Data-basedMechanistic Modelling of Rainfall-RunoffProcessesand Its Application in a ComplexHydr ological

Context

K. Bogner�, B. Hingray

�andA. Musy

�

�SwissFederalInstituteof Technology- Lausanne,

ENAC-HYDRAM, CH-1015Lausanne,Switzerland(konrad.bogner@epfl.ch)

Abstract: Although the inherentuncertaintyassociatedwith rainfall-runoff processesis well known, mostmathematicalmodelsof suchsystemsarecompletelydeterministicin nature. Stochasticmodellingrequiresthat the uncertainty, which is associatedwith both the model parametersand the stochasticinputs, shouldbe quantifiedin somemannerasan inherentpart of the modellinganalysis. To achieve theseobjectives,aData-basedmechanistic(DBM) modellingapproachwill be testedfor theJuralake system(Switzerland).InDBM modelling,the mostparsimoniousmodelstructureis first inferredstatisticallyfrom the availabletimeseries.Statedependentnon-lineardependenciescanbeidentifiedobjectively from therainfall andrunoff dataandwill be usedasthe basesfor the estimationof non-lineartransferfunction modelsof the rainfall-runoffprocesses.After this themodelwill beacceptedif it canbeinterpretedin aphysicallymeaningful,mechanisticmanner. Beforethis approachwill be appliedsomepreprocessingof the datahasbeendoneusingWavelettransformations.Furthermorea simplifiedsnow-melt modelhasbeenappliedin orderto calculateequivalentrainfall. First resultsof this preprocessingandof theDBM modellingwill beshown in this paper.

Keywords: Data-basedmechanisticmodelling;Transferfunctions;Snowmeltmodelling;Wavelettransforma-tion.

1 INTRODUCTION

Rainfall-runoff models are well-establishedtoolsthatarewidely utilized in engineeringpractice,e.g.for waterresourceplanning.Themajorityof modelstructurescurrently usedcan be classifiedas con-ceptual. Usually thesekind of modelshave a lotof parameters,which cannotbe estimatedwithoutimposing prior restrictions. RecentlyBeven andFreer[2001] have describedtheseproblemsusingthe termequifinality. This meansthatmany differ-entparametersetsandmodelstructuresexist,whichareableto explain the observed dataequallywell,so that no unique solution can be obtained. Thedata-basedmodelling methodologyof Young andBeven[1994] triesto solve theproblemsof identifi-cation and over-parametrisationof conceptualhy-drological modelsby the use of simple and par-simoniousmechanisticmodelstructures.Thusnoprior assumptionsaremadeaboutthe form of themodelother thana generallinear transferfunctionapproachcanbe usedto relatean effective rainfall

input to thetotal discharge.This model will be appliedto a very large (about8000km

�) andcomplex catchmentin Switzerland.

The overall objective will be to study the impactsof climate variability and changeon the sustain-ableuseof waterwithin a projectcalledSWURVE(SustainableWater: Uncertainty, Risk andVulner-abiliy in Europe)fundedby the EU andthe SwissFederalOffice of Educationand Sciences. Theaim of SWURVE is theelaborationof a probabilis-tic framework to assessclimatechangeimpactsonthe sustainableuseof water. This methodologicalframework takesinto accountthe combineduncer-taintydueto thenaturalvariability andtheerrordueto incompleteknowledgeof futureconditions.TheJuralake systemis a complex hydrologicalsystemlocatedin the westernpart of Swissplain, whichwill be analysedin detail. This systemof threein-terconnectedlakes(Neuch̀atel, BienneandMorat)wasformed15,000yearago,whenthehugeRhoneglacierhadretired. It wassignificantlymodifiedin

381

Bern

Hagneck

Brienzwiler

0�

50�

KILOMETERS�

Lake Geneva�

Lake Neuchatel�

Lake Bienne

Lake Morat�

Figure1: DEM of theJuralakesystem( c�

SwissfederalOfficeof Topography)

the periods1833-1884and1963-1972by the con-truction of channels. Different regulation water-works have beenbuilt, in orderto reducethe risksof inundationand to transformthe former unpro-ductiveareasinto fertile arableland.In thebeginningof thisprojectthehydrologicalsys-tem hasto be analysedfor the presenttimesandarainfall-runoff modelwill beappliedin orderto cal-culatefuturescenariossubsequently. Themostim-portantinflow to the Juralake systemcomesfromthe Aare River (approximately70%) and it drainsan areaof 5128 km

�up to the station Hagneck,

wheretheriverflows in thelakeof Bienne(seeFig-ure 1). The hydrologicalsystemis influencedbyvariouscomponentslike glaciers,artificial andreg-ulatednaturallakes,whichmakeit difficult to applya more physically basedmodel. Furthermoretheessentialcontribution of the snow-melt have to betaken into consideration.Thussomepreprocessingof the dataarenecessary, which will be explainedin thenext chapter. AfterwardstheDBM modellingapproachwill be describedin more detail and thepreliminaryresultswill beshown anddiscussed.

