Framework for Event-based Semidistributed Modeling that Unifies the SCS-CN Method, VIC, PDM

18 Pages·2017·1.54 MB·English

Marquette University
e-Publications@Marquette
Civil and Environmental Engineering Faculty Civil and Environmental Engineering, Department
Research and Publications of
9-1-2016
Framework for Event-based Semidistributed
Modeling that Unifies the SCS-CN Method, VIC,
PDM, and TOPMODEL
Mark Bartlett
Duke University
Anthony J. Parolari
Marquette University, [email protected]
Jeffrey McDonnell
University of Saskatchewan
Amilcare Porporato
Duke University
Published version.Water Resources Research, Vol. 52 (September 2016): 7036-7052.DOI. © 2016
American Geophysical Union. Used with permission.
PUBLICATIONS
Water Resources Research
RESEARCH ARTICLE Framework for event-based semidistributed modeling that
10.1002/2016WR019084 unifies the SCS-CN method, VIC, PDM, and TOPMODEL
KeyPoints: M.S.Bartlett1,A.J.Parolari1,J.J.McDonnell2,3,andA.Porporato1
(cid:2)Semidistributedmodelsareposedin
event-basedformsthatconsistof 1DepartmentofCivilandEnvironmentalEngineering,DukeUniversity,Durham,NorthCarolina,USA,2GlobalInstitutefor
ready-to-useanalyticalexpressions WaterSecurity,UniversityofSaskatchewan,Saskatoon,Saskatchewan,Canada,3SchoolofGeosciences,Universityof
fortherainfall-runoffresponse
Aberdeen,Aberdeen,UK
(cid:2)Anewcanonicalformoftherainfall-
runoffcurveunifiestheevent-based
semidistributedmodelsandtheSCS-
CNmethod Abstract Hydrologistsandengineersmaychoosefromarangeofsemidistributedrainfall-runoffmodels
(cid:2)Theevent-basedrunoffcurveofeach suchasVIC,PDM,andTOPMODEL,allofwhichpredictrunofffromadistributionofwatershedproperties.
semidistributedmodelisextended
However,thesemodelsarenoteasilycomparedtoevent-baseddataandaremissingready-to-useanalytical
withprethresholdandthreshold-
excessrunoffmechanisms expressionsthatareanalogoustotheSCS-CNmethod.TheSCS-CNmethodisanevent-basedmodelthat
describestherunoffresponsewitharainfall-runoffcurvethatisafunctionofthecumulativestormrainfall
Correspondenceto: andantecedentwetnesscondition.Herewedevelopanevent-basedprobabilisticstorageframeworkand
A.Porporato, distillsemidistributedmodelsintoanalytical,event-basedexpressionsfordescribingtherainfall-runoff
[email protected]
response.Theevent-basedversionscalledVICx,PDMx,andTOPMODELxalsoareextendedwithaspatial
descriptionoftherunoffconceptof‘‘prethreshold’’and‘‘threshold-excess’’runoff,whichoccur,respectively,
Citation:
beforeandafterinfiltrationexceedsastoragecapacitythreshold.Fortotalstormrainfallandantecedent
Bartlett,M.S.,A.J.Parolari,
J.J.McDonnell,andA.Porporato wetnessconditions,theresultingready-to-useanalyticalexpressionsdefinethesourceareas(fractionofthe
(2016),Frameworkforevent-based watershed)thatproducerunoffbyeachmechanism.Theyalsodefinetheprobabilitydensityfunction(PDF)
semidistributedmodelingthatunifies
representingthespatialvariabilityofrunoffdepthsthatarecumulativevaluesforthestormduration,and
theSCS-CNmethod,VIC,PDM,and
TOPMODEL,WaterResour.Res.,52, theaverageunitarearunoff,whichdescribestheso-calledrunoffcurve.Thesenewevent-basedsemidis-
7036–7052,doi:10.1002/ tributedmodelsandthetraditionalSCS-CNmethodareunifiedbythesamegeneralexpressionforthe
2016WR019084.
runoffcurve.Sincethegeneralrunoffcurvemayincorporatedifferentmodeldistributions,itmayeasethe
wayforrelatingsuchdistributionstolanduse,climate,topography,ecology,geology,andother
Received17APR2016
characteristics.
Accepted23AUG2016
Acceptedarticleonline29AUG2016
Publishedonline17SEP2016
1.Introduction
Ideally, a mathematical framework for rainfall-runoff modeling should be applicable to watersheds every-
where.Runoffatapointisclassifiedasoneofthreephysicalprocesses,i.e.,infiltrationexcessoverlandflow,
saturation excess overland flow, and subsurface storm flow, all of which may be considered as a similar
threshold-initiated process once different watershed conditions are taken into account [McDonnell, 2013;
Dunne,1983].Therunoffprocessiscontrolledbytopographicconditions,aswellasgeologicalandecologi-
calcontrols.Bymappingrunoffprocessestothespatiallyheterogeneousvaluesofecologicalandgeological
properties,amodelshouldpredictrunoffaccordingtotheprevailingclimateandlocaltopographiccondi-
tions. Following this notion, Freeze and Harlan [1969] outlined the ‘‘fully distributed’’ modeling approach
where small-scale physics including runoff processes are explicitly mapped to watershed heterogeneities
thatarespatiallyresolvedtothegridcellelementssubdividingthewatershedmodel.
