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Ocean Engineering 34 (2007) 956–961

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On the adequacy of second-order models to predict abnormal waves

P. Petrova, Z. Cherneva, C. Guedes Soares�

Unit of Marine Technology and Engineering, Technical University of Lisbon, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

Received 14 January 2006; accepted 1 June 2006

Available online 16 October 2006

Abstract

The paper is intended to extend the investigations about the nature of abnormal waves that have been reported in the work of Guedes

Soares et al. (Characteristics of abnormal waves in North Sea states. Applied Ocean Research 25, [337–344]). The same dataset gathered

at the oil platform North Alwyn in the North Sea during the November storm in 1997 is used along with the time series from the

Draupner platform, in which an abnormal wave occurred. The data are reanalyzed from the viewpoint of the applicability of second-

order models to fit large waves. The observed results confirm that the second-order approximation is not adequate to describe highly

asymmetric and abnormal waves.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Asymmetric waves; Abnormal waves; Second order models

1. Introduction

For scientific research or engineering-design purposes,wave statistics is very important, especially the statisticalproperties of maximum wave parameters. The linear theorywas found to systematically underestimate the large wavecrests and heights. On the other hand, the second-ordertheory was reported to be in general agreement withlaboratory simulation results and full-scale field measure-ments (Forristall, 2000; Jha and Winterstein, 2000).

The non-linear second-order effects on wave-heightdistribution are weak and sometimes they are welldescribed even by the linear-wave theory (Stansberg,2000a). Nevertheless, the largest wave heights or crestsare usually underestimated, especially in a sea state with anabnormal wave event. Under certain conditions, themaxima in random wave trains are observed to besignificantly higher than the second-order predictions, evenfor moderately steep wave conditions (Stansberg, 2000b).Thus, in order to avoid serious damages to marinestructures, it is necessary to have appropriate models ofwaves applicable to severe sea states in order to assess thestructural response to such wave loads.

front matter r 2006 Elsevier Ltd. All rights reserved.

eaneng.2006.06.006

ng author. Tel.: +351 01 841 7607; fax: +351 01 847 4015.

ss: guedess@mar.ist.utl.pt (C. Guedes Soares).

There is a variety of possible complex mechanisms thatcan generate abnormal waves (Kharif and Pelinovsky,2003). The importance of the observed phenomenonoriginates from the numerous examples for the last 20years showing that the most serious damages are ratherassociated with few very large waves than with largefrequently encountered waves (Kjeldsen, 1997; Faulknerand Buckley, 1997).The criteria to define a wave as an abnormal one are still

not totally clarified. Initially they were defined as beingwaves that would not conform to linear theory and thusDean (1990) assumed a sea state of 20min duration andused the Rayleigh distribution to suggest that the abnormalwave is defined as a single wave of height exceeding twotimes the significant wave height, i.e., Hmax/Hs42. Thisratio became known as the amplification or abnormalityindex, AI.Other commonly used definitions refer instead to the

ratio between the maximum crest amplitude and thesignificant wave height, which could be denoted as crestamplification index CI ¼ Cmax/Hs4R given Cmax is themaximum crest height. Different values have been pro-posed for the factor R that would represent the limit forwaves to be classified as abnormal. Haver and Andersen,(2000) suggested a value of 1.2, while Olagnon and Iseghem(2000) preferred 1.25.

ARTICLE IN PRESSP. Petrova et al. / Ocean Engineering 34 (2007) 956–961 957

Tomita and Kawamura (2000) have adopted a combina-tion of the two ratios in order to define a genuine freakwave, which should have CI41.3 and AI42.

Clauss (2002), on the other hand applies a combinationbetween the abnormality index and some global waveparameters. For him an abnormal wave should haveHmaxX2.15Hs and CrmaxX0.6Hmax.

As seen, various criteria have been used without anytheoretical basis to support them and thus, they can all beconsidered as possible conventions that could be used tostandardize definitions. The one exception is the abnormalindex which resulted from considerations of the Rayleighdistribution. However, even in this case that requirementneeds to be associated with a sea state duration as longerdurations require larger values of the index in order to beconsistent from a probabilistic point of view.

Guedes Soares et al. (2003) have analyzed a data set ofstorm conditions in various locations in the North Sea andstudied the influence of the choice of the different criteriaon the number of abnormal waves identified. They alsodetermined the third-and fourth-order statistical momentsof the time records arguing that the sea states could not befully explained by second-order wave theories as in thiscase the excess kurtosis would need to be zero and this wasnot the value determined in those sea states.

The present study aims at extending the investigations onabnormal wave phenomenon performed by Guedes Soareset al. (2003) providing additional results on the applic-ability of the second-order wave models to describe theobserved large and highly asymmetric waves.

