Stream-scale flow experiment reveals large influence of understory growth on vegetation roughness

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ContentslistsavailableatScienceDirect

Advances

in

Water

Resources

journalhomepage:www.elsevier.com/locate/advwatres

Stream-scale

ﬂow

experiment

reveals

large

inﬂuence

of

understory

growth

on

vegetation

roughness

Koen

D. Berends

a,b

_,

_Un

_Ji

c,d

_,

_W.E.

_(Ellis)

_Penning

b

_,

_Jord

_J.

_Warmink

a,∗

_,

_Joongu

_Kang

c

_,

Suzanne

J.M.H.

Hulscher

a

a Department of Marine and Fluvial Systems, Twente Water Centre, University of Twente, P.O. Box 2017, Enschede, 7500 AE, the Netherlands b Deltares, Boussinesqweg 1, Delft, 2629 HV, the Netherlands

c Department of Land, Water and Environment Research, Korea Institute of Civil Engineering and Building Technology, Goyang 10223, Republic of Korea d Department of Civil and Environmental Engineering, Korea University of Science and Technology, Daejeon 34113, Republic of Korea

a

r

t

i

c

l

e

i

n

f

o

Keywords: Natural vegetation Flow experiment Vegetation roughness Bayesian inference Hydraulic modelling

a

b

s

t

r

a

c

t

Vegetationisakeysourceofflowresistanceinnaturalchannelsandfloodplains.Itisthereforeimportantto ac-curatelymodeltheflowresistancetoinformdecisionmakersandmanagers.However,itischallengingtopredict theresistanceofrealvegetation,becausevegetationmodelsarebasedonrelativelysmall-scalelabexperiments withmostlyartificialvegetation.Experimentaltestsofrealvegetationunderfieldconditionsarescarce.The purposeofthisstudyistomeasuretheflowresistanceofasubmergedwillowpatch,wheresmallherbaceous vegetationwasallowedtogrowinbetweenthewillowstemstosimulatefieldconditions.Detailedflowvelocity measurementswereperformedduringanfullscaleexperimentofflowaroundasubmergedpatchofwillows. Theparametervaluesofthewillowvegetationmodel,aswellasthefrictioncoefficientsofthevegetatedbanks andunvegetatedchannelbed,werecomputedsimultaneouslyusingBayesianinferenceusinga2Dhydrodynamic model.

Resultsshowthatthepresenceofunderstorygrowthgreatlyaffectsflowpatternsandthevalueoftheeffective vegetationdensityparameter.Measuredflowvelocitiesinthepatchwithunderstorygrowthwereverylow,and thepatchhasrelativelyhighdeflection.Afterremovalofthisundergrowth,flowvelocitiesinthepatchincreased anddeflectionofthevegetationcanopydecreased.Weshowthatestimatingvegetationdensityusingan often-usedrigidcylinderestimatorbasedonvegetationsampling,underestimatedtheeffectivevaluebymorethanan orderofmagnitude.Wearguethatproposedextensionstoexistingvegetationmodels,whichcantakeintoaccount understorygrowthandreconfiguration,couldbetestedunderfieldconditionsusingtheapproachfollowedin thispaper.

1. Introduction

Vegetation in rivers and streams is one of the largest sources of flow resistance (Luhar andNepf, 2013) and animportant source of uncertainty in hydrodynamic models used for flood manage-ment (Warmink et al., 2013) and river engineering applications (Berends et al., 2018). The presence of vegetation affects the mor-phologicalevolutionofrivers(vanOorschotetal.,2015),biodiversity (Straatsmaetal.,2017),andwaterquality(Dosskeyetal.,2010). There-fore,agoodrepresentationofvegetationincomputermodelsis impor-tant.

Advancesinremotesensingandclassiﬁcationalgorithmsenable de-tailedmapsof thespatialdistribution of vegetation(Geerlinget al., 2009;Forzierietal.,2011;Hodges,2015).Thelocaleﬀectofvegetation

∗_{Corresponding}_author.

E-mailaddress:j.j.warmink@utwente.nl(J.J.Warmink).

onflowiscommonlymodelledthroughafrictionterminthemomentum equation.Thisterm,alsoknownasroughnessorflowresistance,often representsvariouswaysofenergylossesthatarenotexplicitlymodelled. Wecanbroadlydistinguishtwowaystodeterminevegetativefriction infieldsituations.

Thefirstapproachtodeterminevegetativefrictionisbasedon look-uptablesorvegetationmodels.Thetableoftypical(Manning-type) fric-tionvaluesbyChow(1959)arestillusedtocharacterisetheroughness ofstreamsbasedonimagesordescriptionsofthevegetation(e.g.“light brushandtrees,inwinter”).Look-uptablesareaconvenientwayto coupleremote-sensingtohydrodynamicmodelling(e.g.,Forzierietal., 2011).However,despitethewidespreaduseoftheManningcoefficient asalumpedparametertocharacterisedescriberoughness,theimplicit assumptionoftheManningequationforalogarithmicvelocityprofile

https://doi.org/10.1016/j.advwatres.2020.103675

Received4February2020;Receivedinrevisedform15June2020;Accepted2July2020 Availableonline3July2020

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Fig.1. Left:PhotooftheA1experimentalﬂume(left) withtheadoptedcoordinatesystem.Right:schematic cross-sectionatthelocationoftheﬁrstpatch (cross-sectionD3).

doesnotholdforvegetation(Nadenetal.,2006;Ferguson,2007).For thisreason,various(semi-)empiricalmodelshavebeendevelopedthat computethecontributionforvegetationonﬂowresistancebasedon veg-etationparameters,suchasstemcountandplantmorphology(Klopstra etal.,1997;Baptistetal.,2007;Huthoﬀ etal.,2007;YangandChoi, 2010; Lietal., 2015). Ithasbeen shown thatsuch modelsperform wellagainstalargebodyofdatacollectedfromlaboratoryexperiments (Vargas-Lunaetal.,2015).

Furtherstudieshavebeencarriedouttobettercapturereal-world vegetation dynamicssuchasreconﬁguration(Järvelä,2004;Dijkstra andUittenbogaard,2010;Verschorenetal.,2016)andcontributionof foliage(Baletal.,2011; Västilä andJärvelä,2014).Inpractice, the vegetationparametersnecessarytousetheseformulasshouldbeeither directlymeasured, ortakenfromalook-uptable.Successful applica-tionofthisapproachtoﬁeldconditionsdependsonthevalidityofthe vegetationmodelandtheavailabilityofdataonvegetation.

