ProbabilisticmodellingofextremestormsalongtheDutchcoast
F.Lia,?,P.H.A.J.M.vanGeldera,R.Ranasinghea,b,D.P.Callaghanc,R.B.Jongejand
a
DepartmentofHydraulicEngineering,DelftUniversityofTechnology,Delft,2600GA,TheNetherlandsDepartmentofWaterEngineering,UNESCO-IHE,Delft,2611AX,TheNetherlandsc
DivisionofCivilEngineering,TheUniversityofQueensland,Brisbane,4072,Australiad
JongejanRiskManagementConsulting,Delft,2628DX,TheNetherlands
b
articleinfoabstract
Duetotheunprecedentedgrowthinpopulationandeconomicdevelopmentalongthecoastalzoneallovertheworld,knowledgeaboutfutureextremeoceanographiceventswillassistinensuringhumanandpropertysafety.Thiswillbeataskwithincreasingsigni?canceinthelightofprojectedclimatechangeimpacts.Ajointestimationofextremestormevents'variatesofdeepwaterwaveconditionswasperformed.Itcanbeusedformultivariatedescriptionsofwaveclimatevariates,suchaswaveheight,period,steepness,andstormduration.Thestormsequencescanbesimulatedandextrapolatedfromlimitedobservationaldataforoptimalstructureprotectionstrategiesandvariousdisasterriskanalysis,likeerosionorovertopping.Theanalysisnotonlyshowstheeffective-nessoftheproposedstatisticalapproachesforimprovingmultivariatemodellingofthestormparametersbutalsohighlightsthemostcompatibleapproachfortheDutchwaveclimatedatafrom1979to2009.WeusedtheMonte-Carlomethodandfourmethodstoconstructthedependencystructures,basedoncopulafunctions,physicalrelationshipandextremevaluetheory.Themarginalprobabilisticdistributionfunctionsofwaveclimatevariablesandthejointprobabilitywerethenobtained.Thesimulateddatagroupperformsareasonablesimilaritytothe?eldmeasurementsaccordingtothegoodness-of-?ttest,andtheGaussiancopulamodelwasfoundtobethebestwaveclimatesimulationmethodfortheDutchcoast.
?2013ElsevierB.V.Allrightsreserved.
Articlehistory:
Received23May2013
Receivedinrevisedform12December2013Accepted17December2013AvailableonlinexxxxKeywords:
MultivariateanalysisSeastormsimulationCopulas
WaveclimateDependence
1.Introduction
Inthecontextofcoastalengineering,theprobabilisticdesignofmarinestructuresorseadefencesiscloselyrelatedtothestatisticalpredictionoftheoceanstate,suchaswaveheight,period,etc.Theseoceanwaveclimateelementsarethedatasourceforcoastalhazardanalysisandevaluationofthesafetylevelofcoastalstructuresintheso-calledSource-Pathway-Receptorconcept(Oumeraci,2004).Oneofthechallengesforscenariooreventbasedcoastalriskassessmentisstochasticallysimulatinganddescribingseastormswhichmaybecustomarilycharacterizedintermsofmaximumsigni?cantwaveheight(Hs,max),peakwaveperiod(Tp),peaksealevel(h),wavedirection(θp)andstormduration(D).Whendealingwiththesimulationofcoastalerosion,inter-arrivaltime(I)orcalmtimebetweentwoindependentsuccessivestormsisalsoakeyvariateamongothers.Withthesimulatedevents,thecoastalerosionand?oodingriskwithvariousreturnperiodswillbequanti?edmoreaccuratelycomparedtothemethodbasedonabenchmarkevent(usuallythelargestmeasuredhistoricalevent).Thatisbecause,forsystemsinvolvingtwoormorerandomvariables,the
?Correspondingauthor.Tel.:+31152785013.
E-mailaddresses:fan.li@tudelft.nl(F.Li),p.h.a.j.m.vangelder@tudelft.nl(P.H.A.J.M.vanGelder),r.ranasinghe@unesco-ihe.org(R.Ranasinghe),
dave.callaghan@uq.edu.au(D.P.Callaghan),ruben.jongejan@jongejanrmc.com(R.B.
0378-3839/$–seefrontmatter?2013ElsevierB.V.Allrightsreserved.
returnperiodofoutcomesisnotequaltotheforcingreturnperiodofaparticularvariate(Hawkesetal.,2002).
