THE SOCIETVOF NAVAL ARCHITECTSAND MARINE ENGINEERSOne WorldTradeCenter,Suite1369,New York,N.Y.10048

Pam,,tobe PresentedatExtremeLoadsResPo”seSymPosi”mArlington,VA,October19-20,1981

Principles of Extreme Value Statisticsand their ApplicationMichel K. Ochi, University of Florida, Gainesville, FL

ABSTRACT

This paper presents the theoreti-cal background of recent techniques topredict extreme loads (and responses)pertinent to ship (or marine system)operation in a seaway. The magnitudeof extTeme values, in general , can beestimated through different approaches,each based on a different principleThis paper outlines the three differentapproaches for predicting extremevalues vihicb provide information usefulfor design consideration of a marinesystem. These are,

(1) The initial probabilitydistribution is known; for example,prediction of extreme responses of amarine system in a seaway in the short-term based on the linear superpositionprinciple. In this case, the extremevalues can be sinmlv evaluated b“analytical formul~t;on thr&h rifiplica–tion of order statistics to the initialprobability distrih”tion.

(2) The initial Drobabilitvdistribution is unknow; but obse;ved(or computed) data are available; forexample, prediction of the extreme seastate (significant wane height) orprediction of the extreme responses ofa marine system in the long-term, etc.Estimation of tbe extreme values iscarried out by approximate methodsthrough tbe use of the observed (orcomputed) data.

(3) The initial probabilitydistribution is unknown but observeddata of the maxima only are available;for example, prediction of lifetimeextreme responses of a marine systemfrom daily observed tna.xim”mvalues ,etc.1“ this case, the extreme values areobtained from tbe asvmvtotic extremevalue distributions &~eloped in orderstatistics.

The difference in these threeprediction approaches are clearlyexplained, and practical examples toevalwite the extreme values followingeach method are presented.

INTRODUCTION

This paper is prepared to provideinformation necessary for understandingthe probabilistic prediction of extremevalues, specifically extreme responsesof a marine system in a seaway.

For the design of a marine system,it is highly desirable to obtain themagnitude of the system, s responses invarious seas. Among others, the largestresponse (extreme value) which tbe sys-tem will experience in her lifetime isnecessary to assess possible structuralfailure which may occur as soon as asingle load exceeds the value criticalfor the system, s structural strength.This type of failure is called the @&ttxctii.an@&ULE in stochastic processtheory. The information on firstexcursion failures is necessEYy togetbt?rwith that on fatigue failures forreliability analysis of the system underrandom excitation.

The magnitude of extreme responseof a system which is associated with tbefirst excursion failure can be evaluatedthrough different approaches, each basedon a different principle. Hence, greatcare has to be given in evaluating theextreme responses depending on the pre-diction method one may take.

The CXtiUW w.&e is defined, ingeneral, as the largest value expectedto occur in a certain number of observa-tions or in a certain period of time,It can be defined on a short-term basisin which the sea environment is statis.tically invariant (usually from 30M1nUte S to several hours ) as well as ona long-term basis (usually for manyyears ) In either case, however, thenumber of observations or a period oftime have to be specified in definingthe extreme value. For example, inestime.ting the magnitude of tbe extremeresponse of a marine sy~temr~ lifetime,the number of response cycles expectedin the lifetime ba.sto be clearlyspecified. It is unfortunate that this

L-

15{

important issue of the extreme valuestatistics has often been neglected inestimating the extreme responses of amarine system.

Prior to discussing the extremeresponse 0? a marine system, let usconsider the response at every cycleof encounter with waves irrespectiveof its magnitude. Here,theresponseis a random variable, denoted by X, andit has its own probability densityfunction, f(x), and the cumulativedistribution function, F(x). Thesefunctions are often called the ~~pmbubdl-itg datiity &.utcZionandthe-&&iduundtivtdhtibtion @m.t.ion,respectively,indiscussingtheextremevaluestatistics.

Theextremeresponseinn-waveencounters,denotedbyYn,isalsoarandomvariableandfollowsitsownprobabilitylaw,whichisdifferentfromthatapplicablefortheresponsex. Toavoida possibleconfusion,letuswritetheprobabilitydensityfunc–tionandcumulativedistributionfunc–tionoftheextremeresponseasg(yn)andG(yn),respectively.Here,theprobabilityfunctions,f(x),F(x),g(Yn),andG(yn)havemathematicalrelationshipsiswillbeshownlater.Therefore,theextremeresponsecanbeeasilyevaluatedbyapplyingtheformu–lationinextremevaluestatistics,calledofidtiAttiticb,iftheinitialdistribution,f(x),isknown.

However,thisisnotalwaysthecase. Iftheinitialdistributionisunknown,theextremeresponsecanbeevaluatedeitherthroughapproximatemethodsorbyapplyingasymptoticformulations.Consequently,theesti-mationoftheextremevaluesmaybecategorizedintothreeareasasshowninTable1. Thedetailsareasfollows

(1) Theinitialprobabilitydistribu-tionisknown

Theestimationofextremeresponses

ofa marinesystemintheshort–termisa typicalexampleofthiscase,sincetheinitialprobabilitydistributionisusuallyconsideredtobetheRayleighprobabilitydistribution.Inthiscase,theprobabilityfunctionoftheextremevaluescanbepreciselyderivedbyapplyingorderstatistics.

