Sensitivity and Interaction Analysis Based on Sobol Method and Its Application in a Distributed Flood Forecasting Model

Transcription

1 Water 205, 7, ; do:0.3390/w Artcle OPEN ACCESS water ISSN Sestvty ad Iteracto Aalyss Based o Sool Metod ad Its Applcato a Dstruted Flood Forecastg Model Hu Wa,2, Ju Xa,2, *, Lpg Zag,2, *, Duxa Se,2, Yag Xao,2 ad Le Zou,2 State Key Laoratory of Water Resources ad Hydropower Egeerg Scece, Wua Uversty, No.8 Dogu Sout Road, Wua , Ca; E-Mals: wau208@26.co (H.W.); seduxa@sa.co (D.S.); xaoyag98990@wu.edu.c (Y.X.); zzu_zoule@63.co (L.Z.) 2 Hue Collaoratve Iovato Ceter for Water Resources Securty, Wua Uversty, Wua , Ca * Autors to wo correspodece sould e addressed; E-Mals: xaj@gsrr.ac.c (J.X.); zlpwu@sa.co.c (L.Z.); Tel.: (J.X.); (L.Z.); Fax: (J.X. & L.Z.). Acadec Edtor: Ataasos Loukas Receved: 28 Feruary 205 / Accepted: 8 Jue 205 / Pulsed: 7 Jue 205 Astract: Sestvty aalyss s a fudaetal approac to detfy te ost sgfcat ad sestve paraeters, elpg us to uderstad coplex ydrologcal odels, partcularly for te-cosug dstruted flood forecastg odels ased o coplcated teory wt uerous paraeters. Based o Sool etod, ts study copared te sestvty ad teractos of dstruted flood forecastg odel paraeters wt ad wtout accoutg for correlato. Four ojectve fuctos: () Nas Sutclffe effcecy (ENS); (2) water alace coeffcet (WB); (3) peak dscarge effcecy (EP); ad (4) te to peak effcecy (ETP) were pleeted to te Luxe odel wt ourly rafall-ruoff data collected te Naua Creek catcet, Pearl Rver, Ca. Cotrastve results for te sestvty ad teracto aalyss were also llustrated aog sall, edu, ad large flood agtudes. Results deostrated tat te coce of ojectve fuctos ad o effect o te sestvty classfcato, wle t ad great fluece o te sestvty rakg for ot ucorrelated ad correlated cases. Te Luxe odel eaved ad respoded uquely to varous flood codtos. Te results also dcated tat te parwse paraeters teractos revealed a o-gorale cotruto to te odel output varace. Paraeters wt g frst or total order sestvty dces preseted a correspodg g secod order sestvty dces ad

2 Water 205, correlato coeffcets wt oter paraeters. Wtout cosderg paraeter correlatos, te varace cotrutos of gly sestve paraeters gt e uderestated ad tose of orally sestve paraeters gt e overestated. Ts researc lad a asc foudato to prove te uderstadg of coplex odel eavor. Keywords: sestvty aalyss; teracto aalyss; correlated paraeters; ojectve fuctos; flood agtudes; Sool etod. Itroducto A dstruted ydrologcal odel s a essetal tool ad as ee wdely used ydrology ad water resources aageet due to ts good alty sulatg teporal ad spatal varato of ydrologcal varales [,2]. I practce, te perforace of a ydrologcal odel s gly correlated to te settg of odel paraeters. Because of te exstg peoeo called equfalty [3,4],.e., dfferet paraeter sets resultg te sae odel sulatos, tere s large ucertaty te applcato of ydrologcal odels. Varous selecto of paraeters wll duce a large varety of sulato results; o te oter ad, cosderg tat te dstruted ydrologcal odels always cota coplex structures wt large uer of paraeters, te optzato coce of paraeters s cosdered to e a dffcult ad te-cosug task. However, te sestvty aalyss (SA), wc volves testg te relatve pact of odel paraeters o te ruoff predcto, as ee proved to e a effcet way to detfy te paraeters ydrologcal odels ad as ee appled ay studes [5,6]. Soe studes ave stated tat SA ca prove paraeter calrato effcecy, avod over-paraeterzato [7], reduce odel ucertaty [8 0], ad uderstadg odel structure [,2], ad geerate ore accurate odel outputs [6,3]. I geeral, SA etods ca e roadly grouped to two types: local etods ad gloal etods [6,7,4,5]. Te local etods oly ake oe paraeter cage at oe te, wle te oter paraeters are kept as costat [7,6]. Te advatage of tese etods s tat oly a sall coputato s eeded durg ter applcato, wle te drawack s tat tey caot detere te teracto etwee paraeters, tus ay result a large as of SA. O te cotrary, te gloal etods overcoe te drawacks of local etods y cagg all of te odel paraeters at te sae te [6,7,4,5]. Ts approac ca cosder te lear or olear teracto etwee odel paraeters, ad t s suggested to e ore approprate to coplex ad pyscally-ased dstruted ydrologcal odels ta local etods. May gloal etods ave ee proposed ad used, suc as Regoal Sestvty Aalyss (RSA) [7], Paraeter Estato Software (PEST) [8], Aalyss of Varace (ANOVA) [4,9], Fourer apltude sestvty test (FAST) [5,20], Sool etod [6,2], etc. Aog tese etods, Sool s proved to e oe of te ost roust ad effectve tools to descre dvdual ad cooperatve sestvtes [6,4,5,20 23], ad t s wdely appled ot oly ydrologcal odels [4,2,23 25], ut also evroetal odels [26], cecal odels [27], ologcal odels [28], ecologcal odels [29], etc. Tag et al. [4] ad Ga et al. [5] copared Sool etod wt oter etods to aalyze te sestvtes of te SAC-SMA odel. Tag et al. [4] coducted a cotradstctve study etwee Sool etod ad tree oter etods for two waterseds wt two evaluato etrcs cludg