However, fully distributed models typically have significant data requirements for calibration [Semenova
andBeven,2015;McDonnelletal.,2007].Asaresult,hydrologistsoftenusesimplerspatiallylumpedmodels
suchastheevent-basedSoilConservationService(nowtheNaturalResourcesConservationService)Curve
Number(SCS-CN)methodandsemidistributedrainfall-runoffmodels[e.g.,PonceandHawkins,1996;Beven,
2012].Semidistributedmodelstypicallyoperateincontinuoustime(discretizedintotimesteps),andinlieu
of the variability of the grid cells of fully distributed models, they use a distribution, which is typically of
waterstoragecapacity.Forthedistributionofwaterstoragecapacity,thesemidistributedVariableInfiltra-
tion Capacity (VIC) model [Wood et al., 1992; Liang et al., 1994] and probability distributed model (PDM)
[Moore,1985]assumeaversatileprobabilitydensityfunction(PDF).Alternatively,thesemidistributedTOP-
VC2016.AmericanGeophysicalUnion.
AllRightsReserved. MODEL (TOPography-based hydrologic MODEL) defines similar watershed points according to the
BARTLETTETAL. FRAMEWORKFOREVENT-BASEDSEMIDISTRIBUTEDMODELING 7036
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topographicindex[BevenandKirkby,1979].However,thistopographicindexactsasthebasisforthedistri-
butionofstoragecapacityvalues[Sivapalanetal.,1997].
Previous studies have explored different water storage capacity distributions [Moore and Clarke, 1981],
examinedthephysicalbasisofsemidistributedmodelassumptions[Franchinietal.,1996],comparedsemi-
distributedmodelingapproaches[e.g.,HabetsandSaulnier,2001;Warrachetal.,2002],analyzedtheeffects
fromspatiallyvariableprecipitation[Sivapalanetal.,1997;Liangetal.,1996],andexploitedcombinationsof
different semidistributed assumptions [e.g., Noto, 2013]. In general, most semidistributed models assume
the same functional form for direct runoff, as the product of the rainfall rate and the threshold-saturated
area[Kavetskietal.,2003].Recentmodelingframeworks,e.g.,FrameworkforUnderstandingStructuralErrors
(FUSE)[Clarketal.,2008]andSUPERFLEX[Feniciaetal.,2011],usetheseequivalentfunctionalformsinter-
changeably,recognizingthatcertainmodelstructuresmaybebestsuitedforcertainwatershedcharacteris-
tics. While components of semidistributed models (such as a saturated variable source area (VSA)) have
beenincorporatedwithevent-basedmodelssuchastheSCS-CNmethod[e.g.,ChenandWu,2012],semidis-
tributedmodelshaveyettobegivenanevent-basedrepresentationwherethecumulativestormrunoffisa
functionofthecumulativestormrainfallandantecedentwetnessconditions.
Hereweposeasuiteofsemidistributedmodelsinevent-basedformbyfollowingtheBartlettetal.[2016]
ProStorframeworkwheretherainfall-runoffresponseatapointisupscaledtoawatershedscalebasedona
probabilistic joint distribution of storage capacity thresholds and the antecedent soil moisture status. Fol-
lowingtheframework,eachsemidistributedmodelcanbedistilledtoabasicsetofanalyticalexpressions
fordescribingtheevent-basedrainfall-runoffresponse.Whengiventhetotalstormrainfallandantecedent
watershedwetnessstatus,theseexpressionsdefine,thesourceareas(fractionofwatershed)thatproduce
runoffbyaspecificmechanism,theprobabilitydensityfunction(PDF)thatdescribesthespatialdistribution
ofrunoff,andthe(unitarea)averagerunoff.Theexpressionforaveragerunoff,commonlycalledarunoff
curve[e.g.,Descheemaeker etal.,2006;Wineet al.,2012], accountsforthespatialvariabilityof rainfall and
waterstorage.Notably,forexponentiallydistributedrainfalloverthewatershed,thesemidistributedmodels
andtheSCS-CNmethodareunifiedbythesamecanonicalformoftherunoffcurve(e.g.,equation(10)).
Indevelopingeachevent-basedmodel,weassumethesamepointrainfall-runoffresponsethatextendedthe
SCS-CN method to the SCS-CNx method [Bartlett et al., 2016]. This response consists of a fraction of area
whererunoffoccursbothbeforeandafterinfiltrationexceedsthestoragecapacitythreshold,andacomple-
mentaryareawhererunoffonlyoccursepisodicallywheninfiltrationexceedsthestoragecapacitythreshold.
ThisarearepresentstheoriginalrunoffresponseformulationofVIC,PDM,andTOPMODEL.Incomparison,the
newevent-basedmodelforms(calledVICx,PDMx,andTOPMODELx)areextendedwiththeadditional‘‘pre-
threshold’’runoffresponsethatoccursbeforeinfiltrationexceedsthestoragecapacitythreshold.
2.ExtendedEvent-BasedModels
Thefundamentalunderlyingconceptofsemidistributedmodelingisthatthedistributionofwatershedhet-
erogeneityisthebasisforawaterstoragecapacitydistributionrepresentedwithaprobabilitydensityfunction
(PDF).ThesewaterstoragecapacityPDFsarethebasisforderivingthecorrespondingevent-basedmodels.