(2) Theinitialprobabilitydistribu-tionisunknown

Therearemanypracticalcasesofevaluatingextremeresponsesforwhichtheinitialprobabilitydistributionisnotknown.Inthiscase,theextremeresponsescanbeevaluatedeitherthroughstatisticalestimationoftheinitialdistributionorbyapplicationoftheasymptoticextremevaluestatis–tics.Fortheformerapproach,measured(orcomputed)dataoftheresponsearerequired,whiledataofthedaily,monthly,oryearlymeasured(orobservemaximaarerequiredforthelatterapproach.

(a) Measured(orcomputed)dataareavailable

A typicalexampleofthiscaseisthepredictionoftheextremeresponsesinthelifetimeofa marinesystem.Theinitialdistributionoftheresponscoveringthelifetimeofthesystemisnotpreciselyknown,butaccumulationofsomemeasured(orcomputed)datamaybeavailable.

Anotherexampleistheestimationofthemostsevereseastate(signifi-cantwaveheight)inthelongterm.Again,theprobabilitydistributionforthelong-termsignificantwaveheightisnotpreciselyknown,andhenceitisestimatedapproximatelythroughtheuseoftheobserveddata.

(b) Measured(orobserved)dataofthemaximaareavailable

Itisoftennecessarytoestimatetheextremevaluesina specifiedperiodoftimefromtheaccumulationofdaily,monthly,oryearlymeasured(orobserved)largestvalues,calledthemaximaforbrevity’ssake.Forexample

Table1 Estimationofextremevalues

1=-:2’:N:’E3INITIALPROBABILITYDISTRIBUTION1SKNOWN

INITIALPROBABILDISTRIBUTIONISUNKNOWN

16

estimatetheextremeshipresponseexpectedin30yearsfromthedataofthedailylargestresponsemeasuredover3 years.Inthiscase,theextremeresponsesin30years,forexample,areevaluatedby theasymptoticextremevaluedistributionsw~ichwereorigi-nallydevelopedbyFrechetandlaterweresystematizedbyGumbel(1958).

EXACTEVALUATIONOFEXTREMERESPONSES

Asstatedearlier,theinitialprobabilityfunctionof theresponseandtheextremeprobabilityfunctionaremathematicallyrelated.Thatis,

Probabilitydensityfunctionofextremevalue:

g(yn)=n[f(x){F(x))n-l]wy(1)n

Cumulativedistributionfunctionofextremevalue:

G(yn) = [{F(x)}n]x=yn

(2)

Variousstatisticalpropertiesoftheextremevaluescanbe evaluatedfrom(1)and(2). F~rexample,thepkobablczOWIWJTW value, Yn, definedastheextremevaluemostlikelytooccurinn–observationscanbeobtainedasthemodalvalueoftheprobabilitydensitymctiong(yn)asshowninFigure1.

kII 9(Y”II

Thatis,~n isgivenasthesolutionofthefollowingequation:

+dYn)= on

whichyields,

f’(yn)F(yn)+ (n-l){f(yn)]z= O

(3)

(4)

Therelationshipoftheprobableextremevalue~n totheinitialproba-bilitydensityfunction,f(x),showninFigure1 willbediscussedindetailinthenextsection.

ltisoftenassumedthatwavesareconsideredtobea narrow-bandGaussianrandomprocessandthata marinesystem’sresponsesin waves arelinear.Hence,themagnitudeoftheresponsefollowstheRayleighprobabilitylawwhichcanbewritteninthedimensionlessformby,

_ +2f(~)=~e

(2-—F(~)=l-e2

(5)

where, ~ = x/Jiiio

x = response(amplitude)

mo = areaundertheresponsespectrum

Thus, fortheRayleighprobabilitydistribution,equation(4)becomesinthedimensionlessform,

~2n c:

2 —— .—Ln(ne2 -1)-(e2 -1)=0 (6)

where, <n= YnlG-

Thesecondtermin(6)maybeneglectedfora largen,andthisresultsintheprobableextremeresponseas,

~n=J= (Dimensionlessform)(7)

Fig.1 Explanatorysketchofinitial Yn=mfio (Amplitude)‘“probabilitydensityfunctionf(x)andextremevalueprobabilityfunctiong(yn)

17

In equat ion (7), the extremeresponse is presented as a function ofthe number of wave encounters, n. Forpractical purposes, however, it may bemore meaningful to express the extremeresponse in terms of time rather thanas a function of number of wave encoun-ters. This expression can be made byusing the formulation of the averagenumber of zero–crossing per unit timegiven by,

where, m2 = second moment of theresponse spectrum

Then. the urobable extreme

(8)

response, Yn, i; expressed as a functionT:of time

?“ =

where,

In

T = time in hours

the foregoiEg derivation of theprobable response, Yn, the response isassumed to be narrow-banded. If thisassumption is removed, then it isextremely difficult to find a solutionof (4) since the response no longerobevs the Ravleieh distribution:ins~ead, the”pro~abi lity density func-tion of the response is given as afunction of the band–width parameter,E. That is,

where, E = band–width parameter

found approximately for the band-widthparameter less than O.9. That is, forc < 0.9, it is possible to use thefollowing approximation for a large nun-ber of observations, n (Ochi 1973) :

()1-s2*-T< -o“

()4+’-1(11)

“12

‘2l-—

‘o%

= 4th moment of the spectrum‘0

The sol”t ion of equat ion (4) can be

By neglecting terms of small orderof magnitude, (4) becomes,

E;+ “(1-.2)e + .(cn) = o (12)

Then, the following solution can heobtained as the dimensionless probableextreme response:

Tbe dimensional p~obable extremeresponse (amplitude ), I’n,becomes,

The dimensionless probable extremeresponse calculated from (13) as afunction of tbe hand-width parameter areshown in Figure 2. As can be seen inthe figure, there is no significantdifference in the probable extremeresponse for up to E = 0.9, and theeffect of ~–value of the extremeresponse is noticeable only for greaterthan 0.90, irrespective of the numberof observations.

