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journalhomepage:www.elsevier.com/locate/jhydrol
Simplifiedevaporationmethodfordeterminingsoilhydraulicproperties
A.Peters*,W.Durner
InstituteofGeoEcology,DepartmentofSoilScienceandSoilPhysics,BraunschweigTechnicalUniversity,GermanyReceived26September2007;receivedinrevisedform4March2008;accepted3April2008
KEYWORDS
Hydraulicproperties;Evaporationexperiment;Soilwaterflow;Richardsequation;
Parameteroptimization
Evaporationexperimentsarecommonlyusedtoderivehydraulicpropertiesof
soils.Inthesimplifiedevaporationmethod,asproposedbySchindler[Schindler,U.,1980.EinSchnellverfahrenzurMessungderWasserleitfa¨higkeitimteilgesa¨ttigtenBodenanSte-chzylinderproben.Arch.Acker-u.Pflanzenbauu.Bodenkd.Berlin24,1–7],theweightofasoilsampleandpressureheadsattwoheightlevelsarerecordedatconsecutivetimes.Theevaluationofthesemeasurementsreliesonlinearizationassumptionswithrespecttotime,spaceandthewatercontent–pressureheadrelationship.Inthisarticle,weinves-tigatetheerrorsthatresultfromthelinearizationassumptions,andshowhowsystematicandstochasticmeasurementerrorsaffectthecalculationofwaterretentionandhydraulicconductivitydataandtheresultingfitsofsoilhydraulicfunctions.Wefindthatlineariza-tionerrorswithrespecttotimearenegligibleifcubicHermitesplinesareusedfordatainterpolation.Linearizationsinspaceleadtominorerrors,eveninthelatestageofevap-orationwherestronglynon-linearpressureheadprofilesemerge.Abiasintheestimatedretentionfunctionresultsfromthenegligenceofanon-linearwatercontentdistributioninthesampleatthebeginoftheevaporationprocess,andaffectsprimarilycoarsesandsorsoilswithstructuredporesystems.Thiserrorcanbeavoidedifanintegralevaluationofthemeasurementsisused.Weintroduceanapplicablerejectioncriterionforunreliablehydraulicconductivitydatanearsaturation,basedontheerrorinthehydraulicgradient.Calibrationerrorsoftensiometersleadtobiasedestimatesofhydraulicpropertiesinthewetrange,whereaserrorsintensiometerinstallationpositionsyieldbiasesinthedryrange.Randomerrorsindatacausenosignificantbias,andparametrichydraulicfunctionscanbeestimatedwithsmalluncertainties,ifwaterretentionandconductivityfunctionsarecoupledandtheunderlyingmodelstructureiscorrect.ª2008ElsevierB.V.Allrightsreserved.
Summary
*Correspondingauthor.Tel.:+495315930;fax:+495315637.E-mailaddress:a.peters@tu-bs.de(A.Peters).
0022-1694/$-seefrontmatterª2008ElsevierB.V.Allrightsreserved.doi:10.1016/j.jhydrol.2008.04.016
148Introduction
ModelingwaterandsolutetransportintheunsaturatedsoilbymeansoftheRichardsequationrequiresanaccurateknowledgeofthewaterretentionfunction,hðhÞ,andthehydraulicconductivityfunction,KðhÞ,wherehisthevolu-metricwatercontent,KðcmdÀ1Þisthehydraulicconduc-tivityandhðcmÞisthepressurehead.Thesesoilhydraulicpropertiesmaybeobtainedbyavarietyofmethods(vanGenuchtenetal.,1999;DurnerandLipsius,2005).MostwidelyusedarehydrostaticcolumnexperimentstoderivehðhÞ(DaneandHopmans,2002)andtheestimationofKðhÞfromhðhÞbycapillarybundlemodelslikeMualem’sintegral(Mualem,1976).Alternatively,transmissionandcapacityparameterscanbedeterminedsimultaneously,e.g.,byaseriesofsteady-stateexperiments(Boumaetal.,1983;Dirksen,1991;DurnerandLipsius,2005),butthetimede-mandforthismethodbecomesprohibitivelylargeforlowerwatercontents.Mostsuitableisthereforetheevaluationoftransientflowexperiments,suchasmulti-stepinflow/out-flowexperiments(Durneretal.,1999)orevaporationexperiments(Wind,1968;Simuneketal.,1998a).
Thenumericalinversionoftransientflowexperimentsisthemostaccuratewaytodeterminesoilhydraulicproperties(Simuneketal.,1998a;RomanoandSantini,1999;Hopmansetal.,2002),butthenumericalinversionhasdisadvantagessuchasthehighcomputationalburdenandproblemswiththenumericalsolutionoftheforwardproblemintheinverseframework(Durneretal.,1999).Therefore,simplifyingmethodsthatdependoncertainlinearizingassumptionsareattractiveandwidelyapplied.Themostpopularwaytoeval-uateevaporationexperimentsisWind’smethod(Wind,1968),whichwaslaterautomated(Boelsetal.,1978;HalbertsmaandVeerman,1994)andfurthersimplified(Schindler,1980;Wendrothetal.,1993).
Intransientevaporationexperiments,boththeverticaldistributionsofthewatercontentsandpressureheads,andtheirtemporalevolutionareingeneralnon-linear.How-ever,theycanbelinearlyapproximated(Wind,1968;Schin-dler,1980).Atthebeginningoftheexperimentthepressureheadprofileislinear(RomanoandSantini,1999)andtheevaporationrateisgivenbytheatmosphericdemandofthelaboratoryair.Inthisfirststagetheevaporationrateremainsaccordinglyalmostconstantbecausethedecreaseinunsaturatedhydraulicconductivityduetowaterlossissufficientlycompensatedbyanincreaseofthehydraulicgradient.Inthesecondstage,theevaporationrategradu-allydecreaseswithtime(KutilekandNielsen,1994)andboththepressureheadandwatercontentprofileinthesamplebecomenon-linear.Amongthedifferentevaluationmethodsforevaporationexperiments,themethodproposedbySchindler(1980)isparticularlyattractiveduetoitssim-plicity,easycomputationandsmalldatademand.Itreliesonpressureheadmeasurementsinonlytwodepthsandthetotalcolumnweightsrecordedatseveraltimes,thusassumingthatallnon-linearitiesaresmallandthattheer-rorsintroducedbythelinearizationsarenegligible.
TheaimofthisarticleistoevaluatetheaccuracyandtheuncertaintiesofthesimplifiedevaporationmethodbySchindler(1980),forsoilsofvarioustextureandstructureandtoimprovethemethod.Inasensitivityanalysiswe
A.Peters,W.Durner
investigatethetypeandmagnitudeofsystematicerrorsthatareintroducedbythelinearizationassumptionsandweshowhowerrorsinthecalibrationandpositionofthetensiometersanduncertaintiesinthemeasurementaffecttheresults,similartoMohrathetal.(1997)andBertuzzietal.(1999)fortheoriginalWindmethod.Weshowhowthetemporallinearizationerrorscanbediminishedbyuseofcubichermitesplineinterpolation.Furthermore,oursub-sequentparameterestimationprocedureaccountsforthenon-linearwatercontentdistributioninthefirststageoftheexperimentandthuseliminatesoneofthesystematicerrorsoftheclassicevaluationclosetosaturation.Thisisofparticularrelevanceforstructuredsoilswithasecondaryporesystemduetoaggregation(Coppola,2000;Spohreretal.,2005)orsoilswithmultiplepore-densitymaxima(Mal-lantsetal.,1997),whichrequirebimodalormultimodalretentionfunctions(Othmeretal.,1991;Durner,1994)orfree-formfunctions(Bitterlichetal.,2004;IdenandDur-ner,2007).SincetheestimationofKðhÞisveryuncertainatlowhydraulicgradients(Wendrothetal.,1993;Tamarietal.,1993)weintroducearejectioncriterionforKðhÞdatadependingontheerrorinthehydraulicgradient.
Materialsandmethods
Simplifiedevaporationmethod
Inatypicalevaporationexperiment,pressureheadsatmul-tipleheightsandcolumnweightsaretakenatanumberoftimes,untileitherthetensiometersusedforpressureheadreadingsfail,ortheweightchangebecomesnegligible(Pla-ggeetal.,1999;HalbertsmaandVeerman,1994).UnlikeWind’sapproach(Wind,1968),thesimplifiedmethodofSchindlerusespressureheadmeasurements,h1andh2,atonlytwodifferentdepths,z1ðcmÞandz2ðcmÞ(Schindler,1980).Themeanwatercontent,hi,derivedumnweight,andthemeanpressurehead,fromthecol-h
i,areevaluatedateachtimestepitogetdatafortheretentionfunction.SinceSchindlerproposedtoplacethetwotensiometersatheightsz1¼0:25Landz2¼0:75heightofthecolumn,L,whereLðcmÞisthe
h
iiscalculatedsimplyasarithmeticmeanofthetwomeasuredvalues.
