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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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STATISTICALANALYSISFWAVERECORDS

9

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,whilethecurve representsthedistribution ofpeakheightsactuallypredicted from measured estimatesof 0 " 0 and0aswellasofthe meanordinate / Q.h e verticallevel isindicated inchart divisionson theleftand in< 7 " o - u & i t son theright.

ENERGY DISTRIBUTIONThestatisticalanalysisoftheperiods between theserocros-

sings,or between thepeaksonawave recordi scloselyrelated tothedistribution ofenergyoverthefrequency or periodspectrum.hisas-signmentofenergy toa continuumof superimposed elementary wavesi sspecified bytheso-calledspectraldistribution function orspectrum.Thespectrum providesa meansofderiving and interpreting the parametersof the distributionsof wave-record ordinates and derivativesenteringintothedistribution ofwaveheights.

Thedistributionofenergy i sof fundamentalimportance,ofcourse,forthestudy of thegeneration,propagationand the effectsof waves.T O i i l einformationregarding thefrequencyofoccurrenceof waveheightsorperiodsofgiven magnitudesappearsoccasionally to bedirectly usefulfordesign purposes,aknowledgeofthespectrum canbe usedwheneverthefrequency-responsefunction corresponding toageneralfilteringactionisknown.

Theusualspectrum analysisof agivenwave-recordsample resultsin the assignment ofan amplitudeand,in addition,a phase angle toeachofa certainsetof frequencies.owever,thedirect dependenceofsuchphaseinformation on the timeorigin towhich thespeotrumanalysisisattached,makesi tunsuitableas a possibleparameterforastationaryrandom process.husitis naturalto takeasfundamental the remainingaspect of thespeotrum i.e.,the amplitude,orequivalently,thesquareof the amplitude,theso-cajled powerspectrum(Rioe,1944-5).

Therei safunotion obtainablefrom asample wave recordknownas

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COASTALENGINEERINGthesamplecovariancefunction v r h i o h hasforitsspectrum just thepowerspectrumof the waverecorditself.n manycasesthepowerspec-trum of agivensample - . * a v erecord i s most naturally obtainedby firstcomputing thecovariancefunction forthe wave record whichwasobserved.However,anyproblem whosesolution involvesthepredictionofthepos-

siblesamplewaverecords whichmighthavebeen observedbeforeorafterthegiven one willnaturally leadt o theassociation of a probabilisticmodel with eachsamplewaverecord.his model may bethoughtof ascomprising a i v h o l esetofpossible,infinitely long,waverecords,eachhaving thesamepowerspectrum or covariancefunction.heidentifica-tion ofthiscommon covariancefunctionwith thatdescribed earlierinconnectionwithstationarymultinormalordinatedistributionsisthestep which suppliestherelationbetween theseprobability distributionsandthespectral-energydistribution.hespecification of theco-variancefunction,togetherwith thestationary multinormalassumption,definesaso-calledstationary Gaussian random(orstochastic)function( o r process).(Mann,1953;ee,1960).

Theability ofsuch a theoreticalmodelto describeobservedwaverecordsi sillustratedby comparingasampletaken fromsucha randomprocesswith atypical waverecord.heuppercurvein Pig.1 wasgener-ated afterfirstspecifying threefixed numbersdefining thespectrumandthen consulting,a tableof randomnumbers( i f b l d ,1948).If the mathematical momentsof thespectrumexist(Putz,1953a),certain of thesecanbeshown tobeequivalent to ther.m.s.valuesofthederivativesof theoriginalwaverecord,whilecertain ratiosof thesemomentsareessentially thecoefficientsofcorrelation between thederiva-

tives, l h u sther.m.s.ordinate " 0i sthezero-ordermomentof thespec-trumorthetotalspectralenergy,while0 i srelated tothesecond-orderspectral moment,o r more precisely,tocertain measureof therelativespectralbandwidth when thelatteri ssmall.V&VEPERIODS