Thesecondapproachtodeterminevegetativefrictionisthrough in-versemodelling,whichinvolvesfittingthemodel(inpracticeusuallya subsetofmodelparameters)tomeasurements.Itiscalled‘inverse’ mod-elling,becausethequantityofinterestisnotmodeloutput,butmodel input(such asparameters).Inpractice, inversemodellingis usedto findthevaluesforlook-uptables,becausethelumpedroughnesscan bestraightforwardlycomputedbyinvertingtheManningequation, un-dertheassumptionofsteadyuniformflow,alogarithmicflowprofile andaknownenergyslope(Marjoribanksetal.,2014).Inthecaseof non-uniformflow,theinvertedBélangerequationcanbeusedinstead (Erricoetal.,2018).

Roughness values estimated through inverse modelling are often usedasthe‘measured’roughnesstoprovidetheexperimentalbasisfor vegetationmodels.However,estimatingthefrictioninheterogeneous real-worldsituationsbasedontheenergyslope(e.g.,asingle vegeta-tionpatchwhich doesnotcovertheentirechannel), iscomplicated. Estimatingthefrictionofmultiplesources,basedonasingle observa-tion(theenergyslope),mayresultinnon-uniquesolutionstoparameter values(Beven,2006).Thisproblemisanalogoustothatof underdeter-minationin regressionproblems,whichallowsaninﬁnite numberof acceptablecombinationsofunknownfrictionfactorsiftheparameters exceedtheavailableobservations.Thisseverelylimitsourabilityto es-timatethefrictionvalues,andvalidatevegetationmodels,inreal-world situations.

Incaseswherenon-uniquenessisanissue,probabilisticparameter estimationispreferredtodeterministicoptimisation(Matottetal.,2009; Guillaumeetal.,2019).Formally,thisisachievedthroughBayesian in-ference,whichproducesprobabilitydistributionsofparametervalues giventhe(subjective) likelihood thatthemodel, giventhese values, confirmsexperimentalobservation.AwellknownBayesianinference methodologyisGLUE(generalisedlikelihooduncertaintyestimation), originallydevelopedfornon-uniquenessproblemsinhydrology(Beven andBinley,1992; Fonsecaetal., 2014; Mayotteet al.,2017). How-ever,previousworkhasshownthatestimatingmorethanonefriction sourcebasedonwaterlevelscanresultinunidentifiableparameter dis-tributions,whicharecharacterisedbywideandunconstrainedshapes (Werneretal.,2005).Toincreasetheidentifiabilityofparameter dis-tributionsotherobservationaldatacanbeused,suchasinundation pat-terns(Pappenbergeretal.,2005).Hypothetically,detailedvelocity mea-surementsaroundvegetationpatchescanserveasimilarfunctionto

es-timateflowresistanceusingGLUE.However,theliteratureonflowdata aroundreal-worldvegetationpatchesarescarce(Nadenetal.,2006; Marjoribanks etal.,2017),andcomparisonsbetween laboratoryand fieldin generalarerare(Huthoff etal.,2013; GroomandFriedrich, 2018).

Ouraimistoinvestigatetheflowresistanceofvegetationunder nat-uralconditions.Specifically,weareinterestedintheeffectofsecondary vegetationgrowinginbetweenandunderthebranchesofthedominant vegetationspecies.Suchsecondaryvegetationishardtodetectremotely fromsatelliteordroneimagery,andmaythereforeleadtoan underesti-mationofbiomassinlarge-scalerivermodelsthatrelyonecotopemaps toestimatefloodplainfriction(StraatsmaandHuthoff,2011;Forzieri etal.,2011;Warminketal.,2013;vanOorschotetal.,2015;Berends etal.,2019).Itisexpectedthatunderstorygrowthincreasestheflow resistance,butitisunknownbyhowmuch.Inthisstudy,weperform astream-scaleexperimentwithrealvegetation wheresecondary veg-etationwasallowedtodevelopnaturally.Thisphysicalexperimentis coupledtoadigitaltwinnumericalmodeltocomputefriction parame-tersbyBayesianinference.

2. Methods

2.1. Physicalexperimentsetup

Large-scaleexperimentswereperformedatthesiteoftheKICT-REC (KoreaInstituteofCivilEngineeringandBuildingTechnologyRiver Ex-perimentCenter),whichislocatedinthecityofAndong,SouthKorea. Thisfacilityisdesignedforfullscalephysicalexperimentsandconsists ofthreeseparatechannelsofvariousslopeandsinuosity.Largecapacity pumpscangeneratethemaximumﬂowrateupto10m3_s−1_._The_length

ofeachchannelisapproximately600m.

Theexperimentsforvegetatedﬂow wereperformedinthe down-streamsectionofthe“A1” channel,whichhasatrapezoidalcross-section withabottomwidthof3m,topwidthof11m,bedslopeof0.001m/m andbank-sideslopeofroughly1:1.5(V:H),asshowninFig.1.The bank-sideslopesofthechannelarevegetatedwithgrassesandsmallannuals nativetotheregion.Thechannelbedconsistsofsandymaterialswitha meanparticlesizeof0.8mm.

Seven alternatingwillow patches,eachwitha lengthof 4mand widthof1.5m,wereplantedinthe52msectionofA1channelwhich was located about 125 m upstream from the downstreamweir and 400mdownstreamfromtheupstreamweir.Thepatcheswereplanted intwoconﬁgurations:thefourmostupstreampatchesinadense con-ﬁguration of22treesperm2 _and_the_three_downstream_patches_in_a

sparseconﬁgurationof7.3treesperm2_._The_willow_saplings_were

al-lowedtogrowatthesitefor10monthsbeforetheexperimentsstarted inAugust2015.Theaverageheightofrootedwillowswas0.4mwhen theywereplanted,withaninitialtrunkdiameterof1cm.Inaddition, indigenoussmallscaleroughherbaceousvegetationandgrasseswere allowedtogrowonthebank-sideslopesofthechannels.Thebed it-selfwasrelativelyunvegetatedandmobile.Herbaceousvegetationhad alsospontaneouslydevelopedinbetweenthewillowsinthevegetation patches.Althoughtheflumewasmostlydryduringthisinwhichthe vegetationgrew,althoughperiodsofprolongedflowoccurredwhenthe experimentstookplaceinsome(other)partoftheflume.Thisregimeof mostlydryandoccasionalfloodingistypicalformanyKoreanstreams,

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Fig.2. Top:Photographofthe experimen-talsetuptakenfromthetop.Bottom:layout ofexperimentalchannelandlocationofthe D0andD3cross-sections.

whichareephemeralinnature.Aftertheﬂowexperiment,vegetation samplesweretakentomeasuretheheight,diameterandmorphologyof thewillows.Thebedlevelintheexperimentalareawasmeasuredusing RIEGLLMS-Z390iterrestriallaserscanner.