Duetotheinternalphysicalconnectionsamongthesewaveclimatevariates,generallythereexistsamutualpartialdependencybetweenvariates.Hence,anestimationofthejointprobabilitydistributionofwaveclimatevariatesisrequired,especiallyfortheextremestormevents,whichareofparticularconcernforcoastalengineers.Univariatemarginaldistributionsreceivedconsiderableattentionintheliterature(Borgman,1973;FerreiraandGuedesSoares,2000;Forristall,1978;Krogstad,1985;NerzicandPrevosto,1998).Furthermore,researcheffortsinthepastdecadeshaveledtovariousmethodstostudythebi-variatedescriptionofwaveconditions;forinstance,thejointprobabilityofextremesigni?cantwaveheightsandsealevels(Hs,max,h)(Hawkesetal.,2002;LiandSong,2006),thejointdistributionfunctionofmaxi-mumsigni?cantwaveheightandwaveperiod(Hs,max,Tp)(MyrhaugandHansen,1997;Repkoetal.,2005),andthecorrelationbetweensigni?cantwaveheightsandassociateddurations(Hs,D)(Mathiesen,1994;SoukissianandTheochari,2001).
Recently,copulafunctions,?rstmentionedbySklar(1959),havebeenincreasinglypopularintheirapplicationtovariousmultivariatesimulationstudiesincivilandoffshoreengineering.DeWaalandvanGelder(2005)appliedcopulastomodelextremesigni?cantwaveheightsandwaveperiods(Hs,max,Tp);Wahletal.(2012)appliedcop-ulastostatisticallyanalysepeaksealevelsandmaximumsigni?cantwaves(Hs,max,h);CorbellaandStretch(2012)usedcopulastoconstruct
2F.Lietal./CoastalEngineering86(2014)1–13
atrivariatemodelofsigni?cantwaveheight,stormdurationandwaveperiod(Hs,max,D,Tp);DeMicheleetal.(2007)usedcopulastoprovideafour-dimensional,multivariatefrequencyanalysisof(Hs,max,D,I,θp).
Oneoftheadvantagesofcopulasisthattheyareabletocorrelatetwoormorevariableswithoutchangingtheirmarginaldistributions.Inthispaper,asitwillbedescribedlaterinmoredetails,ArchimedeanandGaussiancopulas(alsoknownasnormalcopula)wereusedtocon-structamultivariatedependencystructureforthesigni?cantwaveheight,stormduration,surgelevelandpeakwaveperiod(Hs,max,D,h,Tp)forthepurposeofwaveclimatesimulations.Thewavedirection(θp)wastreatedindividuallyby?ttingtoanempiricaldistribution.Andbasedonthesimulatedwaveclimate,alargenumberofsyntheticstormsurgeeventscanbeobtainedbytheincorporationofthestormfrequency(Fs),de?nedasthestormnumberpermonth,insteadofthestorminter-arrivaltime(I).ThismodelmaybeutilizedforcoastalriskassessmentandintegratedcoastalmanagementbyusingtheMonte-Carlotechniquetoavoidthedrawbackofusingabenchmarkevent.
Theaimofthispaperistopresentanexampleofafulltemporalsim-ulationforstormeventsalongtheDutchcoastbymeansofastatisticalmechanism.InSection2thestudysiteandrequiredvariatesareintro-duced.Themarginaldistributionfunctionsofthesevariates,andthede-pendencystructureforthedependentvariatesarestudiedinSection3.Section4comparesthesimulateddataandobserveddata.Sections5and6includethediscussionandtheconclusionsoftheworkrespectively.
2.Studysiteanddatapreparation
TheprobabilisticmethodwasadoptedtoestimatestormeventsalongtheHollandcoast(fromtheHoekvanHollandtoDenHelder,Fig.1),TheNetherlands.ThewaveclimatedatawascollectedintheNorthSeaatIJmuidenMunitiestortplaats(YM6,period:1979–1992,location:52°33′00″N,4°03′30″E)andatNoordwijkMeetpost(MPN,pe-riod:1993–2009,location:52°16′26″N,4°17′46″E)byRijkswaterstaat,theexecutivebranchoftheDutchministryofInfrastructureandEnvironment,overaperiodof31years.Theanalysisofthewaveclimatedatain1992and1993ofthetwogaugesindicatesthattheybelongtoahomogenousregionintermsofwaveclimateconditions.Therefore,theobservationsofthetwogaugesweremergedintoonesingledatasetwithoutadjustment.TheYM6stationislocated26kmfromthecoast,wherethelocaldepthis21m.TheMPNstationislocated9.5kmfromthecoast,wherethelocaldepthis18m.Themissingdataiscomplementedandcorrectedbyadjacentgauges,toavoiderrorsandtoensureconsistency.