ThewaterflowbetweentwotimestiÀ1andtithroughaplane,whichislocatedexactlyatthecenterbetweenthetwotensiometricmeasurementlevels,isassumedtobeequaltoqi¼zmÁDhi=Dti,whereDhi¼hiÀhiÀ1isthemeanwatercontentchangebetweenthereadings,causedbytheevaporationlossthroughthesurface,Dti¼tiÀtiÀ1isthetimeincrementbetweentworeadingsandzmðcmÞisthedis-tancefromthebottomofthecolumntotheplanebetweenthetwotensiometers.Thedataforthehydraulicconductiv-ityfunctionarederivedbyinvertingDarcy’slaw:KiðhÞ¼Àqi
i
Dh;
Þ
i=Dzþ1
ð1wherehi
isthemeanpressureheadbetweentimestepiandiÀ1forthetworespondingtodepths,Kiisthehydraulicconductivitycor-hi,Dhi
isthemeandifferencebetweenthetensiometerreadingsandDz¼z2Àz1isthedistancebe-tweenthetensiometers.upofSchindler,Inthesymmetricexperimentalset-hi¼0:25ÁðhiÀ11þhiÀ12þhi1þhi2
ÞandDhi¼Simplifiedevaporationmethodfordeterminingsoilhydraulicproperties0:5Áððhi2À1Àhi1À1Þþðhi2Àhi1ÞÞwherethesubscriptsindicatethespatialpositionandthesuperscriptsthetimesteps.
149
Correctingfornon-linearityinhðzÞ
Thesimplifiedmethodassumesthattheverticalwatercon-tentandpressureheaddistributionsarelinearoverthewholecolumn,andnotonlyoversmallsectionsofthecol-umn,asinWind’soriginalmethod.Forphysicalreasons,thisassumptioncanbeonlyapproximatelyfulfilled.First,thehydraulicgradientmustbelargertowardthesoilsurfacesincethewaterfluxdensityincreasesandthehydraulicconductivitydecreaseswithincreasingdistancefromtheimpermeablebottomboundary.Second,thewatercontentisusuallyanon-linearfunctionofthepressurehead.
Themagnitudeofthelinearizationerrorwithrespecttothepressureheaddistributionisnegligiblysmallduringthefirststageoftheexperiment(Fig.1,left),butmaybecomesignificantinthesecondphase.Thisisexemplarilyillus-tratedbyFig.1,leftforasandysoil,whichshowsthatthepressureheaddistributionisapproximatelylinearforaconsiderabletimeoftheexperiment.
Thelinearizationassumptionwithrespecttothewatercontentdistributioncausesalinearizationerrorthatisthemorepronouncedthehigherthesoilcolumnandthecoarserthematerial(PetersandDurner,2006).Thiserror(Fig.1,right)canbeavoidedbyconsideringexplicitlythenon-line-arityofthewatercontentdistributionwithdepthintheparameterestimationprocedure.Iftheretentionfunctionisknown,themeanwatercontentatanytimestepiinthefirststagecanbederivedbyevaluatingtheintegralofthewatercontentsoverthecolumnheightdividedbythecolumnheight.Thisisequaltotheintegraloftheretentionfunctionoverthepressureheadsfromthelowerboundaryofthecol-umn,hlbi,totheupperboundary,hubi,dividedbythepres-sureheaddifferencehlbiÀhubi(PetersandDurner,2006):^hiðbÞ¼
1
Àhubi
Z
hub
i
oftheexperimentisquasi-hydrostatic(Fig.1,left),itiscal-ÀðzÀzmÞ.DuetotheuncertaintiesofculatedashðzÞ¼h
thetensiometerreadingsthisapproximationwasfoundtobesuperiortothetheoreticallymorecorrectapproximationthathðzÞ¼azþb,whereaistheslopeandbistheinter-ceptoflinearpressureheaddistributionoverthecolumn.Ingeneralterms,theerrorduetothenon-linearityofthewaterretentionfunctionbecomessmallerforcolumnsoflessheight,whichsuggeststousesmallsoilsamplesintheexperiment.However,forcalculatingtheconductivityfunc-tion,themostdecisivefactoristheaccuracyofthemea-suredhydraulicgradientbetweenthetwomeasurementlevels.Sincethedifferenceofthehydraulicheadreadingsdependsdirectlyonthedistanceofthetwomeasurementlevels,andsincetheconductivityestimationisclearlythemorecriticalpartofthedataevaluation,itisadvantageousinpracticetousetallercolumnsandwiderdistancesforthetensiometerpositions.
Interpolationofmeasureddata
Þ,anadditionalForthecalculationoftheconductivity,Kiðhi
linearizationerrorarises,ifpressureheadchangesandwatercontentchangesarenon-linearwithtime.Thiserrorislesspronouncedthesmallerthetimeintervalsformea-surementsare.However,smallmeasurementtimeintervalscorrespondtosmallweightchanges.Assoonasthesediffer-encescomeclosetotheresolutionofthemeasuringdevicethefluxcalculationandthustheconductivitycalculationbecomehighlyuncertain.Ifreadingsofweightsandpres-sureheadstakeplaceatlargetimeintervals,theycanbeinterpolatedbetweentwoconsecutivemeasurementsoraregressionfunctioncanbefittedtothedata.SchindlerandMu¨ller(2006)proposedthereforeasecondorderpoly-nomialforlinearregressionofthewatercontentversustimedata.Inthisstudy,wecomparetheperformanceofpiecewiselinearandcubichermitesplinefunctions(FritschandCarlson,1980)forinterpolation,sothatonlyfewread-ingsarenecessary.Cubichermitesplinefunctionsareanappropriatemeansfordatainterpolation,sinceononehandtheyaremoreflexiblethanlinearinterpolationfunctionsbutontheotherhandmonotonicbetweentwodatapoints.Thisinterpolationmethoddecreasesthecostsformeasure-mentequipmentandlabortime.
0hlbi
^hðb;hÞdh;
ð2Þ
hlbi
where^hðb;hÞistheparametricretentionfunction,bisthe
parametervector,andhlbiandhubiarethepressureheadatthelowerandupperboundaryofthecolumn.Sincethepressureheaddistributionoverthecolumninthefirststage
0–1Column depth [cm]–1Column depth [cm]–2–3–4–5–6–50t=0.0 dt=1.5 dt=3.0 dt=4.5 dt=6.0 dt=7.5 dt=9.0 dt=11.5 d–40–30–20Pressure head [cm]–100–2–3–4–5–600.10.20.30.4Vol. water content [–]t=0.0 dt=1.5 dt=3.0 dt=4.5 dt=6.0 dt=7.5 dt=9.0 dt=11.5 d 0.5Figure1Pressurehead(left)andwatercontentdistributions(right)inatypicalsandatdifferenttimes.150FitofparametricexpressionstothehðhÞandKðhÞ
data
Tobeusedinnumericalsimulationmodelsofwatertrans-port,parametricmodelsforhðhÞandKðhÞarefittedtothedatabyanon-linearregressionalgorithmbyminimizingameasureofmisfit.Inthiswork,wefitthecoupledK–h–hmodelofvanGenuchten/Mualem(vanGenuchten,1980)orthebimodalmodelofDurner/Mualem(Durner,1994)tothemeasureddata,byminimizingthesumofweightedsquaredresidualsbetweenmodelpredictionanddatapairs:UðbÞ¼wh
Xrwhi½hiÀ^hiðbÞ2þwK
XkwK½KiÀbK
iðbÞ2i;ð3Þ
i¼1
i¼1
whererandkarethenumberofdatapairsfortheretention
functionandtheconductivityfunction,respectively,whandwKaretheclassweightsofthewatercontentdataandcon-ductivitydata,whiandwKdatapoints,andhiarethei,^hiðbÞ,Kiandbweightsoftheindividual
K
iðbÞarethemeasuredandmodelpredictedvalues,respectively.InEq.(3),thepre-dictedwatercontents,^h,areeithercalculatedinastandard
mannerasthepointwatercontentsatpressureheadh
i(‘‘classicmethod’’),orasthemeanwatercontentofthewholecolumn,asgivenbyEq.(2)(‘‘integralmethod’’).Sincetheobjectivefunction,UðbÞ,involvesdataofdif-ferenttypeswithdifferentmeasurementfrequency,theresultoftheoptimizationwilllikelybeaffectedbytheweightsofthedata(SimunekandHopmans,2002).There-fore,wenormalizedtheweightsfirstbyafactorforthedatatypeandsecondbyafactorforthedatafrequency.Toaccountforthedifferentmeasurementfrequency,theindividualweights,whiandwKiwerechosensuchthatthecombineddatawithineverylog10ðÀhÞðpF)incrementhavethesameweight,i.e.,theweightforacertaindatapointwasproportionaltoitsdistancetotheneighboringpointonthepFscale.Toaccountadditionallyforthediffer-entdatatypes,theweightsforthedataclasseswerecalcu-latedbywh¼1=ðhmaxÀhminÞandwk¼1=ðlog10ðKmaxÞÀlog10ðKminÞÞ,wherehmax,hmin,KmaxandKminarethemaxi-mumandminimumvaluesofthedatasetstobefitted.Asfittingprocedureweusearobustcombinationoftheshuffledcomplexevolution(SCE-UA)algorithmwithaLevenberg–Marquardt(LM)algorithm.TheSCE-UAalgo-rithm(Duanetal.,1992),isaglobaloptimizerandcon-vergeswithinpre-definedpermissibleparameterrangestowardtheminimum(ifthatexists),notdependingonini-tialguesses.TheLevenberg–Marquardt(LM)algorithmisanefficientlocaloptimizer(Marquardt,1963).Itisusedtospeeduptheconvergence,oncethecloseregionoftheglobaloptimumisidentifiedbytheSCE.Asimilarapproach,combiningaglobalwithalocaloptimizer,fortheestima-tionofsoilhydraulicpropertieswasintroducedbyLambotetal.(2002).