Theflowmeasurementsweretakenafteruniform,steadyflowwas achieved.Thewaterdepthinthechanneldownstreamwasfixedto ap-proximately1.1m.Detailedthree-dimensionalvelocitymeasurements withanSontek16MHzADV(AcousticDopplerVelocimeter)were per-formedinthemiddleand4mupstreamofthefirstdensepatch(atD3, Fig.2).TheADVdeviceswerepositionedpointingdownwardfroman overhangingbridge.Togetasignalwithinthevegetationpatch,the de-viceswereslightlymovedtotheleftorrightwhenneeded.Duringtrial convergencetestingarequiredmeasurementtimeof100swasfound. Boththedischargeandthedepth-averagedflowvelocityprofileare de-rivedfromtheADVmeasurements.Wegathereddatafromthreecases: (i)flowmeasurementsattheupstreamcross-sectionD0,(ii)flow mea-surementsinthecentreofthefirstwillowpatch,D3,(iii)andfinally flowmeasurementsatD3,butwiththeundergrowthremoved(Fig.2). WewillrefertothesecasesasD0,D3a(willowsandundergrowth)and D3b(onlywillows)respectively.Forcase D0andD3aADV measure-mentswerecarriedout every30cm,foratotalof23 locationsin Y direction.ForD3balowerresolutionwasusedof60cm,foratotalof 12locationsinYdirection.Intheverticaldirection,ameasurementwas carriedoutevery5cm.ForD3aaverticalresolutionof2.5cmwasused inthevegetationpatch.

2.2. Numericalexperimentsetup

Adigitaltwinoftheexperimentalﬂumewasconstructedwiththe opensourceDelft3D4modellingsystem(Lesseretal.,2004),1 _using

atwo-dimensional,regular,numericalcomputationalgridof15 m× 171m.Thesizeofindividualgridcellswas0.5m(inflowdirection)by 0.375mwithbedlevelsdefinedinthecentreofeachcell.Aconstant discharge,determinedfromtheflumeADVmeasurements,wasimposed attheupstreamboundary,whilethedownstreamboundarycondition wasdefinedasaconstantwaterleveltoensureadownstreamwater depthofapproximately1.1m.Themodelwasinitialisedwitha con-stantwaterlevel.Eachsimulationranfor30minofmodeltimewitha timestepof0.3s.Theruntimewaschosensuchthattheentiremodel achievedequilibriumconditionbythelasttimestep.Thescaleofthe

1 _The_source_code_of_Delft3D_4,_as_well_as_its_validation_documents,_can_be

retrievedfromhttps://oss.deltares.nl.

model,giventherelativelyshortruntime,smallwaterdepths,ﬁnegrid andsmalltimestep-requiredchangingsome defaultnumericaland physicalparameters.Weusedasmoothingtimeof2minanda thresh-olddepthforﬂooding/dryingof 1cm.Thehorizontaleddyviscosity washeldconstantthroughoutthemodelat10−2_m2_s−1_._To_be_able_to

capturethesharpvelocitygradientsaroundthepatches,lowviscosity valueswereused(Vionnetetal.,2004;Verschorenetal.,2016).Inthe experimentalarea,thegeometricdatawasbasedonthebedlevel mea-surementsof theexperimental setup.Forregions notcoveredbythe bedlevelmeasurements,asyntheticcross-sectionwasdeﬁnedfromthe designspeciﬁcationsofthechannel.

Bedfrictionwasdeﬁnedusingfourdistinctroughnessclasses,two forthewillowpatches(denseandsparse),oneforthevegetatedchannel slopesandoneforthemobilesandchannelbed.Eachindividualclassis assigneditsownroughnessformulaandfrictionparameters.Everygrid cellboundarywasassignedoneofthesefourclasses.Allfriction formu-lasareexpressedintermsoftheChézyfrictioncoeﬃcientC[m1∕2_s−1_].

ForthechannelbedweusetheManningfrictionformula:

𝐶=ℎ1∕6𝑛−1_𝑏 (1)

withwaterdepthh[m]andManningcoeﬃcientnb[sm−1∕3].The

Keule-ganequation(alsoknownastheColebrook-Whiteequation)wasfound tobebestsuitedtoreproducethedepth-averagedvelocityonthe chan-nelslopes: 𝐶=𝛼1log10 ( 𝛼2_𝑘ℎ 𝑠 ) (2) with Nikuradse roughness height k_s [m] and parameters 𝛼1=

18m1/2_s−1_,_𝛼

2=12.Thevaluesofbothnbandksweredeterminedwith

GLUE.

Therearevariousmodelsavailableforresolvingvegetationfriction. Here,weusedthetwo-layerapproachofBaptistetal.(2007),which performsfavourablyagainstlaboratoryexperiments(Vargas-Lunaetal., 2015),andisgeneralisedasfollows:

𝐶= ( 𝐶−2 𝑏 +₂𝜙_𝑔 )−1 2 +𝛼 √ 𝑔 𝜅 lnℎℎ𝑑 (3)

whereCbistheChézyfrictioncoeﬃcientofthebedwithoutvegetation,

𝜙 [−]thevegetationparameter,g[ms−2_]_the_{gravitational}_{acceleration,}_𝜅

thevonKarmanconstant,hd[m]thedeﬂectedvegetationcanopyheight

and𝛼 anindicatorsuchthat𝛼 =0foremergentconditions(h<h_d)and 𝛼 =1forsubmergedconditions(h>hd).