3 Water 205, o-trasfored root ea square error (RMSE) ad trasfored RMSE (TRMSE). Ga et al. [5] also copared Sool etod wt egt oter etods for frst, secod, ad total order sestvtes ased o te ea asolute error (MAE) of te sulated ad oserved flow easureet, te Sout Brac Potoac Rver as te U.S. Reusser et al. [20] exaed te doat paraeter ad paraeter teracto of TOPMODEL ad a coplex WaSM-ETH odel y te coparso of te Sool etod wt FAST ad exted FAST. Za et al. [6] proposed a gly effcet tegrated sestvty aalyss tool o te ass of Sool etod for a DTVGM odel te Huae Rver Bas, Ca. Te fdgs revealed te paraeters sestvtes vared wt regard to tree dfferet evaluato crtera, aely, water alace coeffcet, Nas Sutclffe coeffcet, ad correlato coeffcet. More recetly, Zag et al. [23] proved te uderstadg of te SWAT odel troug Sool sestvty aalyss etod at te Ycu Rver Bas Ca, y explorg odel perforace ased o four etrcs RMSE, Nas Sutclffe, ruoff coeffcet error, ad slope of te flow durato curve error uder dry, oderate, ad wet clate codtos. Prevous studes ave valdated te roustess ad ease of pleetato of Sool etod to geerate gloal sestvty aalyss (cludg frst, secod, ad total order) uerous ydrologcal odels. Lttle atteto as ee pad to te depedet assupto of paraeters Sool etod. Most researcers [6,4,5,23] dd ot cosder te correlato uderlyg te paraeters, eve toug great secod order sestvty dces were dsplayed. For ts case, a geeralzed Sool etod [30] tat cosders correlato s ore sutale for correlated paraeters. Besdes, as Wageer et al. [24] poted out te sgfcace of ultple ojectve fuctos durg te odel perforace assesset, researcers realzed te portace of reducg te ucertaty of sestvty aalyss y ult-etrcs, varous caracterzed catcets, ad clate codtos. However, ost dstruted odels were carred out wt daly or otly data [6,5,23], seldo wt ourly data. I addto, eve toug ultple ojectve fuctos were stregteed, lttle atteto was pad to te peak dscarge tg ad agtude. I ts study, te Morrs screeg etod ad te exteded [3] ad geeralzed [30] Sool etod for ucorrelated ad correlated paraeters are pleeted to te dstruted flood forecastg odel aed te Luxe odel y four ojectve dcators tat ca explctly caracterze te flood processes: () Nas Sutclffe effcecy (ENS); (2) water alace coeffcet (WB); (3) peak dscarge effcecy (EP); ad (4) te to peak effcecy (ETP). Dfferet types of flood agtude (sall, edu, ad large) are cosdered to copare te sestvty ad teracto of paraeters te Luxe odel. Te Naua Creek catcet, wc s located te Pearl Rver as, Ca, s selected as te study area, ad te ourly rafall-ruoff data s used. 2. Metodology Te sestvty ad teracto aalyss ased o Sool etod ca e carred out y te followg steps: () screeg effectve paraeters out y Morrs etod [5]; (2) detfyg te paraeters correlatos troug Geeralzed Lkelood Ucertaty Estato (GLUE) [3,4]; (3) saplg paraeter sets wt ad wtout cosderg te detfed paraeters correlatos; (4) evaluato of te sestvty dces of odel correlated ad ucorrelated paraeters.

4 Water 205, Morrs Screeg Morrs [5] proposed a oe-at-a-te (OAT) desg to scree te ost fluetal paraeters easly ad effectvely wt a sall aout of coputato, wc s ow aed te Morrs screeg etod or Eleetary Effect (EE) etod. Altoug defed as a local SA etod, te Morrs screeg etod s roadly take as a prelary step to separate out te eglgle paraeters ad decrease te uge coputato dead for dozes of odel paraeters [6,32 34]. Te detal of te Morrs screeg etod s gve elow. Let a odel wt paraeters Θ= ( θ, θ2,, θ) varyg across p selected levels te -desoal ypercue space, ad correspodg odel output f ( Θ ). For a specfed paraeter set Θ, te eleetary effect of te t paraeter ca e estated y erely cagg te t paraeter wt a step Δ wle te rest of te paraeters are kept ucaged. Ts process ca e cooly defed as: f ( θ,, θ, θ +Δ, θ+,, θ) f ( θ,, θ, θ, θ+,, θ) d ( Θ ) = Δ () were Δ s a value te set { / ( p ), 2 / ( p ),, / ( p ) }; p s te uer of levels; ad ( θ,, θ, θ, θ+,, θ) ad ( θ,, θ, θ +Δ, θ+,, θ) are a rado ad te desged saple set, respectvely, ot of te wt te predefed -desoal ypercue space. Two sestvty easures are proposed to vestgate te paraeters [5]: μ ad σ, wc evaluate te overall effect of eac paraeter o te odel output, ad te paraeters ger order effects caused y olear or teractve relatosp etwee paraeters, respectvely [35]. As suggested y Morrs, R radoly sapled dfferet trajectores are preferred to assess te eleetary effects. Statstcally speakg, μ ad σ are te ea ad te stadard devato of te R ( + ) eleetary effects. We * te odel s o-ootoc, Capologo et al. [35] proved te estato of μ y μ wt te asolute value of d ( Θ ) as: R k = R k μ = d( Θ ) (2) μ = ( ) (3) R * k d Θ R k = 2 σ = [ ( Θ ) μ ] (4) R R k d k = * A large μ uderles te fact tat te t paraeter s a relatvely fluetal paraeter ad a large σ uderles a strog olear or teractve relatosp etwee te t paraeter ad te oters [6] Sool Metod for Ucorrelated Paraeters Sool etod s oe of te wdely used gloal sestvty aalyss etods ased o varace decoposto [6,4,2,22]. Assue tat te odel paraeters are depedet or ucorrelated, te total varace of odel output s coposed of te varace fro dvdual paraeters, ad te varaces fro cooperatve paraeters. Te proporto of varace resultg fro dvdual ad cooperatve

5 Water 205, paraeters to te total varace s appled to estate te frst order ad teractve sestvty dces of paraeters. Te varace decoposto ca e expressed as: Vy () = V+ V+ + V (5) j = j were V(y) deotes te total varace of odel output; V deotes te frst order cotruto of te t odel paraeter; Vj deotes te parwse teractve effect of te t ad jt paraeters; ad deotes te uer of odel paraeters. Ovously, te frst, secod, ad total order of sestvty dces ca e respectvely expressed as: S = V/V(y) (6) Sj = Vj/V(y) (7) ST = V~/V(y) (8) were V~ deotes te cotruto of all except te t paraeter. Te varaces Equato (5) ca e estated y approxate Mote Carlo uercal tegratos [4]. Te coputato requreet of te orgal Sool etod to evaluate te frst ad total order sestvty dces s (2 + ), were deotes sze of saple sets, ad te secod order sestvty dce caot e calculated suc a case wt treedous odel rus. Saltell [3] proposed a exteded etod tat geerated ore roust ad satsfactory sestvty outputs y doule calculatos of te frst, secod, ad total order sestvty dces wt (2 + 2) odel rus. Te estato for V, Vj, V~, ad V(y) s descred as follows: Ê () y [( f ) f( )] (9) 2 2 = Θr Θr r = ˆV ( y) = [ f( Θ )] [ f( Θ )] 2 2 r r r= r= (0) Vˆ [ f( ) f( )] Eˆ ( y) or [ f( ) f( )] Eˆ ( y) = Θr Θ~ Θ~ Θr r= r= () Ê () y [( f ) f( )] (2) = Θ~ Θ~ r = ˆV2 ( y) = [ f( Θ )] [ f( Θ )] 2 2 ~ ~ r= r= (3) Vˆ [ f( ) f( )] Eˆ ( y) or [ f( ) f( )] Eˆ ( y) c j = Θ~ Θ~j 2 Θ~j Θ~ 2 r= r= (4) c Vˆ j = Vˆ ˆ ˆ j V Vj (5) Ê y f (6) T() = [ ( Θr)] r = ˆ V y f E y (7) T() = [( Θr)] ˆ T() r =