Since the range of c-values of tbemarine system’s response spectra spansfrom the order of 0.35 to 0.80, it maysafely be concluded that tbe effect ofband-width parameter can be ignored asfar as tbe estimation of tbe extremeresponse is concerned.

Next, let us express the probableextreme response given in (13a) whichis applicable for any non-narrow-bandspectrum in terms of time instead of

—-

NUMBER N

Fig. 2 ProbableextremevalueY~ (ampli-tude)a,.a functionof E

number of observations. n. The exDectednumber of posit ~.vemaxima per uni; timeis given by,

Then, the probable extreme responsebecomes,

It is noted that the above equationis exactly the same as that for narrow-band spectra given in (9) This leadsto an important conclusion that themagnitudes of tbe extreme responses i“a specified period of time are the sameirrespective of the band-width parameterof a response spectrum. However, itshould be remembered that the number ofpeaks for a non-narrow-band spectrum islarger than that for a narrow-bandspectrum during the same period of time,

Tbe probable extreme response! ~n,Q&Yo:~unterpreted as being most l~kely

, since it is the value forwhich the probability dens<.ty function

Of the extreme respOnse, g(yn), has apeak (see Figure 1) It should benoted, however, that the chance ofoccurrence of a response higber_thantbe probable extreme response, Yn, israther high ;_hence it is not appropriateto consider Yn for the design of marinesystems in a seaway.

statement, let us evaluate the proba-bility that the extreme response (i”the dimensionless form) will exceed theprobable extreme value. Equat ions (2)and (13), together with the approxima-tion given in (11), yield,

h Pr{ ExtremeresPor,se> Tn} = I - G(~m)“+=

= 1 - ,-1 = 0.632 (16)

Equation (16) implies that there isa 63.2 percent cha”ce_that the largestresponse will exceed Yn. This proba-bility is extremely high; hence, it ishighly desirable fro” a design consider-ation to predict the extreme responsefor which the probability of beingexceeded is very small In other words jchoose a very small number, a, whichmay be called the &h @zhIII@A, andobtain the extreme response C“, (in dimen-sionless form) for which the rollowi”grelationship holds (Ochi 1973) ;

Jeng(c”)dcn= {F($n))n- 1 -u (17)

o

J(n

where ,F($n)= f(c”)dc“

o

Considering that a is small and nis large, we have,

F(<n)- (1-~):- l-:+od) (18)

Then, with tbe aid of the assump-tion given in (11), we have,

(19)

.

In order to amplify the above

19.L

For a narrow–band assumptionc = O, (19) is reduced to,

(20)

The extreme response for designconsideration with a = 0.01 calculatedfrom (19) and (20) is shown in Figure 3as a function of the band-width para-meter, As can be seen in the fi~ure,there is no appreciable difference inthe extreme wave amplitude for E up toO.9. The difference in the magnitudeof extreme response is noticeable forE greater than 0.90; however, thec-value is usually less than 0.8 i“practice, as stated earlier in connec-tion with Figure 2, Hence, the narrow-band assumption is acceptable for theextreme response for design considera–tion of marine systems. Thus j analogousto equation (15), the design extremeresponse can be written as a functionof time by,

for amplitude

(21)

The risk parameter, m, involved in(21) is at the designers discretion,In choosing the a-value, it is noted

that the number of encounters for amarine system with a particular seaseverity in the lifetime has to be con-sidered even though the design extremevalue given in (21) is applied to theshort-term prediction. For example,suppose ve evaluate the design extreme“alue in a particul~r se% se~erity with99 percent confidence, i.e. , e = 0,01.If the system will encounter seas ofthis severity 20 times in her lifetime,then it is necessary to divide the riskparameter a = 0.01 by 20 (i.e.(i= 5 X 10-4) for evaluation in order tomaintain the 99 percent assurance in theprediction.

The effsct of the a-value m themagnitude of design extreme values ofthe midship bending moment of theMARINER i“ seas of significant w’awheight 4.6 m (15 ft) and 10.7 m (35 ft)are shown in Figure 4, The family ofsix-parameter wave spectra consisting ofeleven members in each sea severity(Ocbi 1976) is used in this computation,and the largest values in each seaseverity are plotted in the figure. Thesolid circles are plotted in the figureare the extreme values evaluated withE = o.01 but taking into considerationof the nwnber of occurrences of eachsea state in tbe North Atlantic in 30years. The figure indicates that thedesign extreme values do not increasesubstantially with increasing m-value.

It is of interest to note theeffect of time duration o“ the magnitudeof extreme values, Figure 5 shows anexample of the effect of ship operation

5.,

,.4

u

,..

,.8

,.,

.,,

,,2

...

3.8

3,8., ,8,, 48, ,,. *,,,4,.— —.