I thasbeenseen thattheinstantsof timeat which the waverecordpassesthroughzeroor througha peak enter,by way oftherelativespectralbandwith.intoa relationbetweenthe waveheightsand ther.m.s.ordinate.Fortheanalysisofthespectrum,thedifferencesbetweentwosuccessivezero-crossingsmaybeconsidered thefundamentalindividual waveperiods.According tothetheory,for along recordthelistoftheseindividualwaveperiodsintheorderof theiroccurrencei scompletelyequivalent tothespectrum.eak-to-trougho rtrough-to-peaktimeintervals,corres-pondingtotheperiods betweensuccessivezero-orossingsof the wave-reoordderivative,while moresensitivetotheinfluenoeof noise,arecapableinprincipleof yieldingthesameinformation.nt h eother hand,peak-to-peak ortrough-to-trough timeintervals(Snodgrass,1952),corresponding totheperiods betweenalternatezero-crossingso f thederivative,would beex-pectedto contain lessinformation.

Thesignificanceofthezero-crossingsforour modelliesin thefactthat theabscissasofthese pointsseparatetheintervalsduringwhich the

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STATISTICALANALYSISFWAVERECORDSwaverecordhasconstantsign.sin^theseabscissas,wemay estimate

thecorrelationfunctionforagiven waverecordbyfirstdeterminingt hefractionF(T)fthetimethattheoriginalwaverecordandthewaverecordadanced byatimeinterval aveoppositesign.tisfromthecorrelationfunctionP(T)-oos[TTP( T)] thatthespectralin-formationisobtained.

Pig.6showsacomparisonoftheestimateofyfr(r)obtainedforatwenty-minuterecordby measuringzero-crossingabscissasandtheordinaryestimate(Tukey,1949;Pierson & M f t r k ^ lobtainedbymeasuringordinateseverysecond. Thisagreement betweencorrelationfunctionshasbeenfoundsuffieientlyclosetoresultinarelativelysmalldifferencebetweenthecorresponding estimatesof thewave-recordpowerspectrumfrom thetwomethods. tmay beobservedthatinaddition t o measuring1200ordinatesonthewaverecord,nearly1200 multiplicationsand1200additionsarere-quiredtoobtaineachpointof thesolid-linecurve,whileforeachpointofthedottedlinecurveitissufficienttomeasureabout200 abscissason

the- w a v erecord andperformabout400additionsand200subtractions.vhileeithermethodleadst oasubstantialsavinginlaboroverastraightforwardnumericalFourieranalysisoftheoriginalwaverecord(jam6s,NicholsandPhillips,1947);thezero-crossingmethodfor thecorrelationfunction,requiringessentially no multiplication,maybecarriedou t withonlyanad-d i n g , machine.

Thedegreeofreliabilityofthezero-crossingsexhibited bythisoneexperimentalcheckonatwenty-minutewaverecordmay beofsometheo-reticalinterest. Theclosecorrespondence,particularlyforsmalltimelags,seeninFig,6isadditionalevidenceforthegeneral mathematicalmodel wehavedescribed,andforthe multinormalpropertyinparticular.Itconstitutesevidencealsoforthesamplingadequacyof a wavereoordtwentyminutesinlength. Theindicationsarethattheso-calledindividu-al waveperiods,suitablydefined,canfurnishreliableinformationabouttheFourierspectrumofthewaverecord.

Thiskindofapproximatespectralinformation hasbeenusedtopre-dict,byaleast-squares method(Putz,1952b),thebehaviorofoceanwavesatonetimeandplacefromtheirbehavioratatimeandplacenearby.Theresulting predictionisshowninFig.7,wheretheupperplotrepre-sentsthetimehistory oft heobservedsurfaoeelevation atapoint1570feetseawardofthepointatwhichpredictionwastobe made.helowerpartofthefigures h o v r atheobservedandthepredicted waverecordattheshorewardpointfiftysecondslater. Thefactthattheaccuracyofprediction representedherecomparesfavorablyv v i t hthatbasedupon amoreexactspectrum analysisisafurtherindicationoft he reliabilityoftheinformation provided bythezero-crossings.

SUIvMURYThereexistsanapproximate mathematicalmodelforwaverecords

which maybeprofitablyexploiteduntil refinementsinitarefound ne-cessary.his modelspecifiessimpleverifiablerelationswhich may be

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COASTALENGINEERING

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STATISTICALNALYSISFWAVERECORDSexpectedto holdbetween appropriatequantitieseasily measured onlongwaverecords.heparametersocourring inthe modelmay beestimated fromindividual wave-height and wave-perioddistributions.fewsuch para-meters,easily interpretedintermsofthespectrum,areavailableforthedescription of awaverecord.