6 Water 205, ˆ ˆ V f f E y or f f Eˆ y ~ = [ ( Θr ) ( Θ~)] T ( ) [ ( Θ~) ( Θr )] T ( ) r= r= (8) were deotes te saple sze; Ê () y, Ê () 2 y ad ÊT () y are dfferet estated eas of odel outputs for frst, secod, ad total order, respectvely; ˆV () y, ˆV () 2 y, ad ˆVT () y are estated varaces of odel outputs for frst, secod, ad total order, respectvely; ˆV, ˆV j, ad ˆV ~ are estated varaces of odel outputs results fro te t, t ad jt, ad all except te t paraeters, respectvely. Te syols Θ ad Θ preset sapled paraeters selected fro saple ad 2, respectvely. Te syol r 2 r Θ ~ presets all te sapled paraeters selected fro saple except te t oe fro saple 2. Te coatos of sapled paraeters put fro saple ad saple 2 produce doule tes calculato of odel outputs (see Equatos (), (4), ad (8)). Equatos (9) () ca e used to estate te frst order sestvty dces accordg to Equato (6). Slarly, Equatos (2) (4) ad Equatos (6) (8) ca e used to estate te secod ad total order sestvty dces accordg to Equatos (6) ad (7), respectvely. Before te sestvty aalyss s pleeted, te saplg scee ust e carefully selected sce t as a portat effect o te covergece effcecy of Mote Carlo approxato. As suggested y Za et al. [6], Tag et al. [4], ad Saltell [3], te Lat Hypercue saplg (LHS) [7,2,36,37], ased o Mote Carlo, wc geerates a effcet estato of odel output troug a stratfed saplg etod, s adopted ts study Sool Metod for Correlated Paraeters Te orgal Sool etod ca ota good effcecy te odel of depedet or ucorrelated paraeters, wc gt e usutale for coplex ydrologcal odels wt g olearty ad teracto. Kucereko et al. [30] proposed a ew etod ased o Sool etod to estate te gloal sestvty dces wt depedet paraeters, ad ade t approprate for coplex ydrologcal odels GLUE Metod for Correlato Idetfcato Before te pleetato of te Sool etod for correlated paraeters, correlato paraeters ust e vestgated advace. I ts paper, te Geeralzed Lkelood Ucertaty Estato (GLUE) etod [3,4] s eployed to detect te correlated paraeters [38]. Te detaled procedures are lsted as follows: () radoly saple a uge uer of paraeter sets fro te -desoal ypercue space, te sae as Sool etod. Because o supposto aout te a pror dstruto of tese paraeters s essetal, a ufor dstruto s a coo coce; (2) calculate correspodg odel outputs ad lkelood easures wt tese saple sets; (3) defe a lkelood easure tresold to separate eavoral ad o-eavoral paraeters sets, tered as Θ ad Θ, respectvely; (4) estate te ea ˆμ, stadard dervato ˆσ, ad parwse correlato coeffcet ρ j of eac paraeter vector fro Θ. Te parwse correlato coeffcets te costtute te correlato atrx M c.

7 Water 205, Te frst tree steps are astracted fro te GLUE etod, wc s appled to geerate te eavoral paraeters. Te susequet ucertaty aalyss wt tese eavoral paraeter sets s eyod te scope of ts paper. Detals aout te GLUE etod ca e see Beve ad Bley [3,4]. Te fourt step sows te -dept aalyss of tese eavoral paraeters sets ad relevat works referred to are Horerger ad Spear [7] ad Ratto et al. [38] Geerato of Correlated Saple Sets Based o te detfed paraeters correlatos y te GLUE etod, correlated paraeter sets ca e geerated troug a geeral procedure troduced y Kucereko et al. [30]. Te procedures are pleeted as follows: () radoly geerate a paraeters set fro stadard oral dstruto, tered Φ y LHS. If te saple sze s, ad paraeters uer s, te te deso of Φ s. Sce Φ s produced alessly, te paraeters vector Φ s deftely depedet ad ucorrelated; (2) decopose te correlato atrx M c y a approprate decoposto etod. Colesky s preferred we te correlato atrx s postve defte, wereas ts s ot easy to prose. Terefore, ortogoal decoposto etod, wc s ore uversal, s eployed: M c = QR (9) were Q s te ortogoal atrx ad R s te upper tragular atrx. Bot desos are ; (3) trasfor te ucorrelated stadard oral saple sets Φ to correlated stadard oral saple sets Ψ : Ψ=Φ R (20) were te deso of Ψ s stll ; (4) derve te correlated paraeters sets Θ c = ( θc, θc2,, θc,, θc) fro correlated stadard oral saple sets Ψ= ( ψ, ψ2,, ψ,, ψ ): were θ c ad ψ are te t paraeter vector fro ea ad stadard dervato of te t paraeter vector, respectvely Geeralzato of Sool Metod ˆ ˆ θ c =ψσ+μ (2) Θ c ad Ψ, respectvely; ad ˆμ ad ˆσ are te Cocluded fro Kucereko s etod, te Sool etod to estate te frst ad total order sestvty dces for parwse correlated paraeters s slgtly dfferet fro tat for ucorrelated paraeters, as Zao et al. suarzed [39]. For ucorrelated paraeters, te frst order sestvty dce ca e otaed fro Equatos (6) ad (9) (). Equato () estates te varaces of odel outputs fro te t paraeter wt te explct eag tat te t paraeter saple s replaced y te correspodg t paraeter saple 2 or vce versa. Slarly, te frst order sestvty dce for correlated paraeters ca e derved fro replacg several vectors wc ust cota te t paraeter saple fro tose saple 2. If te parwse correlato s cosdered, te te deso of replaced vectors s equal to 2, wt te followg equato:

8 Water 205, Vˆ [( f ) f( )] Eˆ () y or [( f ) f( )] Eˆ () y = Θr Θ~,k Θ~,k Θr r= r=, (22) were Θ ~,k presets all te sapled paraeters are selected fro saple except te t ad kt oes fro saple 2. Te oters keep te sae eag as troduced aove. Atteto as to e pad to te fact tat for correlated paraeters, saples ad 2 are geerated fro procedures troduced Secto Aalogously, Equato (8) for ucorrelated paraeters expressed te cootato tat te estated varace fro all except te t paraeter s te product of odel output fro saple or 2 ad te t paraeters te saples replaced y tose fro saple 2 or. Terefore, te equato to estate te varace for all except te t paraeter for correlated paraeters ca e preseted as: Vˆ ~ = [( ) ( )] ˆ r ~,k T () [( ~,k ) ( r )] ˆ f Θ f Θ E y or f Θ f Θ ET () y. (23) 2.4. Luxe Model r= r= I ts study, te Luxe odel, wc s a pyscally ased dstruted watersed odel proposed y Ce et al. [40] tat as ee appled successfully to uerous catcets Ca [4 44], s selected as te used dstruted ydrologcal odel. Ts odel coprses fve su-odels: as dgtzato, evapotrasprato, ruoff geerato, ruoff routg, ad paraeter dervato. I ts odel, te dstruted odel paraeters are acqured drectly fro te DEM ad cael cross-sectoal propertes fro avalale free pulc data (also see te detals Ce et al. [40]) Bas Dgtzato Te Luxe odel dvdes te catcet orzotally to tree topograpc cell types, aely llslope, cael, ad reservor. Te catcet s also dvded vertcally to tree layers, aely caopy, sol, ad su-surface. Ruoff geerato ad flow routg s carred out depedetly for eac grd cell Evapotrasprato I te Luxe odel, evapotrasprato s detered y te trplex patter: Es = ve ( θ> ) θ Wp Es= ( v) Ep ( Wp<θ ), (24) Wp Es = 0 ( θ Wp) were E s te actual evaporato (); v s te evaporato coeffcet relevat to vegetato type; s te feld water capacty (%); Wp s te sol water cotet at wltg pot (%); θ s te teporary sol water cotet (%); ad Ep s te potetal evaporato. Cells tat are etrely ope water (reservors ad caels) ave a evapotrasprato coeffcet of, wle oter cells ave a value less ta Ruoff Geerato Te odel calculates ruoff geerato usg te saturato-excess ecas. Rafall excess, or et rafall, s derved fro te dfferece etwee actual rafall ad evapotrasprato. Postve