Forwardmodelingofevaporationscenarios
Modelingscenario
Toanalyzethetypeandmagnitudeoferrorsthatarein-ducedbythelinearizationassumptionsoftheevaporationmethod,andbydifferenttypesofmeasurementerrors,weperformedasensitivityanalysiswithsyntheticmeasure-
A.Peters,W.Durner
mentsforanevaporationexperimentwithaconstantpotentialevaporationrateandfourtypesofsoils.Thesoilhydraulicpropertieswereexpressedbytheuni-andbimodalconstrainedvanGenuchtenexpressionfortheretentionfunctionandMualem’spredictivemodelfortheconductiv-ityfunction.TheretentionfunctionisgivenbySeðhÞ¼
XkwiSei;
ð4Þ
i¼1
whereSe¼ðhÀhrÞ=ðhsÀhrÞistheeffectivesaturation,Seareweightedsub-functionsofthesystem,wtheiiareweightingfactors0 whereai(cmÀ1)andniarecurve-shapeparametersofpore-subsystems.Fork¼1,Eq.(4)representstheretentioncurveofvanGenuchten(1980),fork¼2thebimodalreten-tionfunctionbyDurner(1994). Therelativeunsaturatedhydraulicconductivityfunction,KrðhÞ,asusedinthisstudy,iscalculatedfromsoilwaterretentioncharacteristicaccordingtoMualem(1976) \"RSeÀ1#2 KrðSeðhÞÞ¼SseR0 hdSeðhÞ1À1 ;ð6Þ0hdSeðhÞwheresisafactoraccountingfortortuositythathasinthe originalpublicationofMualemavalueof0.5.FollowingHoffmann-Riemetal.(1999)andSchaapandLeij(2000),wetreatsasafreefittingparameter.ToevaluateEq.(6),theanalyticalsolutionsofvanGenuchten(1980)andPriesackandDurner(2006)wereused. EvaporationfromaverticalsoilcolumnwassimulatedbysolvingtheRichardsequationwiththeconstitutiverelation-shipsasdescribedbyEqs.(4)–(6)usingthefiniteelementcodeHYDRUS-1D(Simuneketal.,1998b).Theheightofthesimulatedcolumnwas6.0cm.Atthelowerboundaryano-fluxboundaryconditionwasÀKapplied,dhdzþ1¼0:ð7Þ andattheupperboundaryafluxboundaryconditionwaschosen:ÀKdhdzþ1¼q0; ð8Þwhereq10was0:2cmdÀ.ThisfluxwasmaintaineduntilthepressureheadattheupperboundaryreachedavalueofÀ105cm.Aftertheheadmetthatvalue,theboundarycon-ditionwaschangedtowardaDirichletconditionandthustheevaporationratebecamenon-linear,indicatingthebeginningofthesecondstageoftheexperiment.Themassbalanceerrorintheforwardsimulationswasinallcasessmallerthan0.5%. Threeunimodalsoils,representingatypicalsand(S),loam(L)orclay(C)wereused.TheparametersforthehydraulicpropertieswereobtainedwiththeneuralnetworkprogramROSETTA(Schaapetal.,1998)assumingtypicaltexturesandbulkdensitiesforthetestedsoils.Asafourth Simplifiedevaporationmethodfordeterminingsoilhydraulicpropertiessoil,bimodalhydraulicfunctionsthatrepresentastructuredclay(BI)soilwerechosen.ThesoilhydraulicparametersofthefourcasesarelistedinTable1. Generationofsyntheticdata Toobtainsyntheticpressureheadmeasurementsthatcouldbeusedtogetherwiththecumulativeoutflowasinputdataforthesimplifiedevaporationmethod,thepressureheadswerereadat1.5and4.5cmunderneaththesurface.TheresultingdatawereanalyzedeitheruntilthematricheadattheuppertensiometerpositionreachedÀ1000cm,whichisthetypicalmeasurementlimitfortensiometers,oruntiltheweightchangebecamenegligible(forthesandafter15d).The‘‘measurements’’weretakenindifferenttempo-ralresolutions(0.1–3d)andwereinterpolatedpiecewiselinear(plin)orbycubichermitesplinefunctions(spline).Fromtheinterpolatedfunctions100pffiffidatawereselectedthatwereequallydistributedonataxis. Inordertoinvestigatetheerrorcausedpurelybythelin-earizationassumptions,wefirstcreateddatawithoutanyadditionalerror.Inasecondpart,weinvestigatedtheerrorthatresultsfromabiasinthetensiometercalibration.Forthis,constantoffsetsofÀ1,0and1cminthereadingoftheuppertensiometerwereimposed.InthethirdpartthepositionoftheuppertensiometerwasvariedtobeÀ0.42,0and0.42cmofffromthecorrectplanetoinvestigatetheeffectscausedbyerroneoustensiometerpositions.Inafourthpart,theinfluenceofstochasticmeasurementerrorswasinvestigated.Thus,thepositionsandthecalibrationswereassumedtobecorrect,butanormaldis-tributednoisewithzeromeanandstandarddeviationrh¼0:2cmwasimposedonthetensiometerreadings.Inthesesimulations,theweightmeasurementswerealsoper-turbedwithanormaldistributedrandomerrorwithastan-darddeviationrw¼0:02g(i.e.4:8Â10À4cmfortheonedimensionalcaseiftheheightandvolumeofthecolumnare6cmand250cm3,respectively).Botherrorsrepresenttypicaluncertaintiesoflaboratorymeasurements.Togetreliableresultsonhowthisrandommeasurementerrorsaf-fectthecalculations,500realizationswerecreatedandevaluated. Dataevaluation Thesyntheticallycreateddatawereanalyzedwiththesim-plifiedevaporationmethodtoderivedatapairsforthehðhÞandKðhÞ-functions.Duetolimitedaccuracyoftensiometricmeasurements,thecalculationofKðhÞatsmallhydraulicheadgradientsbecomeshighlyuncertain(Wendrothetal.,1993;Tamarietal.,1993).Thus,allcalculationsofKdataforgradientssmallerthanacertainthresholdvalue,asspecifiedintheresultssection,wererejectedinthisstudy. 151 Theremainingdatapairswereconsideredreliable,andthewaterretentioncurveandtherelatedconductivitycurvewerefittedbyminimizingEq.(3).Allparameterswereesti-matedsimultaneously.Thus,fortheunimodalsoilssixparameters(hs,hr,a,n,Ksands)andforthebimodalcasenineparameters(hs,hr,a1,n1,a2,n2,w2,Ksands)werefitted. Diagnosticvariables Toquantifythemaximumerrorintroducedbythelineariza-tionassumptionsandthevariouserrorsources,themaxi-mumdeviations,DhmaxandDKmax,betweenthewatercontentsofthefittedandthetrueretentioncurveandthefittedandtrueconductivitycurveweredetermined:Dhmax¼maxjhðhiÞÀ^hiðhiÞj;and biðhiÞÞj;hmin hmin \"#0:5 np1X RMSEh¼½hiðhiÞÀ^hiðhiÞ2;hmin RMSElog10ðKÞ np0:5 P21b¼np½log10ðKiðhiÞÞÀlog10ðKiðhiÞÞ;hmin ð12Þ wherenp¼151isthenumberofdependentvariablevalues intheinvestigatedmoisturerange,forwhichthetrueandfittedfunctionareevaluated.ForEqs.(9)–(12),hmaxwassettoÀ1cm,correspondingtopF¼0,andhminwassettoÀ1000cmðpF¼3Þ.Therangewasrestrictedtoavoidmis-leadingerrorsthatmightoccurintheextrapolatedrangeofthefittedfunctions. Realsoils Totestthesimplifiedevaporationmethodandthesubse-quentparameterestimation,weanalyzedtwotypicalevap-orationexperimentsthatweredescribedbyMinasnyandField(2004).TherawdataoftheirsoilB(sandyloam)andF(clay)werechosenforthiscontribution.TheexperimentswerecarriedoutusingtheoriginalSchindlersetupwitha Table1SoilSLCBI Parametercombinationsforthefoursoilsinthesensitivityanalysishs0.430.430.380.38 hr0.0450.0780.0680.068 a;a1ðcmÀ1Þ0.1450.0360.0080.008 n;n12.681.561.091.09 a2ðcmÀ1Þ–––0.2 n2–––4 w2–––0.2 KsðcmdÀ1Þ720254.8100 s0.50.50.50.5 S:sand;L:loam;C:clay;BI:bimodalsoil. 152 columnheightof6cmandtensiometersinstalled1.5and4.5cmabovethebottom.SeeMinasnyandField(2004)fordetailsoftheexperimentalprocedure. Themeasurementpointsforthepressureheadsinbothdepthsandtheweightsasfunctionsoftimewereinterpo-latedusingcubichermitesplines.Fromtheinterpolation,200datapointswerechosenequidistantonthepffiffi taxis.Thesedatawereanalysedasdescribedabove. Results FiltercriterionforvalidKdata Tensiometershavemeasurementnoise,rh,withminimumerrorstypicallyaround%0:2cm,andthusinduceuncertain-tiesintheKðhÞcalculationinthewetrange,wherethehydraulicgradient,rH¼Dh=Dzþ1,iscloseto0.Hence,KðhÞdataobtainedatverysmallhydraulicgradientshavetoberejected,regardlesswhethertheelaborateWindmethodorthesimplifiedmethodofSchindlerisused.IntheliteraturewefindacceptationlimitsforrHfrom0.2(Wendrothetal.,1993)to5:0cmcmÀ1(Mohrathetal.,1997).ApplyingGauss’slawoferrorpropagation,andassumingDztobeerrorless,yieldsanerrorforthehydraulicgradientofrrH¼rh=Dz.Thus,theerrorinrHispropor-tionaltotheerrorinhandinverselyproportionaltoDz.Atthebeginningoftheexperiment,wherethetruegradientiscloseto0,smallerrorsinthetensiometerreadingscanleadtosmallorevennegativevaluesofrH.Asdescribedabove,inthefirststageoftheexperimentthewaterfluxthroughtheupperboundaryisconstantandequalsthepo-tentialevaporation.SinceaccordingtoEq.