The paper is organized as follows. The applied modelsare introduced in Section 2. The subset of waves classifiedas abnormal with reference to the considered criteria isconsidered. Comparisons with the models are given inSection 3. The obtained results confirm the conclusionderived in the paper of Guedes Soares et al. (2003) thatsecond-order theory is not able to describe well theabnormal waves. However, the models are found toperform better when applied to wave heights than to wavecrests, as has been previously reported in Stansberg(2000a).

2. Theoretical wave models for maximum value statistics

The available models serve to predict theoretically theexpected maximum values of wave crests. They areclassified into two groups: linear and non-linear second-order models. The non-linearity is usually identified eitherby the third and fourth central moments, a3 and a4(Stansberg, 1998, 2000 a, b) or by cumulants. In the presentstudy the third and fourth normalized cumulants—coefficient of skewness, g3, and coefficient of kurtosis, g4,are used. They are related to a3 and a4 by the ratios: g3 ¼a3=a

3=22 and g4 ¼ a4=a22 � 3. Since for a Gaussian process

g3 ¼ 0 and g4 ¼ 0, it follows that the fourth normalizedmoment g4 ¼ a4=a22 is expected to be equal to 3. Thecoefficient of skewness reflects the systematically increasing

vertical asymmetry in the non-linear waves—higher crestsand shallower troughs, while the coefficient of kurtosisstands for the excess of the total crest-to-trough height inrelation to the Gaussian elevation process.

2.1. Linear model

The free surface elevation of linear waves followsGaussian statistics and Rayleigh distributed peaks. Themaximum linear crest amplitude is assumed to be Gumbeldistributed (Gumbel, 1958). The associated statistics aregiven as functions of the number of independent wavecrests, M, in the record

ARmax ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln Mð Þ

pþ 0:577=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 lnðMÞ

pj k¼ E Amax½ � ¼ sZRmax, ð1Þ

sAmax ¼pffiffiffi6p

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln Mð Þ

p , (2)

HRmax ¼ E Hmax½ � ¼ 2ARmax, (3)

where E[y] stands for the expected largest value, s is thestandard deviation of the free surface elevation process andthe factor 0.5774 is the Euler’s constant. ZRmax is the linearcrest according to the Rayleigh model (ARmax) normalizedby the standard deviation and sAmax is the standarddeviation of the maxima following the Gumbel’s distribu-tion. According to the Rayleigh model the largest waveheight is simply equal to two times the Rayleigh amplitude(Eq. (3)).

2.2. Non-linear models

The existing reference models are based on the second-order representation of the surface profile assuming that itusually gives an adequate description of the real sea. Thenon-linear effects are found to increase the crest statistics(Marthinsen and Winterstein, 1992). However, the largestwave heights generally follow the linear theory, because thesecond-order non-linear effects amplify the wave crests tothe same extent that they are seen to decrease the troughs.The coefficient of skewness, g3 was found to dependlinearly on the sea state steepness, Sp ¼ Hs/Lp, while thecoefficient of kurtosis, g4 is increasing in quadratic mannerwith Sp and g3, respectively (Vinje and Haver, 1994). Thecorresponding equations are as follows:

g3 ¼ 5:41Hs

Lp

, (4)

g4 ¼ 1:33 g3� �2

. (5)

Also, the coefficient 1.33 in Eq. (5) has been proposed to bereplaced by

ffiffiffi3p

, thus including some third-order effects.Recently, Guedes Soares et al. (2003) found anotherrelation between g3 and g4 for the data from the NorthAlwyn platform considered here, assuming that the process

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is to second- and third-order non-linear but remainsstationary over the 20min of measurements

g4 ¼ 3:764g23 þ 0:236. (6)

The second-order theory maximum, denoted as As, is givenby the simplified formula of Kriebel and Dawson (1993):

AS ¼ ARmax 1þ1

2kpARmax

� �. (7)

The non-linear term accounts for the contribution of thesea state steepness, Sp, expressed by the product (kpARmax),where kp is the wave number, corresponding to thespectral peak period, Tp and ARmax is calculated by meansof Eq. (1).

Applications of second-order approximation models todescribe experimental or laboratory data shows that theygenerally agree with the full-scale measurements orlaboratory results (Marthinsen and Winterstein, 1992;Stansberg, 1998; Krogstad and Barstow, 2002). However,this is not the case for specific conditions contributing tolarge and steep waves. Thus, a higher order correctionrelated to g4 has been introduced in some of the models. Itwas shown that one of the reasons for abnormal wavegeneration is the non-linear energy focusing in the wavegroups and the coefficient of kurtosis is used as a measureof the increased wave grouping (Stansberg, 2000b). In factGuedes Soares et al. (2003) have observed that therewas a good correlation between the abnormality index,which identifies the abnormal waves, and the kurtosis ofthe sea state.

