Rogue

wavesand

extreme

eventsin

measured

time-series

Alastair

D.Jenkins 1,A

nneK

arinM

agnusson 2,AndreasN

iedermeier 3,

JaakM

onbaliu4,A

lessandroToffoli 4,and

Karsten

Trulsen

5

1BjerknesC

entreforC

limate

Research,B

ergen,Norw

ay2N

orwegian

MeteorologicalInstitute,B

ergen,Norw

ay3Faculty

ofMathem

atics,T.U.M

unich,Germ

any4K

.U.Leuven,B

elgium5D

epartmentofM

athematics,U

niversityofO

slo,Norw

ay

Alastair.Jenkins@

bjerknes.uib.nohttp://w

ww.gfi.uib.no/˜jenkins

Introduction

MA

XW

AV

Eproject

–F

undedbyC

.E.C

.,2000-2003

–W

orkPackage

2,Extrem

ewave

statisticsfrombuoy

records

–R

eportavailable:

–http://w

3g.gkss.de/projects/maxwave/

Work

packagesW

P2

Extrem

ewave

statisticsfrombuoy

recordsW

P2

report(finalversion)

–http://w

3g.gkss.de/projects/maxwave/w

p2rep138pages1-93.pdf

Locations:

–E

kofiskand

Draupner

fields,North

Sea

–B

elgiancoastalw

aters

Analysis

procedures

Extrem

e-valuestatistics

Waveletanalysis

HA

CY

Manalysis

Quasi-Lagrangiancorrection

ofbuoym

otion

Propagation

oftime

seriestootherlocationsusing

linearandnonlineartechniques

Locationofobservations

Ekofisk

platforms

Flem

ishB

anksobservations

0 5 10 15 20 25 30 35 40

10

2015

5

10

5

20

25

3030

25

15’ 30’

45’ 3

oE

15’ 30’

51oN

6’

12’

18’

24’

30’

36’

WH

I

AK

Z

BV

HA

2B

TR

G

OS

T

Draupner

event(1995-01-0115:20U

TC

)��� �

��

0�

20

0

�

40

0

�

60

0

�8

00

�

10

00

12

00

-35

-30

-25

-20

-15

-10 -5

22

02

40

26

02

80

30

0-3

5

-30

-25

-20

-15

-10 -5

������

HA

CY

Manalysis

Firstapplied

tooceanw

avesbyB

uckley(1983

Tech.Rep.D

WT

NS

R&

DC

).

The

heightsofsuccessiveextrem

epoints(w

avecrestsand

valleys)areplotted

againsteachother. aS

mallerw

avesondifferentsealevels

givepoints

nearthebisecting

lineofthe

plottingquadrant.H

ighw

avesyieldpoints

faraway

fromthe

diagonal.The

up-goingw

avesareon

theleftupperside

ofit,thedow

n-goingw

avesonthe

lowerrightside.

The

HA

CY

Mm

ethodshouldprovide

agood

methodofidentifying

typicalshapesofextrem

ewaves,and

itsvalue

would

beenhancedif

thesecanbe

associatedwith

hydrodynamic

phenomena.

aLetx

ii

Ibe

theextrem

aofthesignalf

:RR

.A

sH

AC

YM

analysiswe

considerthepoint

setf

xi

fx

i1

R2;i

i1

Iw

hichis

sometim

esdiscretised

inform

ofatw

odim

ensionalhistogramon

h0 Z

2R

2,whereh

0

issom

ewave-heightquantisation,e.g.h

00

1[m

etres].

HA

CY

Manalysis

plotfrom

theD

raupnerdata

setof

1995-01-0115:20U

TC

09:0012:00

15:0018:00

21:00

2 4 6 8 10 12

H4rm

s: 9.17

Crx:11.85

−T

rx: 5.98

[UT

C]

[m]

H4rm

sC

rx−

Trx

0200

400600

8001000

1200

−10

−5 0 5 10

i=22

[m]

EK

OF

ISK

FLA

RE

SO

UT

H D

AT

E: 25−

Oct−

1998 Start tim

e: 16:00 UT

C

SW

L= −

19.168 mH

4rms=

9.17 m

Crx=

11.85 m

Trx=

−5.98 m

00.1

0.20.3

0

100

200

300

400

Frequency [H

z]

1998−10−

25 16:00 nfft:2395 <N

fa>:5

Hm

0 = 8.8 m

Tm

02= 9.7 s

Tp =

12.6 s

‘Stenfjell’case,sam

plefrom1998

October

25,16:00U

TC

.

09:0012:00

15:0018:00

21:00

2 4 6 8 10 12

[m]

Tim

e [UT

C]

WA

VE

RID

ER

H

4rms

Crx

−T

rx

09:0012:00

15:0018:00

21:00

2 4 6 8 10 12

[m]

FLA

RE

NO

RT

HH

4rms

Crx

−T

rx

09:0012:00

15:0018:00

21:00

2 4 6 8 10 12

[m]

EK

OF

ISK

: 1998−10−

25

FLA

RE

SO

UT

HH

4rms

Crx

−T

rx

Ekofisk

‘Stenfjell’

case,Waveriderand

laserobservations.T

helettercodesare

asfollo

ws:

H4rm

sis

significantw

aveheight(H

sor

Hm

0 ,4tim

esthe

standarddeviationof

theseasurface

displacement);C

rx

isthe

maxim

umcrestelevation;

Trxis

them

aximum

troughdepth;

WR

,W

averiderobservations;FN

,

laserobservationsfrom

Flare

North;

FS

,laserobservations

fromF

lareS

outh;WR

c,quasi-Lagrangian

correctiontothe

Waveriderobservations.