. w .@““mm.0,.=.......’..: .!@

Fig. 3 Extremewave ~n (amplitude)for designconsidera-ticm as a functionof “umberof observationsfor variousband-widthparameterE (c!= 0.01) (fromOchi 1973)

20

I

1

Fig. 4 Designvalue of the midshipbendingmomentof the NARINERas afunctionof coefficienta (fromOchi1977)

time on extreme values. The figure per-tains to a sea of significant waveheight of 35 ft (10.7 m) , and the uppe~bound values using the six-parameterwave spectrum are plotted. As can beseen in the figure, the magnitude ofextreme values, both probable as ~eIl ~Sthe design extreme values, i“crea.sesignificantly during the first severalhours and thereafter increase “cryslowly with time. This is the generaltrend of the extreme values in all seaseverities.

An example of the design extremevalues of the wave-induced bendingmoment for various sea severities areshown in Figure 6, The computationsare made on the MARINER, and the curvesshown in the figure are the upper boundcurves obtained by using the two-parameter and six-parameter families ofwave spectra, As can be seen in thefigure, the results of the comp”ta.tionusing the two families of wave spectraagree well up to seas of significantheight of 30 ft (9.2 m), For seas ofsignificant wave height ovsr 30 ft(9.2 m) , however, the design extremevalues evaluated using the two–parameter spectrum xre higher than thoseusing the six-parameter spectrum. It isof interest to note here that Russo andSullivan (1953) gives the values ofhogging bending moment of 85.3 x 107ft–lbs (116 x 103 m-tons) calculatedfollowing the classic standard procedureof assuming the ship to be staticallysupported on a trochoidal wave equal inlength to the length of the ship andhaving a height equal to one-twentieth

“,.7. ,.mm .103

103

DESIGNVALUE

! 60 w

Ii< mBABLEEXTREMfVALUE 2

>460:

. a~ 57

43

$20

: ZJ

o 10 20 4350TIME(H3UR:I

Fig. 5 Probableextremewalue anddesign.Jalneof the midshipbendingmomentof the MARINERin sea ofsignificantwave height35 ft (10.7M)as a functionof ship operationtime(fromOchi 1977)

Fig, 6 Desifmvalueof the midshipbending nwmentof the NARINER(franOchi 1977)

1.

L21

Ji

of the length.

APPROXIMATE ESTIMATION OFEXTREME RESPONSES

In the previous sect ion, exactinformation of the initial probabilitydistribution was necessary for estima-ting the extreme response by applyingorder statistics. In practice, however,the information for the initial distri–bution is often not precisely know”.For example, the probability distribu-tion of the long-term response of amarine system is not know”. It may betbe Weibull distribution or the log-normal distribution. However, there isno scientific basis for selecting anyspecific probability distribution func-tion to characterize the long-termresponses as contrasted to the short-term responses. In other words, wehave to estimate tbe extreme responsesfrom the accumulation of the observed(or computed) data over a sufficientlylong period of time without preciseinformation of the initial probabilitydistribution.

One way to overcome this difficultyis to evaluate the extreme response hyau approximate method that is applicablefor rmy probability distribution ifcertain conditions are met. Theprinciple of this approximation methodis as follows:

Let the initial curmlative distri-bution function, F(x), be in the formof

,(X) ., _ ,-q(x) (22)

where, q(x) is a positive real-valuedfunction that satisfies the conditionsrequired for F(x) being a c“m”lativedistribution function, Then , equation(4) becomes,

q“(Yn) I -q(Y )

{q’(y”))’,,-, q

-q (Y”)+“, -1=0 (23)

Since tbe first term is small incomparison with other terms for largen, (23) yields,

-q (Y”) 1e=— (24)

n

22

Thus , the probable extreme response,Y“ , for large n is given by,

?n=q -1(LR“)

-1where, q ( ) is tbe inverseq(x).

(25)

function of

For evaluating the extreme responsein practice, however, it is not necessaryto know tbe f“”ction q(x) in (22), Ascan be seen from (22) a“d (24), theprobable extreme response for large nis given as the x-value in (22) forwhich tbe probability of exceeding x isequal to lln. That is,

1 - F(~n) = ; (26)

Equation (26) implies that theprobable extreme response expected tooccur in n–observations can be evaluatedfrom tbe initial cumulative distributionand the number of wave enco””tersin”olved. Although tbe initial cumula-tive distribution function is mtprecisely known, the function can beconstructed from the observed (or com-puted) data, For this, the data areoften fitted by some known distributionsuch as a Weibull distribution or log-normal distribution; etc. , a“d theextreme value in a desired period oftime is estimated based on this presumeddistribution, Sometimes, the extremevalue is determined by simply extendingthe cumulative distribution functionwithout fitting the data to a particularprobabilityy distribut ion function.

Although many examples are avail-able for estimating the extreme responsesof marine systems by applying thisapproximate On method, let us considerthe estimation of the extreme seaseverity (significant wave height ) “singthe data presented in Table 2 (Bouws1978). Tbe table shows the frequencyof occurrence of significant wave heightmeasured in the North Sea. A total of5,412 measurements were made in 3 years.