2.1.ProStorFrameworkforEvent-BasedRainfall-RunoffResponse
Asemidistributedmodelmaybedescribedbyajointdistributionofthesoilmoisturedeficit,c,andwater
storage capacity, w, which describe the potential retention S5cw [see Bartlett et al., 2016]. In an event-
basedrepresentation,candSareconsideredantecedentvaluesimmediatelypriortothestartofthestorm,
whilethepointvaluesofrainfall,R,andrunoff,Q,representcumulativedepthsforthestormduration[see
Bartlett et al., 2016, Figure 1]. The spatial variability of these values may be described by the probabilistic
storage(ProStor)frameworkofthejointPDF
p ðQ;R;c;wÞ5p ðQjR;c;wÞp ðRÞp ðc;wÞ; (1)
QRcw QjRcw R cw
wherep ðQjR;c;wÞistherunoffPDFconditionalonR,c,andw;p ðRÞisthePDFofrainfall;andp ðc;wÞ
QjRcw R cw
is the joint PDF of the antecedent soil moisture deficit and storage capacity. As indicated by these PDFs,
rainfall,R,isreasonablyassumedtobestatisticallyindependentofboththeantecedentsoilmoisturedeficit,
c,andstoragecapacity,w[e.g.,Rodr(cid:2)ıguez-IturbeandPorporato,2004].
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Figure1.Thefractionareawiththreshold-excessrunofffordifferentwatershedaverageantecedentpotentialretentions,(cid:3)S,for(a)VICx/
PDMxand(b)TOPMODELx.SeeTable3forparametervalues.
Ifrunoffatawatershedpointismodeledasadeterministicfunction,i.e.,
Q5QðR;c;wÞ; (2)
thenthejointPDFbecomes
p ðQjR;c;wÞ5dðQðR;c;wÞ2QÞp ðRÞp ðc;wÞ; (3)
QjRcw R cw
where the conditional distribution p ðQjR;c;wÞ is now represented by the point mass of probability
QjRcw
dðQðR;c;wÞ2QÞ,forwhichdð(cid:3)ÞistheDiracdeltafunction.ThePDFdescribingthespatialdistributionofrun-
off is found by integrating equation (3) over the remaining variables (i.e., R, c, and w), while the lumped
(unitarea)runoffresponse(i.e.,therunoffcurve)istheaverageofequation(3)[Bartlettetal.,2016].Frame-
workvariablesandparametersarelistedinTable1.
2.2.TheJointPDFofAntecedentSoilMoistureandStorageCapacity
ForthesemidistributedmodelsofVIC,PDM,andTOPMODEL,itispossibletoshowthatthejointPDFp ðc;wÞ
cw
isapproximatedwithaformthatisexplicitlydependentonthespatialaverage(cid:3)S,i.e.,
8dðcwÞp ðwÞ for 0(cid:4)w<P21½Fð(cid:3)SÞ(cid:5)
>>< w w
p^ ðc;w;(cid:3)SÞ5 ; (4)
cw
>>:d(cid:2)cw2w1P21½Fð(cid:3)SÞ(cid:5)(cid:3)p ðwÞ P21½Fð(cid:3)SÞ(cid:5)(cid:4)w<w
w w w max
whereP21½(cid:3)(cid:5)isthequantilefunctionoftherespectivemodel(Table2),andP21½Fð(cid:3)SÞ(cid:5)representsthewater
w w
storagecapacity,w,attheedgeoftheprestormareaofthresholdsaturation,Fð(cid:3)SÞ.Notethattheconditional
PDFp ðcjwÞ,whichisrepresentedbyp^ ðcjw;(cid:3)SÞ,consistsofthepointmassesofprobabilitydðcwÞandd
cjw cjw
(cid:2)cw2w1P21½Fð(cid:3)SÞ(cid:5)(cid:3) that, respectively, indicate c
w
w50andcw5w2P21½Fð(cid:3)SÞ(cid:5)withprobability1.
w
Table1.ProStorVariablesandParametersa
The fraction of preexisting threshold-saturated
Symbol Description
area,Fð(cid:3)SÞ,isfoundfromtheself-consistentcon-
R Stormeventrainfalldepthatapoint[L]
dition[Bartlettetal.,2016]
w Waterstoragecapacitydepthatapoint[L]
xc AAnntteecceeddeennttssooiillmmooiissttuurreeadteaficpitoaintta,ðp1o2inctÞ,ð12xÞ (cid:3)S5Z Kð(cid:3)SÞZ wmax Sd(cid:2)S2w1P21½Fð(cid:3)SÞ(cid:5)(cid:3)p ðwÞdwdS;
FSt AFrnatcetcioendeonftwpaotteernshtieadlrwetiethnttihornesahtoaldp-oeixncte,sSs5rcuwno[fLf] 0 Pw21½Fð(cid:3)SÞ(cid:5) w w (5)
Q Stormeventrunoffdepthatapoint[L]
QQpt PThrertehsrheoslhdo-eldxcreusnsorfufndoefpftdheaptthapatoainptooivnetroðv1e2rFFttÞ whereKð(cid:3)SÞ5wmax2Pw21½Fð(cid:3)SÞ(cid:5),andtheintegrand
b Fractionofwatershedwithnon-zeroprethresholdrunoff is the second term of the r.h.s of equation (4)
PI Prethresholdrunoffindex,PI5bð12(cid:3)cÞ transformed by a change of variables for S5cw
n ShapeparameterofthestoragecapacityPDFpwðwÞ
[Bendat and Piersol, 2011] and modified accord-
aVariablesinthetextwithanoverlinebarindicateaspatialaverage
ingtothescalingpropertyofthedeltafunction.