2 PREPROCESSINGTHE DATA

Theobserveddischargeseriesof theAareriver aremostly superposedandalteredby variousdifferentimpacts(e.g. hydropower stationmanagementandregulation of lakes), which can be interpretedasnoise. For the hydrological modelling it will be

necessaryto remove this noiseandto selectrunoffgauges,wherethedischargeseriesaremoreor lessnoise-free. The WaveShrink methodologydevel-opedby Donoho[1995] for estimatinganunknownsignal from data has beenapplied to remove thenoisefrom the discharge series. The advantageofthis procedure,which is basedon Wavelet transfor-mations,is that the peaksare preserved, whereastraditionalnoisereductionmethods,suchassplinesor kernelsmootherswould result in somesmooth-ing of the spikes. In particular, the non-decimateddiscreteWavelet transformationhas beenusedinshrinkingthe Wavelet coefficient towardszero,be-causeit leadsto both betterpredictionand fewerartifacts(e.g. Nasonand Sapatinas[2002]). Ap-plying the WaveShrinkalgorithmto threedifferentdischargeseriesobservedatgaugesat theAareriver(shown in Figure2) indicatesthatthegaugelocatedat Bernwill bebestsuitedfor furtheranalysis.Theresiduals(=noise)of this dataseriesaremuchlessin comparisonto the seriesobserved at Hagneck(downstream)and Brienzwiler (upstream),whichare highly influencedby hydro power production.Thusthecatchmentof Bern is chosenrepresentingapproximately2/3of theAarecatchmentarea(2969km�).

To improve modelling capabilities,rainfall seriesaremodifiedhereto accountfor snowfall andsnow-melt, producingseriesof daily equivalent rainfall,which will be the input for rainfall-runoff mod-elling. Theprocessalsoaccountsfor theorographic

382

0 200 600 1000 1400

Resid

Signal

Data

Brienzwiler

0 200 600 1000 1400

Resid

Signal

Data

Bern

0 200 600 1000 1400

Resid

Signal

Data

Hagneck

Figure2: Signalandnoiseseparationof daily dis-charge series(1996-1999)by the use of Wavelettransformations

enhancementof rainfall, calculatingthe equivalentrainfall for different elevation bandsbefore aver-aging theseseries for the whole catchment(seeFigure3). The degree-dayapproachto snow-meltmodelling provides a simple methodologyfor thesatisfactory calculation of snow-melt, relying ondaily averageair temperatureto representthe ma-jor heat fluxes in operation. As found in an in-ternationalcomparisonof snow-melt runoff mod-els WMO [1986], the degree-daymethodhas anaccuracy comparableto morecomplex energy bud-getformulations(RangoandMartinec[1995]). Themostwidely useddegree-daymodelwasdevelopedby Martinecetal. [1983]andis centeredon thefor-mula

��� �����������(1)

where�

is daily snow-meltrunoff (mm),�

is dailymeantemperature( ˚ C),

���is a basetemperature

(usually0 ˚ C), and

is thedegree-dayor melt fac-tor (mm/˚ C/day).As well astheability to optimiseboth

� �andthedegree-dayfactor

, thereis poten-

tial for modificationof thebasicformulato includefactorssuchasalbedo(for examplebasedontheageof thesnowpack).With the availability of detailedtopographicdatafrom Digital Elevation Models(DEMs), the possi-bility of consideringseparateelevationbandswithina catchmentalsoprovidesopportunityfor improve-ment. The main difficulty encounteredwith the

Grimsel

Interlaken

Guttannen

Jungfraujoch

no data

500-800

801-1500

1501-2500

>2501

Figure 3: Elevation bandsof the Bern catchmentand locationsof the temperaturerecordinggauges( c�

SwissfederalOfficeof Topography)

degree-daymethodis the variability of the degree-dayfactor. Work by Schreideret al. [1997] showedthe ability of a degree-daytype model, used to-getherwith IHACRES (Jakemanand Hornberger[1993]), to modelsnow-affectedcatchmentsin theAustralianalpine region to the sameefficiency assnow-freebasins.Althoughthemodelof Schreideret al. [1997] was initially developedfor usein thesouthernhemisphere,its successwith IHACRES,degree-daymethodologybasis,andincorporationoftopographicdatamadeit a suitablestartingpointin the snowmelt modellingprocess.Its relianceondaily temperatureandprecipitationdatamake it ex-tremelyusefulfor modellingsnow processesin re-gionswith a lack of regular snow observations,orhistoricalperiodswith limited data.Thereareseveralpartsof themodelwhichhasbeenalteredby Steel[1999]andhavebeenappliedin thisstudy. Somechangeswerestructuralsuchaschang-ing thedatesusedfor thestartandendof thesnowseason(and albedofactor) to be suitable for thenorthernhemisphere,and changesto the grid cellsystem.Otherchangesrequiredparametervalueop-timisation,notablywherevalueshadbeenbasedonempiricalobservations(for examplethedegree-dayfactorusedandalbedofactor).Thedaydegreefac-tor hasbeenestimatedby MonteCarlosimulationsusingdifferentvaluesfor � rangingfrom 0.3 to 10.Theresultof thecalculationsof theequivalentrain-fall is shown in Figure4 with ������� � .

383

0 500 1000 15000

10

20

30

40

50

60

70

Time [d]

Pre

cip

ita

tio

n [

mm

/d]

Observed rainfall at Bern (1996−1999)

0 500 1000 15000

10

20

30

40

50

60

Time [d]

Ra

in−

Sn

ow

[m

m/d

]

Equivalent rainfall

Figure 4: Comparisonof observed andcalculatedequivalentdaily rainfall (1996-1999)

3 DATA-BASED MECHANISTIC (DBM)MODELLING

The DBM philosophy (Young and Lees [1994])and its application are discussedfully in Young[1993], Young and Beven [1994], Young et al.[1997], Young [1998], Beven [2001] and the ref-erencestherein. The first stepof the DBM mod-elling approachinvolvesthestatisticalidentificationandestimationof a modelrepresentingthestochas-tic, dynamicsystem. The prior assumptionsaboutthe structureare kept at a minimum. The generalDBM modelfor a singleinput (e.g. effective rain-fall) takestheform:

"! �$# �&%('*)+�, ��% ' ) �.- ! '�/1032 ! (2)

wherethetransferfunctionpolynomialsaredefinedas:

, �&% '*) �4� 5 036() % ' ) 08797:7�0;6.< % ' )# �&% '*) �4� = � 0 = ) % ' ) 0879797>0 =+?@% ' ) (3)

and%

is a backward shift operator(i.e.%�'�A "! � "! '�A ). �CBD� is the measuredrunoff; - �CBD� is the ob-

served input, which is in our casethe equivalentrainfall; E is a time delay and 2 ! is the stochasticnoise.Theorderof theassociatedtransferfunctionmodelis definedby thetriad F G1HJIKHLE�M . Theresiduals

2 ! aredefinedasfollows,

N2 ! � ! �N# ��%�' )+�N, �&% '*) � - ! (4)

whereN, ��%�' )O�

andN# ��%�' )+� aretheestimatesin (3)

and representmeasurementnoiseandunmeasuredeffects.

3.1 Model structur e identification

Thefollowing identificationandoptimisationmeth-ods are implementedin the CAPTAIN Matlabc

�Toolbox1. More informationaboutthis toolboxcanbefoundathttp://www.es.lancs.ac.uk/cres/captain.Appropriate model structures identification, i.e.finding the bestvaluesof F G1HDI�HJE�M and the corre-spondingcoefficients,will bedoneby a processofobjective statisticalinference. Thus in a first stepthe input-outputdatawill be analysedand the pa-rametersin a constantparametertransferfunctionmodel will be estimatedbasedon optimal Instru-mentalVariable(IV) algorithm(seeYoung[1984]andYoungandBeven[1994]). Thiswill oftenresultin a first or secondordermodel(seeFigure5). Theevaluationof the identifiedmodelstructurecanbedoneby applyingdifferentcriteriafor thegoodnessof fit like thecoefficientof determination( P � ), YICand AIC, which are definedin Young and Beven[1994]. Comparingthe two modelswith differentinput (observed and equivalent rainfall) indicatesthattheapplicationof thesnow-meltmodelin frontof thetransferfunctionmodelimprovesthesimula-tion resultsignificantly(seeTable1).

