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  • 7/28/2019 Statistical Analysis of Wave Records- Rr Putz - Chap2

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    CHAPTERSTATISTICALANALYSISFWAVERECORDS

    H.R.PutzUniversityofCaliforniaBerkeley,Calif.

    ABSTRACTTheestablishmentofquantitativerelationshipsbetweenrecordedwave-systemcharacteristicsandotherphenomenarequiresnumericaldescrip-

    tionofthewaverecord. Currentconceptsappliedtotimehistoriesofwaveactivity atapointarediscussed.haracteristicstatisticalregularitiesfoundinwavemeasurementsaredescribed.xamplesgivenshow theapplica-tionofstatisticaltechniques'tothedescriptionofwavesystemsintermsoft hedistributionofspectralenergy aswellasthedistributionsof"individual"waveheightsandperiods.esultsfrom theprediction,fromonepointtoanother,ofsurfacetimehistoriesillustratetheapplicationofapproximatespectralinformation.

    INTRODUCTIONInthestudy ofwavesandtheirinteractionwiththeirenvironment,

    therehasbeenaneedforeffectivedescriptionofobservedwavesystems.Convenientdata(Snodgrass,1951,1952)isprovided bytherecordedtimehistory,takenabovesomefixedpointont hebottom,ofthefluctuatingpressurebelowthesurfaceoforthewaterlevelatthesurfaoe.t willbeunderstoodthatthelengthofax v a v erecordselectedforanalysisislongrelativetotheperiodsofthefluctuationsofinterest,bu tshortcomparedtot hevariationsinmeteorologicalconditions.

    Thequestionsweareconcernedwitharej *hatinformationiscon-tainedin awaverecord?Andhowmay thisinformationbeconveniently ex-tracted? Weshallfirstdescribethestatisticalregularitiessuggest-ingaparticular mathematicalmodel(Tukey,1950;Pierson1952a)foundusefulforinterpreting wave-recorddata. Thetwocomplementaryaspectsofthismodeleachinvolveadistribution-oneastatisticaldistributionof wave-recordordinates,theotheraspectraldistributionofenergy.Theoreticalrelationsconnectingvariousmodelparameterswillbeillus-trated with actualdata.lthoughtheexperimentalresultstobepre-sentedhavebeenobtainedwithpressurewavesintheoceanfort hemostpart,manyofthemethodsandresultsareapplicabletootherkindsofdata.

    InthelowerhalfofFig.1isshownashortsegmentof atypicalwaverecord,givingthepressurefluctuation64feetbelowthesurfaceoff theCaliforniacoast, 4 i etotaltimeintervalshown representsaboutfourandone-halfminutes,duringwhichfromtwenty totwenty-fivewavespassthereoordingpoint. It willbeseenthatonthetime-history curvetheapparentslopeandcurvature,aswellastheordinate,varyirregular-lyfrom oneinstant tothenextandbearlittleobviousrelationtooneanother. Themostnoticeablefeature may bet hetendency forcurvatureandordinate,measuredfromit saveragelevel,tohaveoppositesigns.

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    COASTALENGINEERINGTheupperhalfofFig.1showsasamplesegmentgeneratedbythetheore-ticalmodelin whichsuitableparameterswerechosent oyieldanartifi-cialwaverecordcomparablet otheobservedone.

    DISTRIRTTIONSOF ORDINATESANDDERIVEDQTIANTITIESSupposeahorizontallineisdrawn cuttingthewave-recordcurve

    atanarbitrarylevel.hefractionof t hetimethecurvespendsbelowsuchalineisafunction(Birkhoff andKotig,1953;Putz,1953b)in-creasing withtheheightofthelineknownast hedistributionfunctionforthecurve.plotofthisfunctionforatypicaltwenty-minute waverecordisshowninFig,2 .erethenumberontheverticalscalerepre-sentsthearbitrarychartlevel,thehorizontalscale,thepercentoftimethatthecurveliesbalowthatlevel. Thenotionof t heprobabilityoffindingthecurvebelowagivenlevel maybeintroducedifonethinksof choosingatrandom aninstantoftimeont hechart. Thelocationoft heplottedpointson anapproximatestraightlineischaracteristic(Puti,1953b)andresultsherefromthechoiceofthehorizontalscalewhichisthefamiliarGaussian-distri^tion ornormal-probabilityscale.

    Theinterestingthingaboutwaverecordsistheapparentability ofthenormalprobabilityscaletorectify very nearly notonly t hedistri-butioncurvesfort heordinateandthefirstderivative,which havebeenexperimentallychecked(Ruanick,1951;Putz,1953b),bu talso,accordingtothetheory,thedistributioncurvesforderivativesof allorders. Ineachcasethemean valuebelowwhichthecurveorderivativespendsjustfifty percentofit stimewillbezero. Fortheordinates,thiswillbetruebecauseofourconventionofmeasuringthemfromtheirmean.inceinthiswayonepointisfixedoneachcurve,thestraight-lineplotsforthevariousderivativesof thewave-record willdifferamongthemselvesonlyin theirslopes.convenient measureoftheslopeisthedifferencebetweentheheightofthelineatt he84.1percentlevelandit sheightatthe50percentlevel. Thisdistance,calledthestandarddeviation,orroot-mean-square(r.m.s.)ordinate,whenmeasuredforthek^hderiva-tive,isdenotedby"^