(unitarea)depthvalue,e.g.,forthepointrainfalldepth,R,theaverage
(unitarea)valueisdenotedbyR(cid:3).Allvalueshavedimensionsoflength The delta function of equation (5) is evaluated
exceptc,x,b,PI,n,andFt,whicharedimensionless. using the property presented in Au and Tam
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Table2.VIC/PDMandTOPMODELExpressions
Expression VIC/PDM TOPMODEL
Storagecapacitydist. Pareto MirroredExponentialb
SIntvoerarsgeeCcDapFacityPDFa pPw2w1ðwðFÞÞ55ðwðwwmmwmmaaaxxax2x22wwÞ1wÞ=nnmaxð12FÞn pPww2ð1wðFÞÞ55Cw1mnwaxmnalnxe(cid:2)21w1mnaFxð(cid:2)wemna2x21w(cid:3)Þ(cid:3)
(quantilefunction)
Avg.storagecapacity w(cid:3)5w1m1axnn w(cid:3)5wmax(cid:4)C121n(cid:5)
Avg.potentialretention (cid:3)S5w(cid:3)ð12FÞn11 (cid:3)S5C1wn(cid:3)21(cid:4)F2C1ln(cid:4)FCe1n11(cid:5)(cid:5)1w(cid:3)
Antecedentsat.areac Fð(cid:3)SÞ512(cid:4)w(cid:3)(cid:3)S(cid:5)1=ð11nÞ Fð(cid:3)SÞ512C12C1W 2e2(cid:4)11(cid:3)SðCw(cid:3)1Cn121Þ(cid:5)!
Avg.soilmoisturedeficitd (cid:3)c5ð12Fð(cid:3)SÞÞ (cid:3)c512Fð(cid:3)SÞ
(cid:3)(cid:4)122F1(cid:4)1;1;111n;11ð12Fð(cid:3)S1ÞÞn21(cid:5)(cid:5) 1C1e2nln(cid:4)FðC(cid:3)S1Þen11(cid:5)(cid:4)li(cid:4)FðC(cid:3)S1Þen11(cid:5)2EiðnÞ(cid:5)
C15aF1o2r1ev2anl(uAepspoefn0d<ixBw).<wmax andnistheshapeparameteroftherespectivePDF.InthecaseofTOPMODEL,n5jmaxj2sjminand
bDerivedfromatruncatedexponentialPDFrepresentationofthetopographicindexdistribution(AppendixB).
cWhereWð(cid:3)ÞistheLambertW-function[Olveretal.,2010,p.111]giveninMatlabbylambertw((cid:3))andinMathematicaby
ProductLog((cid:3)).NotethatExcelrequiresanadd-intocalculatetheLambertfunction.
dWhere2F1ð(cid:3);(cid:3);(cid:3);(cid:3)Þisthehypergeometricfunction[Olveretal.,2010,p.384]giveninMatlabby2F1ða;b;c;dÞ5hypergeom([a,b],c,
d)andinMathematicaby2F1ða;b;c;dÞ5Hypergeometric2F1[a,b,c,d].lið(cid:3)ÞandEið(cid:3)Þaretherespectiveexponentialandlogintegrals
[Olveretal.,2010,p.150].
[1999].ForeachassumedstoragecapacityPDFp ðwÞ,equation(5)issolvedforthefunctionalformofFð(cid:3)SÞ
w
(seeTable2).NotethattheFð(cid:3)SÞforVIC/PDM(Table2)waspreviouslyfoundbyKavetskietal.[2003]byinte-
gratingthequantilefunctionofTable2basedonassumptionsfortheconditionalsoilmoisturedistribution,
whichareexplicitlystatedinequation(4).
Based on the joint PDF p^ ðc;w;(cid:3)SÞ of equation (4) and the Bartlett et al. [2016] spatial description of
cw
threshold-excessandprethresholdrunoffatapoint,wederiveanevent-basedmodelrepresentationthatis
general to the PDFs of rainfall, p ðRÞ, and storage capacity, p ðwÞ (see Appendix A). This point runoff
R w
descriptionofBartlettetal.[2016]consistsofbothprethresholdandthreshold-excessrunoffoverthefrac-
tion of watershed area b, while over the complementary area, 12b, the point response only consists of
threshold-excess runoff. The fraction b may represent riparian and lower hillslope areas with a persistent
hydrologicconnectiontothestreamthatfacilitatesaprethresholdrunoffbeforerainfallinfiltrationexceeds
the storage capacity threshold. The complementary fraction of area 12b may represent upslope areas
whereprethresholdrunoffiszerobecausetheconnectiontothestreamepisodicallyoccurswheninfiltra-
tionfillsandthenspillsoverthestoragecapacitythreshold[Bartlettetal.,2016].
2.3.RainfallandStorageCapacityPDFAssumptions
Eachnewevent-basedmodelassumesaspecificformofthestoragecapacityPDF,p ðwÞ(seeTable2).Typ-
w
ically,VICandPDMbothassumeaParetodistributionforthePDFp ðwÞ,andsowerefertotheextended
w
event-basedversionsofVICandPDMcollectivelyasVICx/PDMx[Moore,2007;Liangetal.,1994].Differently,
in the case of TOPMODEL, p ðwÞ has a physical basis in the slope and contributing area at a point that
w
define the topographic index, j (see Appendix B). Over a watershed, the spatial distribution of the topo-
graphicindexmayberepresentedbythePDFp ðjÞ.Herewerepresentp ðjÞwithanexponentialPDF,and
j j
thisPDF typically providesa goodfit to theempiricaldistribution of thetopographic index,especially for
larger values that are representative of lowland areas where the majority of runoff production occurs
accordingtotheTOPMODELconcept(AppendixB).StartingfromthisexponentialPDF,wederivethemir-
roredexponentialPDF(Table1)thatreasonablyrepresentsthedistributionofTOPMODELstoragecapacity
(seeAppendixB).