(1)thecalcu- Figure2Uncertaintiesofthescaledconductivity,KÃðq¼1Þasafunctionofthehydraulicgradient,rH,andtheerrorofrH,rrH.Redline:truescaledconductivity.Grayshadedarea:95%confidenceintervalforthecasethatrrH¼0:067.Lines:95%confidenceintervalsforthecasesthatrrH¼0:133and0.267(correspondingtovaluesoftensiometerdistanceDz¼1:5and0.75cmifrh¼0:2).NotethatnegativevaluesforKarenotshown.(Forinterpretationofthereferencesincolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)A.Peters,W.Durner latedKisinverselyproportionaltorH,thisleadstoanunderestimationifrHisoverestimated,ortounboundedoverestimationsorevennegativevaluesofK,ifrHisunderestimated.ThisisillustratedinFig.2forthescaledconductivity,Kü1=rHðq¼1Þ.ThelinesofrrH¼0:067representthe95%confidenceintervalforrh¼0:2andDz¼3cm. SincerrHisinverselyproportionaltoDz,theSchindlerevaluationgivesmorereliableconductivitydataatlowgra-dientsincomparisontoWind’sapproach,wherethedis-tancebetweenthetensiometersissmallerforagivencolumnheight. ToavoidmisleadingKðhÞdataforthesubsequentparam-eterestimation,weproposetoapplyarejectioncriteriumforconductivitydatathatdependsonboth,themeasure-menterrorandthespatialdistanceofthetensiometers.Asaresultfromourstochasticanalysis,werejectalldatathatarecalculatedfromgradientssmallerthan6ÁrrH.Thus,inourstudyðDz¼3cm;rh¼0:2cmÞthehydraulicgradienthadtobeP0:4cmcmÀ1.WeapplythiscriterionforthefilteringofKdatainallsubsequentdataanalysis,regardlesswhetherthedatawereassumedtobeerror-less(sections‘Analysisoflinearizationerrors’and‘Analysisofoffseterrors’)orwhetherdatacontainmeasurementerrors(sections‘Uncertaintyofestimatedhydraulicfunctionsduetorandommeasurementerrors’and‘Analysisofevapora-tionmeasurementsofrealsoils’). Analysisoflinearizationerrors Effectsoftemporaldatainterpolation Fig.3showshðhÞ(leftcolumn)andKðhÞdata(rightcolumn)obtainedbyapplyingthesimplifiedevaporationmethodtothesyntheticallygeneratedmeasurements.Forcomparison,thetrueunderlyinghydraulicfunctionsaredepictedaslines.ThefourrowsofFig.3representthefoursoilsinves-tigated,fromthecoarsesoilatthetoptothestructuredclaysoilatthebottom.Theeffectsoftwointerpolationtechniquesandfourdifferentreadingintervals,from0.1to3d,arecompared. Asageneralresult,themethodyieldsverygoodesti-matesfortheretentionfunctioninthewholemeasuredh-range,evenifreadingsaretakenatverylargetimeinter-vals.Inparticularforthesandandloam,evenameasure-mentintervalof3dyieldsdatathatareremarkablyclosetothetruefunction.Fortheclaysoilandthestructuredsoil,wherepressureheadschangerapidlywithtime(inthisstudy,h1¼À1000cmwasreachedafter1.5d),muchshorterread-ingintervalsarerequired(Dt60:1drecommended).Ifasoilisstructured,frequentreadingsaremandatorytogetreliableresults,becausetheinterpolationwouldsmearoutthenon-linearresponseofthetensiometricreadingsthatarecausedbythebimodalwatercapacityfunction. Comparingthepiecewiselinearinterpolationmethod(pluses)withthecubichermitesplineinterpolation(circles)showsthatbothinterpolationmethodsgivealmostsimilarresults.However,inmostcasesthedataobtainedbyher-mitesplineinterpolationwereclosertothetruefunctions.Incontrast,polynomialfunctions(datanotshown)per-formedratherpoor.Since,withareasonablemeasurementresolutionðDt60:5dÞ,cubichermitesplineinterpolation Simplifiedevaporationmethodfordeterminingsoilhydraulicproperties spline 0.1dspline 1.0dspline 2.0dspline 3.0dplin 0.1dplin 1.0dplin 2.0dplin 3.0dtrue func32log10 (K in cm d–1)10–1–2–300.511.5pF [–]22.53–400.511.5pF [–]22.53spline 0.1dspline 1.0dspline 2.0dspline 3.0dplin 0.1dplin 1.0dplin 2.0dplin 3.0dtrue func153 0.4vol. water content [–]0.350.30.250.20.150.10.05020.4vol. water content [–]0.350.30.250.20.150.10.0500spline 0.1dspline 1.0dspline 2.0dspline 3.0dplin 0.1dplin 1.0dplin 2.0dplin 3.0dtrue func0.511.5pF [–]22.53log10 (K in cm d–1)10–1–2–3–4–5–60spline 0.1dspline 1.0dspline 2.0dspline 3.0dplin 0.1dplin 1.0dplin 2.0dplin 3.0dtrue func0.511.5pF [–]22.530.40.380.360.340.320.3spline 0.1dspline 0.33dspline 0.67dspline 1.0dplin 0.1dplin 0.33dplin 0.67dplin 1.0dtrue func00.511.5pF [–]22.53log10 (K in cm d–1)0–0.5vol. water content [–]–1–1.5–2–2.5–3–3.5–40spline 0.1dspline 0.33dspline 0.67dspline 1.0dplin 0.1dplin 0.33dplin 0.67dplin 1.0dtrue func0.511.5pF []22.530.45vol. water content [–]log10 (K in cm d–1)0.40.35spline 0.1dspline 0.33dspline 0.67dspline 1.0dplin 0.1dplin 0.33dplin 0.67dplin 1.0dtrue func43210–1–2–3–4spline 0.1dspline 0.33dspline 0.67dspline 1.0dplin 0.1dplin 0.33dplin 0.67dplin 1.0dtrue func0.30.2500.511.5pF [–]22.53–500.511.5pF [–]22.53Figure3Outputofthesimplifiedevaporationmethod(withoutparameterestimation)forthefourinvestigatedsoilswithdifferentinterpolationmethodsandtimeresolutioninmeasurements.Leftcolumn:retentionfunctions;rightcolumn:hydraulicconductivityfunctions.Firstrow:sand;secondrow:loam;thirdrow:clay;fourthrow:structuredsoil.154 wasinallcasesslightlysuperiortothepiecewiselinearinterpolation,wealwaysusedcubichermitesplinesforthedatainterpolationinthesubsequentanalysis. Effectsofspatiallinearizationassumptions CloserinspectionofthedatainFig.3revealsthat,eveniftheyareobtainedwithsufficienttemporalresolution,theyshowsmallbutsignificantsystematicdeviationsnearsatura-tion.Thisismostdistinctforthesandandthestructuredsoil,wherethespatiallinearizationleadstoanunderestima-tionofthewatercontentsincomparisontothetruepointwatercontentsatthewetendofthemeasurementrange.Thissystematiceffectiscausedbylinearizingthenon-linearwatercontentprofileinthecolumnatquasi-hydrostaticcon-ditions.Theerrorappearsinsignificantatafirstsight,butsincetheretentiondataare,inasecondstep,usuallyfittedwithaparametricmodelwhichinturnisusedtoestimatetheshapeoftheconductivityfunction,itcanbegreatlyamplifiedinthisstep(Durner,1994).WewilldiscussthishðhÞlinearizationerrorinthenextsubsection. Towardstheendofanevaporationexperiment,thepres-sureheadprofileandthecorrespondingwatercontentpro-filebecomenon-linear(Fig.1).Thisismostpronouncedclosetotheevaporatingsurfaceandforcoarsesoils.Quiteremarkably,ouranalysisoftheSchindlerevaporationmeth-odshowedinnocasethatthespatiallinearizationassump-tionatthislatestageoftheexperimentcausessignificanterrors(Fig.3). Incontrasttowaterretentiondata,conductivitydata(Fig.3,rightcolumn)canbereliablydeterminedonlyuptomagnitudeswherethesoil’sunsaturatedconductivityisinthesamerangeastheappliedevaporationflux,inourcase0:2cmdÀ1.Therefore,thepressurerangewherereli-abledataareavailableisratherrestricted.Asdescribedabove,thisisaknowngenerallimitationoftheevaporationmethod,andisduetothenecessityofmeasuringahydraulicgradientwhichissignificantlydifferentfromzero.Giventhetensiometerdistanceof3.0cmandtheaccuracyoftensi-ometersinourstudyðr¼0:2cmÞ,ahydraulicgradientof%0:4mustbepresentinthesampletogetreliableesti-matesofKðhÞ.Animprovementinthemeasurementaccu-racybyoneorderofmagnitudereducestheerrorinthedeterminationofthehydraulicgradientalsobyoneorderofmagnitude(seeabove).Accordingly,thisenablesonetomeasureconductivitiesthatreachoneorderofmagnitudehigher.Forthesand,weseethatthiswouldexpandthepressurerangeinwhichreliableconductivitydatacanbeobtained,onlyslightly.Fordirectdeterminationsofcon-ductivitiesnearsaturation,theevaporationmethodisunsuited(MinasnyandField,2004;Wendrothetal.,1993).Theerrorsintheretentiondataarereflectedandampli-fiedintheconductivitydata.Largetemporalinterpolationintervalsofthetensiometerdataleadinparticularforthefinetexturedclaysoilandforthestructuredsoiltosignifi-cantandsystematicdeviationsfromthetruefunction.Again,theproblemislesspronouncedforthecoarsersoils.Effectofnon-linearwatercontentdistributionnearsaturation