9 Water 205, rafall excess results fltrato to te sol. Rafall excess ecoes surface ruoff oly after te sol ecoes saturated. Iterflow te sol layer s calculated accordg to water ass coservato: Qlat θ + L = r Qper x t (25) Qlat = Vlat Lz were s te sol tckess (); L s te cell wdt (); Qlat s te terflow per ut wdt ( 2 /s); r s te rafall excess vertcally ad adjacet terflow put per ut wdt ( 3 /s); vlat s te terflow velocty (/s); ad Qper s te percolato per ut wdt ( 2 /s). Accordg to Darcy s law, te terflow velocty vlat s: lat 0 { vlat 0 v = Ktaα= KS θ> = θ (26) were α s te llslope cell slope degree ( ); S 0 s te llslope cell slope radas; K s ydraulc coductvty at water cotet (/); ad θ s derved accordg to Capell [45]: K θ = 2+ 3 (27) were s te saturato water cotet (%); s te saturated ydraulc coductvty (/); ad s te sol porosty caracterstc coeffcet Ruoff Routg Te Luxe odel assues tree types of flow routg accordg to te tree water sources: surface flow, terflow, ad groudwater. Also, tere are tree dfferet flow routg etods correspodg to te tree dfferet grd cell types of llslope, cael, ad reservor. Surface flow occurs te llslope cells; tere are total fve dfferet routg categores. For llslope routg, te keatcal wave approxato (te -D Sat Veat equatos) s adopted wt te assupto tat te ydraulc cell slope deotes te llslope: Q + L = q, (28) x t were Q s te surface flow ( 3 /s); s te rafall excess (); ad q s te lateral flow per ut wdt ( 2 /s). I addto, accordg to te Mag equato, Q ca e derved y: were a 3/ 5 / 2 = S0 L ad = 3/5. Q x Q t + aq q = 0, (29) For rver routg, te -D dffusve wave etod s used. Te odel assues a trapezod cross-secto sape of area A ( 2 ), water dept (), rver otto wdt w (), ad sde slope β. Te cotuty equato s:

10 Water 205, A Q + = q t x. (30) Sf = S0 x Accordg to te Mag equato ad cotuty equato, rver routg ca e calculated y: were c = χ S / 3 / 2 f 3/ 5 Q x Q t + cq q = 0, (3) ad χ s te wetted pereter of te cross-secto (). For reservor routg, flow to te reservor s calculated as te su of all flow cells to te reservor ased o Ce s dea tat dscarge o suc cells ca e regarded as flow wtout lapsg. For terflow routg, te odel deteres te terflow y Equato (25) fro adjacet sol cells sce te ruoff geerato process already takes te terflow routg to cosderato. For groudwater routg, te lear reservor etod s calculated as: Q = g,t t KgQ + g,t ( +Δ Kg) Q, per (32) were Qg,t ad Qg,t + t are groudwater flow at te t ad t + t, respectvely ( 3 /s); ad Kg s te groudwater recesso coeffcet Paraeters Dervato Based o te evapotrasprato, ruoff geerato, ad routg of te Luxe odel troduced aove, tere are 2 relevat paraeters ts odel (see Tale ). Tese 2 paraeters ca e categorzed to fve dfferet types related to varous aspects, cludg clate, lad use, sol type, terra, ad oters. A vtal ad callegg task efore te sestvty ad correlato aalyss s te deterato of approprate rages ad dstruto for te paraeters [46,47]. Tog ad Graza [48] ad Za et al. [6] suggested tat te rages of paraeters sould e decded advace cosderg ter sgfcat effect o te sestvty results. Fortuately, te Luxe odel, as oe of te pyscally ased dstruted ydrologcal odels, developed a reasoale etod to derve paraeters fro terra data. Te paraeters ca e derved fro te followg two procedures,.e., tal paraeter deterato ad paraeter adjustet. Sol related paraeters (Tale ) were pre-fxed y Sol Water Caracterstcs Hydraulc Propertes Calculator [49]. Oter paraeters were defed accordg to refereces. After te tal paraeters values were pre-decded, te paraeters were adjusted ldly tur y ultplyg a cage factor f (f > 0). Te est-perforg paraeters set was te selected judgg fro te odel perforace. It s oteworty tat te Luxe odel s pyscally ased, so te predetered paraeters ca perfor well to soe degree. A slgt adjustet s aed at provg te perforace. Detals aout Luxe odel paraeters dervato ca e otaed fro Ce et al. [40]. After te calrato troug te aove predesg ad slgt adjustet processes, te large saple sets for sestvty aalyss ased o Sool etod are geerated y ultplyg a cage factor atrx wt te calrated paraeters. Te cage factor atrx s sapled fro te oral dstruto ad esures te correspodg sapled paraeters are wt eac rages lsted Tale.