Foreachstormevent,weassumethespatialdistributionofrainfallfollowsanexponentialPDF,i.e.,
1
p ðRÞ5 e2R=R(cid:3): (6)
R R(cid:3)
This PDF has been used to represent the spatial distribution of rainfall in small watershed areas of about
40km2[Yu,1998;Schaakeetal.,1996],aswellasinclimateandlarge-scalehydrologicmodels[Thomasand
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Henderson-Sellers,1991;Liangetal.,1996;Tangetal.,2007;Sivapalanetal.,1997;Shuttleworth,1988;Arnell,
2014,p.279].Inthenextsections,weassumetheexponentialrainfallPDFofequation(6)andthestorage
capacityPDFsofTable2andpresenttheanalyticalexpressionsoftheextendedmodelsofVICx/PDMxand
TOPMODELx.
2.4.FractionofAreaWithThreshold-ExcessRunoff
Foreachevent-basedmodel,thefractionofareaF representsthegreatestextentofthreshold-excessrun-
t
offattheendofastormevent.Itisafundamentalquantitythatformsthebasisofthesubsequentevent-
basedmodelexpressions.Followingequation(A4),forVICx/PDMxthefractionofareawiththreshold-excess
runoffis
Fð(cid:3)S;R(cid:3)Þ5Fð(cid:3)SÞ1exp 2wmaxð12Fð(cid:3)SÞÞn!1(cid:6)2 wmax (cid:7)21n C(cid:6)1(cid:7)2C 1;2wmaxð12Fð(cid:3)SÞÞn!!; (7)
t R(cid:3)ð12PÞ n R(cid:3)ð12PÞ n n R(cid:3)ð12PÞ
I I I
where Cð(cid:3)Þ and Cð(cid:3);(cid:3)Þ are the gamma and lower incomplete gamma functions, respectively [Olver et al.,
2010](seeAppendixC).ForTOPMODELx,thefractionofareawiththreshold-excessrunoffis
Fð(cid:3)S;R(cid:3)Þ5Fð(cid:3)SÞ1 C1R(cid:3)ð12PIÞn (cid:6)Fð(cid:3)SÞ1e2n(cid:7)R(cid:3)ðw1m2aPxIÞn2(cid:6)Fð(cid:3)SÞ1e2n(cid:7)!: (8)
t R(cid:3)ð12PIÞn2wmax C1 C1
FortheSCS-CNxmethodthefractionofareaFð(cid:3)S;R(cid:3)ÞisgivenbyBartlettetal.[2016,equation(24)],where(cid:3)S
t
5(cid:3)cw(cid:3) becausecandwarestatisticallyindependent.
Forequations(7)and(8),thefirsttermisthefractionofareawiththresholdsaturationpriortothestorm,
andthesecondtermisthefractionofareawherethreshold-excessrunoffdevelopsduringthestorm.Over
thecomplementaryfractionofareað12Fð(cid:3)S;R(cid:3)ÞÞ,prethresholdrunoffoccursoverbð12Fð(cid:3)S;R(cid:3)ÞÞbutiszero
t t
overð12bÞð12Fð(cid:3)S;R(cid:3)ÞÞ.ThefractionofFð(cid:3)S;R(cid:3)Þvarieswith(cid:3)S (Figures1aand1b).Notethatanincreasein
t t
the prethreshold runoff area (an increase in b) decreases Fð(cid:3)S;R(cid:3)Þ and thus the extent of threshold-excess
t
runoffoverthewatershed.WhenP 50,equations(7)and(8)ofVICx/PDMxandTOPMODELxdescribethe
I
extentofthreshold-excessrunoffwhentheprethresholdrunoffiszero.
2.5.RunoffPDF
Foreachstormevent,therunoffPDFrepresentsthespatialvariabilityofrunoffdepthsthatarecumulative
values for the storm duration. Following equation (A5), this PDF is the weighted sum of the prethreshold
and threshold-excess runoff PDFs of equations (C1–C4). For the exponential rainfall PDF and no prethres-
holdrunoff(b50andP 50),therunoffPDFhasaremarkablysimpleformforallmodels,i.e.,
I
Fð(cid:3)S;R(cid:3)Þ
pQðQÞ5ð12Ftð(cid:3)S;R(cid:3)ÞÞdðQÞ1 t R(cid:3) e2R(cid:3)Q; (9)
whereFð(cid:3)S;R(cid:3)Þisthatofeitherequations(7)and(8)orBartlettetal.[2016,equation(24)]fortheSCS-CNx
t
method.Inequation(9),runoffiszerooverthefractionofarea12Fð(cid:3)S;R(cid:3)Þasrepresentedbythepointmass
t
of probability ð12Fð(cid:3)S;R(cid:3)ÞÞdðQÞ, while the runoff depths are exponentially distributed over the fraction of
t
areaFð(cid:3)S;R(cid:3)ÞasrepresentedbyPDFofthesecondterm.
t
Forthegeneralcaseofb6¼0,theprethresholdrunoffcontributestotheprobabilitydensityforsmallerrun-
offvalues(Figure2,dashedlineandbar),whilethethreshold-excessrunoffcontributesmoretotheproba-
bilitydensityforlargerrunoff values(Figure2,grayline).Asa result,theexpressionsarenotassimpleas
equation(9),butarestillexactexpressionsasshowninAppendixC.NotetheblackbarsofFigure2repre-
sent the fraction of area that produces zero runoff as indicated by the atom of probability
ð12Fð(cid:3)S;R(cid:3)ÞÞð12bÞdðQÞ, which is the first term of prethreshold runoff PDFs weighted by 12Fð(cid:3)S;R(cid:3)Þ (see
t t
equations(C1)and(C2)).