Asmentionedintheprevioussection,thewatercontentdis-tributionwithheightinthesoilcolumnattheinitialstageoftheexperimentcanbesignificantlynon-linearforcoarseor A.Peters,W.Durner structuredsoilmaterials(Fig.1).Thisleadstoasystematicdeviationbetweenthetruewaterretentionfunctionandthedata,whichshowmeanwatercontentsassignedtomeanpressureheads(Fig.3,firstrow,left).Fittingareten-tionfunctiondirectlytotheobservedmeandata(‘‘classicfit’’)wouldaccordinglygivebiasedparameterestimates.Iftheconductivityfunctionfitiscoupledtotheretentionfunctionfit,thisalsoinfluencestheoutcomeoftheKðhÞdetermination. AmorecorrectfitisachievedbyinsertingEq.(2)intoEq.(3).Fig.4showsthatthisintegralfiteliminatesthebiasthatisintroducedbytheclassicfitandleadstoaperfectmatchbetweenthefittedandthetrueretentionfunction,ifnoadditionalnoiseorothererrorsourcescorruptthedata.Ifwecomparea(point)retentionfunctionwithmeasured(integral)datapoints,inordertojudgeonthequalityofthefit,wearefacedwithapsychologicaldrawback:thebestfit-tedfunctionseemsnottomatchtheobserveddata,becausethedatarepresentadifferentquantity.Forameaningfulcomparisonofthefittedfunctionwiththemeasuredwethereforemustcalculateanintegralfunction,hðdata, h Þ,thatisadoptedfromEq.(2).Thisnewfunctiondependsnotonlyonthemodelparametersbutalsoonthecolumnlength.Wecallit‘‘meanretentionfunction’’todistinguishitfromthepointretentionfunction(seeFig.4).Notethatpointandmeanretentionfunctioncoincideatlowpressureheads.Intherestofthisarticle,wewillcomparemeanretentiondataalwayswithmeanretentionfunctions. Table2summarizesthecalculatederrormeasuresasde-scribedbyEqs.(9)–(12)forallsimulatedcases.Asex-pected,thedifferencesbetweenclassicandintegralfitwerenegligiblefortheloamandclay. Analysisofoffseterrors Effectofoffseterrorsintensiometercalibration Theconductivitycalculationsintheevaporationmethodaresensitivetoincorrectestimatesofthehydraulicgradient.Therefore,correctmeasurementofpressureheaddiffer-encesarerequired,aslongastheseareclosetohydrostaticconditions.Smalloffseterrorsofthetensiometercalibra-tionarecommon,andmayleadtoerroneousestimatesinparticularfortheconductivitydata.Also,theretention 0.45]–[0.4 tnetnoc r0.35etaclassic fit pointw classic fit meanl.ov0.3integral fit pointintegral fit meantrue function pointtrue function mean0.2500.20.40.60.81pF [–]Figure4Influenceofneglectingorincludingthenon-linearwatercontentdistributionontheestimationofhydraulicpropertiesforthesand.Simplifiedevaporationmethodfordeterminingsoilhydraulicproperties 155 Table2DiagnosticvariablescalculatedfromEqs.(9)–(12)fortheerroranalysisCaseSoilDhmaxDlog10ðKmaxÞRMSEhRMSElog10ðKÞcla S0.01770.39530.00700.1914L0.00330.03350.00160.0252C0.00060.14310.00040.0630BI0.01330.44200.00550.1939int S0.00080.37430.00050.1767L0.00340.03340.00150.0279C0.00060.14310.00040.0629BI0.00650.18280.00270.1084insth S0.00580.49730.00310.3316C0.00190.23260.00090.1653L0.00200.08810.00090.0683instl BI0.00250.53880.00120.2753S0.00680.69690.00360.3901L0.00650.23650.00300.1951C0.00160.21980.00070.1825BI0.00960.21720.00360.1575offsþ S0.01590.87030.00690.4730L0.00370.23590.00210.1675C0.00190.20340.00070.1396BI0.00650.13730.00320.0844offsÀ S0.01521.81270.00600.9654L0.00460.63080.00300.4067C0.00120.17140.00050.1015BI 0.0044 0.3910 0.0023 0.1980 cla:classicapproach;int:integralapproach;insth:installationerrorofuppertensiometer=+4.2mm;instl:installationerrorofuppertensiometer=À4.2mm;offsþ:offsetincalibrationofuppertensiometer=+1cm;offsÀ:offsetincalibrationofuppertensiometer=À1cm. functiondataclosetosaturationwillbeaffectedbyerrorsintensiometercalibrationbutthiseffectislesspro-nounced,sinceitrelatestotheaverageoftwomeasure-ments,whereastheconductivityestimatesdependondifferences. Fig.5showshowconductivitydataandthefittedpara-metricconductivityfunctionsareaffectedbyanoffseterroroftheuppertensiometerofÆ1cm.AnoffsetofÀ1cmleadstoanunderestimationofKðhÞandviceversa.Thisiscausedbythecorrespondingunder-andoverestimationofthehydraulicgradientthatisinverselyproportionaltoKðhÞ(Eq.(1)).AnoverestimationofrHleadstolargesys-tematicerrors,whereasanunderestimationbasicallyleadstolessinformationinthewetrangeoftheconductivityfunction.Thisisaresultofthefiltercriterionthatdiscardsdataattoosmallhydraulicgradients,whereastheoveresti-matedgradientsleadtobiaseddata.Thesameresultholdsforcorrespondingoffsetsinthecalibrationofthelowersiometer.Inthedryrange,theinfluenceofbiasedten-h ivaluesduetocalibrationerrorsbecomesnegligible,duetothegreatlyincreasedhydraulicgradients. TheestimatedKðhÞ-functionsshowthatthemoderatedataerrorsfromtheunderestimatedhydraulicgradients areamplifiedintheextrapolationrangetowardsaturation.Theerrorsforthesandarelargerthanforthefinersoils.Thisisononehandaresultofthewideextrapolationrange,wherenoinformationaboutthefunctionisavailable,butisfurthermorecausedbysystematicdeviationsofthereten-tionfunction,asshownbytherelativelyhighvaluesofDhmaxandRMSEh(Table2).AnoffsetofÆ1cminthetensi-ometercalibrationleadstoanoffsetofÆ0:5cminh iforacertainhi.Similartothepreviouslydiscussednon-linearwatercontentdistribution,theseoffseterrorsaffectthecalculationofthewaterretentiondataparticularlyforcoarsesandsandstructuredsoilsinthewetrange.Thecom-binedeffectsofaslightlybiasedretentioncurveestimateandthebiasedconductivitydataleadstothedifferencesintheestimatedconductivitycurveintheinterpolatedrange. Effectoferrorsintensiometerposition Slightdeviationsinthepositioningoftheinstalledtensiom-etersareunavoidableinpractice.Wethereforeinvesti-gatedtheconsequencesofplacementerrorsoftheuppertensiometerofÆ0:42cm,whichyieldedsomeinterestingaspects.Ingeneral,wefoundonlysmalldeviationsbetweenretentiondataandthetruefunctions.Theestimatedreten-tionfunctionforthesand(notshown)hassomedeviationsinthewetrange,whereastheestimationoftheretentionfunctionfortheclay(Fig.6,firstrow)andloam(notshown)exhibitthedeviationsinthedryrange.Thedeviationinthewetrangeofthesandmaybeexplainedinthesamewayastheoutcomeoftheoffsetinthetensiometercalibration.Infact,aslongasthehðzÞdistributionislineartheeffectsofwrongtensiometerpositionandwrongtensiometercalibra-tionareidentical. Thedeviationinthedryrangeisduetotheverystrongnon-linearityofthepressureheaddistributioninthesecondstageoftheevaporationexperiment(Fig.1).Inthisstagethepressurehead0.42cmaboveorbelowtheproperuppertensiometerlocationmaybeoneortwoorderstudelowerorhigher,leadingtolowerorhigherofmagni-h