Harmonic tidal analysis methods on time and frequency ... · 3 CEM/UFPR, Pontal do Paraná, Brazil...

Bollettino di Geofisica Teorica ed Applicata Vol. 54, n. 2, pp. 183-204; June 2013

DOI 10.4430/bgta0068

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Harmonic tidal analysis methods on time and frequency domains: similarities and differences for the Gulf of Trieste, Italy, and Paranaguá Bay, Brazil

E. MaronE1,3 , F. raicich2 and r. MosEtti1

1 OGS,Trieste, Italy2 CNR-ISMAR, Trieste, Italy3 CEM/UFPR, Pontal do Paraná, Brazil

(Received: June 13, 2011; accepted: February 16, 2012)

ABSTRACT Two years of sea level data obtained atTrieste, Italy, and Paranaguá, Brazil, wereused to compare the performances of two tidal analysis methodologies, one in thetime domain (HMB) and other in the frequency domain (HMF). For each station,the first year was analyzed to estimate the tidal constituents while the second year wasusedtocompareobservationsagainstforecastedsealevels.Withrespecttotidaldynamics,resultsfrombothmethodsshowedthatthesemi-diurnalatmospherictideS2pisdisturbingtheseaconstituentS2aswellasotherfrequencies,particularlyinthediurnalspeciesinthesubtropicalcaseofParanaguá.Theseresultsareinagreementwiththeoreticaldevelopmentandnumericalsimulationsavailableintheliterature.Onthemethodologicalside,bothmethodsshowedequivalentperformancesalthoughHMFismoreuser-friendlyandoffersbetterandmorecomprehensiveresults.Themainreasonseemstobelinkedtothedeepenexploitationofastronomicalandphysicalaspectsofthetidalconstituentsgenerationanddiscrimination,consideringthatHMFcanestimatemorethan170whileHMBestimatesaround110.Also,HMFshowedbetterresultsforshallowwaterandlongtermcomponents.However, theresidualsshowedthatasignificant amount of oscillating energy is left behind by both methods, suggesting that otherdeterministicsignalsnotpresentintheastronomictidalfrequenciesofseawaterhavetobeconsidered.Itisconcludedthatabetterstochasticmodelfortidalanalysisandforecastneedstobeformulatedinordertobetterrepresentthephysicsofsealevel:whiletidalforecastwiththeusualmethodsseemstoworkwellinmanypracticalcases,thehighdependenceofnumericalmodelsoninitialandboundaryconditionssuggeststhatsealevelharmonicconstituentsestimationhastobeimproved.

Key words: tides,analysis,prediction,harmonicanalysis,Trieste,Paranaguá.

1. Introduction

Theastronomical tideshasbeenstudied in the last twocenturies indifferentwaysbut thebackgroundofalltheanalysisandforecastingmethodologiesisbasedontheearlydevelopmentbyDarwin(1898,1907),improvedbyLaplace,LordKelvinandotherscientists,mostlyinthe19thcentury.Thisdevelopmentisbasedontheprinciplethatthesealevelheightsinagivenplace

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Boll. Geof. Teor. Appl., 54, 183-204 Marone et al.

canberepresentedbyasumofNharmonicterms(fromi=0toN-1),eachonehavingauniquepairofamplitude (Hi)andphase (Gi),oscillatingataparticular frequency (ωi) defined by the astronomicaltidegeneratingpotentialand,whenpresent,alsobynon-linearcombinationoftheastronomicalconstituents(shallowwatertidalcomponents).

From the revolutionary work of Newton, the tides offered a great scientific challenge, but at thebeginningofthe20thcentury,thegreatutilityofagoodknowledgeoftheastronomicaltidesinagivenlocationmovedtheinteresttotheapplicabilityoftheanalysisandpredictiontechniques,which evolved from the use of complicated analysis tables to determine a restricted numberofHandGpairs,passingthroughtheanaloguetidepredictionmachines,tothepresentdigitalcomputerbasedmethodologieswithalmostnolimitations,butthephysics,fortheirdetermination.However,mostifnotallthepastandpresenttidalanalysisandforecastingmethodologieskeptthebasicDarwinprincipleunchanged.

Nowadays,agiantnumberofdifferentdigitalcomputerprogramsarewidelyusedtoanalyzeandpredicttheastronomicaltides,usingdifferentalgorithmsandapproaches.Inaway,allanalyzesealeveldata(evenlyorunevenlytimespaced),inthetimeorfrequencydomains,determiningasetofHandGpairsforthegivenplace,allowingtheastronomicaltidetobeforecasted.AllthecurrentmethodologiesworkproperlyforpredictionpurposesbutnotnecessarilywhentheHandGvaluesareusedforotherpurposes.

However,whenwecomparethesetsofHandGobtainedforasameplaceusingthesamesealeveldatabutdifferentanalysismethodologies(Marone,1991),wenotethatthetidalconstituentsdifferinboththeirparametervalues,butmainlyinG,notwithstandingtheiruseforforecastingpurposesdonotshowsubstantialdifferences,whichhasnotbeenyetwellexplained.

Theextensiveusesofnumericalmodellinginoceansciencesbringintoplaythetideasoneofthemostimportantdynamicalforcingand,inmostcases,thereisstillaneedforimprovementregardingthetidalconstituentsusedtofeedthemodels,whichseemstoworkbetterifobservedsea level data are used instead (Hendershott, 1977; Camargo and Harari, 2003; Harari et al., 2006;Lyardet al.,2006;Moreiraet al.,2011).Particularly,coastalandshallowwatersnumericalmodelresultsarestillnotfullysatisfactory,mainlybecauseofthereducednumberofusedtidalconstituents,whichcoverslessthat95%oftheastronomictidalenergy(Padmanet al.,2008).Also,tidalanalysiscouldleadtoassignamplitudesandphasesatastronomicaltidalfrequencieswhicharenotallofdirectastronomicorigin(ChapmanandLindzen,1970;Arbic,2005).Evenwhenmethodsconsiderexplicitsignal-to-noiseteststoacceptconstituentswithamaximumerrorof,say,5%inbothamplitudesandphases,theseerrorestimatesareofstatisticalorigin.Anaccuratetidalanalysiscould,inthebestofthecases,includealsosomeerrorestimatesforobservations(MaroneandMesquita,1995,1997),butoncewehavedeterminedthespectralamplitudesandphasesforagivenacceptedastronomicaltidalfrequency,thereareothertypesofuncertainty(duetononlinearinteractions,air-seatidalcoupling,etc.)whichcouldleadtomisunderstandingsinbothtidalforecastsanmodeling.