Thedimensionlessvegetationterm𝜙 modelsthecontributionof veg-etationtothetotalfrictionofthevegetatedpartofthecross-section.

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Intheoriginalformulationof Baptistetal.(2007),itiscomputedas 𝜙 =𝑐𝐷𝑚𝐷ℎ𝑣,withdragcoeﬃcientcD [−], stemspersquaremeterm

[m−2_],_stem_diameter_D_[m]_and_vegetation_height_h

v [m].This‘stick

model’for𝜙 isideallysuitedforvegetationthatmaybeapproximated asrigidcylinders.

Aﬁxedvalueforhdwasused,whichwasestimatedfrom

measure-ments.Furthermore,weassumedthatthefrictioninthepatchwas dom-inatedbythevegetationandthatthebedtermhadaneglibleimpact. ThereforewechoseaconstantvalueofCbat60m1/2s−1,whichledto

negligibleadditiontothetotalroughness.Thevegetationparameter𝜙 wasconsideredunknown,andestimatedwithGLUE.

2.3. GLUEmethod

Thesetofunknownparametersin thenumericalexperimentwas givenby𝜃 ={𝑛_𝑏,𝑘_𝑠,𝜙},i.e.thefrictioncoefficientsofthechannelbed andslopeandthevegetationparameter.Theproblemconsideredhere washowtochoosethevaluesforparametersinsuchaway,thatthe measuredflow velocitieswerereproduced bytheflowmodel. While anoptimalsetofvaluesmaybefoundusingoneofvarious optimisa-tionstrategiesavailableinliterature,Beven(2006)arguedthat multi-pleparametersetsmaywellbefoundthatallproduceacceptableresults. Thiscreatesuncertaintyregardingtheparametervaluesthusobtained throughinversemodelling.TheaddedbenefitofGLUEaboveregular calibrationproceduresisthatthisuncertaintyisformallytakeninto ac-count.Toachievethis,GLUEusesBayesianinference,inwhich(usually verylimited)priorknowledgeabouttheparametervaluesisupdated giventheprobabilitythatthoseparametervaluesresultedinmodel out-putthatcomparedfavourablywithmeasurements.Thisistechnically achievedthroughBayes’theorem:

𝑝(𝜃|𝑢)∝(𝑢|𝜃,𝜖)𝑝(𝜃) (4)

where𝜖 istheerror betweenmodelledandmeasuredflowvelocities andumeasuredflowvelocities.Thepriordistributionp(𝜃)thatencodes ourpriorassumptionsofprobableparametervaluesforparameterset 𝜃.Theposteriordistributionp(𝜃|u)isourgoal,becauseitexpressesthe probabilityofparametervaluesin𝜃 afterhavingseenthemeasuredflow velocitiesu.Thelikelihood(𝑢|𝜃,𝜖)isafunctionofthemodel(𝜃,𝜖)and observations(u),andrepresentstheprobabilityofugiven𝜃 and𝜖.

Todeterminethefunctionalform ofthelikelihood weneededto assumeastatisticalrelationshipbetweenobservedandmodelledﬂow velocities.Here,werelatedthemeasuredﬂowvelocityvectorucat

cross-sectionctothesimulatedﬂowvelocitŷ𝑢𝑐,𝑦(𝜃)asfollows:

𝑢𝑐,𝑦=̂𝑢𝑐,𝑦(𝜃)+𝜖𝑐 (5)

where𝜖cisanormalandindependentlydistributederrortermwith

vari-ance𝜎2

𝜖,i.e.𝜖𝑐∼(0,𝜎𝜖2).Theassumptionofindependenterrorsand

zerobiasexpressesourassumptionthatthehydraulicmodelshouldbe abletoreproduceobserveﬂowvelocityproﬁles,suchthatany remain-ingdiscrepancyis adequatelydescribed byanormaldistribution.As iscommonlydoneinGLUEapplications,weadoptedauniformprior distributionforeachparameter.Theupperandlowerlimitsofeach dis-tributionweredeterminedbyexploratorycomputationswiththemodel. Forouradoptederrormodel(5),thecorrespondinglikelihoodfunction is: (𝜃,𝑢)=(2𝜋𝜎2_𝜖)−1∕2exp ⎛ ⎜ ⎜ ⎝ − ( 𝑢𝑐,𝑦−̂𝑢𝑐,𝑦(𝜃))2 2𝜎2 𝜖 ⎞ ⎟ ⎟ ⎠ (6)

Theunknownsintheinverseproblemareboththemodel parame-ters𝜃 andthevarianceoftheresidualerrors𝜎2

𝜖.Toestimate𝜎𝜖2weused

themaximumlikelihoodestimate𝜎2

𝜖,𝑀𝐿𝐸=𝑛−1∑𝑛𝑖=1(𝑢𝑖−𝑢𝑀𝐿𝐸𝑖 )2,with

nthenumberofobservationsand𝑢𝑀𝐿𝐸

𝑖 themodelresultsforthebest

performingparameterset(Stedingeretal.,2008).Sincethelikelihood functionreturnstheprobabilityoftheobservedﬂowvelocitiesgiven themodelresultsevaluatedwithagivenparameterset𝜃,modelresults

thatdeviatesignificantlyfromobservationswillreturnlikelihoods ap-proachingzero.Thelikelihoodfunctionusedherewasformallyderived fromtheadoptedstatisticalmodel(5).WenotethatGLUEallowsfor greatflexibilityinchoosingfromavarietyofinformallikelihood func-tionsaswell,whichneedanadditionalbehaviouralthresholdto differ-entiatebetweenbehaviouralandnon-behaviouralparametersets.Since (6)tendstozeroforveryimprobableparametersets,suchabehavioural thresholdwasnotneededhere.