iforacer-tainhiandthustoanover-orunderestimationoftheretentionfunctioninthedryrange.Sincethebimodalsoilcombinesacoarsewithafineporesystem,botherrorsarecoupledinthebimodalsoil(Fig.6,secondrow). Thecalculatedconductivitiesareonlyslightlyaffectedbytheoffseterrorintensiometerposition.Intherangewherethedatapairsarenotrejectedduetosmallhydraulicgradientstheerrorsintheconductivitypredictionareacceptableforallsoils,althoughtheerrorismostpro-nouncedintheclaysoil,wherethepressureheaddistribu-tionbecomesnon-linearinaveryearlystageoftheexperiment.Inthewetrange(closetoKs),thedeviationsappeargreater,whichiscausedbysmallergradientsinthatstage,andthuslargerrelativeerrorsinthegradient,asout-linedabove. Uncertaintyofestimatedhydraulicfunctionsduetorandommeasurementerrors Weexpectedthatnoiseinthemeasurementswillincreasetheuncertaintyofparameterestimates,butwillnotinduceasystematicbiasintheresults.Thiswasconfirmedbyour 156 43log10 (K in cm d–1)210–1–2–3–4A.Peters,W.Durner tensup +1tensup corr.tensup –1true func.21log10 (K in cm d–1)0–1–2–3–4–5–600.511.5pF [–]22.5332log10 (K in cm d–1)10–1–2–3–400.511.5pF [–]22.53tensup +1tensup corr.tensup –1true func.00.511.5pF [–]22.530–0.5log10 (K in cm d–1)–1–1.5–2–2.5–3–3.5–4tensup +1tensup corr.tensup –1true func.tensup +1tensup corr.tensup –1true func.00.511.5pF [–]22.53Figure5Estimatedhydraulicconductivityfunctionsforthefoursoilswithdifferentoffsetsinuppertensiometerreadings(+10mm,correctandÀ10mm).Firstrow–left:sand,right:loam;secondrow–left:clay,right:structuredsoil.MonteCarloanalysis.Fig.7showstheresultsofthe500MonteCarlosimulationswithsyntheticmeasurementsthatweredisturbedwithindependentnormaldistributedmea-surementerrors.Thetemporalresolutionoftheunderlyingdatawas0.1d.ThefigureshowsforallsoilmaterialsthatthehðhÞdataareidentifiedwithsmalluncertainty,whereastheerrorsareslightlymorepronouncedintheKðhÞdata,inparticularatthewetanddryendofthedatarange.Thecor-respondingfittedfunctionsareremarkablyclosetothetruefunctions(shownasredlines),evenintheextrapolationrangeoftheconductivityfunction.Obviously,thesmalluncertaintiesoftheretentionfunctionsstabilizetheoverallfit,andthusreducetheuncertaintiesintheestimationoftheKðhÞfunctions.Summarizing,allfunctionsarewelliden-tifiedinthemeasuredmoisturerange. Table3quantifiestheaccuracyoftheparameteridenti-ficationbylistingthemeansandstandarddeviationsofthediagnosticvariablesthatdescribetheerrorsoftheidenti-fiedfunctions.Despiteagoodoverallidentificationofthehydraulicfunctions,estimationofsingleparametersmaystillbeuncertainandbiased,inparticulariftheyarehighlycorrelatedorinsensitivetotheavailablemeasureddata(IdenandDurner,2007).ThisisshowninFig.8.Thecirclesindicatethemeanrelativeparametervalueandtheerrorbarsindicatethe95%confidenceinterval,calculatedfromthemarginaldistributionsofthe500estimatedparameters.Sincetheevaporationmethodyieldsreliableestimatesoftheretentioncurveinthewetrange,uptopF=3,the estimationoftheparametershs,nanda,whichdeterminetheshapeofhðhÞinthismoisturerange,wasverygoodforallsoils.Ontheotherhandthemodelfitwasquiteinsen-sitivetotheparameters,whichleadstoacorrespondinglylargeuncertaintyofthisparameterestimate.Fortheclayandthebimodalsoiltheestimationofhrwasalsoveryuncertain,becausewithinthemeasurementrangeðhPÀ1000cmÞ,wedonotgetinformationforthedryendofthehydraulicfunctions.Thus,theexperimentalsetupdoesnotprovidesufficientinformationabouttheparametersthatgoverntheshapeofthefunctionsinthedryrange.Thisisalsoevidentbythegreateruncertaintiesoftheaandnparametersofthestructuredsoil. Analysisofevaporationmeasurementsofrealsoils Forthesyntheticdatathetruemodelandmodelparame-tersareknownandtherearenolimitationsinthenumberof‘‘measured’’data.Forrealmeasurementsneitherthetruemodel,northeparametersareknownandweusuallyhavealimitedamountofdata. Fig.9showstheresultsoftheanalysisofthetworealsoils.Thedataareapproximatelyequallydistributedonthelog10ðhÞ-scale,pffiffithusthedataselectionprocedure(equi-distantonthet-axis)seemstobeanappropriatechoice.Aspointedoutinthesensitivitystudytheestimationoftheretentionfunctionwasnotverysensitivetomeasure- Simplifiedevaporationmethodfordeterminingsoilhydraulicproperties 0.40.38log10 (K in cm d–1)0–0.5157 vol. water content [–]–1–1.5–2–2.5–3–3.5tensup hightensup corr.tensup lowtrue func.00.511.5pF [–]22.530.360.340.320.3tensup hightensup corr.tensup lowtrue func.00.511.5pF [–]22.53–40.43tensup hightensup corr.tensup lowtrue func.2log10 (K in cm d–1)10–1–2–3vol. water content [–]tensup hightensup corr.tensup lowtrue func.0.350.30.2500.511.5pF [–]22.53–400.511.5pF [–]22.53Figure6Estimatedhydraulicpropertiesfortheclay(firstrow)andthestructuredsoil(secondrow)withdifferentoffsetsinuppertensiometerinstallation(4.2mmhigh,correctand4.2mmlow).Leftcolumn:retentionfunctions;rightcolumn:hydraulicconductivityfunctions.menterrors.Thisresultcanalsobeseenintheevaluationofthetwosoils,wheretheretentiondataarerathersmoothandwelldescribedbytheunimodalvanGenuchtenfunc-tion.However,thesmallbutsystematicmisfitofthesandyloamdata(Fig.9,top)indicatesthatthevanGenuchtenmodelcannotperfectlydescribethissoil. Again,themorecrucialpartwastheestimationofthehydraulicconductivityfunction.Duetolowgradientsinthebeginningoftheexperiment,conductivitydataforthesandyloamwhereonlyavailableforpressureheads<À100cm.Fortheclay,thegradientincreasedalreadyatlowpressureheadsandthusconductivitydatawereavail-ableathigherpressureheads(<À60cm).TheconductivitydataofthesandyloamcouldbewelldescribedwiththevanGenuchten/Mualemmodel,whereastheconductivitydataoftheclayshowdeviationsinthewetandinthedryrange,indicatingeitherabiasinthemeasurements,i.e.anunderestimationofthehydraulicgradientinthewetrange(seeabove)orawrongmodelassumption.Thelatterisobservedfrequentlyinthedryrangeoftheconductivityfunction,wherethecapillarybundlemodelssuchastheoneofMualem(1976)donotwelldescribetheflowbehav-ior.Onepossibleexplanationforthisisthenegligenceofthefilmflowleadingtoanunderestimationoftheconduc-tivityinthedryrange(TullerandOr,2001). Theindependentlymeasuredsaturatedhydrauliccon-ductivity,Ks,was%1300cmdÀ1forthesandyloam,thatisabouttwoordersofmagnitudehigherthanestimated. Thisreflectstheuncertaintyoftheparametricextrapola-tionofKðhÞinthewetrange.FortheclaynoindependentKsmeasurementwasavailable. Althoughthemodelfitsarenotperfectthelargestdevi-ationontheKðhÞ-curveishalfanorderofmagnitude,andthusinmostapplicationsstilljustifiable.Neverthelessthemeasurementresultsshowthatthemodelstructure(espe-ciallytheKðhÞ-model)needstobeimprovedtobetterde-scribethenaturalsoils. Discussion Quiteremarkably,ouranalysisoftheSchindlerevaporationmethodshowedinnocasethatthespatiallinearizationassumptionatthelatestageoftheexperimentcausessig-nificanterrors(Fig.3).Incontrast,thenon-linearityoftheretentionfunctionturnedouttoyieldsmallbutsystem-aticerrorsatthebeginningoftheexperiment.Thesearenon-negligibleforcoarsesoilsorsoilswithastructuralporesystem,inparticular,iftheretentiondataareusedinaparameterestimationproceduretoderivesimultaneouslythecoupledretentionandconductivityfunction.Wecouldshowthatthiserrorcanbefullyavoidedbyusingtheinte-gralapproachofPetersandDurner(2006)intheparameterestimation.Intheearlyphaseofevaporationexperiments,themostseverehandicapistheuncertaintyindeterminingthehydraulicgradient.Sincethisuncertaintyisinversely 158A.Peters,W.Durner Figure7MonteCarlosimulationsandparameterestimationsforthebimodalsoil.Errorintensiometerreading,rh,was0.2cmanderrorinweightreading,rw,was0.02g.Everycasewasrealized500times.Firstrow:sand;secondrow:loam;thirdrow:clay;fourthrow:structuredsoil. Simplifiedevaporationmethodfordeterminingsoilhydraulicproperties Table3DistributionofdiagnosticvariablescalculatedfromEqs.