Ifthetidalconstantsarenotsoconstant,notonlybyphysicalreasonsbutalsoaccordingtothe used method of analysis, the efficiency of numerical models is compromised, as well as the interpretationofmanyoceanphenomena.Ithasbeenprovedthattheoceanmicro-structurehashighcorrelationwiththeturbulentdissipationandthetidalcycles(Polzinet al.,1997;Ledwellet al., 2000). In coastal areas, tides contribute significantly with the vertical mixing, redistributing nutrientsandoxygen,whichimpactmarinelife(Romeroet al., 2006) and, also, with the flow of

Harmonic tidal analysis methods on time Boll. Geof. Teor. Appl., 54, 183-204

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CO2betweenseaandair(Bianchiet al.,2005).Thebasicequationsoftidaldynamics(MunkandCartwright,1966;Godin,1972;Hendershott,

1972,1977;Pugh,1987,2004;MunkandWunsch,1998;Franco,2009)arerelativelysimpleandthey are well known from Laplace times. However, in numerical modelling for instance, theneedofhigherprecisionoftidalconstituents,particularlyincoastalregions,isbeingindicatedas a relevant scientific challenge (Lefevre et al.,2000;Lyardet al.,2006;Simionatoet al.,2009;Moreiraet al.,2011).

Inrecentyears,withthegreatevolutionofsatellitealtimetry,thecombinationofsatellitedataassimilationintonumericalmodelling(Matsumotoet al., 1995) requires, for tidal fields to be well representedinalimitednumberoflocationsinsidethemodeldomain,thatconstituentsinsuchplaceshavetobewellandaccuratelyknown(Kantha,1995;EgbertandErofeeva,2002).

Itisthenclearthattheneedforabetterunderstandingoftheprinciplesandqualityofthetidalconstituentsestimatedbydifferentmethodologieshave tobeachieved.Also,whenexaminingmanyof the actual tidal analysismethodologies, one facesother small but not less importantproblem: thenamesgiven to the tidal constituents are not yet fully standardized, anddiversemethodologiesidentifythesamewiwithdifferentnames,makingtheuseofthem,thecomparisons,and the physical interpretation, more difficult. Also, the names of the different methodologies are somanyandwerecreatedusingsuchdiverseapproachesandstylesthattheyhelpalsotocreateconfusion.

Inthisworkweapplytwomethodologies,bothbasedontheHarmonicMethod(HM),onesolvingtheequationsinthetimedomain(HMB)andtheotherinthefrequencydomain(HMF),totwosetsofoneyearof30-minutesampledsealeveldataobtainedintheGulfofTrieste,Italy(Fig.1,top),andtheParanaguáBay,Brazil(Fig.1,bottom),theresultsoftheanalysiswerecomparedandusedforpredictionpurposes.Havingasubsequentyearofsealeveldataforbothplaces(Fig.2),theforecastedsealevelswerecomparedwiththeobserveddataandtheresidualswerestudiedtoclarifythecausesofthedifferences,bothphysicalandmethodological,insearchofanswersregarding themethodologies quality and,mainly, about the reasons of suchdifferences.Suchobjectivedoesnotintendsolelytosolvesomejustformalissues,likethestandardizationofnamesor,worst,whatmethodologyisthemostaccurate.

2. Methods

Asalreadymentioned,weusedheretwoanalysisandpredictionmethodologiesbasedintheHM for our purposes, one solving the equations in the time domain (HMB) and other in thefrequency domain (HMF). HMB method is based in the developments by Godin (1972) andForeman(1977)anditisactuallyamuchevolvedcomputerprogram(Bellet al.,1999),whosefundamentalswillbeexplainedbelow.Also,theHMFisanevolutionoftheworkdevelopedbyDoodson(1921)duetoFrancoandRock(1971)anditsfurtherupdates(Franco,1997,2009),anditisshortlyexplainedatthefollowingsection.

OurapproachissimilartotheoneusedbyMunkandCartwright(1966)whencomparingtheResponseMethod(RM)withothertidalanalyses.WedidnotincludetheRMbecauseitisnotconsideredaccurateenoughtodrawconclusionsfromtheanalysiswhencomparedwithHMs.SuchlimitationswereaddressedbyMarone(1996)andMaroneandMesquita(1997).

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Fig.1-Half-hourlysealevelrecordfor2004at(up)theGulfofTrieste(45°38.82’N–13°45.54’E)andfor1997at(bottom)theParanaguáBay(25°30.1’S–48°30.2’W)usedforanalysis.

Fig.2-Hourlysealevelrecordsfor2005attheGulfofTrieste(top)andfor1998attheParanaguáBay(bottom)usedforpredictioncomparisons.


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3. Harmonic analysis in the time domain - HMB

ThebasicassumptionfortheapplicationoftheHMBmethodisthatthetidalvariationscanberepresented by a finite number Nofharmonictermsoftheform:

Nζ(t) = Hncos(ωn t–Gn);n=1,…,N. (1) 1

Hnisanamplitude,ωnisanangularspeedandGnisaphaselagontheEquilibriumTide.PhaselagsareconventionallyexpressedrelativetotheGreenwichmeridian.Usually,angularspeedsareexpressedindegreespersolarhourandphaselagsindegrees.EachangularspeedΩn=2πσn/360(inradianspersolarhour)isdeterminedasalinearcombinationoftheangularspeedsΩ1,…,Ω6 ofthetidalcomponentsrelatedtosolarandlunarmotions,namelymeanlunarday(1),siderealmonth(2),tropicalyear(3),Moon’sperigee(4),regressionofMoon’snodes(5)andperihelion(6).ω0 is the angular speed related to the mean solar day. The linear combination coefficients aresmall integers.Themostcompleteworkonthesubjectaswellasthedeterminationofthecoefficients mentioned above was performed by Doodson (1921). Details on the mathematical procedurescanbefound,e.g.,inPugh(1987).