Theposteriorparameterdistributions,wereobtainedthroughMonte Carlo simulation using theSobol’ low discrepancysequence (Sobol’, 1967).Wesampledfromthepriordistributionstocreatealargenumber ofpossibleparametersets.Here,weusedasamplesizeof5000model evaluations,witheachevaluationusingadifferent,randomlysampled setof parametervalues.Thesameensemble wasused forcasesD3a andD3b,usingthevalueforh_dderivedfromcaseD3a.Afterwards,the ensemble valuesof𝜙 forD3bwerecorrectedtoaccountfora differ-entheightofthedeflectedvegetationcanopy.Theprobability distribu-tionfortheparametervector𝜃 wasobtainedbyapplicationofBayes’ theorem.Thisproceduregaveustwovaluablesourcesofinformation onuncertaintyrelatedtomodelandobservation.First,theestimateof 𝜎2

𝜖,orthe‘predictiveuncertainty’.Thisistheresidualvariancebetween

modelandmeasurement,whichcannotbeexplainedbychoosing diﬀer-entparametersettings.Thesecondisp(𝜃|u),i.e.theposteriorparameter distributionor‘modeluncertainty’.Thisexpressestheuncertaintyinthe parametervalues,giventhemeasurements.

3. Results

3.1. Flowmeasurements

BasedontheADVmeasurements,theflowfieldscoveringthevertical (Y,Z)planewereconstructed.Tocomputedischargefromvelocity mea-surements,itisusuallynecessarytoassumeavelocityprofile(e.g. log-arithmic,seeBoiten,2000).However,duetotheirregularflowprofiles expectedintheflowthroughvegetationandthedensityofvelocity mea-surements,weinsteadlinearlyinterpolatedpointsinbetweentheADV supportpointstoaregular,cross-sectioncoveringgrid,whileassuming zeroflowvelocityatthebed(Fig.4b,d,f).Thetotaldischargewasthen computedbysummingtheproductofthegridcellarea.Wefound dis-chargesof2.91m3_s−1_(D0),_2.66_m3_s−1_(D3a)_and_2.54_m3_s−1_(D3b).

Thediﬀerencesareattributedtouncertaintyinﬂowvelocity measure-mentsandinterpolationinaccuracies.

Themeasurementsattheunvegetatedcross-section D0showthat thedepth-averagedﬂowislowontheslopesandincreasestowardthe centreofthechannel,reachingabout0.5ms−1₍_Fig.₄_a)._At_the_right

handsideofthechannel(y=6m)alocalincreaseinﬂowvelocityis measured.Giventhatthehighvelocitieswereconsistentlymeasuredby diﬀerentADVdevices,weassumetheyarenotduetoinstrument prob-lems.Therefore,wedidnotrejectthesemeasurements,butleftittothe numericalmodeltoexplainwhetherthesemeasurementsareexpected. WeusedthedischargeatD3aastheupstreamboundaryconditionin thenumericalmodel.

Twoseriesofmeasurementswerecarriedoutforthecross-sectionin thewillowpatch(D3);oneforthenaturalsituationincludingorganic debrisandundergrowth(D3a)andtheotherforwhichthepatchwas cleaned(D3b).

AnalysisofmeasuredflowvelocitiesforD3arevealedthatoneADV measurementdevicereturnednear-zeroresultsforallflowdepths, sug-gesting malfunction. Measurements from this device (at𝑌 =6.4m), werediscardedfromtheresults.SimilartoD0,aregionofhigherflow velocitieswas observed attherighthandside ofthepatch near the watersurface.Inthepatchitself,boththeflowfield(Fig.4d)andthe depth-averagedresults(Fig.4c)showslowervelocities.However,the flow velocitiesnearthewatersurfacearesimilartotheunvegetated partofthechannel,suggestingsubmergedflow.Thisisclearfromthe verticalvelocityprofileinthepatch(Fig.3),whichshowslowflow

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ve-Fig.3.Verticalflowvelocityprofilewithinthewillowpatchesasfunctionof elevation(z)Thedashedlineat𝑧=−3.4mmarkstheassumeddeflectedcanopy heightforcaseD3a.

Table1

Measuredparametersofwillowsinthedensepatchesoftheexperimentalsetup.

Parameter Average st.dev. n Unit

Measurements

Willow height 1.06 ± 0.10 66 m

Stem height 0.43 ± 0.01 66 m

Undergrowth height 0.29 ± 0.15 41 m

Stem diameter 10 ± 3 66 mm

Stem diameter incl. organic debris 15 ± 8 66 mm

Undergrowth diameter 4 ± 4 41 mm

Branch diameter 3 ± 1 50 mm

Nr. of stems 22 m −2

Nr. of branches per stem 15 −

locities(<0.3ms−1₎_and_large_variation_up_to_𝑧₌₋₃_.₄_m,_above_which

thevelocitiesrapidlyincreasetomorethan0.6ms−1_and_variation_is

signiﬁcantlyreduced.FromFig.3,weassumedthatthevegetationwas deﬂectedsuchthatthecanopyheightwas0.8m.

CaseD3b— afterremovalofallorganicdebrisandundergrowth— broadlyshows(Fig.4e,f)similarpatternstotheD3aresults,withsome keydifferencesin theflowthrough thepatch.Intheverticalprofile (Fig.3)velocitiesnearthebed(𝑧<−3.6m)arehighercomparedtoD3a, whichisattributedtoremovalofundergrowth.Forhigherwaterdepths (𝑧>−3.6m)velocitiesshowaninitialdecrease,whilethevariationover thepatchincreases.Thisisattributedtoflowthroughthebranchesand leaves,whichaddcomparativelymoreresistance.IncontrasttoD3a, clearsubmergenceisnotobserved.Thereforeweassumethatinthecase ofD3b,thevegetationwasjust-submerged,meaningthatthedeflected canopyheightwasequaltothewaterdepth.

3.2. Vegetationmeasurements

Aftertheinitialflowexperiment(casesD0andD3a),several sam-plesweretakentodeterminetheheightanddiameterof thewillows (Table1).Thestemsweremeasuredupuntiltheupperknot,fromwhich pointseveralbranchessprouted.Weobservedthatorganicdebriswhich hadattacheditselfaroundthewillowstemsincreasedtheeffective di-ameteroftheplantsby50%onaverage.However,thedistributionwas notuniform— someplantsexperiencedasignificantlylargerincrease indiameterwhileothershadlittleattacheddebris.Thiswasreflected intheincreasedvarianceoftheobservations.Theundergrowthhadan averageheightof29cmandgrewinbetweenthewillowstems(Fig.5) andconsistedofvariouswaysofherbaceousvegetation.

Table2

Priordistributionsofmodelparameters.