11 Water 205, Tale. Suary of te Luxe odel paraeters. ID Syol Descrpto Classfcato Processes Rages Ut Saturated ydraulc coductvty Sol type related Ruoff geerato 00 / 2 llslope rougess Lad use related Ruoff routg M Cael rougess Oter related Ruoff routg Sol tckess Sol type related Ruoff geerato Sol porosty caracterstc Sol type related Ruoff geerato 0-6 Ep Potetal evapotrasprato clate related evapotrasprato 0-7 Kg Groudwater recesso coeffcet Oter related Ruoff routg d 8 Saturato osture cotet Sol type related Ruoff geerato/evapotrasprato % 9 Feld water capacty Sol type related Ruoff geerato/evapotrasprato 0 50 % 0 v Evapotrasprato coeffcet Lad use related evapotrasprato 0 - Wp Wltg pot Sol type related Ruoff geerato/evapotrasprato 0 30 % 2 S Rver slope Terra related Ruoff routg % 2.5. Ojectve Fuctos Te assesset of paraeters sestvty depeded greatly o te coce of ojectve fuctos. I ts study, four wdely used etrcs were used to test odel perforace: () Nas Sutclffe effcecy (ENS) [50]; (2) water alace coeffcet (WB); (3) peak dscarge effcecy (EP); ad (4) te to peak effcecy (ETP). Tese four etrcs were calculated accordg to te followg equatos: E = NS t= WB = ( Qt ( ) Qt ˆ ( )) t= ( Qt ( ) Q) t= t= ˆQ () t Qt () 2 2 (33) (34) E ˆQ Q p p P = (35) Qp ETP = Tˆ p Tp, (36) were Q(t) was oserved or easured flow; ˆQ () t was sulated or forecast flow; Q was te ea of oserved flow; ˆT p ad T p were te sulated ad oserved te to peak, respectvely; ad ˆQ p ad Q p were sulated ad oserved peak dscarge, respectvely. Te ENS s wdely used to assess overall ydrograp goodess-of-ft [6,7,23,24,5,52]. Te WB sows te gap etwee te total sulated flood volue ad total easured volue. Ts ojectve fucto s erely flueced y ruoff geerato process ad s extesvely appled to te water

12 Water 205, resources aageet [6,24,53]. Te EP ad ETP are te oter two essetal eleets flood forecastg, ad tese two eleets are also crucal factors flood preveto, wc s adopted te Forecastg or for ydrology tellgece (GB/T ) of Ca. 3. Study Area ad Data 3.. Study Area Te Naua Creek catcet (Fgure ) s a or trutary of te upper Pearl Rver Ca. It s located etwee 2 37 E ~ 3 05 E ad N ~ 25 6 N wt a draage area of 20 k 2 ad a legt of 82 k. Te Naua Creek catcet elogs to te osoo clatc zoe, wt cold ad dry wters ad ot ad ud suers. Te log-ter ea aual precptato te catcet s aout , te ajorty of wc occurs suer. Te steep slope of te catcet, coed wt cocetrated rastors, akes ts area proe to flas ruoff respose Data Fgure. Te locato, elevato, rver etworks, ad dstruto of ra gauge ad ydrologc statos of te Naua Creek catcet. Te Luxe odel requres elevato (DEM), lad use, ad sol type data. I ts study, DEM was otaed fro NASA Suttle Radar Topograpc Msso (SRTM, ttp://srt.cs.cgar.org/) at a

13 Water 205, resoluto of Lad use data cae fro te USGS (ttp://ladcover.usgs.gov/) ad sol type data were fro te FAO/UNEO (ttp:// Egtee stor floods wt ourly easured rafall data of tree ra gauge statos ad dscarge data of te catcet outlet ydrologcal stato (Fgure ) were collected fro 980 to 987 to calrate te odel paraeters. Te stor floods fro 990 to 995 were appled to valdate te odel. I order to vestgate te respose of Luxe odel to varous flood codto, tree dfferet types of floods wt sall, edu, ad large agtudes were selected fro te avalale data set. Tese typcal flood evets wt dfferet agtudes were selected wt flood frequecy of 75%, 50%, ad 25%, respectvely. Te caracterstcs of tese floods were lsted Tale 2. Tale 2. Caracterstcs of te selected varous flood codtos. W0 deoted te sol osture cotet efore te flood evet ega. Evet Magtude Precptato Te Step Durato Peak Dscarge W0 Flood Frequecy () () () ( 3 /s) () (%) 9806 Sall % Medu % Large % 4. Results ad Dscusso 4.. Results of Calrato ad Valdato Te average of te four ojectve fuctos of te 8 stors te calrato perod were 0.82,.08, 0.0, ad 0.7 respectvely (see Tale 3). Accordg to te Forecastg or for ydrology tellgece (GB/T ) of Ca, tese satsfactory results et te secod grade of te or, wc requres 0.7 < ENS < 0.9, 0.8 < WB <.2 ad EP < 0.2. Te calrated results deostrated tat te Luxe odel was sutale for flood forecastg te Naua Creek catcet ad te paraeters were reasoale. Slar results were also otaed fro te odel valdato perod. Te average of te four ojectve fuctos of te 0 stors te verfcato perod were 0.74,.06, 0.2, ad.7, respectvely. Eve toug te perforace of valdato was slgtly worse ta te calrato perod, te forer stll et te secod grade of te or, wc stregteed te cofdece of te Luxe odel ad calrated paraeters. A coparso of oserved ad calculated flow for te tree represetatve stors s llustrated Fgure 2. It ca e see fro tese ydrograps tat te Luxe odel geerated good results te rsg ls, flood peaks, ad recesso ls. Tale 3. Goodess of ft statstcs etwee Luxe predcted ad oserved ydrograps te Naua Creak catcet. Perod E NS WB E P E TP () Calrato Valdato

14 Water 205, Dscarge ( 3 /s) Te (dt=) Precptato Oserved Flow Sulated Flow Precptato () Dscarge ( 3 /s) Te (dt=) (a) () (c) Precptato () Dscarge ( 3 /s) Te (dt=) Precptato () Fgure 2. Coparsos etwee te oserved ad te Luxe sulated ydrograps for te tree represetatve stor floods. (a) 9806; () 98505; (c) Results of Morrs Screeg Te Morrs screeg etod, or Eleetary Effect (EE) etod, was eployed to qualtatvely detfy te uportat paraeters. Te detfed relatvely portat paraeters were te vestgated for te followg gloal sestvty aalyss. Two paraeters sould e predefed efore te applcato of Morrs screeg,.e., level uer p ad trajectores uer R. As suggested y Capologo et al. [32,35], p s preferetally eve ad R s geeral etwee 0 ad 50. I ts paper, level uer p was detered to e 0 ad R was 40. Susequetly, 520 paraeters sets were geerated radoly y ufor dstruto wt te ratoal rages. Te Morrs screeg results ca e preseted te for of rakg tale [33,34], Torado dagras [33], ad scatter plots [6]. Scatter plots are cofred to e te ost effcet ad clearest wt * odfed eas μ as te x-axs ad stadard devatos σ as te y-axs, wle te forer two faled to llustrate te caracterstcs of stadard devatos. Te scatter plots of te 2 paraeters te Luxe odel were copared tree dfferet flood codtos as well as four dfferet ojectve fuctos, as sowed Fgure 3. O te ass of te fact tat a larger μ * value reveals ger sestvty, t s terestg to see tat eve uder varous codtos ad ojectve fuctos of terest, te sestve rakgs could all e categorzed to tree roug classes: gly sestve, sestve, ad sestve; te classfed paraeters also gave cosstet results. Aog all te paraeters, ad were gly sestve;, M,,, ad were sestve; ad Ep, Kg, Wp, v, ad S were sestve. Explct rakg aog te 2 paraeters dffered wt varous stuatos. Te top sestve paraeters were foud aog, ad M fro te ost to te least frequet. Te least sestve paraeters covered were S, v, ad Wp, fro te ost to te least frequet. Our results were accordace wt tose of Xu et al. [54], wo stated tat ad were frst-rate; ad were secod-rate; Wp,, ad were trd-rate; ad S, Ep, M, Kg, ad v were sestve.