Figure 7 shows fhe cumulative dis–tribution function of tbe significantwave height plotted on the log-normalprobability paper, while Figure 8shows tbe same data plotted on theWeibull probability paper. It appearsfrom comparison of Figures 7 and 8,that tbe data are better represented bythe Weibull distribut ion, and hence theestimation of the extreme significantwave height in 50 years or 100 yearsis made most commonly by extending the

_—

Table2 Significantwave heightdata obtainedfrommeasurementsin the North Sea (53.5°N, 4°E)(fromBouws 1978)

SIGNIFICANT NUMBER OFWAVE KIGHT (M) OBSERVATIONS

o –0.5 1,2800.5– 1.0 1,549lo- 1,5 1,088l.5–2. o 628

2.0- 2.5 4022.5– 3.o I923.0- 3,5 1153.5-4.0 63

4.0 -4,54.5 – 5.05.0– 5.5

5.5-6,06.0 – 6.56.5- 7.o7.0– 7.5 I

TOTAL 5,4I2 in3 YeWS

straight line given in Figure 8,

However, extreme care has to begiven in interpreting the data plottedon the probability paper. If we takea close look at the data plotted inFigures 7 and 8, the data are notsatisfactorily represented over theentire range of significant wave

0.9999.

0.99s. /

./ ‘

099

LOQ-Nan!alDlstrhtimQ95

0,93

Om /

0.60

040 / r

02L-

alo / — —0.5 6 80SIGNIFl~ANTWA2W HEIG;T(M)

Fig. 7 Cumulativedistributionof.i&nificantwane heightplottedonlog-nom.mlprobabilitypaper (datafzomBou.,m1978)

5+$

~E

SIGNIFICANTwAvE HEIGHT (M)

Fig. 8 Cumulativedistributionfunctionof significantwave heightplottedon Weibullprobabilitypaper(datafrom Bows 1978)

height by either the log-normal proba–bility distribution or the Weibullprobability distribution, That isevident in Figure 9 in which the compar-ison between the histogram a“d the twoprobability distribution functions areshown.

As can be see” in Figure 9, theextreme significant wave height may beestimated from neither the log-normalor Weib”ll distribution b“t from theextension of data points by applying

SIGNIFICANTWAVE HEIGHT IN METERS

Fig. 9 Comparisonbetweenhistogramof significantwane heightand log-normaland Wefbullprobabilitydensityfunctions(datafrom Bouws197S)

23

t-

“r1==10

8

+--

-52L-

‘?!0.6,

3?!50 YEARS

— 10YEARS

++468

SIGNIFICANTWAVE HEIGHT IN METERS

Fig. 10 Estimationof extremesigni–ficantwave heightfrom the extensionof data points (datafran Bmnrs 1978)

equat ion (26) The extension of dataDoint is shown in Fie’ure 10. The&ertical scale of th~ figure is theinverse of the lefthand side of (26) ,which is often called the k@LULnpex,iod,in logarithmic form. Since the numberof measurements of significant waveheight is 5,412 in three years, thenumber of significant wave heightsexpected in ten yea?s will be 16,040,By taking tbe logarithms of this n“m-ber, it is obtained from the figure thatthe extreme significant wave heightexpected in 10 years will be 7.5 meters,Similarly, the extreme significant waveheight expected in 50 years ca” beestimated as 8,0 meters

The approximate method presentedin the foregoing can be applicable forany probabilityy distribut ion; however,the method has a drawback. That is,tbe extreme wzve height expected tooccur in the future is determined byextending the line, taking intoaccount the higher wave heights thatare extremely unreliable data, For amore precise estimation, it is highlydesirable to extend the cumulativedistribution function representing thedata over the - range of thevalues, Emphasis should not be givento the representation of data pointshigher than the cumulative distributionof 0.99.

To achieve a precise representation

of the data, a function q(x) in (22) maybe expressed as a combination of an ex-ponential and a power of the wave height(Ochi and Whalen 1980):

kq(x) = axm e-px (27)

The parameters in q(x) are deter-mined numerically by a nonlinear leastsquared fitting procedure. The formused in this minimization procedure isgi”en by,

G = &\-&l (l-F)}= I?na+.n!nx -pxk

(28)

The parameters are optimized suchthat the sum of the squared values ofthe difference between G in (28) andthe corresponding data values becomesminimal Once the parameter values aredetermined, the extreme wave height canbe evaluated from (26) This method isapplied to significant wave height d&tashown in Table 2. For this example,values of four parameters involved i“(27) are obtained as,

a = 0,980

m = 1.101

P = 0,161

k =-1.328

The cumulative distribution func-tion obtained by using these values areplotted o“ log-normal probabil;{ty paperas shown in Figure 11, together withdata points. .4similar presentationplotted on Weibull paper is shown inFigure 12. As can be seen in thesefigures , this cumulative distributionfunction represents very well the dataover the entire range of the values, andtherefore it can be used for estimatingthe extreme significant wave height Theresult is shown in Figure 13, The com-parison between tbe extreme significantwave height thus estimated and thoseestimated based on the log–normal andthe Weib”ll probability distributionsshows that the log-normal probabilitydistribution substantially overestimatesthe extreme significant wave height jwhile the Weib”ll probability distribu–tiOn underestimates it on the order of10 percent for this example.

For design consideration of a marinesystem, a certain margin above theprobable extreme value is required, andthis can be obtained following the sameconcept as was discussed in the develop-ment of (20) That is, by choosing asmall ., tbe extreme value for design

24

.