Parameter Distribution Lower bound Upper bound n b Uniform 0.01 sm −1∕3 0.15 sm −1∕3

k s Uniform 0.1 m 5 m

𝜙 Log-uniform 10 −2 m −1 ₁₀2 m −1

Itshouldbeexpectedthatboththeundergrowthandthediameter increaseduetoorganicdebrisincreasetheeﬀectivefrictionofthe veg-etationpatch.Theexpectedvegetationparameter𝜙 isestimatedbased ontheBaptistvegetationmodel(𝜙 =𝐶𝑑𝑚𝐷ℎ𝑑)andthevegetation

mea-surementsfromTable1.Tocompute𝜙,weassumedthebranchheight tobeequaltothewillowheightminustheaveragestemheight. Further-more,weassumedrigidbendingatthebed(followingVerschorenetal., 2016),suchthatthedeflected stemandbranchheightscan be com-putedfromthedeflectedwillowheight(seeSection3.1)usingstandard trigonometry.Weassumedacasedragcoefficientof𝐶𝑑=1,whichisa

commonassumptionforsubmergedflow(Wunderetal.,2011).Here, wedidnotaccountfortheeffectoffoliage.Theestimatedvegetation pa-rameterforcaseD3a,i.e.includingdebris,undergrowthandadeflected willowheightof0.8m,isapproximately𝜙𝐷3𝑎=1.33±1.044.Thelarge standarddeviationintheexpectedparameterismainlyduetothe vari-anceinthediameteroftheherbaceousvegetationintheundergrowth. ForcaseD3b,i.e.nodebris,noundergrowthandadeflectedheightequal tothewillowheight,is𝜙_𝐷3_𝑏=0.76±0.20.

3.3. GLUEresults

AnimportantstepwithinGLUEistodeterminetherangesofthe uni-formpriordistributions.Theserangesshouldbegenerous,sinceitis im-portantthatthesepriordistributionscovertherangewherethe(apriori unknown)posteriordistributionswillbe.Assumingatrapezoidal chan-nel,thelumpedManningcoeﬃcientforthegivendischargeandwater depthisapproximately0.11sm−1∕3_._Based_on_this_relatively_high_friction

factorandexploratorycomputationswiththehydrodynamicmodel,we chosesuitablylargerangesforthepriordistributionsandalog-uniform distributionfor𝜙 (Table2).

Forallthreeparameters(n_b,k_s,𝜙)wethencomputedthejoint poste-riordistributionsusingGLUE.Theposteriordistributionsgivethe like-lihoodthatacertainparametervalueleadstogoodmodelresults.The widthofthosedistributionsisameasureoftheparameteruncertainty regardingthepossible‘true’valueoftheparametersgiventhemodel andthemeasurements.Themarginalposteriordistributionsofall pa-rametervaluesareshowninFig.6.

Thedistributionof𝜙 forD0isverywideanddoesnotshowaclear peak(Fig.6a).Thisindicatesthatthe𝜙 isaninsensitiveparameterfor cross-sectionD0.Therefore,𝜙 cannotbeidentifiedfromflow measure-mentsatthatcross-section.Thiswasnotunexpected,asD0islocated 4mupstreamfromthefirstpatchandisthereforenotdirectlyinfluenced bythepatch.

TheothertwodistributionsinFig.6ashowthevaluesfor𝜙 thatlead togoodmodelresultsforD3aandD3b.D3a,whichhadsigniﬁcant un-dergrowthanddebris,showssigniﬁcantlylargerinferredvaluesfor𝜙 comparedtoD3b.Themedianvalueof101.6₍_{~ 39.8)}_is_an_order_of

magnitudehigherthanwouldbeexpectedbasedplantonparameters usingtheBaptistmodel.Forthecleanedwillows,caseD3b,thevalues aremuchlower, withamedianvalueof 𝜙 =1.56. Thisismore than twiceashighaswasexpectedbasedontheplantparameters.The un-certaintyoftheestimatedparametervaluesfor𝜙 issigniﬁcant,ascan beobservedbythewidthoftheposteriordistributionsinFig.6aandthe standarddeviationinTable3.ForcaseD3aespecially,thevaluesfor𝜙 varybetween20and100,withsomeofthebestsimulationsfoundnear theupperboundary.Itisimportanttonotethatatsuchhighvalues,the sensitivityof𝜙 totheroughnesscoeﬃcientCisverymuchreduceddue

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Fig.4. ThedepthaveragedflowvelocitiesattheADVlocations(a,candd)andtheinterpolatedflowfields(b,d,andf)fortheunvegetatedcase(D0),vegetated casewithundergrowth(D3a)andcleanedvegetationcase(D3b).

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Fig.6. Theposteriorprobabilitydistributions;theseshowthedistributionof likelyparametervaluesaftermodelresultswereweighedagainstmeasurements.

Table3

Thenumberofobservationsnusedforparameterestimation,themedian andstandarddeviationofthemodelparametersin𝜃 andthemaximum likelihoodestimationof𝜎𝜖. D0 D3a D3b unit n b 0.062 ± 0.007 0.098 ± 0.008 0.104 ± 0.010 sm −1∕3 k s 0.99 ± 0.28 2.70 ± 0.332 4.0 ± 0.44 m 𝜙 – 10 1.6±0.2 ₁₀0.19±0.12 ₋ 𝜎𝜖,MLE 0.039 0.024 0.030 m ms −1 n 23 21 12 −

totheinversesquarerootin(3),whichmaypartlyexplainwhysuch highuncertaintyisfoundforthiscase.Therefore,valueshigherthan theupperboundaryarenotexpectedtomeaningfullyimproveresults.

EstimationofManningcoeﬃcientofthechannelbednb and

Niku-radseroughnessheightoftheslopesksresultedinwell-deﬁnedposterior

distributionsforallcases(Fig.6b,c),withstandarddeviationsaround 0.01sm−1∕3₍_Table₃_).

Interestingly,thedistributionsarenotthesameforthethreecases. Tomeasuretheroughnessofvegetationfromwaterslopemeasurements, theroughnessofthebediscommonlyassumedtobeindependentfrom theroughness ofthevegetation (e.g.,Verschoren etal., 2016). How-ever,theresultsshowninFig.6suggestthattheparametervaluesfor bothnbandksareaﬀectedbythevegetationpatch.Forbothnbandks,

thepresenceofapatch(D3a,D3b)resultsinhigherparametervalues. Therefore,theassumptionofindependencebetweentheparameterof thevegetationpatch(𝜙)andthoseofthechannelbedandslope(n_band ks)wouldnothavebeenvalidinourcase.