In practice, in the harmonic analysis in the time domain we fit a tidal function:

NT(t)=Z0 + Hn fncos[ωn t–Gn+(Vn+un)] (2) 1

where theunknownparameter areZ0,Hn andGn (n =1,…, N), to a time series of observedvaluesO(t).Thetermsfnarethenodalfactorsandunthenodalangles,whilethetermsVnaretheequilibriumphaseanglesfortheconstituents(forsolarconstituentsfn=1andun=0).Again,theconventionisadoptedtotakeVn as for the Greenwich meridian (Pugh, 1987). The fit is performed usingtheleast-squareprocedure,insuchawaythatthequantityS2 = [O(t)–T(t)]2isminimum;thesumismadeoverallthetimesoftheobservations.

4. Harmonic analysis in the frequency domain - HMF

4.1. AnalysisTheharmonicmethodusedherewasoriginallydevelopedbyFrancoandRock(1971)andwas

permanentlyupdatedfromtheiroriginalroutinesinFORTRANtomodernlanguagescapabletoworkinpersonalcomputers(Franco,1997,2009).ItisthemethodologyusedbytheBrazilianNavy,whichisthenationaltideauthority,toanalyseandforecasttheastronomicaltidesallalongthemorethan8,000kmofBraziliancoastline.

The HMF is based on the assumption that the observed sea level in a given place can beaccuratelyrepresentedbyastochasticmodelwithaharmonicpart[mostlytheastronomicaltide,asinEq.(1)]andanondeterministicresidual(noise).

Departing fromadeepknowledgeof theastronomical tidalpotential, ituses theharmonicoscillationof thegenerating forces to identify the tidal frequencies ωi thatmaybepresent in

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thedatarecordforagivenplace.Thedataarethenanalysedinthefrequencydomainvia,forinstance,theFastFourierTransformorotheralgorithm,astheWattsmethod(JenkinsandWatts,1968;Marone,1991)ortheDirectFourierTransform(Marone,1991).Oncetheenergyspectrumisobtained, thetrickyapproachof theHMFis theuse, toseparateveryclosebutwellknownastronomic tidal constituents, using the angular frequency differences among neighbour tidalconstituentsinsteadoftheRayleighprinciple(Pugh,1987).

In a classical spectral analysis, a set of 2N harmonic equations is solved (in the time orfrequencydomains)inordertodetermineNpairsofHandGunknowns.IntheHMF,theangularfrequenciesdifferencesarealsousedinthearrayofequationstobesolved,generatingaredundant(moreequationsthanunknowns)groupwhichenhancetheprecisionandincreasethenumberoffrequenciesthatcanbesolvedfor.Thisispossiblebecauseparticularspectralpeaksofsealeveldataarenot“contaminated”byenergyoftheothertidalspecies(diurnal,semi-diurnal,etc.),allowingthemethodtotreateachtidalspeciesinseparatesetswitharedundantnumberofequationswithrespect to the unknowns.The split into sub-systems allows, for instance for the diurnal band(290<n<380cyclesper 8,192hours), to have90 equations to solve just 20 tidal constituents,whichcanbeeasilysolvedusingtheleastsquaremethod.TheHMFoffersanotheradvantagewithrespecttoshallowwaternon-linearconstituents,whicharestraightforwardlyrepresentedascombinationoftwoormoreastronomictidalcomponentsinbi-linearortri-lineararrays(Marone,1991).Also,themethoddoesnotrequiresealeveldataseriescorrespondingtoexactmultiplesofhalflunation(FrancoandRock,1971).Finally,theHMFusesasimpleapproachtoestimatethequalityoftheresults,calculatingthesignal-to-noiserateforeachtidalspeciesbasedinthestochasticityofthemodelandontheenergypresentinfrequencieswhichdonotcorrespondtoastronomicaltidalforcing(residualspectrum),andcalculatedfromthevariancesoftheresidualenergypresentoneachtidalband(species).

5. Forecast

Asmentioned, theHMF represents the sea levelζ(t) at anygiven instant t as a stochasticsumofNharmonicterms(withamplitudeH,phaseGandafrequencyω)plusa“noise”ε,asfollows:

Nζ(t) = [ Hicos(ωi t+Gi)]+ε(t) (3) 1

ThemoreNtermswehave,byknowingthepairsHandGobtainedwiththeHMBorHMF(or any other), the better will be the fit. The tidal prediction is then obtained solving the sea level equation [theharmonicdevelopmentofEq. (1)]of themodel for theperiodof interest,reconstructing the astronomical tide as the sum of the contribution of each obtained and significant tidalconstituentωi,characterizedbytheirHandGpairsinEq.(3),disregardingthenoise.IthastobenotedthatthephasesGhavetobereferredtoacommont=0,whichintheHMFandHMBis00:00hourUTofJanuary1,1900.


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6. Studied areas

In order to help the readers with the geographical onset, Fig. 3 depicts a couple of mapscorrespondingtotheGulfofTrieste,Italy,andtheParanaguáBay,Brazil.

6.1. Gulf of TriesteTheGulfofTriesteliesinthenorthernmostpartoftheAdriaticSeaatabout45°40’Nlatitude

and13°40’Elongitude.Itisapproximatelya20×20km2,connectedwiththeAdriaticontheSWsideandsurroundedbylandonthethreeothersides.Itisashallowbay,withmaximumdepthofabout25m.