    Theordinatedistributionconcept maybeextendedbyconsideringmorethanoneordinateatatime. Givenn arbitrarily-selectedchartlevels,then-dimensionalgeneralization(Cramer,1946)oftheGaussiandistributionisthen applicabletothefractionofthetimethat,simul-taneously,nordinates,chosenatdifferenttimeinstantsonthewaverecordwillliebelowtheircorrespondingchartlevels.heassumptionthatforeachfiniten,t heseleotionof nordinatesactuallyresultsin amulti-dimensionalGaussiandistributionisknown ast hemultinormalhypothesis.Suohadistributionischaracterized by asetofparametersknown asthecovarianoes(Cramer,1946).heseparametersmay bereadilyinterpretedinterms"ofthedegreetowhichthevariousordinatesdetermine,oraredeterminedby,eachother.omputation(Rioe,1944-5)tellsusthatt hecovarianoeofany t woquantitiessodistributedisproportionaltoos(TT.p) ,wherep istheprobabilitythatthetwoquantitieshaveoppositealgebraicsigns,

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    STATISTICALANALYSISFWAVERECORDS

    =ii6IBfegr-fe=7 pi1iluiiL._^- f=== -^RTT"3gttr~f-^4:f~ ^p =rp^\f\ f\ (fi~rjTt T^4#% =xr~'Vpff ^Ef"|H"DLlXm" ^ -=feH=E T ~LI j| r; 4H\ Br^r \~V~\=3p TTWCAL OCEAN WAVE RE C>R 0 = ^3Fig. 1

    60

    70 Q2

    LJ -1Ozi 5 0

    if3"< n oUJ 5

    LEGEND0 Databtained byanualeasurement*

    to ofrdlnatesfheavest-second intervals ^ x Databtainedyheav enalyzerhichmeasurestherdlnatest0I-secondntervals

    'S aw u a 0 o n e0 9 0 40 50 2 1 0 ) ! 09 0 2 0 I 005 0 PERCENT OF TESTNTERVAL THAT ORDINATES O F THE WAVESXCEED TIMING LEVELS

    Fig.2 . Cumulativedistribution function ofordinates.

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    COASTALENGINEERINGAveryconvenientproperty of t h e s e - multipleordinatedistritutions,andone whichis physically reasonableexceptfor relativelylong records,

    i sthestationary property( L e e ,1949}Putz1953a),namely thatthedis-tribution of anysetof ordinatesdependsonly upon thedifferencesin timebetween thecorrespondingabscissas,ie then havedistributions with equalcovarianoes wheneverthecorrespondingtwoordinatesareseparatedbythesametimelag, i.e.,theoovariance/betweenany twoordinatesse-lected attimes andt+ T dependsonlyuponT 4 i eoovariancei sthusafunction " o fthelagr ,hefunctionyT )being knownastheco-variancefunction,(Mann,1953),and,when dividedbyits maximum valuey o ) ,asthecorrelationfunction,pT),thelatter being thefamiliarproduct-momentcorrelationcoefficient.nceassumedforvhe ordinates,thestationary andmultinormalpropertiesfollowforderivativesof allorders.

    V J A V f i f f i S I C H T SInterpretationsformany oftheparametersoftheseunderlyingprobabilitydistributions may befoundint h edistributionsof certainquantitieswhich may begraphicallymeasured on thewaverecord.If N j j -denotesthenumberof timesthe k * * 1 derivativepassesthrough

    thezerolevel,andif Tisthetotallengthofthewaverecordinseconds,then theaveragenumberof zero-levelcrossings ( u po rd o w n ) peraecondwillbeN f e / r .natural definition oft h e mean(undirectional)zero-crossingfrequency isthen fk=\/(.2T) in cyclespersecona,or U )k- 2 7 r H j c/(2T)7TNk/Tinradianspersecond.heorypredictsthat ik=crv+l/0" kwhichservesto relatether.m.s.valuesoftheordinateandof thefirstandsecondderivativestotheobserved maanzero-crossingfrequenciesoftheordinateandthefirst derivative,whenki staken tobezero andone.

    Furtherrelations appearif weoonsidersimultaneous valuesofor-dinateandsecond derivative. T h e valueof thecoefficientof correlationbetween thesetwoquantitiesi sgiven byp0- cos(TT-PO), whereP0i sthefraction of thetimethat they haveoppositesigns.heory( R i c e ,1944-5)predictsfurtherthat >0s-(wo/

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    Average heightfa\I avesO" . m valuefrdinatttN0 umberfimesrdinateseroN( *slopesero

    MEAN WAVEEIGHT VERSUSPRODUCT OFRDINATEISPERSION,t> ,TIMESATIOFUMBER OF ZEROS

    OFRDINATE AN DLOPE,o/Ni60246 (N/N,)

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    COASTALENGINEERINGvaS ---

    ft,=-1001

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    STATISTICALANALYSISOFWAVERECORDSagood estimateof ther.m.s.ordinate when combinedwithacount ofthenumberof timestheordinateand theslopepassthroughzero.

    Theentiredistribution o f peak(ortrough)heights(Rice1944-5)givenby thetheory,dependsonlyuponc r0andpQt andi sshown in Fig.4 .T h e verticalscalecorrespondstothechartlevelmeasuredfrom themean in T0-nitsequal to ther.m.s.ordinate.he percentof peakheightsnotexceedingthis verticallevel appearson the horizontal,thescalein thiscase being chosen tocorrespond to theso-calledEayleighdistribution(Longuet-Higgins1952;Lawson andUhlenbeck;1950,Knudtzon,1949) which thepeak h eightsfollow more andmoreclosely as the param-eterp0tendstothevalue minusone.tmay be observed that thepro-bability ofalow peaksituated below the mean levelisjust i( l+/ >0),which tenastozeroasp0tendsto minusone,corresponding to a rela-tivelynarrow-bandspectrum.

    Pig.5 showsatypicalobserved distribution ofpeakheightsforatwenty-minute waverecord. ' A i estraightlinerepresentsthe asymptotioRayleighdistribution,w