Modelledﬂow velocities,given theposteriorprobability distribu-tionsoftheparameters,arecompared withthedepth-averagedADV

measurements(Fig.7).Thedepicteduncertaintybandsshowtherange ofthemodeluncertainty.Variationwithinthemodeluncertaintycanbe explainedfromuncertaintyintheparametervalues.Measurementsthat falloutsideofthesebandsareexplainedthroughtheresidualerrorterm 𝜖 inEq.(5).FromFig.7aweobservethattheregionofhigherflow ve-locityincross-sectionD0cannotbeexplainedbythenumericalmodel. Thisindicatesthatthismustbecaused,eitherbyanunmodelledprocess orfeature,orbymeasurementerror.However,ingeneralthevelocity profilesforallcasesarewellexplainedbythenumericalmodel,which isreflectedinthesmallstandarddeviationoftheresidualerror(𝜎𝜖,MLE)

forallthreecases(Table3).

Finally,theuncertaintybandsforcasesD0andD3aare compara-tivelysmallcomparedtotheothercases,bothinFigs.6and7.Thisis duetoahigherdensityofADVmeasurements.Ingeneral,alarger num-berofmeasurementshelpstodecreasemodeluncertaintyandincrease theidentiﬁabilityoftheindividualparameters.

4. Discussion

4.1. Identiﬁabilityoffrictionparameters

In this case study we identified three unknown friction parame-ters,relatedtothechannelbed,vegetatedchannelslopesandthe wil-lowpatch.Usingonlywaterlevelmeasurements,itisnotpossibleto uniquelydeterminethevaluesoftheseparameters.Ourfindingsshow thatdetailedmeasurementsofthetransversedepth-averagedvelocity profile,incombinationwithaprobabilisticinversemodellingapproach (here,GLUE),allowsustoestimatetheparametervaluesandquantify theuncertaintyofthoseestimations.Theinversemodellingapproach resultsintwodifferenttypesofuncertainty.Thefirstismodel uncer-tainty,whichistheuncertaintyof̂𝑢_𝑐,𝑦(𝜃)in(5).Thisuncertaintycanbe decreasedbyincreasingthenumberofobservations(i.e.ninTable3), oralternativelyifahigheruncertaintyisacceptablethenumberof ob-servationsmaybedecreased.Ingeneral,“dataprovidesinformationand moredataprovidesmoreinformation” (Stedingeretal.,2008).The sec-ondtypeofuncertaintyispredictiveuncertainty,givenby𝜖cin(5).This

uncertaintycannotbedecreasedwithoutchangingmodels,onlymore preciselyestimated.

Inliteratureitisoftenassumedthatthetotalroughnessisconstituted ofalinearsumofindependentconstituentterms(e.g.bedroughnessand vegetation roughness).Thetotalroughnesscanbe estimatedwithout detailedﬂowmeasurements.Therefore,ifthebedroughnessisknown (e.g.fromrepeatingtheexperimentwithoutvegetation)thevegetation roughnesscanbedetermined.However,ourresultsshowthatthe as-sumptionofindependencebetweenthetermswouldnothavebeenvalid forourstudycase.Forbothvegetatedcases,theestimatedfrictionofthe bedandtheslopewashighercomparedtotheunvegetatedcase.A po-tentialexplanationforthiscouldbethatturbulentandshearstresses notsuﬃcientlymodelledbytheparameterisationofeddyviscosityare compensatedbyahigherbedroughness.

Theestimatedparametervalues for𝜙 werefoundtobegenerally higherthanwasestimatedbasedonthe“rigidcylinder” estimator.This isespeciallytrueforcaseD3a,wherethepresenceofundergrowth con-tributestotheoverallfailureoftheestimator.Inthecleanedcase,D3b, thevaluesarewithinthesameorderofmagnitude,althoughstillafactor twohighercomparedtotherigidcylinderestimation.Apartial expla-nationcouldbefoundinthepresenceoffoliage,whichinotherstudies isreportedtoaccountfor60%oftotaldrag(Baletal.,2011),aswellas unmodelledstressesinthehorizontalexhangelayerbetweenvegetated andunvegetatedﬂow(Truongetal.,2019).

4.2. Uncertaintyofvegetationmodelparametersunderﬁeldconditions In this study we used the well-known two-layer model of Baptistetal.(2007)forsubmergedﬂow.Whilethismodelconsistently

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Fig. 7. The modelled velocities (colored bands) and depth-averaged ADV measure-ments(markers)forthethreecases.The col-ored uncertaintybandsconveythe resultof themodeluncertainty,whilethedashedlines showthetotalpredictiveuncertainty.

comparesfavourablytoexperimentaldata,applicationtofield condi-tionsismetwithseveralchallenges.Animportantlimitationofthe Bap-tistmodelisthatreconfigurationisnotexplicitlytakenintoaccount. Reconfigurationcanbetakenintoaccountinthetwo-layerapproachby modellingtwodifferenteffects.Thefirsteffectisstreamliningofstems andfoliagetodecreasethetotaldragofthevegetation.This directly effects thevegetation parameter 𝜙. The modelproposed byJärvelä (2004)canbeusedtoaccountforthis.Inthismodel,thevegetation parameterisdefinedby𝜙 =𝐶𝐷,𝜒𝐿(𝑢𝑢𝜒𝑣)

𝜒_,_where_L_[m]_is_the_leaf_area

index,u_v[ms−1_]_the_ﬂow_velocity,_u

𝜒[ms−1]aconstantand𝜒,CD,𝜒[−]

arespecies-speciﬁcparameters.ThevalueoftheVogelexponent𝜒 is negative,andthusdecreases𝜙 withincreasingvelocity.