15 Water 205, ENS WB EP ETP σ σ σ σ Sall flood Medu flood Large flood 0.2 M v Wp Ep Kg 0.0 S μ* M EpKg 0.0 S Wp v μ* Wp v M S EpKg μ* 5 v M Kg S Ep 0 Wp μ* σ σ σ σ Kg Wp M Ep v 0.0 S 0.0 μ* Ep v Kg M S Wp μ* Wp M 0.0 v S Ep Kg μ* M 4 v S 2 EpKg Wp μ* σ σ σ σ v Wp M 0. S Ep Kg μ* S Ep v Kg Wp M μ* S Wp v EpKg M μ* Ep Kg S Wp v M μ* Fgure 3. Scatter plots of te Morrs screeg results ased o dfferet ojectve fuctos easures uder sall, edu, ad large flood agtudes. * Alog wt tese sestvty rakgs reflected y μ, te olear ad teractve effects etwee paraeters reported y σ were also rearkale. All te suplots Fgure 3 dcated tat ad ad a larger σ ta te oters, wc ferred tat te olear relatosp etwee paraeter values ad odel outputs was ore otceale or tat ter teracto wt oter paraeters was ore evdet. Slar results ca also e otaed fro we takg WB as te ojectve fucto ad M we ETP s te ojectve fucto of terest. It s plausle to suspect tat paraeters wt ger odfed ea values ( μ * ) ad a ger posslty of presetg a correspodgly larger stadard dervato value ( σ ). Za s work [6] supported ts suspco wt aoter dstruted ydrologcal odel (DTVGM). Terefore, atteto ust e pad to tese g rakg paraeters wt large σ sce te g olearty ad teracto wt oter paraeters decreased te relalty of g sestvty classfcato. Ts was also oe of te doatg dsadvatages of te Morrs screeg etod: te olearty ad teracto etwee paraeters caot e dfferetated. O te oter ad, te

16 Water 205, sestve paraeters wt lower odfed ea value μ * were fortuately ale to dsplay a correspodgly lower stadard dervato value σ (see Fgure 3), wc aceved te orgal goal of screeg out sestve or o-eavoral paraeters wt g relalty. Terefore, te sestve paraeters of te Luxe odel ca doutless e detered eve uder dverse flood codtos ad ojectve fuctos. Tese were S, v, Wp, Ep, ad Kg. Tese paraeters were te kept costat te susequet GSA wt eglgle flueces o odel output. Paraeters passg te valdato screeg, cludg,,,,,, ad M, were cose for te furter vestgato of GSA ased o Sool etod. Te orgal 2 paraeters te Luxe odel were reduced to seve for GSA, reducg te coputato y alost alf. Tat s te reaso for te road applcato of te Morrs screeg etod as a frst step to lower coputato dead for GSA [6,32 34] Results of Correlato Idetfcato Sce te Sool etod s a varace-ased decoposto etod wt te assupto tat all paraeters are depedet, te valdato of depedece s deftely a prerequste. However, ts procedure was ofte gored. I ts paper, te GLUE etod was eployed to caracterze te correlato relatosp etwee te Luxe odel paraeters efore te pleetato of te Sool gloal sestvty aalyss etod. Te GLUE etod ust defe te lkelood easure ad ts tresold for dvdg eavoral ad o-eavoral. As Beve ad Bley cocluded [4], te Nas Sutclffe effcecy (ENS) (see Equato (33)) was te ost extesvely used lkelood easure. For te sake of splcty wtout loss of geeralty, ts paper also set ENS as te lkelood easure. I addto, accordg to te Forecastg or for ydrology tellgece (GB/T ) of Ca, te tresold was decded to e 0.7 order to eet te requreet for te secod grade. Based o te udeadg ypotess aout te pror dstrutos of te odel paraeters, a saple of 5000 sets was geerated y te ufor dstruto wt te rages of paraeters. Te Pearso correlato aalyss ad sgfcace level etwee te two paraeters are lsted Tale 4. Te ger te correlato etwee te paraeters, te darker gray Tale 4. Te poorly correlated paraeters are sowed wt o color. Hgly correlatg paraeters were suggested to cut paraeters ypercue space. Postve values desgated tat te relevat paraeters ad te sae effect o odel perforace ad worked as a quotet or dfferece, wereas egatve values desgated opposte effects fro two relevat paraeters o odel perforace ad worked as a product or su [38]. Te sae or opposte effects fro paraeters ca e easly otaed fro te prcples of te odel Secto 2.4. Iterestgly, te ad dsplayed te gest postve Pearso correlato, wc was extreely close to at te sgfcace level of 0.0. Oter paraeters, lke ad, M ad, ad, M ad, ad ad wt Pearso correlato coeffcets of 0.32, 0.34, 0.30, 0.26, ad 0.28, respectvely, were also relatvely correlated at te sgfcace level of 0.0. Tese gly or relatvely correlated paraeters were all classfed as te gly or relatvely sestve paraeters te Morrs screeg desg Secto 4.2. Idvdual paraeters deostratg a larger stadard devato te Morrs screeg scatter plots Fgure 3 exted a ger correlato relatosp wt oter paraeters (e.g., ad ). Tese two etods stregteed te realty tat te correlated paraeters te Luxe odel were attetve ad te stragtforward applcato of te Sool etod was approprate. Dfferet fro te fguratve llustrato y te Morrs screeg etod, te GLUE

17 Water 205, etod glgted te correlato aalyss y drectly talg te correlato coeffcets, wc lad a fudaetal ass for te susequet sestvty aalyss accoutg for correlated paraeters. Tale 4. Correlato aalyss etwee effectve paraeters. Dark grey sadow deoted gly correlated paraeters. Lgt gray sadow deoted relatvely correlated paraeters. Wte cells deoted poorly correlated paraeters. Values laeled ** eat ts sgfcace level was 0.0. Values laeled * ad a sgfcace level of M * ** ** * 0.7 M 0.34 ** ** ** * 0.96 ** 4.4. Results of Frst ad Total Order Sestvty Aalyss I order to copare te dfferece of te sestvty aalyss wt ad wtout cosderg paraeters correlato, te geeralzed ad exteded Sool etod [30,3] were pleeted to estate te frst ad total order sestvty dces of te seve paraeters (.e.,,,,,,, ad M) te Luxe odel. I ts study, as suggested y Ga et al. [5], 2000 paraeter sets were geerated fro oral dstruto y te Lat Hypercue Saplg (LHS) etod [7,2,36,37] for te exteded Sool etod, gorg correlatos. As for te geeralzed Sool etod accoutg for paraeter depedecy, for te sake of cosstecy, 2000 paraeter sets were geerated fro oral dstruto y LHS as well. Te dfferece wt te latter was tat te paraeters were correlated wt correspodg coeffcets detfed fro Secto 4.3 ad were geerated troug te procedures Secto Terefore, te coparso aalyss requred a total of 32,000 odel rus, alf for ucorrelated paraeters ad alf for te correlated. Idepedet ad depedet frst ad total order sestvty dces ased o te four dfferet ojectve fuctos uder te tree dfferet flood agtudes are sow coparatvely Fgure 4. Te red ad lue ars preseted te frst order sestvty dces for ucorrelated ad correlated paraeters, laeled wt UCS ad CS, respectvely. Te gree ad yellow ars exted te total order sestvty dces for ucorrelated ad correlated paraeters, laeled wt UCST ad CST, respectvely. Te dfferece etwee total order ad frst order dces ca dcate te teractve dces etwee te paraeters of terest ad oter paraeters. Te dfferet egt of te ars revealed te varous levels of sestvty. Te ger te ar, te ore sestve te paraeter.