Detaileddescriptionsofthephysicalcharacteristicsandofthecirculationandwatermassesof the Gulf of Trieste can be found in Malačič and Petelin (2001). The water body is generally stratified from spring to autumn, with transitory exceptions related to the occurrence of NE (offshore) wind, namely Bora, which causes homogenization by up-welling. Occasionally,in late autumn the stratification can also be broken by vertical mixing related to dense water formation caused by surface heat losses; however, this process is typical of winter. Stratification isenhancedbyrelativelyhighsurfacetemperatureandfreshwaterrun-off,mainlyinthenorthofthebay,wherethemeanannualriverIsonzodischargeisabout200m3/s.Duringwinterthewatercolumnisgenerallyhomogeneous,asaconsequenceofthesurfacecoolingandfrequentup-wellinginducedbyBora.Onaverage,temperaturecanrangefrom8°to12°Cinwinterand20° (in the bottom layer) to 26° (near the surface) in summer. Away from the direct influence of freshwaterrun-off,salinityrangesare33-38inwinterand32-36(surface),upto37(bottom)inspring-summer.

Themeancirculationismostlycyclonic;howeveritdependsverymuchonthepresenceofstratification and wind activity, in such a way that different patterns may be observed along the vertical.Particularlyinlateautumnandwinter,coldairspellscausehugesurfaceheatlossesandtheformationofdensewater,whichcontributestotheNorthernAdriaticDeepWater(Artegianiet al.,1997).

ThemainwindregimesarecharacterizedbyBoraandSirocco.BorablowsfromtheNE-EastsectorandcausesthedecreaseofsealevelatTrieste.SiroccoblowsfromSEalongtheAdriaticbasinanditisamajorfactorresponsibleforstormsurgesinthenorthernAdriatic,whoseimpactislargerintheareaaroundVenice,butisnotnegligibleintheGulfofTrieste.However,themostseverestormsurgesatTriestearegenerallyconnectedwithSWwind(Libeccio).

The astronomic tide is prevailingly semi-diurnal (Malačič et al., 2000), the dominantconstituentsbeingM2,K1andS2.IntheGulfofTriestetheamplitudeisthelargestobservedin theAdriaticSeawitha theoreticalmaximumofabout80cm,and thesecondlargest in theMediterraneanSeaaftertheGulfofGabes,Tunisia.IntheAdriaticSeathetidalsignalpropagatescounter-clockwise,i.e.,fromtheCroatiancoasttowardstheGulfofTriesteandthenalongtheItaliancoast,withaperiodslightlylongerthan12hours.

Systematic sea levelobservationswere started in the autumnof1859,whena tidegaugewasinstalledonMoloSartorio(Schaub,1860),fewmetresawayfromthepresenttidegaugeposition.Originaldiagramsandmanuscriptsareavailableonlyfrom1905onwards,whereassealeveldataforthe1859-1904periodcanbeonlyfoundintheliterature.Since1905thesealeveltimeseriesischaracterizedbyfewmajorgaps(January-December1916andJanuary1925-June

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Fig.3-MapsoftheGulfofTrieste,Italy(top)andtheParanaguáBay,Brazil(bottom).

BRAZIL

Paranaguá

Estuarine

Atlantic

Paranaguá


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1926)duetomissingdataortidegaugeoperationinterruptionformaintenance(Raicichet al.,2006;Raicich,2007).

ArichliteratureexistsaboutboththeoreticalandexperimentalstudiesontidesintheGulfofTriesteandtheAdriaticSea,whichissummarizedbyCushman-Roisinet al.(2001)andreferencestherein.

AtpresentthesealeveldataareavailableeveryminutefromtwoOTTThalimedesinstruments,withnear-real-timetransmission,aswellasacontinuousanaloguerecordfromaBüsum-OTTinstrument. All of them are float tide gauges.

6.2. ParanaguáTheParanaguáBayestuarinecomplex is located inaQuaternarycoastalplain in southern

Brazil(Bigarellaet al.,1978;AnguloandLessa,1997;Aielloet al., 2005); it has been classified asapartiallymixedestuary(typeB),withlateralheterogeneity(Knopperset al.,1987;Maroneet al.,1997),meandepthof5.4m,withmaximumupto30m,totalwatervolumeof14,109m3andaresidencetimeof3.49days(Maroneet al., 2005). As circulation patterns and stratification vary betweenseasons,meansalinityandwatertemperatureinsummerandinwinterare12-29°and25-30oCand20-34°and18-25oC,respectively.Asalinity-energygradientfromfreshwatertomarineconditionsalongtheE-WandN-Smainaxesdividesthebayintoahighenergy,euhaline(averagesalinity~30)outerregion,amiddlepolyhalineregion,andoligo-andmesohaline(averagesalinity0-15)low-energyinnersectors.Lateralgradientoriginatesfromthefreshwaterinputofriversandtidalcreeksandcreatesseveral‘micro-estuaries’intheeuhalineandpolyhalinesectorsofthebay.Atemporalgradient,withdaily,seasonalandinter-annualcomponentsissuper-imposedontheabovegradients(Lanaet al.,2001).

Thehydrodynamicisdrivenbytidalforcingandriverrunoff(Knopperset al.,1987;MaroneandCamargo,1995;Maroneetal.,1997,2005;Lessaetal.,1998;Lanaetal.,2001).Tidesaresemidiurnal with diurnal inequalities. Tidal amplitudes increase towards the head of the bay,being amplified near to two times. The tidal phase and amplitudes indicate that the tidal wave propagatesinamixedform,withaprogressiveformattheouterregionandastandingwaveformintheupperbay.Duringneapcycles,strongnon-linearinteractionsallowfortheformationofupto6highandlowtidesperday.Also,thedoublehighandlowwaterphenomenon(Godin,1972)isconspicuous(MaroneandCamargo,1995;Maroneet al.,2005).Springtidesrangefrom1.7matthemouthto2.7mintheupperbay.Themeantidalrangeis2.2m,withatidalprismof1.34km3andatidalintrusionof12.6km.

Followingcoldfrontforcingorextra-tropicalcyclones,stormsurgeselevatewaterlevelsupto 80 cm above astronomical tides (Marone and Camargo, 1995). Current velocities increaseupstream,withmeanmaximaof0.8-0.85ms-1 at flood and 1-1.4 m s-1atebb.