(9)–(12)fortheMonteCarlosimulationsCasel SoilSLCBISLCBI Dhmax0.00260.00340.00060.00650.00120.00020.00010.0016 Dlog10ðKmaxÞ0.400.04330.14790.23570.21970.02180.01620.0945 RMSEh0.00140.00150.00040.00260.00060.00010.00000.0007 RMSElog10ðKÞ0.20140.03180.06590.12820.09650.01550.00930.0412 159 r listhearithmeticmeanandristhestandarddeviationofthediagnosticvariable. proportionaltothedistancebetweentensiometers,andtheWindevaporationmethodrequiressmallerdistancesbe-tweentensiometersforacertaincolumnheight,wecon-cludethattheSchindlerevaporationmethodissuperiortoWind’smethodinthewetmoisturerange.BybasingtherejectioncriterionforKðhÞdatainthewetrangeontheer-rorinthehydraulicgradientweintroducedanapplicablefil-terforreliabledataranges. Errorsinthecalibrationoftensiometersleadtosomeinterestingeffects.Anegativeoffsetoftheuppertensiom-eterorapositiveoffsetofthelowertensiometerwillcauseasignificantunderestimationoftheconductivityclosetosaturation.Theoppositeoffsets,however,donotcauseabiasbutonlyleadtoalackofinformationinthatpressureheadrange,duetotherejectioncriterionbasedonthemin-imummagnitudeofthehydraulicgradient.Anoffsetinthetensiometerinstallationhasprincipallythesameeffectastheoffsetinthecalibrationforthewetrange.Sincetheinstallationerrorinproperlyperformedexperimentsisonlyintherangeofmillimeters,whereasanoffsetintensiome-tercalibrationcaneasilyreachoneormorecentimeters,theinstallationerrorhasmuchlesssevereimpactsinthewetmoisturerange.Inthedryrange,wefoundthataninstallationerrorcanalsocausesystematicerrorsinbothhydraulicfunctionsduetothenon-linearityofthepressureheaddistribution. Theresultsforthelinearizationerrorsintimeindicatethatthemeasuringintervalforthetensiometerreadingsshouldbesmallð60:1dÞforfinetexturedmaterials.Thissuggeststhattensiometerreadingsshouldberecordedbyautomaticdatalogging.Sampleweights,ontheotherhand,canberecordedmanuallyatverylargetimeintervals,be-causetheweightchangeduringthefirststageofevapora-tionislinearandthesecondstageofevaporationisnot 22rel. parameter value [–]1.5rel. parameter value [–]1.5110.50.50α2nθrθsτKs0α2nθrθsτKsrel. parameter value [–]1.5rel. parameter value [–]o.o.R.1.5110.50.50αnθrθsτKs0α1n1θrθsτKsα2n2w2Figure8Relativeestimatedparametervaluesandtheir95%confidenceinterval.Graydashedline:truerelativeparametervalue;circles:meanofestimatedrelativeparameter;bars:95%confidenceintervalofestimatedrelativeparameterdistribution.(A)Sand;(B)loam;(C)clay;(D)bimodalsoil.160 0.40.35vol. water content [–]A.Peters,W.Durner 21log10 (K in cm d–1)0–1–2–3–40.30.250.20.150.10.05000.511.5pF [–]22.5300.511.5pF [–]22.5330.55vol. water content [–]2log10 (K in cm d–1)10–1–2–3–400.511.5pF [–]22.530.50.450.40.350.30.250.2–500.511.5pF [–]22.53Figure9Outputofthesimplifiedevaporationmethodandestimatedhydraulicpropertiesforthesandyloam(firstrow)andtheclay(secondrow).reachedforthesematerials(KutilekandNielsen,1994).Incontrasttothat,coarsematerialsdoreachthesecondstageofevaporation,andthusthesampleweightsdecreasenon-linearwithtime.However,evenforacoarsesandamea-surementresolutionof2dwassufficient.Thus,thelineari-zationerrorswithrespecttotimearenegligibleifaproperselectionoftimeintervalsandinterpolationofthemea-surementsischosen. Ourfinalanalysisofnoisymeasurementdatathatwerefittedwithparametricmodelsshowedthatmostoftheretentioncurveandconductivitycurveparameterscanbeestimatedwithsmalluncertainties.Eveniffinermaterialsareinvestigated,wherethedryendoftheretentionfunc-tionisnotreachedinthecourseoftheexperiment,onlytheestimationofhrbecomesuncertain.Duetolackofcon-ductivitydataclosetosaturation,theparametersthatde-scribetheKðhÞfunctioncanonlybeestimatedbytrustingthecouplingoftheretentionandconductivitymodel.Inourinvestigatedcases,thesmalluncertaintiesinthereten-tionfunctionleadtorelativelysmallerrorsinthefitsoftheconductivityfunction. Weliketostressatthispointthatinallcasesofouranal-ysis,weassumedthattheunderlyingretentionandconduc-tivitymodelswereexactlyknownandthusthecouplingofbothfunctionsindeedcontributedtoareliableparameterestimation.Forrealsoils,thisassumptionwillneverbeper-fectlymet,andtheanalysisoftworealdatasetsconfirmedthatinparticulartheextrapolationofestimatedconductiv-itycurvestowardsaturationmustbeinterpretedwithcare. Conclusions Acomprehensiveerroranalysisofthesimplifiedevapora-tionmethodshowedthatitisafast,accurateandreliablemethodtodeterminesoilhydraulicpropertiesinthemea-suredpressureheadrange.Themethodreliesonfourdif-ferentlinearizationassumptionsintimeandspace.Byusingtheintegralapproachforparameterestimationweeliminatedoneoftheselinearizationerrors,whichim-provestheestimationofhydraulicpropertiesofcoarsesoilsorsoilswithasecondaryporesystem.Theremainingthreelinearizationassumptionsareofminoreffectfortheestimationoftheretentionfunctionforsoilsofanytex-ture,ifthemeasurementintervalforthepressureheadsisreasonableshortandasuitabledatainterpolation,suchascubichermitesplines,isused.Thedeterminationofhydraulicconductivitydataandthesubsequentestimationoftheconductivityfunctionwasalsoveryreliableinthedryrange.However,withtheevaporationmethodwedonotgetdatafortheKðhÞ-functioninthewetrange,sothatatleastanadditionalmeasurementofKsshouldbeprovided.Also,theshapeoftheconductivityfunctioninthewetrangemustbeinterpretedwithcare.Toimprovethecharacterizationoftheconductivityfunctionnearsat-uration,alternativemethods,suchasmulti-stepoutflowmethodsincombinationwithfree-formparameterestima-tionwillberequired. Inadditiontotheinvestigationofthelinearizationer-rors,weinvestigatederrorsourcesintroducedbyoffsets Simplifiedevaporationmethodfordeterminingsoilhydraulicproperties161 inthetensiometercalibrations,offsetsintensiometerinstallationorrandomerrorinthemeasurementreadings.Altogether,theerrorsintheestimateddatawereremark-ablysmall,inparticularfortheretentioncurve.Tensiome-tercalibrationerrorsleadincoarsematerialstoerroneousestimationsofthehydraulicfunctionsinthewetrange.Especially,theestimationoftheconductivityfunctionclosetosaturationbecomesseverelybiased.Errorsduetowrongtensiometerpositionarelesssevere.Thus,themostcrucialpartofapplyingthemethodistheuseofreliabletensio-meters. Acknowledgements WethankBudimanMinasny(UniversityofSydney)forpro-vidingdataofevaporationexperimentsfortwosoils,andGeorgvonUnold(UMSGmbH–UmweltanalytischeMess-Systeme)andSaschaIdenforfruitfuldiscussions. References Bertuzzi,P.,Mohrath,D.,Bruckler,L.,Gaudu,J.C.,Bourlet,M.,1999.Wind’sevaporationmethod:experimentalequipmentanderroranalysis.In:vanGenuchten,M.T.,Leij,F.J.,Wu,L.