeventwater with

concentrations

and uncertainties

in Table

3 were

usedto evaluatethe partial derivativesof (8)-(10) and computethe mixingfractions(equations(8)-(10)) andtheir uncer-

(3)). Thet valueschosen werethosefor70% •sO.Assumed values of theuncertainty in isotopic compositiontainties(equation confidence, to facilitate comparison with the 68% confidence wereWcs= 0.2, Wc = Wc = 0.4; AC isthedifference Ce intervalspresentedby Bazemoreet al. [1994].The asymptotic t - Cp.Thetriangular •reasatethe right andleftsides represent

areaswhere the uncertaintyin the mixing fractionsis larger

valueat 70% confidence(1.036)wasusedfor the streamsolute

thanthesmaller of thetwomixingfractions (fp ontheleftside concentrations, because(as discussed in section5) the uncerof the figure,orfe on the rightside);in theseareas,the smaller tainty in thesevalueswas taken to be the laboratoryprecision mixing fraction can not be distinguishedfrom zero with the of the analyses,determinedfrom a largenumberof laboratory statedconfidencelevel (95%). replicates.Results(Table 4) showsubstantialuncertaintyin the computedmixingfractions,with little actual or practical difference between the values generated by the present methodand thosefrom Bazemoreet al. [1994].In somecases

ing•80 basedseparations of stormflow.Eachcurverepresentsboth methods lead to confidence intervals which, if taken literally,extendbelow zero or above1, somethingnot physically thevariability in Wtp= Wtefor oneassumed valueof Ce Cp. The symmetric concave-up shapeof eachcurvedemon- andmathematicallypossible.Clearly,with eitherapproach,the stratesthat the most certain separationsare thosewith equal proportionsof event and preeventwater. Clearly, the uncertaintyin mixingfractionsbecomeslesssensitiveto the valueof

the mixingfractions asCe - Cp increases.

Table 3. Tracer Data Used in ComputingMixing Fractions and Their

Uncertainties

for Peak Flow Stream Water

Sample,June 1992 Storm,ShaverHollow, Virginia [Bazemoreet al., 1994] Concen-

Water Event water Preevent soil water Preevent groundwater Peak flow streamwater

t

tration a

Meanb

oc

As Bs A2 B2 A3 B3 As Bs

-8.1 4.0 -6.1 26.6 -7.6 26.0 -6.7 23

0.16 0.8 0.2 8.3 0.26 3.6 0.19 0.7

confidenceintervalsshouldbe taken as approximatequantitative guidesto uncertaintyin mixingfractions. In addition, the individual terms on the right-hand side of (3) were comparedto quantitativelyassess the significance of the eight different sourcesof uncertaintyin each computed mixingfraction (i.e., the eight tracer concentrations used to computeeachmixingfraction).ConcentrationB 2 (the C1concentrationof soil water) accountsfor the majorityof uncer-

taintyinf• (97%) andf3 (79%), but a muchsmallerproportion of the uncertainty in f2 (22%; Table 5). While the significance of the uncertaintyin B 2 is not surprising,given that it is the largestof the eight uncertaintyvalues(Table 5,

W

g/d (70%)e (70%)f 8 8 11 11 31 54 1 1

1.119 1.119 1.093 1.093 1.055 1.047 1.036 1.036

0.18 0.90 0.22 9.07 0.27 3.77 0.20 0.73

aTracerconcentration, A for •80 andB for C1.

Table 4. Comparisonof Three-ComponentSeparation Resultsfor Peak Flow StreamWater Sample,June 1992 Storm, ShaverHollow, Virginia [Bazemoreet al., 1994] Mixing Fraction(f) _+Uncertainty (lo- Confidence)

Component 1 (eventwater)

bMeantracerconcentration, 8•80 relativeto SMOW (standard 2 (soilwater) meanoceanwater)for •so, and/aM for C1. 3 (groundwater) CThestandarddeviationof the samplesusedto define the mean.

dThenumberof valuesusedto compute themean. eThe appropriatet statisticfor 70% confidence.

fThepropagated uncertainty, equalto thet statistic for 70%confi-

Bazemore et al. [1994] a

ThisPaper b

0.12(+0.28, -0.18) 0.65(+0.23, -0.17) 0.23 (+0.36, -0.46)

0.15 _+0.28 0.65 + 0.19 0.20 +_0.41

aAsymmetricconfidenceintervalsmeasuredfrom Figure 6 of Bazemoreet al. [1994,p. 59].

busingconcentrations at peakflow, as measured from the eigh-

dencemultipliedby the standarddeviationof the tracerconcentration. teenthsamplefrom the left in Figure4 of Bazemoreet al. [1994,p. 57].

GENEREUX:

UNCERTAINTY

IN TRACER-BASED

Table 5. Percentageof the Total Uncertaintyin Eachf Value Arising From the Uncertaintiesin Eight Tracer Concentrations,for Peak Flow Stream Water Sample,June 1992 Storm, ShaverHollow, Virginia [Bazemoreet al., 1994] Uncertainty in fx Due to Each Tracer Concentration,c % Tracer

W

Concentrationa(70%)b A1 B1 A2 B2 A3 B3 As Bs

0.18 0.90 0.22 9.07 0.27 3.77 0.20 0.73

fl

f2

f3