The average annual freshwater input from the coastal plain catchment area (about 1,918km2)andfromthesmallandsteepdrainagebasinsof theSerradoMar ishigher than200m3s-1(Mantovanelli et al.,2004;Maroneet al.,2005).Groundwatermaycontributeupto10%ofthe total surface freshwater runoff (Marone et al., 1997; Suresh Babu et al., 2008). Seasonalvariationsoffreshwaterinputcorrespondtoaround30%ofmeanannualvaluesduringthedryperiod(May/October)and170%duringtherainyperiod(November/April).

Tidal data are regularly collected in the Paranaguá area from the 1970’s with continuousanaloguerecordfromOTTtypeinstrumentsand,morerecently,withbottom-pressuresensors.

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Fig.4-ObtainedtidalconstituentsforTrieste:amplitudes(top-cm)andphases(bottom-degrees).

Fig.5-ObtainedtidalconstituentsforParanaguá:amplitudes(top-cm)andphases(bottom-degrees).


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Theanaloguedatawereusuallydigitizedatintervalsofhalfanhouroreach60minutes;whiledigitaldataarenowcollectedwithaoneminutesamplinginterval.Olderdata,collectedirregularlyfromtheendofthe1800’sandalongthe20thcenturyexists,butwerenotusedhere.

7. Data sets

From the existing datasets, we selected two years of sea level data for both Trieste andParanaguá,consideringperiodswithgooddataqualityandnogaps.Also,inviewoftheobjectiveofcomparingtheperformanceofthetwomethods,sealeveldatawerere-sampledwithonehourinterval for bothTrieste (2004 and2005) andParanaguá (1997 and1998).Datawerequalitycontrolledbeforetheanalyses.

8. Results

8.1. HMB analysisInthepresentworkthetidalanalysisinthetimedomainisperformedusingtheTASK-2000

package(Bellet al.,1999).Onecompleteyearofobservations,namely1997forParanaguáand2004forTrieste,isanalysedupto104tidalconstituents,withperiodsrangingfromsixth-diurnaltodiurnalandincludingsomelongperiodtoo.Upto60estimatedtidalconstantsareusedwiththesamepackagetoforecasttheastronomictidefor1998forParanaguáand2005forTrieste.

TheresultingtidalconstantsareshowninFigs.4and5forTriesteandParanaguá,togetherwiththeHMFresults.Onlycomponentswithamplitudeshigherthan1cmareshown,consideringthat oscillations below this limit cannot be estimated by the analysis (Marone and Mesquita,1997).

8.2. HMF analysisOntheotherhand,thetidalanalysisinthefrequencydomainisperformedusingthePACMARE

package(Franco,2009).Thesamecompleteyearsofobservations,namely1997forParanaguáand2004forTrieste,areanalyzedforupto176constituents,withperiodsrangingfromtwelfth-diurnaltodiurnalandalsoincludingsomelongperiodconstituents.Theacceptedestimatedtidalconstantsareusedwiththesamepackagetoforecasttheastronomictidefor1998forParanaguáand2005forTrieste.

TheresultingtidalconstantsareshowninFigs.4and5forTriesteandParanaguá,togetherwiththeHMBresults.Apartofthebelow1cmconstituentslimit,thesignal-to-noisecriteriaoftheHMFwasappliedtodisregardtidalcomponentswhenevertheconstituentwasnotshownwithsimilarvaluesinbothmethods.

8.3. ForecastInordertoperformcomparisonsbetweentheforecastingperformanceofbothmethodsand

places, theobservedsea levelheights for thesubsequentyears,1998 forParanaguáand2005forTrieste(Fig.2)werecomparedwiththehourlypredictedsealevelusingboth,theHMBandHMFtidalconstituents.Thesedatasetswere thenanalyzed,comparing the residualsbetween

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Fig.6-Observedandforecastedsealevelheight(verticalaxis–cm)forTrieste(top)andParanaguá(bottom)withbothmethods constituents for the first two weeks of the respective forecasted year (abscissas in hours).

Fig.7-Sealevelheightresiduals(verticalaxis–cm)calculatedfromobservedandforecasteddatawithbothmethodsfor Trieste (top) and Paranaguá (bottom) for the first two weeks of the forecasted year (abscissas in hours).


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theobservedsealevelandtheforecastedbytheHMBandHMFtidalconstituents.Fig.6depictsonly the hourly data of the first two weeks of each predicted year for a better representation for both,Trieste(top)andParanaguá(bottom),butthefulldataandforecastedyearswerefurtheranalyzed.

8.4. Residual analysisInordertocomparemethodologicalperformancesandthequalityoftheresultingforecasts,a

residualanalysiswasperformedfollowingthreeapproaches.I) The first one was to study the residuals produced by the differences between the forecasted

andthecorrespondingobservedsealevelseries.Fig.7depictsthesametwoweeksofFig.6,forreadiness,ofthecorrespondingresidualsbetweenobservationsandHMBandHMFforecasts,aswellasthedifferencesbetweenbothforecasts(HMB-HMF),forbothTrieste(top)andParanaguá(bottom).

II) Inthesecondcase,weanalyzedthespectrumoftheresidualsofthefullyearseries,usingaFFTroutineandaBartlettwindowwitha37datalength.ThesecalculationsgiverisetotheresultsdepictedinFig.8forTriesteandFig.9forParanaguá,wherethespectracorrespondtotheresidualsbetweentheobservationsandtheforecastedhourlydataforbothHMBandHMFandthespectrumofthedifferencesbetweenthepredictedsealevelsofbothHMB-HMF.

Fig.8-SpectraoftheresidualsforTrieste(forecastsvs.observations-abscissasinhours).

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Fig.9-SpectraoftheresidualsforParanaguá(forecastsvs.observations-abscissasinhours).

Fig.10-SpectralresidualsofHMFtidalanalysisfornontidalfrequenciesforTrieste(top)andParanaguá(bottom).