18 Water 205, Sall flood Medu flood Large flood ENS WB EP ETP Fgure 4. Coparsos etwee te frst ad total order sestvty dces of seve paraeters te Luxe odel wt ad wtout accoutg for correlatos ased o dfferet ojectve fuctos easures uder tree dfferet flood agtudes (UCS ad UCST desgated te ucorrelated frst ad total order sestvty dces; CS ad CST desgated te correlated frst ad total order sestvty dces). For te ucorrelated ad correlated cases, te overall cotruto of to te four ojectve fuctos uder te tree flood codtos vared fro 0.23 to 0.5, wc was te gest ad dcatve of te ost sestve. Te, te partal effect of was te secod largest wt te rage of Te

19 Water 205, oter paraeters deostrated sestvty dces wt te rage of As Tag et al. [4], Za et al. [6], ad Zag et al. [2] etoed, paraeters wt sestvty dces larger ta 0. ca e regarded as gly sestve; sestvty dces etwee 0.0 ad 0. ca e classfed to sestve. Tose paraeters wt lttle cotruto to odel outputs varace (e.g., sestvty dces less ta 0.0) ca e cosdered as sestve. Terefore, ot ad were dcated to e gly sestve wt te largest frst ad total order dces for ucorrelated or correlated cases, wereas,, M,, ad were regarded as sestve paraeters. Te Sool results dd ot classfy ay paraeters to sestve, wc revealed tat te paraeters fxed as costat y te Morrs scree Secto 4.2 were reasoale. Te Sool etod stregteed te Morrs screeg results ad quatfed te sestvty dces of eac paraeter wt detaled values. I addto, our results were accordace wt Lao et al. [44], wo vestgated te Luxe odel paraeters ased o te E-FAST algort usg ENS as ojectve fucto y 560 odel rus. Despte tese slar outcoes otaed y ucorrelated ad correlated paraeters, t s useful to dstgus te dffereces etwee te two cases as well. For te ucorrelated case, te geeral order of average frst order sestvty dces of te seve paraeters studed, fro te gest to lowest, was 0.300, 0.292, 0.047, 0.039, 0.036, 0.035, ad for,, M,,,, ad, respectvely. For te correlated case, te order of frst order sestvty dces, fro te gest to lowest, was 0.380, 0.377, 0.054, 0.045, 0.034, 0.032, for,, M,,,, ad, respectvely. A large crease (larger ta 0.08) of te sestvty dce was oserved fro te ucorrelated case to te correlated, referred to ad. A slgt creet (aout 0.006) was dcated fro te ucorrelated to te correlated we t cae to M ad. Fally,,, ad preset a sall decrease (aout 0.003) for te correlated case. Slar results ca also e otaed for te total sestvty dces coparg te ucorrelated ad correlated cases. Tese results were agreeet wt te results gve y Pa et al. [55], wo revealed tat te sestvty of va Geucte to te percolato flux of terest proved we te va Geucte was correlated wt va Geucte α ad te sestvty dce of perealty decled we o teracto was cosdered. Furterore, te proved sestvty of va Geucte creased as te correlato coeffcet wet ger, wc eat tat paraeters correlatos played a sgfcat effect o te sestvty aalyss ad sould ot e overlooked. I ts study, a early lear regresso etwee ad was exted Secto 4.3. Correspodgly, te varato of correlated aalyss copared wt te ucorrelated was very large (eve larger ta te sestvty dces of te fve sestve paraeters) te case of gly correlated paraeters. For te poorly correlated paraeters (M ad ), te creets were dstct ad we t cae to o-sgfcatly correlated paraeters (,, ad ), te sestvty dces evetually decreased. Ts ay e caused y te saple sets, cosderg te correlato, dses te paraeters space [38]. I oter words, te correlated saples are ore cocetrated. Terefore, te fractoal cotrutos of correlated paraeters to odel output varace are glgted copared wt tat of ucorrelated paraeters. I addto, gly correlated paraeters cast a ore profoud fluece o te glgtg ta te poorly correlated. Fro ts pot of vew, as dcated y te results fro Morrs screeg ad te GLUE etod, t s very lkely tat te gly sestve paraeters wt large odfed eas reflected g correlato coeffcets. Hece, te sestvty aalyss of tese paraeters ay e uderestated f te teracto s ot cosdered. O te cotrary, te sestvty aalyss of sestve paraeters wt o-sgfcat correlato ay e overestated we oter gly correlated paraeters were ot cosdered. Aoter terestg result

20 Water 205, sould also e oted: te dffereces etwee te total ad frst order sestvty dces, dcatve of teracto effect, srak copared wt te ucorrelated case. Sce oly parwse correlatos were take to accout, te sruke yet retaed gaps etwee te total ad frst order sestvty dces ca e cosdered to e caused y te ultvarale teractos ad sould e vestgated te future. Te sestvty results also vared aog dfferet flood agtudes. I ot ucorrelated ad correlated cases, te sestvty dces decreased as te agtudes creased. For exaple, te total order sestvty dce of decreased fro for sall floods to for edu oes ad 0.37 for large oes, ased o te ENS easure for te ucorrelated. I addto, te gap etwee gly sestve ad sestve paraeters decreased as agtudes creased. Ts ca e explaed y te fact tat ost sall floods occurred after a log drougt, durg wc te sol osture was kept at a very low level [56] ad a ydrograp was vulerale to te dsturace ad atteuato of te uderlyg surface. Terefore, a slgt fluctuato of odel paraeters would play a oteworty role deterg te ydrograp of a watersed outlet. I ts way, te sestvty dces ad ter varace for sall floods were ore sgfcat. O te cotrary, large floods aly appeed after log precptato, durg wc te sol was ost eoug ad te dsturace of te uderlyg surface cast a less portat effect o outlet ydrograps. Terefore, te sestvty dces ad ter varato for large floods were less sgfcat. Addtoally, atteto sould also e pad to te varatos of sestvty dces related to dfferet ojectve fuctos. Takg ENS as te ojectve fucto, te ost sestve paraeters, ad, were sgfcat ad fve sestve paraeters were at aout te sae level. Wle takg WB as te ojectve fucto, te dces of ad M decled copared wt oter sestve paraeters. However, ts stuato was reversed we takg EP ad ETP as te ojecto fuctos. Tese dffereces cae fro te dverse epaszed ydrograp process, agtude, ad tg fro te four ojectve fuctos. ENS focused o te varato etwee sulated ydrograp ad te average easured oe. WB cocetrated oly o te gap etwee te sulated ad easured volue. Moreover, EP ad ETP erely pad atteto to te effcecy of te sulated peak dscarge ad te te to peak. Terefore, flow cocetrato paraeters suc as ad M tat detere te flow velocty would ave a ore oetous pact o te peak dscarge preseted y EP ad ETP, wle lttle pact o te flood volue preseted y WB. O te cotrary, ruoff geerato paraeters suc as,, ad would play a ore portat role fluecg WB ta EP ad ETP copared wt ad M. Bot ruoff geerato ad flow cocetrato paraeters affect te sape of a ydrograp; ter copetg cosequeces result te sae sestvty dces level for ENS. Tese results pled tat te frst ad total order sestvty dces of odel paraeters were greatly affected y te selecto of ojectve fuctos as well as te coce of flood agtudes. Prevous studes [6,7,4,23,24,5,53] tat took ENS or RMSE as te ojectve fucto would ot classfy te ruoff geerato ad flow cocetrato paraeters ad ota deep sgt to te odel eavor Results of Secod Order ad Iteractve Sestvty Aalyss I ts study, furter sgt of te secod order dces etwee te seve paraeters wtout cosderato of correlato relatosp was also llustrated. Tale 5 sowed tat te su of te frst order sestvty dces of tese seve paraeters was less ta. Za et al. [6] explaed tat te su