REFERENCESBirkhoff,G.and Kotik, . ,(1953).ourieranalysisof wavetrains:Proceeds.Nat.3 u .of StandardsSemicentennialSymp.on Gravity, a v e s

(1950)pp.221-234,Govt.Print.Off.ash.D.C.Cramer,H.,(1946).athematical methodsofstatisticsjrincetonUniver-

sityress,Princeton,N.J.James,H.M.,Nichols,N . B . ,and Phillips,R.S.(1947),heory of servo-

mechanisms,cGraw-HillBook Co.Inc,NewYork,N.Y.Knudtzon,N.,(1949)Experimentalstudyof statisticalcharacteristicsoffiltered randommoise,ech.Report No.1 1 5 ,ResearchLab.ofElec-tronics,ass.Inst.ofTech.Lawson,J.L.andUhlenbeck,G.E.(1950).hresholdsignals,cGraw-HillBook Co.Inc.,NewYork,N.Y.Lee,Y.W.(1950).ommunicationapplicationsof correlation analysis*Symposium on applicationsofautocorrelation analysisto physical

problems(1949),Off.of Nav.Research,Wash.D.C.Longuet-Higgins,M.S.(1952).n thestatistical distribution of theheightsof sea waves.our,of MarineRes.,v.1 1 ,pp245-266.Mann,H.B.(1953),ntroductiontothetheory ofstochasticprooessesde-

pendingon acontinuousparameter.pplied Math.SeriesNo.2 4 ,Nat.Bu .Standards,i V a s h .D.C.Piersont f f . J . ,J r .(1952a),unifiedtheoryfortheanalysis,propaga-

tionand refraction ofstorm-generatedocean surface waves,P t .I .Res.Div.,CollegeofEngr.,NewYork University,N.Y.

Pierson W.J.,Jr.and Marks,w .(1952) .hepowerspectrumanalysisofocean wave records. Trans.Amer.Geophys.Union,.3 3 ,p p .834-844.Putz,R.R.(1952a),Statisticaldistributions forooean wavesjrans.Amer.Geophys.Union,v.3 3 ,p p .685-692.Putz,R.R. (1952b).redictability ofwavetransformationsinearleastsquaresprediction.ection G , T e c h .Beport3er.2 9 , Issue5 4 ,

Inst.of Engr.Res.,Universityof Calif.,Berkeley,Cal.Putx,R.R.1953a)he analysisof waverecords as randomprocessesi

Presentedat PaoificS.W.Regional Meeting,Am.Geophys.Union,Feb6-71953.alsoTech.ReportSeries3 ,Issue352,IER.,UniversityofCalifornia,Berkeley.

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COASTALENGINEERINGPutz,R.R. (1953b).Ocean-waverecord analysis-ordinatedistribution

and wave heights, Tech.Report Ser.3ssue351,IER.University ofG&lif.Berkeley,Cal.Rice,S.O.(1944-5),Mathematical analysisof randomnoise,PartsI-IV,

Bell SystemT e c h .Journal vol.2 3 ,pp.282-332,v .2 4 ,p p * 46-156.Rudnick,P .(1951)orrelogramsforPaoificOceanwaves.roceedingsofthe SecondBerkeleySymp.onMathematicalStatistics andProbability,

pp.627-638.niv.of Calif.Press.erkeley,C a l .Snodgrass,F.E.(1951)Waverecorders;roceedingsof theFirstConf.

onCoastal Engr.(1950),Councilont f a v e .R e s . , T h eEngr.Foundation,J.W.Johnson,e d . , Univ.ofCalif.,Berkeley.

Snodgrass,F.E.(1952)ceanwavemeasurements;Symposium on OoaanographicInstrumentation(1952),Nat.ResearchCouncil.

Tukey, . T i T . ,(1950).hesamplingtheoryof powerspectrum estimates.Symp.onApplicationsofAutocorrelationAnalysisto PhysicalProblems(1949).Off.of Nav.Research,f l a s h .D . C .

Wold,H .(1948)andomnormaldeviates*ambridgeUniversity Press.

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