SIGNIFICANTWAVE HEIGHT(M)

Fig. 11 cumulativedistributionfunc-tionbasedon Equation(27)plottedonlog-normalprobabilitypaper (datafromBouwi 197S)

zQ

:

~

z

SIGNIFICANTwAVE HEIGHT(M)

15-1 I

10 IOYEARS

8

6 4

—4 “!;

—r—

:2 /. I328

1401-0.1axq (X)= O.908x e

I

0$2 4 681015

SIGNIFICANTWAVE HEIGHT IN METERSFig. 13 Predictionof extremewaveheixhtby usinx the ~xobabilitvfunc-tio~bas~d on &mti& (27) “(datafrom Bouws 1978)

various heading to waves, Pj(iii) Frequency of encounter with

v=iy)wave spectral shapes, ‘kExpected number of cycles of

response for each given sea, heading,and wave spectral shape.

Then, the probability densityfunction for the long–term response ca”be written as follows:,

Fig. 12 Cumnlati”edistributionfu”c-tionbased on Equation(27)plottedonWeibullprobabilitypaper (datafromBouws 1978)

consideration, $n , can be obtained from

1 - F($n)= + (29)

As a“ application of the approxi-mate methods discussed in this sect ion,let us evaluate the design extremevalues of the transverse wave-inducedforce acting on the cross-beam bridgestructure on a semi-submersible off-shore platform shown i“ Fig”m 14, Forevaluating the lifetime response of tbeocean platform, the following variousparameters have to be considered:

(i) Frequency of occurrence ofvarious sea severities, P.

(ii) Frequency of occurrence of

where, f*(x) is tbe probability densityfunct ion for the short-term response,n* is the average number of responsesper unit time of the short–term response.

The total number of responsesexpected in the lifetime is given by,

where, T is the total exposure time tosea.

Computation of the extreme responsegiven i“ (30) is carried out by usingthe family of six-parameter wave spec-tra with the risk parameter m = 0.01.Since the magnitude of the transverseforce acting o“ tbe cross–beam Structureis maximom i“ beam seas, it may be ofinterest to see the difference, if any,

25

z

I 671M(220,) I

I 27.lfl I 32.3* I 271M i(8V1 (106’) (89’)

+7%4=I

57.9*(191X

~

l“-l”t”-0

fi:;.:...i,j...i, j--------

I rI1

83.5*(274’) I

Fig. 14 Seniisubmerslble.typeoceanplatformused for computations

between the predicted long-term designextreme values includins all headings,sea severities, and spectral shapesand those predicted in various seaseverities and spectral shapes in beamseas only. The significant differencein these two long-term computations isthe number of responses expected inthe lifetime; 6.146 x 107 for allheadings versus 2,078 x 107 for beamseas only in 20 years operation.

Figure 15 shows the design extremevalues evaluated for these two long–terms (Ochi and Wang 1979) As can beseen in the figure, the design extremevalue evaluated from the lifetime prob-ability distribution including allheadings and that evaluated from thebeam seas only are both 4,800 tons, ifthe difference in the number of encoun–ters for each case is taken intoconsideration.

Next, the extreme responsesevaluated through the long–term predic–tion approach will be compared withthose evaluated through the short-termpredict ion approach. It may appear thatestimation through the long-term predic-tion approach is superior to thatthrough the short-term prediction ap-proach, since it deals with theac-lat ion of all responses. However,im rnlity, the long-term estimation

26

includes a considerable percentage ofsmall responses in relatively mild seas,which do not contribute to the extremedesign value. Tbe results of computa-tions have shown that the design extremevalue obtained through the short-termapprOach is 4,700 tons (Ochi and Wang1976 ) in seas of significant wave heightof 16 m (52.5 ft.), the value is veryclose to that obtained through the long-term approach. As this example showsthe predicted extreme values throughthe long-term and short-term approachesare nearly equal if the difference inthe number of e“cou”ter with waves foreach case is taken into considerateion.It is noted that the estimation proce-dure of the extreme values through theshort–term prediction approach isextremely simple in comparison with thatthrough the long-term approach. SIence,the short-term prediction approachaPPears to be adequate as far as esti-mation of extreme values is concerned.

ESTIMATION OF EXTREME RESPONSESBY ASYMPTOTIC FORMULAE

As was stated in the Introduct ion,it is possible to predict the largestresponses expected to occur in the life-time of a marine system from the me&–sured (or observed) maxima. The mea-sured (or observed) maxima defined hereis the largest “alue that is measured(or observed) during a certain periodof time; every 6 hours, every 12 hours,a day, etc. Tbe estimation is based onthe asymptotic distribution of theextreme values developed by Gumbel(1958) The significant feature of theasymptotic distribution is that theprobability function for the maxima

F6RCE1#TONSx103Fig. 15 Comparisonof designextremeval.eeof the transverseforcecom-puted includingall headingsto wavesand that computedin beam seas only(fromOchi and Wang, 1979)

—-

reduces to the same type irrespectiveof the initial probability distribution.The underlying principle of the asymp-totic distribution is as follows:

Let us assume that the initialcumulative distribution function is inthe form given in (22). By using therelationship given in (24), we ca”write the initial cumulative distribu-tion function as,

q(;”)- q(x)F(x) =l-$e (32)

Then, from the defi”itio” given in(1), the mmn”lative distribution f“nc.t,io”of the ext~eme value for large nbecomes,

- .w{q(Fn) - q(yn}}=. (33)

Since the probability distributionfunction of extreme values, I’n, is muchnm;:ecogcentrated around its modal