21 Water 205, of frst order dces less ta was caused y te teractve effects aog paraeters, wc could ot e cosdered te sgle paraeter sestvty aalyss. I addto, ts outcoe dcated tat te teractve pacts aog paraeters caot e eglected. Ts dcato was reforced y te gaps etwee te frst ad total order sestvty dces (see Fgure 4 ad Tale 5). O te oter ad, te su of total order sestvty dces was slgtly larger ta. Ts peoeo was explaed y Tag et al. [4]: te trucato ad Mote Carlo tegrato pleeted te Sool etod ca result slgt uercal errors. However, Tag et al. [4] ackowledged tat ts uportat error would ot affect te sestvty rakg of paraeters. Baro ad Taratola [57] also vestgated te roustess of sestvty dces uder te cage of saple sze. Te total order sestvty dces proved to e ore stale ta te frst order oes. Terefore, te total order sestvty dces of te ost ad least sestve paraeters (.e., ad ) were teratvely explored y creasg te saple sze y 200 at a te fro 200 to Results Fgure 5 sowed tat ot total order sestvty dces of (left) ad (rgt) fluctuated wt a wde rage we te saple sze was less ta 200. Furterore, te varato of gly sestve paraeters was ore sgfcat ta tat of sestve paraeters. Te total sestvty dces of wt EP ad ETP as ojectve fuctos were see to coverge at a very low coputato cost. Eve toug te total sestvty dces caged ore sgfcatly we takg ENS ad WB as ojectve fuctos, te dffereces etwee te axu ad u were ad 0.05 for, respectvely. However, fro te perspectve of, oly we takg EP as ojectve fucto dd te total sestvty dce vary slgtly, ad te dffereces etwee te axu ad u values were 0.047, 0.08, ad for ENS, WB, ad ETP, respectvely, wc were uc larger ta tose of. Tese results were cosstet wt te researc of Baro ad Taratola [57]. However, as log as te saple sze creased to a large value, e.g., 600, te total sestvty aalyss ega to coverge ad geerate steady results for ot gly ad geerally sestve paraeters cosderg varous ojectve fuctos. Tese results uderled tat te saple sze of 2000 set te Sool aalyss Secto 4.4 was reasoale. As a atter of fact, t s ovous tat te su of te frst ad secod order sestvty dces accouted for early all te total order (see Tale 5), wc dcated tat te teractve effects aog paraeters ca e aly explaed y parwse teractos. I oter words, te secod order sestvty dces of te Luxe Model ca deote ost paraeter teractos. More precsely, as gve Tale 5, te average secod order sestvty dces were 0.32, 0.25, 0.30, ad 0.27, respectvely, ased o ENS, WB, EP, ad ETP easures, wt correspodg frst order dces of 0.75, 0.77, 0.73, ad 0.76, respectvely. Tale 5. Suatos of te frst, secod, ad total order sestvty dces ased o four dfferet ojectve fuctos. Ojectve Frst Order Secod Order Total Order Fuctos Sall Medu Large Sall Medu Large Sall Medu Large E NS WB E P E TP

22 Water 205, Total Sestvty Idce for E NS WB Ep Etp Saple Sze (a) Total Sestvty Idce for E NS WB Ep Etp Saple Sze () Fgure 5. Varatos of te total order sestvty dces alog wt creasg saple sze. (a) For te least sestve paraeter ; () for te ost sestve paraeter. I order to etter aalyze te two-paraeter teractos, te parwse varato-duced secod order sestvty dces (see Fgure 6) provded a ore exaustve uderstadg of te secod order dces lsted Tale 5. Tese fgures dsplayed te lower tragle atrx of parwse paraeters teractos were dark gray sadow deoted ger teractos ad lgt sadow deoted lower teractos. Rearkaly, te parwse teractos ecae ore sgfcat as te paraeters grew ore sestve. For stace, te teractve sestvty dce etwee ad was as g as 0.09 ased o te WB easure. Furterore, te parwse teractos etwee ad te reag sestve paraeters, ad etwee ad te reag sestve paraeters, were also relatvely ger for all ojectve fuctos uder te tree flood agtudes. However, te two-way teractos etwee sestve paraeters were relatvely lower (e.g., te secod order sestvty dce etwee M ad was less ta 0.0 eve toug ot paraeters descred te process for flow routg). Our results were good agreeet wt te results provded y Tag et al. [4], Za et al. [6], ad Zag et al. [2], eve toug te ydrologcal odels were dfferet. Tag et al. [4] poted out tat te lower zoe storage paraeter ad percolato factor paraeter te odel coed y SAC-SMA ad SNOW-7 preseted te gest uer of teractos, wle te two paraeters ot also sowed te gest fluece o odel output te frst order sestvty aalyss. Za et al. [6] cocluded tat te secod order dce etwee sol osture paraeter ad tckess of upper sol layer te odel aed DTVGM was 0.2 g for te ojectve fucto WB ad ter correspodg frst order dces were ad 0.39, respectvely, rakg sde te top tree. Based o te Xajag odel, Zag et al. [2] cocluded tat te parwse teracto etwee areal ea free water storage capacty paraeter (S) ad recesso costat for cael routg (Cs) ad a profoud effect o odel perforace, wle S ad Cs were te two ost sestve paraeters te total order sestvty evaluato te eate. Te applcato of Sool etod to varous odels sestvty aalyss dcated tat te paraeters tat sowed g sestvty dces te frst ad total order aalyss were proe to presetg a correspodgly sgfcat parwse teracto wt oter paraeters te secod order aalyss. Ts was a expected result, as te cages te sestve paraeter, coupled or ot coupled wt oter paraeters cages, would deftely cotrute to te great varace of odel