Thestormeventswhichwillcauseamorphologicalchangewerecon-sideredandde?nedasperiodswheresigni?cantoffshorewaveheightexceeds300cmandwheresimultaneouslythesurge(TA,de?nedasac-tualwaterlevelminustheastronomicaltidelevel)ishigherthan50cm(Quarteletal.,2007).Toguaranteetheindependencyofselectedstorms,theminimumtimeintervalbetweentwostormswassetas6h,any
two
Fig.1.Localitymapfor?eldmeasurements.
F.Lietal./CoastalEngineering86(2014)1–133
stormswithtimeintervalslessthan6hwereconsideredasonestormevent.
Therawobservationswereselectedandprocessedtoobtainatimeseriesofindependentstormeventsde?nedby(Hs,max,D,Tp,h,θp).Andthetimeserieswasalsocollectedfromthe?eldmeasurementsandrepresentedbyFs.Fig.2showsthede?nitionofastormeventandtherelatedparameters.AndFig.3showsthestormtimeseriesfrom1979to2009.
Inthispaper,theHs,maxisde?nedasthemaximumsigni?cantwaveheightduringthede?nedstormduration,whilethestormdurationDwasde?nedastheperiodwhentheHs,maxandTAbothsatisfytheconditionmentionedabove.Ifthetimeintervalbetweentwostormsissmallerthan6h,Dwillbetheperiodfromthestarttimeofthe?rststormtotheendtimeofthesecondstorm(Fig.2).ThepeakwaveperiodTpistheconcomitantwaveperiodofHs,max,andthepeaksealevelhisthehighesttotalwaterlevelduringastorm.AndthewavedirectionismeasuredatthetimewhenHs,maxoccurs.
Oneaspectintheestimationofanextremewaveclimatethathasroutinelybeenignored,istheseasonaleffect.Inthestudysite,duringtheoceanographicwinterperiod(OctobertoMarch),thestormeventsaremorefrequentandheavierthanthestormsduringthesummerperiod(ApriltoSeptember).Inthispaper,seasonalitywasmanagedbyseparatelysimulatingthewaveclimateparametersinwinterandsummerperiod.
3.1.Estimationofmarginalprobabilitydistributionsofunivariatewaveclimateparameters
3.1.1.FittingtheGPdistributiontoHs,max,D,Tpandh
Therearethreetypesofextremevaluedistributions,widelyknownastheGumbel,FréchetandWeibullfamiliesrespectively.Thesefamiliesgivequitedifferenttailbehaviourandtheselectionamongthemisquitesubjective.Onceaparticularfamilyischosen,subsequentinferencesdonotconsidertheuncertaintyofthedistributiontype.Thegeneralizeex-tremevalue(GEV)distributioniscapableofcombiningthosethreefam-iliesintoonesinglefamily.Howevertheblockmaximumalgorithmcanbewastefulofdataincaseofextremevalueanalysisifotherdataonex-tremesareavailable(Coles,2001).TheweaknessescanbeavoidedbyusingtheGPdistributioninthe?eldofcoastalengineering(Callaghanetal.,2008;Hawkesetal.,2002).However,athreshold,u,shouldbese-lectedbefore?ttingthedistributionfunctiontothedatasample.Selectingaproperthresholdissometimesverysubjective,signifyingatrade-offbetweenbiasandvariance.Atoolowuwillleadtoalargedif-ferencebetweenthedistributionofthresholdexcessesandGPdistribu-tion,resultinginbias.Atoohighuwillleadtoasmallnumberofthresholdexcesses,resultinginlargevariance.Normally,thethresholdvaluecanbeselectedbyanalysingthemeanresiduallifeplot,butthein-terpretationofameanresiduallifeplotisnotalwayssimple,anditcan-notgiveade?niteanswer.Anotherwaytoselectthethresholdisbasedonthetheory,thattheGPshapeparameterandmodi?edscaleparam-etershouldbeconstantforanythresholdaboveu.Butthismethodwillnotgiveaclearanswereither,seedetailsinColes(2001).Therefore,weproposedaroot-mean-square-error(RMSE)analysis.Forcoastalhazardassessments,whatwearemostconcernedwitharetheextremeconditions.Thus,itisreasonabletochoosetheusothattheGPdistribu-tion?tstheextremedatabest,ratherthanathresholdthatprovidesthebest?tofthedistributiontothewholedatasample.Themeasurementsthatexceedthevalueswithareturnperiodofaboutoneyear(forinstance,Hs,maxN600cm,DN40h,TpN8sandhN2mforwintersea-son)wereselectedtocarryouttheRMSEanalysis.Theerrorrepresentsthedistancebetweenextremeobservationsandthemodelledcurve(Eq.(1)),andthethresholdu,whichcanminimizetheerror,wascho-sen(Table1).