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III) Finally,theHMFmethodoffersacomplementaryoutput,whichcorrespondstothespectralamplitudes calculated out of the astronomical tidal frequencies, which are used for thesignal-to-noiseratiodetermination,andthatareagoodmeasureofhowmanynontidalbutharmonicenergyispresentinagivendataset.Fig.10showsthespectralresidualsforTrieste(top)andParanaguá(bottom).

9. Discussion

9.1. Tidal analysis methods: implementation and reachBothmethodsarefullyoperationalinbothcountries,areeasytouseandcananalyzelongdata

seriesbasedondesktopcomputersinaveryshorttime.DatapreparationisverysimpleinbothcasesbutHMBneedsmoreoperatorinterventionprovidedtheanalyzedtidalconstituents(onlyuptothe6thdiurnalspecie)needtobemanuallyintroduced,whileHMFhasalltheastronomicalcomponentsalreadyincludedanduptothe12thdiurnalspecie.TASKpackagecananalyzeupto104tidalconstituents(otherscanbeincluded,butthecodeassumesthattheyhavenonodaldependence)whileHMFarrives tomore than176.Also,HMFhas the capability to combineandsuggestnewshallowwaterconstituentsiftheyappearinagivenanalysis.MostoftheextraconstituentspresentinthePACMAREpackageareofnonlineartype,suggestingitsusecouldbemore appropriate for places were shallow water tidal components are expected to be significant. TheHMFpackagehasmanyotheroptionssuchas:highresolutionanalysis,cross-analysis,longtimeseriesanalysisandofanalysisextreme,sea levels.However,weused thebasicanalysis,which includesa fulloutputof the spectral results fornonastronomical tidal frequenciesandcriteria toacceptorreject tidalconstituents.ThissuggestedadvantageofHMFisanumericalcriteriatoacceptordisregardtidalcomponentsiftheydonotepassthesignal-to-noiseratiotest.Evenif it isanobjectivewayoftestingthequalityoftheoutput, theacceptanceofverylittleconstituents(below1cm),whichit isprovedcannotbeestimatedinthesecases(MaroneandMesquita, 1997), and the rejection of some significant ones (with around 4 cm amplitudes), which inturnappearswithverysimilarvaluesattheHMBshowingadeterministicreiteration,suggestsithastobeappliedcarefully.

9.2. Tidal analysis methods: results comparisonHMBandHMFproduceverysimilarresultsforTrieste(Fig.4),withclosedamplitudevalues

andsimilarphases,withexceptiononlyforM1.Allthemajortidalcomponentswereestimatedbybothmethodsuptothe6thdiurnalspecie,catchingalsosomelongperiodones.

IntheParanaguácase(Fig.5),tidalestimatedamplitudesareverysimilarinbothmethods,butitispossibletonotethatabsolutevaluesdiffer,alwaysatverylowlevels,morethanintheTriestecase.Theseresultdifferencesaremoremarkedatthephasevaluesbut,inbothcalculatedamplitudes and phases, we can say the differences are not significant. On the other hand, the non linear complexity of the co-tidal phenomena at Paranaguá seems to be better fixed by HMF. With the estimation of significant constituents with amplitudes of more than 3 cm (2KN2S2, SP3andS3)whichdonotexistintheanalyzableconstituentsetofHMB,HMFseemstobethemost efficient. Also, HMB does not include outputs for smaller constituents as M(Nu)4, 2MTS4, 3MN4andSL4(allwithvalueshigherthan1cmandbelow2cm).Ontheotherhand,HMB

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hasthecapabilityofanalyzingtheconstituentMVS2(27.4966873deg/h)whichisnotpresentintheHMFset.

Another difficulty when comparing the results was the use of different names for the same tidalfrequencies:NO3inHMFisMQ3inHMB;MA2andMB2inHMBareMTS2andMST2,respectively,inHMFwhileM(Nu)4inHMFisnamedMV4atHMB.

9.3. Tidal prediction accuracyFromFig.6,butalsoforthefullforecastedseries,itwaspossibletoconcludethatthepredictions

ofbothmethodsworkbetterintheParanaguácasethanintheTriesteone,except,forobviousreasons,whenmeteorologicaleventsdisturbthesealevel.Inbothcases,ParanaguáandTrieste,thehighandlowwatertimesarepredictedalmostequallybyHMBandHMF,andwithprettygoodagreement.Comparingthehighandlowwatertimeswiththeobservations,itispossibleto see that both forecasting methods predict in few cases with some minutes in advance theoccurrence of the tidal extremes, but in most of the cases there is a very good fit, more accurate for theHMFpredictionthanforHMB,butwithverylowdifferences.Somesystematicdifferencescanbeobservedamongobservationsandforecasts,andwillbediscussedfurther.

9.4. Tidal residuals9.4.1. Observation vs. forecast

AnalyzingFig.7,itispossibletoseethatforTrieste(top),theresidualsbetweenHMBandHMFagainst theobserved tide arevery similarbutgreater in theHMBforecast.Also,whencomparingHMBagainstHMF,thereisaclearpositivedifferenceforthisperiod,thatcouldreachmaximaof ten centimetres, and it is alsonotedandoscillation in the residualsmostlywith adiurnalperiod.Itisimportanttonotethatmorelong-periodconstituentsareestimatedbyHMF.

Thedifferencebetweenobservationsand(both)forecastsinFig.7turnsouttobegenerallynegativebecauseofapersistenthigh-pressureregimeovertheTriestearea,withdailyaveragesbetween4and18hPahigher than theclimatological Januarymean.At theverybeginningofthemonthseicheoscillationscanbeobserved,connectedwithstormyweatherconditionsinlateDecember2004.

IntheParanaguácase(bottomofFig.7)bothresidualsbetweenHMBandHMFpredictionsagainst observations are extremely similar, even if HMF forecast fits slightly better. Also, the residualsamongboth forecastsareproportionally lower than in theTriestecase (maxima lessthan10cmfortheperiod),buttheyoscillatearoundzerowithperiodicitiesneardiurnalandsemi-diurnalones.