2.6.AverageRunoff(theRunoffCurve)
ThekeyresultisanexpressionoftheaveragerunoffQ(cid:3) asafunctionofaveragerainfall,R(cid:3),andtheaverage
antecedentpotentialretention,(cid:3)S,theso-called‘‘runoff-curve.’’WhiletheextendedSCS-CNxmethod,VICx/
PDMx,andTOPMODELxarebasedondifferentassumptions,eachmodelhasthesamebasicrunoffcurve
expression,i.e.,
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Figure2.TherunoffPDFsfor(a)VICx/PDMxand(b)TOPMODELx,respectively,when(cid:3)S570andR(cid:3)525mm,whichconsistsofacontinuous
probabilitydensity(rightaxis,solidline)andanatomofprobabilityforQ50(leftaxis,blackbar).InbothcasesthePDFisthetotalofthe
(wgeraigyhltineed),pir.ee.t,hFrteðs(cid:3)Sh;oR(cid:3)lÞdpQrutðnQoÞff(APpDpFe(nddasixheCd).lSineeeaTnabdlebl3acfokrbpaar)r,aim.e.e,tðe1r2svFatðl(cid:3)uS;eR(cid:3)s,ÞaÞpnQdphðQeÞre,pwluesatshseumweeigbh5te0d:25th.reshold-excessrunoffPDF
Q(cid:3)5R(cid:3)Fð(cid:3)S;R(cid:3)Þ1R(cid:3)ð12Fð(cid:3)S;R(cid:3)ÞÞP: (10)
t t I
TheexactformsofeachrunoffcurvedifferbasedonthefunctionFð(cid:3)S;R(cid:3)Þ.Therunoffcurveofequation(10)
t
is the first of its kind for describing an event-based semidistributed runoff response consisting of both
threshold-excess runoff generated onthe fraction of area Fð(cid:3)S;R(cid:3)Þ and theprethreshold runoff originating
t
from the complementary fraction of area 12Fð(cid:3)S;R(cid:3)Þ. When P 50, the runoff curve defaults to R(cid:3)Fð(cid:3)S;R(cid:3)Þ,
t I t
whichdescribestherunoffresponseoftheoriginalmodels,butinanevent-basedformat.NotethatforP5
I
bð12(cid:3)cÞ;(cid:3)cmaybegivenintermsof(cid:3)S(seeTable2).
Followingequation(A8),theaveragerunoffofequation(10)istheweightedsumoftheaverageprethres-
holdandthreshold-excessrunoffcomponentsgivenbyequations(C5–C8).Astheaveragerainfallincreases,
the average runoff also increases (Figure 3), with higher amounts of runoff being produced as thewater-
shed nears saturation, i.e., (cid:3)S approaches 0 (Figure 3). The behavior of the prethreshold runoff augments
threshold-excessrunoffandincreasesthetotalaveragerunoff,especiallyforsmallervaluesofaveragerain-
fall (Figure 3). Although the majority of the runoff is threshold-excess runoff (Figure 3), the area of
threshold-excessrunofftypicallycomprisesasmallareaofthewatershed(e.g.,lessthan50%).Thisisespe-
ciallytrueastheamountofprethresholdrunoffincreases(i.e.,asbapproaches1)becausetheprethreshold
runoffactsasadrainagemechanismforthewatershedthatdecreasesthesoilwatercontentandthusthe
occurrenceofthreshold-excessrunoff.
3.AnalysisoftheEvent-BasedModels
3.1.CaseStudyAreaandData
Toanalyzethemodelbehavior,weusedatafromtheDavidsonriverwatershed,whichconsistsofapproxi-
mately104km2nearBrevard,NorthCarolina,USA.Atthewatershedoutlet,streamflowisrecordedbythe
Figure3.Theaveragerunoff,Q(cid:3)(solidlines)versustheaveragerainfall,R(cid:3)for(a)VICx/PDMxand(b)TOPMODELx.Theaveragerunoffisthe
weightedsumoftheaveragethreshold-excessrunoff,Ftð(cid:3)S;R(cid:3)ÞQ(cid:3)t(dashedlines),andtheaverageprethresholdrunoffð12Ftð(cid:3)S;R(cid:3)ÞÞQ(cid:3)p;see
AppendixC).Casesareshownfor(cid:3)S535(blacklines),and(cid:3)S5123(graylines).SeeTable3forparametervalues.
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Figure4.FortheDavidsonRiver,(a)therainfall-runoffdata(graydots)andrank-orderdatarunoffcurve(reddashedline)and(b)a
comparisontherank-orderdatarunoffcoefficients(graydots)tothemodels(lines)forthe(cid:3)SandPofTable3.FortheparametersofTable
I
3contoursindicatetheratioofQ(cid:3)fordifferentw(cid:3) (andPI)toQ(cid:3)forconstantvaluesof(c)w(cid:3)5122mm(andPI50:061)forVICxandPDMx
and(d)w(cid:3)5153mm(andPI50:088)forTOPMODELx.Notethatw(cid:3) andPIsatisfy0.125Fð(cid:3)SÞ1PIwhere0.12istheinitialrunoffcoefficientof
thedataandFð(cid:3)SÞisdependentonw(cid:3) (Table2).