3.Method
Themarginaldistributionsofthemultivariaterandomvariableswereestimatedaccordingtotheirstatisticalcharacteristics.TheGeneralizedPareto(GP)distributionwas?ttedtothewaveclimateparameters(Hs,max,D,Tp,h),whilefortheothertwoparameters(θp,Fs),theempiricaldistributionwereutilized.Thenextstepistheconstructionofthedependencystructuresforthefour-dimensionalvar-iables(Hs,max,D,Tp,h).Gibbssamplingmethod(GemanandGeman,1984)wasintroducedtosimplifytheArchimedeancopulamethodwhenextendingittohighdimensions.MarginaldistributionswerenormalizedbeforeconstructingthedependencystructurebyusingtheGaussiancopula.Inthethirdmethod,thephysicalrelationshipbetweenHs,maxandTpwastakenintoaccount.Anotheroptionalmethod,theLogisticmodelwhichrequiresthevariablestobetransformedintoFréchetscalewasappliedtotheDutchdataset.Afterthat,thescatteredsimulatedstormeventswereorganizedbythesimulatedFstoformatimeseriesofstormevents,andintheend,thegoodness-of-?ttechniqueswereusedtotesttheunivariateandmultivariate?ttingqualitiesofthefourdependencystructures.
e1T
wherethe?ttedvalueisthevalueextractedfromthe?tteddistributioncurvewhichhasthesameprobabilitywiththeobservedvalue,nisthenumberofthevaluesabovetheselectedthresholdintheRMSE
analysis.
Fig.2.De?nitionofindependentstormevents.
4F.Lietal./CoastalEngineering86(2014)1–13
Fig.3.Monthlynumberofstormsfrom1979to2009.
BesidestheGPdistributionandtheGEVdistribution,thelogisticandthelog-normaldistributionswerealsotested.TheresultsindicatethattheGPdistributionisthebestoneto?ttheobservedHs,max,D,Tpandh.TheGPdistributionhasthreeparameters,anditscumulativedistribu-tionfunctionisgivenby:
valuesofθpandFsislessimportant.Fortheotherstormparameters(Hs,max,D,Tpandh),theextrapolationfromobservedleveltounobservedlevels(i.e.thesmallexceedanceprobabilities)arevitalforcoastalprotectionandengineeringdesign.Fortheθp,andFs,theyare,ononehand,bothlimited,andontheotherhand,theirextremevalueshavelittleimpactsontheconsequencesinducedbytheextremestormevents.Therefore,theempiricaldistributionisaplausibleoptionforwavedirectionandstormeventfrequencysimulation.3.2.Constructionofthedependencystructure
e2T
whereσN0isthescaleparameterandξistheshapeparameter.WhenXNu,theGPdistributionwas?ttedtothedata,whilethedatabelowthethresholdwererepresentedbytheempiricaldistribution.Thedistri-butionparameterswereestimatedbymaximumlikelihoodestimation(MLE)method.
3.1.2.DeterminationoftheempiricaldistributionofθpandFs
Thewavedirection(withrespecttomapnorth)dataareavailableinashorterperiod(1989–2009).Fig.4,leftpanel,indicatesthatstormsoccuratα∈[200°,383°]andabout50%oftheeventsconcentratebe-tween200°and250°.Thetestforstatisticallysigni?cantdependencybetweenθpandHs,maxshowsthatthecorrelationisweak(Table1),con-sequently,θpwasassumedtobeindependenttoHs,max.AnempiricalCDFcurveofθpandabivariate(θp,Hs,max)plotwereshowninFig.4,rightpanel.