Thesecondeffectofreconfigurationisdeflectionofthestems,which effectivelydecreasestheheightofthevegetationcanopyheighthd.This

directlymodifiesthesecondtermintheBaptistmodelwhichaccounts forsubmergedflowand,dependingonthechosenmodel,𝜙.For sub-mergedflow,theestimationof vegetation frictionissensitivetothe decrease ofthevegetation canopy.This isespecially soforlow sub-mergenceratios(<0.75)andhighvaluesof𝜙 (>0.2),forwhichthe contributionofvegetationfriction((𝜙

2𝑔)−1∕2)andcontributionofthe

ve-locityproﬁleabovethevegetation(

√

𝑔

𝜅 ln(ℎℎ𝑑))tothetotalfrictionare ofthesameorderofmagnitude.Undertheseconditions,assumingrigid vegetationmayleadtolargeerrorsintheestimationofvegetative fric-tion.

Theprimaryeffectofundergrowthisanincreasedblockageratio, whichcanbemodelledbyahighervaluefor𝜙.Theinherentweakness oftherigidcylinderestimatorsfor𝜙 (usedbyKlopstraetal.,1997; Bap-tistetal.,2007;Huthoff etal.,2007;YangandChoi,2010)isthat inclu-sionof undergrowthis contrived,whileneglectingundergrowthmay leadtosignificantunderestimationoftheeffectiveroughness.Frontal blockageareaestimatorsfor𝜙 (e.g.Västilä andJärvelä,2017)donot sufferfromthislimitation,asundergrowthwouldbeonepartofthe to-talblockagearea.ComparingthetwovegetatedcasesD3aandD3b,we observedlowerflowvelocitiesandgreaterdeflectionofthestems.This maybeseenasasecondaryeffectofundergrowth,i.e.anincreasein

to-taldragofthepatchandthereforegreaterreductionincanopyheight. Thissuggeststhatmodellingthistypeofreconfigurationdoesnotonly dependon plant-specificparameters, buton theconfigurationof the entirepatch,includingsecondaryvegetation.Relativelysimple predic-tivemodelsbasedonsingle-plantbehaviour,orevenlumpedmodels withavelocitydependent relationshipwithplant-specificparameters (Verschorenetal.,2016),arethereforelikelytoincludeuncertainty re-gardingnon-plantspecificdrivers.

FutureeffortmaybedirectedtotestanextendedBaptist(orsimilar two-layer)modelwhichwouldincludetheeffectsofstreamliningand deflection undernaturalconditions. Ideally,these flumetests should coveraseriesofdifferentdischargestotestundervaryingwaterdepths andflowvelocities.Topromoteidentifiability,thenumberof param-etersoftheextendedmodelshouldbelimited.Anotherargumentfor vegetative frictionmodelswitharelativelysmallnumberof parame-ters,apartfromtheissueofidentifiability,isfoundintheusually data-limitedproblemsin practice.Vegetationmodelswithfewparameters forwhichtheuncertaintycanbequantifiedareperhapsbettersuited forlargerscalefieldapplicationsthancomplexmodelswhichrequire intimateknowledgeofvegetationconfiguration.

5. Conclusions

Theobjectiveofthisstudywastoinvestigateflowresistanceunder naturalvegetatedconditions,inwhichsecondaryvegetationgrows un-derandbetweenthedominantspecies.Resultsshowthatthepresence ofundergrowthsharplyincreasesthevegetationparameter𝜙,aswellas increasethedeflectionofthewillows.Overall,thevegetationparameter valuewasfoundtobehigherthanexpectedbasedonaprioriestimations ofarigidcylinderestimator.Thisisattributedtoaspectsofreal vege-tationthatdeviatefromtherigid-stickbasedapproach,namelyfoliage, undergrowthandreconfiguration.

Thispapershowsthatvelocitymeasurementsinnaturalchannelscan beusedeﬀectivelytoestimatetheparametervaluesofmultiplesources ofuncertaintyundernaturalconditionsusingprobabilisticinverse mod-elling.Resultsshowwell-deﬁnedposteriorparameterdistributionsand

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modelresultscomparefavourablytomeasurements.Theparameter val-uesfoundinthenon-vegetatedcross-sectionwerefoundtodiﬀerfrom thevegetatedcross-section,whichsuggeststhatforthegivenmodel, thepresenceofavegetationpatchrequireshigherfrictionvaluesinthe surroundingnon-vegetatedpartaswell.Anexplanationforthishasnot yetbeenfound.

Futurechallengesforpracticalapplicationofvegetationmodelsin large scale,2D applicationsmayrequire relativelysimplemodelsin whichthepresenceofundergrowthandtheeffectsofreconfiguration aretakenintoaccount.Inversemodellingoffullscaleexperimentshas beenshowntobeanappropriatetoolforprovidingtheexperimental basisforfieldvalidationofthesemodels.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnancial interestsorpersonalrelationshipsthatcouldhaveappearedtoinﬂuence theworkreportedinthispaper.

CRediTauthorshipcontributionstatement

KoenD.Berends:Conceptualization,Methodology,Software, Val-idation,Formalanalysis,Writing-originaldraft,Visualization.UnJi: Conceptualization,Methodology,Investigation,Resources,Data cura-tion,Writing -originaldraft,Writing-review& editing,Project ad-ministration,Funding acquisition. W.E.(Ellis)Penning: Conceptual-ization,Resources,Writing-review&editing,Supervision,Project ad-ministration, Funding acquisition. JordJ. Warmink: Conceptualiza-tion,Methodology,Writing-originaldraft,Writing-review&editing, Supervision. JoonguKang: Validation, Investigation, Datacuration, Writing -review & editing, Supervision. SuzanneJ.M.H.Hulscher: Writing-review&editing,Supervision,Projectadministration,Project administration.

Acknowledgements

This research is part of theRiverCare research programme, sup-ported by the Dutch Technology Foundation TTW (project-number 13520),whichispartoftheNetherlandsOrganisationforScientic Re-search(NWO),andwhichispartlyfundedbytheMinistryofEconomic AﬀairsundergrantnumberP12-14(PerspectiveProgramme).This re-searchwasalsosupportedbytheInternationalMatchingJointResearch ProjectofKICT.WeareverygratefultoWijnandIJzermansandthe mea-surementgroupofKICT-RECandNNT(Nature&Technology)ofKorea fortheirhelpduringtheexperimentin2015.Weacknowledgethepeer reviewoftwoanonymousreviewers,whosevaluablecommentshelped improvethestyleandclarityofthispaper.

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