UnitedStatesGeologicalSurvey(USGS)gauge03441000.Dailyrainfallvaluesaremeasuredbythenearest
USGSraingauge(03455773)thatislocated19kmawayatLakeLogan.A3mresolutiondigitalelevation
model(DEM)downloadedfromtheNationalElevationDataset(ned.usgs.gov)describesthetopographyof
the watershed. From this DEM, we computed the topographic index with the TauDEM D-Infinity tools
[Tarboton and Mohammed, 2013], and clipped the resulting values to the Davidson River area using the
WatershedBoundaryDataset(nhd.usgs.gov,HUC12,060101050202).
We examined Davidson river streamflow and rainfall data between 1 December 1998 and 11 November
2015,andonlyconsideredstormeventswhererainfallexceeded2mm.Rainfallvaluesarecumulativetotals
foreachstormevent,andweassumeeachstormmayconsistofuptothreeconsecutivedailyvalues.For
eachstorm,weseparatedstormflowvaluesfromtheUSGSmeasuredstreamflowusingthehydrographsep-
arationprogram(HYSEP)[Sloto andCrouse,1996;Eckhardt,2008]. Runoff valuesalsoarecumulativetotals
that are found by integrating the stormflow over the duration of each storm event. The period of record
(6196days)contained1212rainfall-runoffeventswithanaveragerainfallofhR(cid:3)i515:5mmandaveragerun-
offofhQ(cid:3)i54:1mm(Figure4a).
Inaddition,werank-orderedtherainfall-runoffdatabypairingrespectivelistsoftherainfallandrunoffval-
uesthathavebeensortedindescendingorder.Inthisway,therainfall-runoffdataarematchedbyfrequen-
cy, andconsequently runoff typically is no longermatched with thecausativerainfall event. However,for
theperiodofrecord,thisrank-orderdataapproximatelyrepresentstherainfall-runoffresponse(i.e.,therun-
offcurve)foraveragewatershedconditions,i.e.,(cid:3)S (cid:6)h(cid:3)Si(Figure4a,dashedredline)[seeBartlettetal.,2016,
AppendixD].WereducethenumberofparametersbysubstitutingP50:122Fð(cid:3)SÞwhere0.12istheinitial
I
runoff coefficient for small rainfall events between 2 and 5 mm (Table 3). The remaining parameters (i.e.,
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Table3.ModelParametersandRMSEComparisonfortheDavidsonRiverRank-OrderData
VICx/PDMx TOPMODELx SCS-CNx Unit Description
(cid:3)S 68a 71a 198a mm Averageantecedentpotentialretention
P 0.06a 0.088a 0.12a Prethresholdrunoffindex
I
wmax 137a 182a 1 mm MaximumpointstoragecapacityandPDFscaleparameter(Table2)
n 8.42a 6.3b ShapeparameterofstoragecapacityPDF(Table2)
jmax 12.5c Max.topographicindex
jmin 3.2d Min.topographicindex
js 1.48d ParameteroftopographicindexPDF(seeequation(B4))
RMSE 0.012 0.011 0.033 Rootmeansquareerrorbetweendataandmodelrunoffcoefficients
aNonlinearleastsquaresfitofequation(10)torainfall-runoffdataaftersubstitutingPI50:122Fð(cid:3)SÞwhere0.12istheinitialrunoff
coefficientforrainfallbetween2and5mm.
bParameterisequalton5jmax2jmin(seeAppendixB).
cValueforwhich95.5%topogrjasphicindexvalues,j,arebetween0<j<jmax.Valuesbetweenjmax <j<1representstream
channelareasthatoccupy0.5%ofthewatershedbasedonadrainagedensityofD56.26mi21[Carlston,1963]andanaveragestream
channelwidthofCw54:5ft[Chapman,1996,p.248],i.e.,0:0055DCw. (cid:4) (cid:5)
dNonlinearleastsquaresfitofthecumulativedistributionfunction(CDF)C12C1exp 2j1jsjmin totheempiricalcumulativedistribu-
tionofthetopographicindex.TheCDFcorrespondstothePDFofequation(B4).
w , n, and (cid:3)S) are based on a nonlinear least squares fit of each model runoff curve to the rank-order
max
(Table3).InthecaseofTOPMODEL,thePDFshapeparameternisderivedfromtopographicindexparame-
ters that are found from a nonlinear least squares fit of an exponential cumulative distribution function
(CDF)totheempiricaldistributionofthetopographicindex(AppendixBandTable3).
3.2.DataandModelBehaviorComparison
Weinitially compare each model runoff curveto therank-ordereddata and discusshow therunoff curve
mayrelatetodifferentbasinattributes.Second,wecomparethemodelrunoffcurvesunderdifferentrainfall
andantecedentwetnessscenarios,andwerelatethemodeldifferencestothespatialvariabilityoftherun-
offresponse.Themodelrunoffcurves(equation(10))andrank-ordereddataarecomparedintermsofrun-
off coefficients, Q(cid:3)=R(cid:3) (Figure 4b). Each of the fitted models reproduces the rainfall-runoff event behavior
withasmallrootmeansquarederror(RMSE)(Figure4andTable3).BothVICx/PDMxandTOPMODELxpro-
videabetterfittothedatathantheSCS-CNxmethod.IncomparisontoVICx/PDMxandTOPMODELx,the
SCS-CNx method predicts a larger runoff coefficient for smaller rainfall events, while predicting a smaller
runoffcoefficientforlargerrainfallevents.ThisdifferenceresultsfromdifferentPDFsp^ ðc;w;(cid:3)SÞassumed
cw
forwatershedvariability.