Thestorminesscanbeexpressedby?ttingaPoissondistributiontothemonthlynumberofstorms,by?ttinganon-homogenousPoissonprocesstotheinter-arrivaltimebetweenconsecutivestorms(Callaghanetal.,2008),orbysimplycomputingtheprobabilityofstormnumbersineverymonth.Theanalysisindicatesthattheobserveddataandthe?ttedPoissondistributiondonothaveanacceptableagree-ment,whilethenon-homogenousPoissonprocessimprovesthesimu-lationresultbutnotsigni?cantlycomparingittothethirdapproach,i.e.?ttinganempiricaldistributiontothenumberofstormswithineachmonth.Thethirdapproachissimpleandeffective;withinamonth,thestormeventswereassumedtobedistributedrandomlywiththesmallestinter-arrivaltimeof6h.
Theempiricaldistributionwillgivethemostpreciseestimationwithintherangeofobservedvalues.Althoughproblemsinusingtheempiricaldistributionwillariseinestimatingexceedanceprobabilitieswhichareverycloseto0and1,theaccuracyinsimulatingextreme
Table1
Linearcorrelationcoef?cientsbetweenHs,maxandothervariates.
D
WinterSummer
0.5440.611
h0.5830.526
Sp0.0790.052
θ0.2460.281
Tp0.8620.840
Thefour-dimensionmultivariategroupofvariables(Hs,max,D,Tp,h)wassimulatedbyfouralternativemethodsinthispaper.TherearetheArchimedeancopulas,Gaussiancopulamethod,physics-combinedGaussiancopulamethodandtheLogisticmethod(Coles,2001;Tawn,1988).Thedifferencebetweenthe?rsttwomethodsisthewayofsimulatingthepeakwaveperiod.TheTpcanbesimulateddirectlyasavariableofthemulti-dimensionalcopulaorbytheindependentwavesteepnessanditsintimatephysicalconnectionwithHs,maxandTp.3.2.1.Archimedeancopulas
TheArchimedeancopula:
e3T
whereφ(x)isthegeneratorfunctionandφ(x)?1istheinversefunc-tion.Threeclassicalgeneratorfunctionsarefrequentlyused:φ(x)=x?θ?1(Claytoncopula),φ(x)=?log[(e?θx?1)/(e?θ?1)](Frankcopula)andφ(x)=(?logx)θ(Gumbelcopula),theθparame-tersummarizesthedependencybetweenmarginalcomponents.ForthebivariatecopulabasedonEq.(3),theArchimedeancopulasare:?????1=θ
?θ?θ
Ceu;vT?utv?1
eClaytoncopulaT;
e4T
e5T
??&hi'??
θθ1=θ
eGumbelcopulaT:Ceu;vT?exp?e?lnuTte?lnvT
e6
T
F.Lietal./CoastalEngineering86(2014)1–135
Fig.4.Left:wavedirectionroseduringthestormevents,thelengthofthedirectionaltriangleillustratestheproportionofthedetermineddirection.Right:empiricalCDFandobservationofθcorrespondingtoHs,max
.
Fig.5showsthedifferenttaildependencesfortheArchimedeanfamily.TheClaytoncopulahaslowertaildependence,andtheFrankcopulahasnotaildependence,whiletheGumbelcopulahasuppertaildependence.
TheArchimedeancopulasareusuallyusedforbivariatecases,whiletheybecomeproblematicwithhigherdimensions.Ontheonehand,asingleparameterisnotsuf?cientfordescribingthedependencystructureofmorethantwodimensions.Ontheotherhand,thenumberofparametersquicklygrowswithdimension,whichwillincreasethecomplexityofcomputation(RenardandLang,2007).
AnimportantpropertyofArchimedeancopulas,whichconcernstheconditionalprobability,makestheapplicationpracticalforhigherdimensionalvariables.Let(U,V)beuniformrandomvariableson[0,1],joinedviaacopulafunctionC.Theconditionaldistributionfunctionscanbeobtainedbypartialdifferentiation(Nelsen,2006):
?Ceu;vT
:Thisapproachwasputintopracticewiththefollowingsteps,usingxfortheCDFofHs,max,yfortheCDFofD,zfortheCDFofTpandsfortheCDFofh:
1.CalculatetheempiricalCDFofeachstormparameterviakernelden-sityestimation(KDE),FX(X≤x),FY(Y≤y),FZ(Z≤z)andFS(S≤s);2.FitthedependencyparametersviaCanonicalMaximumLikelihood(CML)methodforeachcopula,themaximumlikelihoodfunctionis:?Ceu;vT????