Table1: Criteria of goodnessof fit for 2 different[1,1,1]models

Model-Criteria YIC P � AICobservedrainfall -7.462 0.492 0.998

equivalentrainfall -8.727 0.740 0.329

3.2 Non-linear modelestimation

If standard statistical tests indicate some non-linearity thefixed interval smoothing(FIS) methodwill beusedto obtainnon-parametrictime variableparameterestimates(Young and Beven [1994]).Consideringthecaseof a singleinput variable(e.g.rainfall) a nonlinearmodelcanbeapproximatedbyQthanksto Prof. PeterC. Young,Departmentof Environmen-

tal Science,LancasterUniversity, Lancaster, LA1 4YQ, UnitedKingdomfor makingtheCAPTAIN toolboxavailable

384

0 500 1000 15000

5

10

15

20

Time [d]

Ru

no

ff [

mm

]

observedsimulated

0 500 1000 15000

5

10

15

20

Time [d]

Ru

no

ff [

mm

]

observedsimulated

Figure 5: Simulatedrunoff with observed rainfallasinput (top panel)andequivalentrainfall asinput(lowerpanel).Thestructureof thetransferfunctionmodelis [1,1,1]

the following transferfunction modelwith time orstatedependentparameters:

�CBD�R� # �CB H % '*) �, ��B H % ' ) �.- ! '�/S0T2 !� =U�9V ! 0 = ) V ! %�' ) 0879797>0 = ? V ! %�' ?5 0;6() V ! % ' ) 0879797>036W< V ! % '�<X - ! '�/1032 ! (5)

The model (5) is a time/statedependentparame-ter versionof the standardtransferfunction modelgiven in (2), wherethe polynomialparametersareassumedto be possiblefunctionsof the time

B. In

Figure6 and7 the resultsof this time variablepa-rameterprocedureare shown. The gain parame-ter

=U�is the only parameter, which shows signif-

icant variation over the observation period, whichis likely to be causedby the soil moisture non-linearity.

In Figure 8 the standarderror of this time vari-able parameterare shown also. The advantageofthismethodologyis thepossibilityof evaluatingthestandarderrors,which canbeusedfor thephysicalinterpretationof the model. The final stepwill betheevaluationof thenatureof theparametervaria-tion in relationto othervariablesin orderto identifyanappropriaterainfall filter. Furthermorethetrans-fer function model will be decomposedby partialfractionexpansion.This decompositionin fastand

0 500 1000 15000

5

10

15

20

Time [d]

Ru

no

ff [

mm

]

observedsimulated

0 500 1000 15000

5

10

15

20

Time [d]

Ru

no

ff [

mm

]

observedsimulated

Figure6: Simulatedrunoff (transferfunctionmodelstructureis [3,3,1]) with constantparameters(toppanel)andtimevaryingparameters(lowerpanel)

0 500 1000 1500−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time [d]

FIS

est

imat

es o

f the

tim

e va

riabl

e pa

ram

eter

s

1996−1999

a1 + 0.5

a2

a3

b0

b1

b2

Figure7: Estimatedtime variableparameters

0 50 100 150 200 250 300 350 400−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time [d]

FIS

est

imat

es o

f TV

P b

0 +/−

se

Year 1996

b0

b0 + standard error

b0 − standard error

Figure8: Time variableparameterandstandarder-ror boundsof theyear1996

385

slow pathwayshasaninterestinghydrologicalinter-pretation,which couldbeconsideredasa represen-tationof acontributingareafunctionfor thefastre-sponsesin thecatchment,whetherthefastresponsesbedueto surfaceor subsurfaceflow processes.

4 CONCLUSION

In this paperthepreliminaryresultsof theapplica-tion of theDBM approachin a complex hydrologi-cal catchmentareshown. Preprocessingof thedatahasbeendoneby theuseof Wavelettransformationin orderto selectnoise-freerunoff dataseries.Theobservedseasonalityin therunoff series,influencedby snow andsnow-melt processes,have to betakeninto accountalso.Thereforeasnow-meltmodelhasbeenappliedto calculateanequivalentrainfall. Thispreprocessingimprovedthe transferfunction mod-elling significantly. Applying a time variablepa-rametertransferfunction model indicatesthe non-linearity of one parameter, which hasto be inter-pretedin a physical(mechanistic)way at next. Inconsiderationof the complexity of the investigatedcatchmentthis preliminaryresultsarequitesatisfy-ing demonstratingvery well the applicability of aparsimonioustransferfunctionmodel.

ACKNOWLEDGMENTS

Thisstudyis partof theSWURVE projectfundedbythe EU Energy, Environmentand SustainableDe-velopmentProgrammeandby theSwissFederalOf-fice of EducationandSciences.The datahasbeenmadeavailableby the SwissInstituteof Meteorol-ogy andSwissFederalOffice for WaterandGeol-ogy. The authorsacknowledgetheseinstitutesfortheir support.

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