Statisticalparametersoftheresidualsbetweentheobservedsealevel(Obs)andthecorrespondingforecastbythetwomethods(HMBandHMF)areshowninTable1togetherwiththedifferencesbetweentheforecastedsealevelsHMB-HMFforbothsites.Ithastobenotedthatthestandarddeviations are lower at the HMF residuals in both ports, being similar for Trieste (σHMB=12.794cm > σHMF = 12.744 cm) and around a half for Paranaguá (σHMB = 44.210 cm >> σHMF=23.692cm).Residualmeansareverysimilarinbothcases,TriesteandParanaguá,fortheobservedminusforecastedsealevel.WhileforTriesteitisasuper-estimationofmorethan2cm(negativevaluesforObs-HM),forParanaguábothmethodssub-estimatethesealevelbyaround4cm.ThebestresultsofHMFforecastscanalsobeobservedon themaximumandminimumresiduals (Obs-HMB>Obs-HMF):whileverysimilarforTrieste,theyshowmarkeddifferencesforParanaguá.


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TRIESTE PARANAGUÁ

Residual Obs-HMB Obs-HMF HMB-HMF Obs-HMB Obs-HMF HMB-HMF

Mean -2.3861 -2.3925 -0.00639 4.2764 4.2866 -0.01027

Median -2.0000 -2.0000 0.0000 0.0000 3.0000 -4.0000

Minimum -55.000 -56.000 -8.000 -140.00 -111.00 -61.000

Maximum 66.000 64.000 8.000 187.00 120.00 80.000

Standard Deviation (σ) 12.794 12.744 2.0363 44.210 23.692 27.946

Table1-Statisticalparametersoftheresidualscalculatedfromobservation(Obs)minusforecastedwithHMBandHMFandamongHMB-HMF.

IthastobenotedthatwhiletheParanaguáforecastsshowbetteragreementwiththeobservations,theTriesteresidualswereproportionallygreaterandbothforecastssuper-estimatethesealevelwhencomparedwiththeobservedone.

9.4.2. Spectrum of residualsWhenanalyzingthespectraoftheresiduals(Fig.8forTriesteandFig.9forParanaguá),some

interestingfeaturesappear.IntheTriestecase,bothmethodsleftbehindgreatandsimilaramountofenergyforthelongperiodspecie.However,atParanaguá,HMBseemstoleavebehindmorelongperiodenergythanHMF,evenifitisasmallamount.

On the diurnal frequencies forTrieste there is a great amount of energy not estimated bybothHMFandHMB,beingslightlygreateratHMF.Atthesefrequencies,HMFdoesbetterforParanaguáwhileHMBleavesalittleamountofdiurnalenergyoutoftheforecast.

TheenergyinthediurnalbandintheresidualsisprobablyduetotheeffectoftheprincipaluninodallongitudinalseicheoftheAdriaticSeawhoseestimatedperiodisabout21.2-21.5hours(Mancaet al.,1974).Itissupposed(Ceroveckiet al.,1997;Cushman-Roisinet al.,2005)thatthisperiodicitycouldaffecttheestimateoftidalanalysisinthediurnalbandwhentidalrecordsarenotlongenoughtoincreasethefrequencyresolution.

Examiningthesemidiurnalband,itispossibletoseethattheenergyleftbehindbybothmethodsissmallforTrieste,whileforParanaguáis thebandwithgreaterresidualenergyafter thelongperiod.AtTriesteforecasts,HMFshowstwosemidiurnalpeaksofthesamemagnitudearoundM2,while HMB also shows both peaks with the first one greater (for frequencies lower than M2).

Forhigherfrequencies(3rd,4th,5thand6thdiurnal),TriesteforecastsinbothHMBandHMFshow non significant residual energy, while Paranaguá forecast still lack spectral energy in the forecastsforthe3rdand4thdiurnalperiodicities,inparticularthelastone.

Thespectrumoftheresidualsshowsthat,exceptforthediurnalcaseofTrieste,HMFforecastsleftbehindlessenergythanHMB.

9.4.3. Residual of non-tidal spectral energyThe above results can also be confirmed after examining Fig. 10, which shows at top the non

astronomicharmoniccomponentsextractedbyHMFforTrieste,wheremanycomponentshigherthan1cm(andupto5cm)areoutsidethetidalfrequenciesinthelongperiodanddiurnalbands.

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This,asexplainedabove,isprobablyduetotheperturbationoftheprincipalseicheonthediurnaltidalgroup.

AtthebottomofFig.10,onecanseethecaseofParanaguá,wherenonastronomicalfrequenciesshowcomponentshigherthan1cm(andupto7cm)inthelongperiod,semidiurnal,3rddiurnaland4thdiurnalspecies.

9.4.4. Atmospheric tidal spectral energyInorder totrytounderstandthisextraharmonicenergy,weanalysedthetidalspectrumof

meteorologicaldata, as atmosphericpressure, usingHMF,which results exposed little energyin the semi-diurnal band for Trieste (only a significant S2pwasfoundwith0.47hPa–weusethepsubscripttoindicateatmospheric/airtidalconstituentsfromnowontodifferentiatethemfromtheastronomicalsea tidalcomponents),whileshowingsomeextraenergy in thediurnalspecie (RO1p,S1pandK1pamountingup to0.5hPa),while thenonastronomical frequenciesrevealed that atmosphere has no significant energy at those frequencies present in the sea level butatperiodicitieswhichdonotcorrespondtowellknownastronomicalforcing.However,whenanalysing one year of pressure (half hourly) data from Paranaguá, a significant tidal energy in theatmospherewasfoundinthesemi-diurnalband(2SK2p;S2p;T2pandR2p,addingupto1.51hPa; plus another non astronomical semidiurnal frequencies with up to another 0.32 hPa). InParanaguá, also the diurnal band was populated with significant energy at PI1p,P1p,S1pandK1p(upto0.65hPaamongall).Also, ithastobenotedthatwhileinTriestethepressureanalysisdidnotgetanylongtermatmospherictidalcomponent,inParanaguáitwasthehigherone(atSapperiodicitywithasmuchas4.13hPa).Anotherdifferenceintheresultswasthenumberofacceptedatmospherictidalconstituents,whichforTriesteamountedupto30,whileParanaguáshowedonly17.