Leu;vjθT?∏uivi;eui;viT∈e0;1T;
i?1N
e8T
3.GenerateauniformrandomnumberA~(0,1)andB~(0,1),andlet
?1
(A);y0=FY
4.Solvetheconditionaldistributionfunctionofx1giveny0,B?P
x1;y0T
eX≤x1jY?y0T??CXYe,and?ndthesolutionofx1;
PeU≤ujV?vT?
e7T
5.Generateuniformrandomnumbers(C,D,E)=U(0,1),andsolvetheconditionaldistributionfunctionsC?PeY≤y1jX?x1T?
?CXYex1;y1T
;
1
?Cex;zT
D?PeZ≤z1jX?x1T?XZ11and
1
?CXSex1;s1T
E?PeS≤s1jX?x1T?
1
Thankstothisproperty,aGibbssamplingapproach(GemanandGeman,1984)ispossibletoapplywithArchimedeancopulastogeneratefour-dimensionstormstateparameters.
Basedonavisualinspectionofthehistoricalmeasurements(Fig.6)andthetailbehaviourofeachcopula,theClaytoncopula,FrankcopulaandGumbelcopulawere?ttedto(Hs,max,D),(Hs,max,Tp)and(Hs,max,h)respectively.
for(y1,z1,h1
).
Fig.5.TheprobabilitydensityfunctionsofClayton,FrankandGumbelcopula,fromlefttotheright.AfterSchoelzelandFriederichs(2008).
F.Lietal./CoastalEngineering86(2014)1–13
7
Table2
Dependenceparameter(β)betweenvariatesandHs,max,95%con?denceintervalsareinthebrackets.
D
Tp
h
Winter0.70(0.64,0.77)0.42(0.32,0.51)0.62(0.54,0.70)Summer
0.67(0.55,0.80)
0.41(0.21,0.62)
0.72(0.61,0.84)
6.Repeatstep4and5forntimes(ndependsonthemonthlystormi-nessandthenumberofthesimulationyears)byiterationstogetthesimulatedwaveclimate,{(x1,y1,z1,s1),…,(xn,yn,zn,sn)};
7.TransformtheCDFstothephysicalscalebytheirinversemarginalCDFs.
Hereinto,
?CXYeu;vThθXY????i?e1tθT=θ
?1tu
v?θXY
?1XYXY;e9T
?CXZeu;vT?uθ???vθ??.h???eXZeXZ?1
e?vθXZ?1????e?uθXZ?1??te?θ
XZ?1i;e10T?CXSeu;vT
e?lnuTθhe?lnuTte?lnvTiXH?1θXHθXHe1?θXHT=θXH
?uexpe?lnuTθ1=θXH:e11T
XHte?lnvTθXH
3.2.2.Gaussiancopula
TheGaussiancopulaisamemberofEllipticalcopulasfamily,anditwasappliedtosimulatethedependencybetweenvariatesandgeneratethesimulatedmultivariablewaveclimatedata(Hs,max,D,Tp,h).Thiscopula,whichisadistributionovertheunitcube[0,1]d,canbeeasilygeneralizedtoahighernumberofdimensions.ItisconstructedfromamultivariatenormaldistributionoverRdusingtheprobabilityintegraltransform.ForagivencorrelationmatrixΣ∈Rd×d,theGaussiancopulacanbeexpressedasfollows:CeuT?Φ??ΣΦ?1eu?1
??R1T;…;ΦeudT;
e12T
whereΦΣisthejointdistributionfunctionofthed-variatestandardnor-maldistributionfunction,andΦ?1denotestheinverseoftheunivariate
standardnormaldistribution.HencethemultivariateCDFcanbewrit-tenas:
e13T
Inpractice,theGaussiancopulamethodisimplementedasfollows:1.TransformtheobserveddatatothescaleofthecopulausingakernelestimatoroftheCDF;
2.FitaGaussiancopulatothecomputedCDFandderivea[d×d]ma-trixoflinearcorrelationcoef?cientsfortheGaussiancopula,i.e.ΣinEq.(12);
3.Generaterandomsamplesfromthe?ttedGaussiancopula;
4.TransformthegeneratedsamplesbacktotheoriginalscalebyusingitsinverseCDFfunctions.AnadvantageoftheGaussiancopulaisthatitcanbeeasilyextendedtohigherdimensioncomparedtotheotheralternativecopulas.3.2.3.Physics-combinedGaussiancopula
Amethodbasedonthefunctionofwavesteepness(Sp)wasusedfordependencymodellingforHs,max,andTp(VrijlingandBruinsma,1980),meanwhilethedependencestructurefortherestofthevariateswasconstructedinthesamewayasmentionedin3.2.1.Abasicassumption
forthismethodisthattheSpisindependentonHs,max,whichisplausiblefortheDutchoffshorewaveclimatedata(Table1).Accordingtothelinearwavetheory,wavesteepnessSpisde?nedas:Sp?Hs;max=L0?2πH.??