Stansberg (2000a, b) has proposed an empirical third-order correction for the biggest crests and wave heightsgiven second-order approximation of the free surfaceelevation

ASmax

s¼

ARmax

s1þ

o2p

2gAR

!þ 1:3g4, (8)

Hmax

2s¼

HRmax

2sþ g4 � 0:25� �

, (9)

where ARmax and HRmax are the linear equivalentscalculated from Eqs. (1) and (3), respectively.

Winterstein (1988) proposed a more general representa-tion of the largest wave crests based on the Hermitetransformation model. The transformation expands a non-Gaussian process, say Z(t), as a function of a standardGaussian random process, say U(t)

Z� Z ¼ gðUÞ ¼ ks U þ c3 U2 � 1� �

þ c4 U3 � 3U� �� �

.

(10)

The model input parameters are the first four statisticalmoments: mean (Z), standard deviation (s), coefficient ofskewness (g3) and coefficient of kurtosis (g4) calculatedfrom the given records. The mean is usually set to zero. Thethird and fourth order statistics enter the model by means

of the coefficients c3 and c4

c3 ¼g3

6ð1þ 6c4Þ, (11)

c4 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1:5g4

p� 1

18, (12)

k ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 2c23 þ 6c24

q . (13)

Then, the non-Gaussian, non-linear maximum is simplyestimated from ZR, introduced in Eq. (1)

AW ¼ ks ZR þ c3 Z2R � 1

� �þ c4 Z3

R � 3ZR

� �� �. (14)

3. Analysis of results

The wave data used in the present study were collected atthe fixed oil marine platform North Alwyn in the northernNorth Sea. The platform is located at 130m water depth,which allows considering the conditions as deep water,except for the largest waves. During the considered stormfrom 16 November to 22 November 1997, 421 20-minrecords of the surface elevation at 5Hz sampling frequencywere stored. The instrumentation included wave laseraltimeters mounted on the platform. The interval betweenrecordings was 2min, thus it is possible to assume that thestorms have been registered continuously. Every recordconsists of 6000 samples of water surface elevations.For comparison, the record with the New Year wave

from the Draupner platform has been also included. Thejacket platform Draupner is situated in the central NorthSea at water depth of 70m. The performed registrationshave been furnished by down-looking laser devices eachhour for 20min. As a result, 10 records are available,covering the period from 31 December 1994 to 1 January1995. Each file consists of 2560 free surface elevationordinates, given the sampling frequency of 2.13Hz. TheNew Year wave refers to the record on 1 January at 15:20pm. This wave has been a subject of study in many works,dedicated to the problem of rogue waves and their betterunderstanding.The records from North Alwyn are processed assuming

that process keeps stationary over the 20-min interval.A linear correction of the mean water level (Goda, 2000) isfirst applied to the time series. The waves are definedfollowing the down-crossing and up-crossing definitions.The maximum positive extrema were taken to represent thewave crests and the maximum by absolute value negativeextrema—the wave troughs. Different statistical andspectral parameters have been also obtained in GuedesSoares et al. (2003, 2004).From all 421 records of North Alwyn 23 waves have

been identified as abnormal ones since the ratio CID ¼CrmaxD/HsD is larger than 1.3, where CrmaxD is the crestheight corresponding to the maximum down-crossingheight in the record. Subsequently, eight of them along

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6

8

10

12

14

16

Observed Crest Extreme [m]

The

oret

ical

Mod

el E

xtre

me

[m]

Perfect fitWinterstein (1988)Second orderLinear waves

Fig. 2. Comparison between the observed maximum wave crests and the

predictions of the linear and second-order theory including the second-

order model of Winterstein (1988).

P. Petrova et al. / Ocean Engineering 34 (2007) 956–961 959

with the New Year wave are classified as genuine freakwaves according to the definition of Tomita and Kawamura(2000). Moreover, these waves fulfill the requirement ofAI42 for both down- and up-crossing definitions. Allwaves are highly asymmetric with vertical asymmetry largerthan 0.65, except for one case (Guedes Soares et al., 2003)and have maximum down-crossing height HD410m. TheNew Year wave has the largest one HmaxD ¼ 25m withcorresponding maximum crest CmaxD ¼ 18.5m.

The abnormal and genuine freak waves have beeninvestigated in the view of the reference models presentedin Section 2. For each case, the theoretical expectationshave been found and subsequently compared with theobserved maximum statistics.