Yn, in cOmP&ri SOn with theinitial pro~ability density function,the term q(yn) - q(y”) may be expandedby the Taylor series. Then jby neglect-ing higher order terms, (33) becomes

-w {-q’(jn)(yn -~”)}G(y”)= e (34)

Here, neither q’(Y”) nor y“ areknown in reality; hence, let us express(34) i“ tbe following form,

–exp{-z}G(z) = e (35)

where,

z = q’(Jn)(y”-;”) (36)

The mean and variance of therandom variable Z ca” be obtained from(35) as,

EIz]= Y (Eulertsco”.sta”t,0.577)

Var[z]= ?r216(37)

Then , from (36) a“d (37), we canderive the following relationship

;n=E[y].LII q’ (;”)

(38)

since the mean and variance, EIY”land Var[yn] , respect ively, can beobtained from the data, the cumulativedistribution function of the extremevalue given in (34) can be evaluatedwith tbe aid of (38)

In summary, the cumulative distri–bution function of the extreme valuecan be written in tbe following form:

-~ (Yn–u)-e

G(Y”) = e _m<y<m“

(39)

where,

Equation (39) is the so-calledGumbel’s L4fmp.totict@@mt WI&e L&tkibu.tion.

The method to estimate the extremeresponses expected in the lifetime of amarine system from equation (39) is tbesame procedure as discussed in connec-tion with equations (26) and (29). Sincethe concept of the asymptotic distribu-tion of tbe extreme value deals with themaxima, Yn, observed in certain periodof time, the cumulative distributionfunctiOn, G(Yn), given in (39) can beconsidered as the initial probabilitydistribution. Hence, the lifetimeprob~ble extreme value is determined astbe yn-value which satisfies

1- G(;).~ (40)n“

where, n is tbe number of the maximaexpected to occur in the lifetime, andit is estimated from the observed data(see the example shown in Figure 18).Similarly, the design value with the

!-- —j\,

27

risk.parameter, a , can be evaluated asthe yn-value which satisfies

1 - G(4n)= : (41)

The sample space of tbe distribu-tion given in equat ion (39) isunlimited; however, there is anotherdistribution called A4ynp.toticalhtnww&t dL&tzibu.tionw&k uppm &mi.t which mayalso be applicable for the analysis ofobserved maxima. The distribution isgiven by (Gumbel 1958) ,

k

-(–)

w-y”

w–vG(Yn)= e --< yn <w (42)

where, w = upper limit

k = positive constant linkedto the limit value, w

“ = y“_”aI”e for wbicb G(Y”) =

e-l = 0.368.

Tbe derivation of tbj.s asymptoticdistribut ion will not be given here;however, it may suffice to state thatit is derived by applying the Taylorseries exvansion to the initial proba-bility di&ribution function in ~heneighborhood of w.

The values of the two parameters jw and k, can be determined from thefollowing two moments of Yn:

E[yn]= “ - (“-V) ~(2+ +)

EIY~l= W2-2W (W-V) ~(1+~

+ (W-V)2 r(l+~) (43)

Tbe values of w and k can also bedetermined eraDbicallv bv choosine thecumulative ~isiributi; n ;unction, ‘G(yn) ,for two diffei-e”tyn-values. It shouldbe noted that the upper limit value, w,thus determined from the data is merelythe limit value of Yn for n + ‘-,whichis unrealist ically large value, andhence it cannot be used for design con-

Tbe probable extreme value as wellas the extreme va.l”e for design shouldbe evalu~ted by the same procedure aspresented i“ con”t?ction with theasymptotic extreme value distribution

with unlimited sample space.

The method to estimate the lifetimedesign value through the asymptoticextreme value formulae is applied to theresults of extensive full scale trialscarried out on eight SL–7 vessels onSEA-LAND service (Fair and Booth 1979)During these trials, the strain gagerecorder was installed at midship ofeach ship, and the maximum peak tomaximum trough stresses which occurredduring every 4-hour sampling period wererecorded.

As an example of analysis which canbe made from these data, tbe extremeresponses expected in the lifetime ofthe SL-7 will be estimated based on thefollowing two data:

(a) One-year data obtained onSEA–LAND MCLEAN in tbe Atlantic. Atotal of 1,802 observations.

(b) Five-data-years obtained inthe Atlantic. A total of 12,319 obser–nations.

It is noted that the results ofanalysis have shown that the responsesobserved in the Atlantic service routesare larger than those in tbe Pacificservice routes. Therefore, the designextreme value may be estimat<?d fromanalysis of data in tbe Atlantic.

Figures 16 and 17 show the compari-sons of the cumulative distributionfunctions of these data and the asymp–totic extreme value formulations givenin (39) and (42). Figure 16 shows thecomparisons using tbe one–year data,while Figure 17 shows those for thefive–data years.

The lifetime extreme responses canbe estimated from the relationship givenin equation (40) together with tbeinformation on the number of observa-tions, The results are shown in Figures18 and 19. As can be seen in thesefigures , the lifetime extreme responseof the SL–7 estimated from the one-yeardata is very close to that estimatedfrom the five-data-years (in which manysister sbipsc data are involved) , if thedifference in tbe number of observationsin each case is take” into consideration.

For instance, from equation (39),the design extreme value (peak-to-troughstress) in 30 years with the risk Para–meter a = 0,01 is 64,0 kpsi(45.4 kglmmz) from the one-year data ascompared with 60.0 kpsi (42.6 kglnunz)from tbe five-data-years.