s;max
gT2??p
;
e14T
whereL0isthedeepwaterwavelengthandgisthegravityacceleration.Therefore,TpcanbeexpressedbyrearrangingEq.(14):r????????????????????????????????????T.????
p?2πHs;maxgSp:
e15T
Spcanberandomlygeneratedfromits?ttedprobabilitydistribution.
AndHs,maxwasgeneratedthroughtheGaussiancopulaof(Hs,max,D,h)inSection3.2.2.Forthestudysite,alog-logisticdistributionis?ttedtowavesteepnessaccordingtoitsstatisticalcharacter.AndtheCDFofSpis:
Fh
lnexT??????μ;σi?f1texp??elnx?μT=σ??g?1
;e16T
whereμandσarethelocationandscaleparameters.
3.2.4.Thelogisticmodel
TheLogisticmodelisusuallyusedtoconstructthejointprobabilitydistributionforthebivariatecase.
e17T
wherex′andy′arerescaledFréchetvariates(convertedfromphysicalscale,xandy)andβisthedependenceparameter.Whenβ→1inEq.(17),F(x′,y′)=exp[?(x′?1+y′?1)],correspondingtoindepen-dentvariables;whenβ→0,F(x′,y′)=exp[?max(x′?1,y′?1)],corre-spondingtoperfectlydependentvariables.Hence,thebivariateextremevaluedistributionsgeneratedbytheEq.(17)coverallthedifferenttypesofdependencestructure,fromindependencetoperfectdependence.
Inordertoobtainthedependenceparameterβ,thelikelihoodfunc-tioncanbewrittenas:
Làα;àx′1;y′á1
;…;àx′n;y′áán?∏nψàβ;àx′i;y′ááie18T
i?1
where,
?2F??????
????x′′
i;yiifex;yT∈R1;1;ψàβ;à?F??????
x′′áá????x′i;u′yifex;yT∈R1;0;i;yi?e19T
?F??
????????u′′
ifex;yT∈R0;1;F??x;yiu′x;u′??y
ifex;yT∈R0;0;
Table3
Probabilitydistributionparameters.VariateWinterSummerHs,max(cm)(u,ξ,σ)=(470,?0.18,107.2)(u,ξ,σ)=(300,?0.49,147.7)D(h)(u,ξ,σ)=(15,?0.05,13.1)(u,ξ,σ)=(14,?0.01,13.6)Tp(s)(u,ξ,σ)=(6,?0.23,1.12)(u,ξ,σ)=(5.5,?0.69,1.85)h(cm)
(u,ξ,σ)
=
(190,?0.04,23.7)
(u,ξ,σ)
=
(150,?0.17,24.6)
8F.Lietal./CoastalEngineering86(2014)1–13
Fig.7.Empirical(×)and?ttedCDF(blackcurves)forHs,max(cm),D(h),Tp(s)andh(cm)intheoceanographicwinter(left)andsummerseason.
F.Lietal./CoastalEngineering86(2014)1–139
Fig.8.Monthlyaveragestormfrequencyofsimulationsandobservations.
u′xandu′y
arethethresholdparametersofthemarginaldistributionsandconvertedtotheFréchetscaleandR?e?∞;u????;∞T????∞;ui
0;0xT??∞;uy;R1;0??uxR0;1
?e?∞;uh???uh??
y;
x???uy;∞;R1;0?x;∞T?uy;∞:
Thetransformation
"
(
????!1???1=ξ)#?X′
??lnX?u1
u1tξ
e20T
inducesthevariablewithFréchetmarginsforXNu.Xisoneofthewave
climateparameters,?un=Pr{u}.Aftermaximizing∑ln?XψàNβ;àx′ái;y′
á?i
,thedependencyparametersfor(Hs,max,D),(Hs,max,1
Tp),and(Hs,max,h)werefoundandlistedinTable2.