Fig. 1 shows the comparison between the observedlargest wave crests and the predictions of the linear theory,the simple second-order theory (Eq. (7)) and the second-order theory with third-order correction of Stansberg(2000a, b) (Eq. (8)). The solid line corresponds to the caseswhen the predictions and observations are in agreement.The full triangles represent the comparison with themaxima following the Gumbel distribution. The simplesecond-order theory is designated by full squares and theconsidered second-order corrected theory—by full circles.The points associated with the observed freak waves areoutlined by light signs and the New Year wave is given witha crossed sign. The regression lines for all comparisons arealso drawn.

As can be seen from Fig. 1, two cases are found to bewell represented by the theory of Stansberg (2000a, b).The smaller observed crest pertains to a genuine freakwave case.

In Fig. 2 the largest observed crests are plotted versus thecorresponding expected values by the linear theory, the

7 9 11 13 15 17 194

6

8

10

12

14

16

Observed Crest Extreme [m]

The

oret

ical

Mod

el E

xtre

me

[m]

Perfect fitStansberg (2000)Second orderLinear waves

Fig. 1. Comparison between the observed maximum crests and the

predictions of the linear and second-order theory, including the corrected

theory of Stansberg (2000a, b).

simple second-order theory and the second-order model ofWinterstein (Eq. (14)). The comparison with second–thirdorder predictions are given with full diamonds. Theconclusion is that the corrected model fails to describewell the observations, except for one case—the wave thathas already been found to agree with the model ofStansberg.Both plots show that the theories systematically under-

estimate the observed large crests even for the models whena fourth order correction is introduced. The New YearWave statistics corresponds to the largest deviations.Subsequently, the models are applied to the heights of

the abnormal waves. The linear predictions (Eq. (3)) arecompared with the second-order corrected theory ofStansberg (2000a, b) given by Eq. (9). The results areplotted in Fig. 3 with full triangles representing the linearmodel and full circles for the higher order model. Thegenuine freak waves are given with light signs and a crossedsign, respectively, for the case of the New Year wave.Fig. 3 shows that the largest observed waves are

systematically underpredicted by the linear model. On theother hand, the second order non-linear approximationincreases the expected maximum values, thus improvingthe predictions. It is seen that the model is in generalagreement with observations, although most of theabnormal waves are underestimated. Few waves are foundto be overestimated by the second-order model. Twoextremely large waves of height exceeding 17m areregistered in the North Alwyn data.

4. Conclusions

The study is based on continuous full-scale measure-ments conducted at the platform North Alwyn in the

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10

12

14

16

18

20

22

24

26

Observed Wave Height Extreme [m]

The

oret

ical

Mod

el E

xtre

me

[m]

Perfect fitStansberg (2000)Linear waves

Fig. 3. Comparison between the observed maximum wave heights and the

predictions of the linear and second-order theory of Stansberg (2000a, b).

P. Petrova et al. / Ocean Engineering 34 (2007) 956–961960

North Sea during the November storm in 1997. The recordwith the New Year wave from the Draupner platform hasbeen also included for comparison. This study aims atextending the analysis in the field of the freak wavesreported in Guedes Soares et al. (2003) providingadditional results on the applicability of second-ordermodels to describe full-scale large wave data.

The 24 cases of high and steep waves, studied here havebeen distinguished as abnormal, since they satisfy themaximum crest criterion CID41.3. Subsequently, ninewaves are classified as genuine freak waves given they alsosatisfy the abnormality index limit for both down-crossingand up-crossing definitions.

The comparisons between the observed large crestsand theoretical model predictions have shown that thelinear and second-order models systematically underesti-mate the measurements. The model of Stansberg(2000a, b) gives slightly higher predictions for the wavecrests as compared to the model of Winterstein (1988).The most pronounced deviation refers to the case of theNew Year wave where extremely high crest has beenregistered.

Similar comparisons have been performed for the heightsof the abnormal waves. The second-order model ofStansberg (2000a) was found to be in better agreementwith the large wave heights than with the maximum wavecrests. However, the majority of the cases are under-predicted by the models. There are few cases when thesecond-order model is found to overestimate the largestheights.

The observed results confirm that the second-orderapproximation, even when fourth-order correction termsare included, is not adequate enough to describe sea stateswith observed abnormal phenomenon.

Acknowledgments

The data used in this work was obtained during theproject Rogue Waves–Forecast and impact on MarineStructures (MAXWAVE), which was partially financed bythe European Commission, under the contract EVK3-CT2000-00026.This work was financed by the Portuguese foundation for

science and technology (FCT) under the Pluriannualfunding to the Unit of Marine Technology and Engineering.

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