It can also be seen in thesefigures that tbe extreme design estimatebased on the asymptotic formulation withupper bound is approximately 17 percent

28

Fig.16 Comparisonbetweenasymptoticdistributionwith unlimitedboundsandthatwith upperboundusing SL-7mm-yeardata

Asymptotic withupper bound

20/

‘$ ‘Design extreme

15— (~ =001) “ ,7

Probableextreme

unlimitedbounds

5~

PEAK TO TROUGH STRESS (KPSI )

Fig. 18 Evaluationof tbe SL-7lifetime(30years)probableextremevalue anddesignextremev.I.. basedon the .ne.year data

less than that estimated based on theasymptotic formulation with unlimitedbound,

CONCLUSION

This paper presents the theoreticalbackground of recent techniques topredict extreme values., specificallyextreme loads and responses of marinesystems in a seaway. There are threedifferent approaches to evaluate extremevalues depending on the information(data) available in the prediction,

K’

n

p

s~ + -–+($trwht line)—~.——

%% 9392CUMULATIK PROBABILITY

Fig. 17 comparisonbetweena.ymptoti.di.tributicmwith unlimitedboundsandthatwith upperbound for SL-7 five-datayears

30 r

/ ,“Asymptoticwith j

20 ‘— upper boundY i

Designextreme~15— ‘— (q= 0.01)

Asymptoticwithunlimltedtmucds

520 40 60 80

PEAK TO TROUGH sTREss (KpsI)

Fig. 19 Evaluationof the SL-7 lifetime(3OYear.) probableextreme“ale ba$edon the five-data-year$

These are (a) the exact evaluation ,(b) tbe approximate estimation, and(c) the estimation by “sing theasymptotic extreme value formulae Theunderlying principle and practicalapplication of each approach are dis–cussed. From the results of numericalexamples, the following conclusions aredrawn:

One way to evaluate tbe designextreme value of the responses of amarine system is to calculate theextreme response through the short-termapPrOach by using the family of wavespectra in each sea. Tbe computations “L-.

29

should be made for the severest loadingand heading of the response, at theattainable highest speed of the system,if applicable, in each sea. From theresults of the largest response in eachsea severity, the design extreme valuecan be obtained with a specified riskparameter, a.

In the short-term prediction, theeffect of the spectral band-width para–meter on the magnitude of extreme valuesappears to be negligibly smallirrespective of the number of observa-.tions.

The extreme values of the long-term response of a marine system canbe evaluated through the approximateestimation method. In this ease, theestimation can be made based on thecumulative distribution function whichrepresents the data over the entirerange of values. Emphasis should notbe given to the representation of datapoints higher than the cumulativedistribution of 0.99.

The extreme value through thelong-term and short-term approachesare nearly equal if the difference intbe number of encounters with wavesfor each ease is taken into considera–tion. It is noted that the estimationprocedure of the extreme values throughthe short–term prediction approach isextremely simple in comparison withthat through the long–term approach.

The lifetime extreme response ofa marine system ca” be estimated fromdata of the observed maxima by applyingtbe asymptotic extreme value formulae.Tbe lifetime (30 years) extremeresponse of the SL-7 estimated from tbeone-year data is very close to thatestimated from the five-data-years(in which many sister ships, data areinvolved), if the difference in thenumber of observations in each case istaken into consideration.

REFERENCES

BOUWS, E. (1978): “Wind and WaveClimate in the Netherlands Sector ofthe North Sea Between 53 and 54 NorthLatitude, ,,Koninklijk NetherlandsMeterologisch Institute, Report W.R.78-9.

Gumbel, E.J. (1958): “Statistics ofExtremes, “ Columbia University Press,New York.

Fain, R.A. and Booth, E.T. (1979):“Results of the First Five Data-Yearsof Extreme Stress Scratch Gauge DatzCollected Abroad SEA-LAND, S SL-7,1.ship StrUCtUre Committee, SSC-286.

Ochi, M,K. (1973): ‘>OnPrediction ofExtreme Values, ,,Journal of Ship Res. ,Vol 17, No. 1, pp 29-37.

Ochi, M.K. and Wang, S. (1976): ‘pred-iction of Extreme Wave-Induced Loadson Ocean Structures, ,%Proc. Sympo.Behavior of Offshore Structures, Vol 1,pp 170-186.

Ochi, M.K. and Hubble, E.N. (1976): ,,OnSix-Parameter Wave Spectra.,‘rProc. 15thConf. Coastal Eng. , Vol 1, pp 301-328,

Ochi, M.K. (1977): “Concept ofProbabilistic Values and its Applicationto Ship Design, ,$Proc, Symp. Pratt icalDesign in Shipbuilding, PP 51-57.

Ochi, M.K. and Wang, S. (1979): “TheProbabilistic Approach for the Designof Ocean Platforms, ,,Proc. conf.Probabilistic Mechanics and StructuralReliability, Ame. Sot. Civil Eng.pp 208-213.

Ochi, M.K. and Whalen, J.E. (1980):,,prediction of the Severest significant

Wave Height,,, 17th Conf. Coast; l Eng. ,Sydney, Australia (in Printing)

Russo, V.L. and Sullivan, E.K. (1953):“Design of the MARINER-Type Ship, ‘fTrans. Soc. Nav. Arch. & Mar. Eng. ,Vol 61.