Tosimulatethefour-dimensionaljointdistributionofwaveclimate,weestimatedtheD,TpandhbyusingtheirconditionalprobabilitiesgivenHs,max(ColesandTawn,1991)andGibbssamplingmethod(GemanandGeman,1984).Thus,theLogisticmodelcanbeputintopracticeinthefollowingway(Callaghanetal.,2008),usingx′forFréchetHs,max,y′forFréchetD,z′forFréchetTp,h′forFrécheth:
1.generatearandomnumberA~U(0,1),andlety0′=?(lnA)?1;
2.solvetheconditionaldistributionfunctionofx′1
giveny0′:A?PrnX′≤x′1jY′?y′
o
???x′?1=β′?1=β??β?1′e1?1=βT exp???x′?1=β′?1=β??
β!1ty0y01ty0ty′?1
0;
e21
T
Table4
K–Stestresultsfortheunivariatesimulation,whereWforwinterseasonandSforsummerseason,theRomannumeralsindicatetheArchimedeancopula(I),Gaussiancopula(II),physics-combinedGaussiancopula(III)andtheLogisticmethod(IV)respectivelyusedinSection3.2.
p-valueI
IIIIIIVHs,maxW0.960.770.960.94S0.830.20.210.46DW0.940.880.740.97S0.090.090.100.13hW0.720.800.590.90S0.750.900.780.73TpW0.200.100.030.07S0.15
0.21
0.480.17
Sp
W0.53S
0.75
withthedependenceparameterof(Hs,max,D),and?ndthesolution
ofx′1
;3.generaterandomnumbers(B,C,D)~U(0,1),solvetheconditionaldistributionfunctionsB?PrnY′≤y′??1????X′?x′
o1
???x′?1=β′?1=β??β?1′e1?1=βT ??′?1=β′?1=β??
β′?!1
tyty1
1x1exp?x11tx1e22T
C?PrnZ′≤z′??1????X′?x′
o1
???x′?1=βtz′?1=β??β?1′e1?1=βT exp???x′?1=β′?1=β??
β′?1
!11x11tz1tx1
e23T
e24T
withthedependenceparametersof(Hs,max,D),(Hs,max,Tp),and
(Hs,max,h)fory1
′(initialvalueforthenextiteration),z1′andh1′;4.repeatstep2and3forntimesbyiterationsandgetthesimulated
Fréchetwaveclimate,{(x′1
,y1′,z1′,h1′),…,(xn′,yn′,zn′,hn′)};5.transformtheFréchetwaveclimatetothephysicalscalebyusing
e25T
whenxbu,xwascomputedfromitsempiricaldistribution.
3.3.Stormsequencesimulation
Sincethejointstatisticaldistributionofthefour-dimensionalwaveclimatehasbeenstudiedandthesimulationshavebeengeneratedintheabove-mentionedsections,thestormstimeseriescanbesimulatedbyaddingtheindependentwavedirectionparameter,θp,andthestorminessparameterFs.
Theempiricaldistributionofstormfrequencywasusedtoobtainthenumberofstormsforeverymonth.Foreachmonth,basedonthesim-ulatednumberofstorms,thestormeventsofcorrespondingnumbercanbesimulated.Thesestormeventsarethendistributedrandomlyinthismonth.Byrepeatingthisprocessforktimes,astormsequenceofk/12yearswillbegenerated.
Withinthetimescaleofonemonth,itisplausibletoassumetheHs,maxoccursrandomly.But,theinter-arrivalbetweenadjacentstormsshouldbenoticed,itshouldbelongerthan6haccordingtothede?nitionofindependentstormsinFig.2.3.4.Goodness-of-?ttest
Fortheunivariatemarginaldistribution,theKolmogorov–Smirnovtest(Massey,1951)andtheChi-squaretest(Pearson,1900)aremostlyusedtoassesswhetherthe?tteddistributionisasuitablemodelfortheobservationaldata.TheK–Stestisanonparametrictestandcanbeusedtodecideifasamplecomesfromapopulationwithaspeci?cdistributionbycomparingtheempiricalcumulativefrequencywiththeassumeddistributionfunction,Dn?maxjFexT?FeexTj;
e26T
where,Dnisarandomvariablewhosedistributiondependsonsam-plesizen,F(x)andFe(x)arethe?ttedCDFandempiricalCDF,respec-tively.Foraspecialsigni?cancelevelα,ifthediscrepancyissmaller
10
Table5
Chi-squaretestfortheunivariateandthejointsimulation.
IC
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