Eachmodel(Figure4b)representstherainfall-runoffresponsecurveforastormeventdefinedbytheTable
3valuesofn,w ;(cid:3)S,andP.Ineachmodel,theinitialrunoffcoefficientofthedata(i.e.,0.12)isequalto
max I
Fð(cid:3)SÞ1P.SinceFð(cid:3)SÞisdependentonw(cid:3) (seeTable2),manycombinationsofw(cid:3) andP resultinapproximately
I I
similar average runoff values (Figures 4c and 4d), which is the so-called problem of ‘‘equifinality’’ [Beven,
2006]. Consequently, the same storm event runoff Q(cid:3) could result from a shallower basin (smaller w(cid:3)) that
producesmorethreshold-excessrunoffbutlessprethresholdrunoff,oritcouldresultfromadeeperbasin
(largerw(cid:3))thatproduceslessthreshold-excessrunoffbutmoreprethresholdrunoff.WhileQ(cid:3) wouldbesimi-
lar, the spatial description of runoff would be different. Thus fitting each model to a single storm event
requiresbothrainfall-runoffdataandotherprocessinformationsuchasthemappednearstreamsaturated
area.
For small rainfall events over the Davidson river basin, the SCS-CNx method produces more runoff than
either VICx/PDMx (Figure 5a) or TOPMODEL (Figure 5b). The greatest differences between the SCS-CNx
methodandeitherVICx/PDMxorTOPMODELxoccursforrainfallbetween0<R(cid:3) <50mm,andthemagni-
tudeofthedifferencesdecreaseswith(cid:3)S(Figure5).Inwetterconditions(e.g.,(cid:3)S <50Þ,therunoffpredictions
of the SCS-CNx method, VICx/PDMx, and TOPMODELx become more similar, but under drier conditions
(e.g.,(cid:3)S>50),theSCS-CNxmethodmaypredict4timesmorerunoffthanTOPMODELxandnearly5times
morerunoffthanVICx/PDMx(Figures5aand5b).
Differencesin theaveragerunoff response (Figures 5a and 5b) aremainly attributed to differences in the
fraction of area with threshold-excess runoff, Fð(cid:3)S;R(cid:3)Þ. The behavior of Fð(cid:3)S;R(cid:3)Þ with rainfall is different for
t t
each model because of difference in the joint PDF p^ ðc;w;(cid:3)SÞ. Specifically, within this joint PDF, the
cw
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Figure5.ContoursindicatetheratioofSCS-CNxmethodrunoff,Q(cid:3)SCSx,to(a)VICx/PDMxrunoff,Q(cid:3)VICxPDMx,and(b)TOPMODELxrunoff,
Q(cid:3)TOPMODELx.Thefractionofareawiththreshold-excessrunofffor(c)(cid:3)S530and(d)(cid:3)S590.SeeTable3forparametervaluesforwhich
thefractionbisequalto0.085and0.16forVICx/PDMxandTOPMODELx,respectively,and0.24fortheSCS-CNxmethodassuming
w(cid:3)5400mm.
marginal storage capacity PDF, p ðwÞ is skewed toward smaller storage capacity values for the SCS-CNx
w
method,butfortheDavidsonrivercasestudyarea,thedistributionisskewedtowardlargervaluesforVICx/
PDMxandTOPMODELx(Figures5cand5d).Underwetterconditions(e.g.,(cid:3)S530Þ,themodelstendtohave
similar values of Fð(cid:3)S;R(cid:3)Þ, while under drier conditions (e.g., (cid:3)S590Þ, the differences in Fð(cid:3)S;R(cid:3)Þ tend to be
t t
greater(Figures5cand5d).
4.Discussion
4.1.UnifiedandExtendedSemidistributedModels
Themaincontributionofthisworkisthedevelopmentofageneralexpressionfortheaveragerunoff(i.e.,
therainfall-runoffcurve)thatunifiesthedifferentmodelingapproachesoftheSCS-CNmethod,VIC, PDM,
andTOPMODEL.Thesegeneralexpressionsincludesbothprethresholdandthreshold-excessrunoffcompo-
nents over the respective fractions of area ð12Fð(cid:3)S;R(cid:3)ÞÞb and Fð(cid:3)S;R(cid:3)Þ. Each runoff area is modeled with a
t t
distributionofstoragecapacitythresholdswhereeachlocalthresholdmayactasaproxyformanydifferent
rainfall-runoff mechanisms, e.g., subsurface flow and overland flow by saturation of infiltration excess
[Bartlett et al., 2016; McDonnell, 2013]. Aside from the expression for Fð(cid:3)S;R(cid:3)Þ, thespatial variability of pre-
t
thresholdandthreshold-excessrunoffisalsorepresentedbytherunoffPDFp ðQÞ.Thisspatialcharacteriza-
Q
tionofrunoffisfurtherenhancedbythefactthateachmodelaccountsforthespatialdistributionofrainfall
withtheexponentialPDFofequation(6).Forallmodels,thefractionofareabandtheaverageantecedent
potentialretention,(cid:3)S,governthemagnitudeandspatialextentoftherunoffresponse.
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