Thesurfacepressuresignalof thesemi-diurnal tide in theTropics iswellestablishedboththeoreticallyasinChapmanandLindzen(1970)andempirically.HaurwitzandCowley(1973)calculateditfromstationdata,gettingamplitudeofthesurfacepressuresignalassociatedwiththesemi-diurnaltideof1.05mb.Also,HsuandHoskins(1989)inananalysisofECMWFdatadetectedasemi-diurnal tideofsimilarmagnitude.Theseobservationshavebeensupportedbyinrecentstudies(DeserandSmith,1998;Arbic,2005)andareconsistentinbothmagnitudeandstructurewiththetheoreticalpredictions,evenifrecentinvestigationssuggestthatthetheoreticalpredictionsofChapmanandLindzen(1970)over-estimatethecontributiontothesemi-diurnaltideoftheozoneheatingandunder-estimatethatofthewatervapourheating.

10. Concluding remarks

BothmethodsproducesimilarresultswhileHMFseemstooffersomeadvantagesbecausethegreateramountofconstituentsitcananalyzeandthecorrespondingbetterforecast.Inanycase,thedifferencesaresosmallthatitisnotnecessarytodiscardHMBinfavourofHMF.

HMBhasthecapacityofofferingupto104andmorepurelyastronomical(includingsomeshallowwater)constituentsmostlyuptothe6thdiurnalspecie,withsomeextraeffortintheinputdatapreparation,andgivesnoopportunitytoanalyseotherthantidalfrequencies,whileHMFanalysesupto176,includingmanynonlinearones,uptothe12thdiurnalperiodicities,withno


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extraoperatoreffort.Onthissense,HMFseemstobemostuserfriendlythanHMB.Wesuggest that further investigationshave tobeperformedcomparingother tidalanalysis

methodologies,consideringweused just two,verysimilar,amongmany todayoperationalallaround,asTideforMatlab(Pawlowiczet al.,2002),etc.

Thesignal-to-noisetestimplementedinHMFseemsnottobeveryaccurate,becauseitacceptverysmallcomponentswhicharenotobserved(lessthan1cm)anddiscardsothers(evengreaterthan5cm),whichappearswiththesamevaluesforHandGalsoinHMB.

Thenatureofthenonastronomicestimatedharmonicconstituentsdeservesanotherdiscussion.In a first approach, it has to be noted that these constituents contribute to the “noise” in the signal-to-noiseratioanalysisatHMF,butbeingharmonicstheyshouldnotbetreatedasrandomnoise,asintheHMFtest.ThisproblemmightexplainwhyHMFrejectstidalconstituentswithhighamplitudes,particularlyatlongperiods,butalsoatdiurnalandsemidiurnalfrequencies.

ParanaguáislocatedattheborderlineintheSubtropicsandthecouplingofatmospherictideS2pwithsealevelcouldexplaintheenergyleftbehindparticularlyatthesemi-diurnalspeciebybothHM.Inparticular,ifweconsiderthatthereisarelationshipofaround7cm/hPa(Arbic,2005)forthesemi-diurnalatmospherictidetothesealevelatthesamespecie,mostoftheParanaguáspectralresidualscouldbeexplainedbythiscoupling.ItisimportanttonotethatatmosphericS2phasaphaselagofaround109owithrespecttotheastronomicalseatideconstituentS2(Arbic,2005).Thus,algorithmsastheonesusedbyHMFandHMBwillaliasthisenergyaroundS2,contaminating other semi-diurnal constituents. It has to be noted that these effects cannot berelatedwiththesocalled“radiationaltide”,whichhasbeenproventobealsoofnonlinearorigin(MaroneandMesquita,1995).

Evenifnotaswellstudiedasthesemi-diurnalS2p,diurnalandotheratmosphericconstituentscould also contribute to the spectrum of the sea level (Arbic, 2005). By now, we can onlyhypothesizethattheatmosphericcomplexityatTrieste,whichislocatedatatemperatelatitude,couldalsocontributetothelessaccuratesealevelforecastweperformedwithbothHMBandHMF,whichconsideronlyastronomicaltidalforcingonthesea.

The harmonic contribution of the atmosphere to the sea level could also explain, at leastpartially, the discrepancies obtained when comparing field data analysis with numerical models (D’Onofrioet al.,2009).Also,consideringthewideuseofnumericalmodelsforcedwithtides,itwouldbewiseifabetterrepresentationoftheoscillatingsealevelisusedinsteadofthepurelyastronomicone.

Aswecannotdisregardtheevidencethatnontidaloscillatingsignalsareclearlypresentinthesealevel,wesuggestreformulatingtheanalysisandforecastingmethodologiesconsideringabetterstochasticmodelforthesealevelatagiveninstanttas:

N Nζ(t)=[ Hicos(ωi t+Gi)]+[ hicos(νi t+gi)]+ε(t) (4) 1 1

whereH,Gandωrelatetopurelyastronomicaltidalconstituents(includingshallowwatersones);h,g andν correspond tootheroscillating signalspresent in the sea level (as theatmosphericinduced) and, finally, with ε(t) asatrulyrandomnoise.

Thenatureofthenonastronomicalestimatedharmonicconstituentsisstillaquestiontobedetailed,andwillbemotiveforfurtherinvestigations.

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Acknowledgements. The authors want to acknowledge the support of CAPES (BEX 1186/10-8) to E.Maroneaspost-doctoralfellowatOGS,Trieste,Italy.Also,thanksareduetothereviewers,whocontributedto improve the final version.

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Corresponding author: Eduardo Marone CEM/UFPR P.O. Box 50002, 83255-000 Pontal do Paraná, Brazil Phone: +55 41 35118643 ; fax: +55 41 35118648; e-mail: [email protected]

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