(Eds.),ProceedingsoftheInternationalWorkshoponCharacterizationandMeasurementoftheHydraulicPropertiesofUnsaturatedPorousMedia,UniversityofCalifornia,Riverside,CA,pp.323–328. Bitterlich,S.,Durner,W.,Iden,S.C.,Knabner,P.,2004.Inverseestimationoftheunsaturatedsoilhydraulicpropertiesfromcolumnoutflowexperimentsusingfree-formparameterizations.VadoseZoneJ.3,971–981. Boels,D.,Gils,J.B.H.M.v.,Veerman,G.J.,Wit,K.E.,1978.Theoryandsystemofautomaticdeterminationofsoilmoisturechar-acteristicsandunsaturatedhydraulicconductivities.SoilSci.126(4),191–199. Bouma,J.,Belmans,C.,Dekker,L.W.,Jeurissen,W.J.M.,1983.Assessingthesuitabilityofsoilswithmacroporesforsubsurfaceliquidwastedisposal.J.Environ.Qual.12(3),305–311. Coppola,A.,2000.Unimodalandbimodaldescriptionofhydraulicpropertiesforaggregatedsoils.SoilSci.Soc.Am.,1252–1262.Dane,J.,Hopmans,J.,2002.Waterretentionandstorage/labora-tory.In:Dane,J.,Topp,G.(Eds.),MethodsofSoilAnalysis.Part4:PhysicalMethods.SSSABookSeries:5.SoilScienceSocietyofAmericaInc.,pp.675–720. Dirksen,C.,1991.Unsaturatedhydraulicconductivity.In:Smith,K.A.,Mullins,C.E.,Dekker,M.(Eds.),SoilAnalysisPhysicalMethods.NewYork,pp.209–270. Duan,Q.,Sorooshian,S.,Gupta,V.,1992.Effectiveandefficientglobaloptimizationforconceptualrainfall-runoffmodels.WaterResour.Res.28,1015–1031. Durner,W.,1994.Hydraulicconductivityestimationforsoilswithheterogeneousporestructure.WaterResour.Res.30,211–223.Durner,W.,Lipsius,K.,2005.Chapter75:determiningsoilhydraulicproperties.In:Anderson,M.,McDonnell,J.J.(Eds.),EncyclopediaofHydrologicalSciences.JohnWiley&SonsLtd.,pp.1021–1144.Durner,W.,Schultze,B.,Zurmu¨hl,T.,1999.State-of-the-artininversemodelingofinflow/outflowexperiments.In:vanGenuchten,M.T.,Leij,F.J.,Wu,L.(Eds.),ProceedingsoftheInternationalWorkshoponCharacterizationandMeasurementoftheHydraulicPropertiesofUnsaturatedPorousMedia,UniversityofCalifornia,Riverside,CA,pp.661–681. Fritsch,F.N.,Carlson,R.E.,1980.Monotonepiecewisecubicinterpolation.SIAMJ.Numer.Anal.17,238–246. Halbertsma,J.,Veerman,G.,1994.Anewcalculationprocedureandsimpleset-upfortheevaporationmethodtodeterminesoilhydraulicfunctions.Tech.Rep.,DLOWinandStaringCentre,Wageningen,TheNetherlands. Hoffmann-Riem,H.,vanGenuchten,M.T.,Flu¨hler,H.,1999.Generalmodelforthehydraulicconductivityofunsaturatedsoils.In:vanGenuchten,M.T.,Leij,F.J.,Wu,L.(Eds.),Pro-ceedingsoftheInternationalWorkshoponCharacterizationandMeasurementoftheHydraulicPropertiesofUnsaturatedPorousMedia,UniversityofCalifornia,Riverside,CA,pp.31–42. Hopmans,J.W.,Simunek,J.,Romano,N.,Durner,W.,2002.Simultaneousdeterminationofwatertransmissionandretentionproperties.Inversemethods.In:Dane,J.,Topp,G.(Eds.),MethodsofSoilAnalysis.Part4:PhysicalMethods.SoilScienceSocietyofAmericaInc.,pp.963–1004. Iden,S.,Durner,W.,2007.Free-formestimationoftheunsaturatedsoilhydraulicpropertiesbyinversemodelingusingglobaloptimization.WaterResour.Res.43,w07451.doi:10.1029/2006WR005845. Kutilek,M.,Nielsen,D.,1994.SoilHydrology.CatenaVerlagGeoEcologyPublications,Cremlingen-Destedt,pp.188–1.Lambot,S.,Javaux,M.,Hupet,F.,Vanclooster,M.,2002.Aglobalmultilevelcoordinatesearchprocedureforestimatingtheunsaturatedsoilhydraulicproperties.WaterResour.Res.38(11),1224.doi:10.1029/2001WR001224. Mallants,D.,Tseng,P.H.,Toride,N.,Timmerman,A.,Feyen,J.,1997.Evaluationofmultimodalhydraulicfunctionsincharac-terizingaheterogeneousfieldsoil.J.Hydrol.195,172–199.Marquardt,D.,1963.Analgorithmforleast-squaresestimationofnonlinearparameters.SIAMJ.Appl.Math.11,431–441. Minasny,B.,Field,D.,2004.Estimatingsoilhydraulicpropertiesandtheiruncertainty:theuseofstochasticsimulationintheinversemodellingoftheevaporationexperiment.Geoderma126,277–290. Mohrath,D.,Bruckler,L.,Bertuzzi,P.,Gaudu,J.C.,Bourlet,M.,1997.Erroranalysisofanevaporationmethodfordetermininghydrodynamicpropertiesinunsaturatedsoil.SoilSci.Soc.Am.J.61,725–735. Mualem,Y.,1976.Anewmodelforpredictingthehydraulicconductivityofunsaturatedporousmedia.WaterResour.Res.12(3),513–521.Othmer,H.,Diekkru¨ger,B.,Kutilek,M.,1991.Bimodalporosityandunsaturatedhydraulicconductivity.SoilSci.152(3),139–149.Peters,A.,Durner,W.,2006.Improvedestimationofsoilwaterretentioncharacteristicsfromhydrostaticcolumnexperiments.WaterResour.Res.42,w11401.doi:10.1029/2006WR004952.Plagge,R.,Haeupl,P.,Renger,M.,1999.Transienteffectsonthehydraulicpropertiesofporousmedia.In:vanGenuchten,M.T.,Leij,F.J.,Wu,L.(Eds.),ProceedingsoftheInternationalWorkshoponCharacterizationandMeasurementoftheHydrau-licPropertiesofUnsaturatedPorousMedia,UniversityofCalifornia,Riverside,CA,pp.905–912. Priesack,E.,Durner,W.,2006.Closedformexpressionforthemulti-modalunsaturatedconductivityfunction.VadoseZoneJ.5,121–124. Romano,N.,Santini,A.,1999.Determiningsoilhydraulicfunctionsfromevaporationexperimentsbyaparameterestimationapproach:experimentalverificationsandnumericalstudies.WaterResour.Res.35,3343–3359. Schaap,M.G.,Leij,F.J.,2000.ImprovedpredictionofunsaturatedhydraulicconductivitywiththeMualem–vanGenuchtenmodel.SoilSci.Soc.Am.J.,843–851. Schaap,M.G.,Leij,F.J.,vanGenuchten,M.T.,1998.Neuralnetworkanalysisforhierarchicalpredictionofsoilhydraulicproperties.SoilSci.Soc.Am.J.62,847–855. Schindler,U.,1980.EinSchnellverfahrenzurMessungderWasser-leitfa¨higkeitimteilgesa¨ttigtenBodenanStechzylinderproben.Arch.Acker-u.Pflanzenbauu.Bodenkd.Berlin24,1–7. 162 Schindler,U.,Mu¨ller,L.,2006.Simplifyingtheevaporationmethodforquantifyingsoilhydraulicproperties.J.PlantNutr.SoilSci.169,623–629.doi:10.1002/jpln.2005215. Simunek,J.,Hopmans,J.,2002.Parameteroptimizationandnonlinearfitting.In:Dane,J.H.,Topp,G.C.(Eds.),MethodsofSoilAnalysis.Part4:PhysicalMethods,vol.5.SoilScienceSocietyofAmericaBook,Madison,WI,pp.139–157. Simunek,J.,Wendroth,O.,vanGenuchten,M.T.,1998a.Parameterestimationanalysisoftheevaporationmethodfordeterminingsoilhydraulicproperties.SoilSci.Soc.Am.J.62,4–905. Simunek,L.,Huang,K.,Sejna,M.,vanGenuchten,M.T.,1998b.Thehydrus-1dsoftwarepackageforsimulatingtheone-dimen-sionalmovementofwater,heatandmultiplesolutesinvariably-saturatedmedia,version2.02. Spohrer,K.,Herrmann,L.,Ingwersen,J.,Stahr,K.,2005.Appli-cabilityofuni-andbimodalretentionfunctionsforwaterflowmodelinginatropicalacrisol.VadoseZoneJ.5,48–58. Tamari,S.,Bruckler,L.,Halbertsma,J.,Chadoeuf,J.,1993.Asimplemethodfordeterminingsoilhydraulicpropertiesinthelaboratory.SoilSci.Soc.Am.J.57,2–651. A.Peters,W.Durner Tuller,M.,Or,D.,2001.Hydraulicconductivityofvariablysaturatedporousmedia:filmandcornerflowinangularporespace.WaterResour.Res.31(5),1257–1276. vanGenuchten,M.T.,1980.Aclosed-formequationforpredictingthehydraulicconductivityofunsaturatedsoils.SoilSci.Soc.Am.J.44,2–8. vanGenuchten,M.T.,Leij,F.J.,Wu,L.,1999.Characterizationandmeasurementofthehydraulicpropertiesofunsaturatedporousmedia.In:vanGenuchten,M.T.,Leij,F.J.,Wu,L.(Eds.),ProceedingsoftheInternationalWorkshoponCharacterizationandMeasurementoftheHydraulicPropertiesofUnsaturatedPorousMedia,UniversityofCalifornia,Riverside,CA,pp.1–12.Wendroth,O.,Ehlers,W.,Hopmans,J.W.,Klage,H.,Halbertsma,J.,Wo¨sten,J.H.M.,1993.Reevaluationoftheevaporationmethodfordetermininghydraulicfunctionsinunsaturatedsoils.SoilSci.Soc.Am.J.57,1436–1443. Wind,G.P.,1968.Capillaryconductivitydataestimatedbyasimplemethod.In:Rijtema,P.E.,Wassink,H.(Eds.),WaterintheUnsaturatedZone,vol.1.ProceedingsoftheWageningenSymposium,19–23June1966.Int.Assoc.Sci.Hydrol.Publ.(IASH),Gentbrugge,TheNetherlandsandUNESCO,Paris.
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