09:0012:00

15:0018:00

21:00

2 4 6 8 10 12

Crx [m]

[UT

C]

MA

X C

RE

ST

WR

FN

FS

WR

c

09:0012:00

15:0018:00

21:00

2 4 6 8 10 12

Trx [m]

MA

X T

RO

UG

HW

RF

NF

SW

Rc

09:0012:00

15:0018:00

21:00

2 4 6 8 10 12

[m]

EK

OF

ISK

wave observations 1998−

10−25

WA

VE

RID

ER

QLcorr

H4rm

sC

rx−

Trx

Ekofisk

‘Stenfjell’

case,Waveriderand

laserobservations.T

helettercodesare

asfollo

ws:

H4rm

sis

significantw

aveheight(H

sor

Hm

0 ,4tim

esthe

standarddeviationof

theseasurface

displacement);C

rx

isthe

maxim

umcrestelevation;

Trxis

them

aximum

troughdepth;

WR

,W

averiderobservations;FN

,

laserobservationsfrom

Flare

North;

FS

,laserobservations

fromF

lareS

outh;WR

c,quasi-Lagrangian

correctiontothe

Waveriderobservations.

440442

444446

448450

452454

456458

−6

−4

−2 0 2 4 6

EK

OF

ISK

WA

VE

RID

ER

AT

1998−10−

25 14:20

[sec]

[m]

WR

−O

bsW

RQ

L−corr

2piH/(gT

**2)=0.095

H=

12.99m

The

quasi-LagrangiancorrectionofM

agnusson,Donelan

&D

rennan(CoastalE

ngng1999)

appliedtoa

wave

ofheight12.99m.T

hecorrection

hasonlya

smalleffecton

theoverall

steepness2πH

gT2

,trough-to-crestwave

height,orwave

period.The

crestdoesbecome

narrower,increasing

thefrontsteepness.T

hesam

etendencyw

asseeninothersteeporhigh

wavesfrom

therecordsanalysed.

34

56

78

910

1112

3 4 5 6 7 8 9 10 11 12

Laser FN

[m]

WR(OBS)2048

[m]

MA

X C

RE

ST

, EK

O−

1998−10−

25

Y =

0.592.X +

1.86

<Y

> =

5.8 m

<X

> =

6.7 m

34

56

78

910

1112

3 4 5 6 7 8 9 10 11 12

MA

X C

RE

ST

, EK

O−

1998−10−

25

WR

−O

bs (2048) [m]

WRQL−corr

[m]

Y =

1.038.X +

−0.11

<Y

> =

5.9 m

<X

> =

5.8 m

Ekofisk

‘Stenfjell’

case,regressionplotsofm

aximum

crestheightbetweenthe

laser

instrumentatF

lareN

orth(F

N),the

observedW

averiderdataWR

(OB

S),and

the

quasi-Lagrangiancorrectionofthe

WaveriderdataW

RQ

Lcorr .T

hetim

eseriesusedin

this

analysiswas2048

sampleslong.T

hered

linesshow

theleast-squaresfit

andthe

bluelines

haveslope1.0

andgo

throughtheorigin.

21

02

20

23

02

40

25

02

60

27

02

80

29

03

00

31

0

−1

0

−5 0 5

10

15

20

se

co

nd

s

surface elevation[m]D

rau

pn

er tim

ese

ries a

t 01

−Ja

n−

19

95

15

:20

:00

− e

vo

lve

d to

−0

m

Orig

ina

lL

inX

−0

mze

tab

lze

tab

n

26

22

63

26

42

65

26

65

10

15

20

X =

−0

m

Sim

ulationoftheD

raupner‘1995

New

Yearw

ave’time

series‘evolved’atdifferentdistances

(seetitlefor

distance)fromoriginalobservation

site.The

originalDraupnertim

eseriesis

the

thickred

line.The

dottedline(‘Linx’)

shows

resultsfromthe

linearmethodofM

agnusson,

Donelan&

Drennan

1999.The

blacklines

areresultsprovided

byK

arstenTrulsen,w

herethe

fulllines

(‘zetabn’)showthe

resultsfromnonlinearcalculations,and

dottedlines(‘zetabl’)

showthe

resultsfromthe

samecalculationsw

ithoutnonlinearterms.

21

02

20

23

02

40

25

02

60

27

02

80

29

03

00

31

0

−1

0

−5 0 5

10

15

20

se

co

nd

s

surface elevation[m]D

rau

pn

er tim

ese

ries a

t 01

−Ja

n−

19

95

15

:20

:00

− e

vo

lve

d to

−2

0m

Orig

ina

lL

inX

−2

0m

ze

tab

lze

tab

n

26

22

63

26

42

65

26

65

10

15

20

X =

−2

0 m

Sim

ulationoftheD

raupner‘1995

New

Yearw

ave’time

series‘evolved’atdifferentdistances

(seetitlefor

distance)fromoriginalobservation

site.The

originalDraupnertim

eseriesis

the

thickred

line.The

dottedline(‘Linx’)

shows

resultsfromthe

linearmethodofM

agnusson,

Donelan&

Drennan

1999.The

blacklines

areresultsprovided

byK

arstenTrulsen,w

herethe

fulllines

(‘zetabn’)showthe

resultsfromnonlinearcalculations,and

dottedlines(‘zetabl’)

showthe

resultsfromthe

samecalculationsw

ithoutnonlinearterms.

21

02

20

23

02

40

25

02

60

27

02

80

29

03

00

31

0

−1

0

−5 0 5

10

15

20

se

co

nd

s

surface elevation[m]

Dra

up

ne

r time

se

ries a

t 01

−Ja

n−

19

95

15

:20

:00

− e

vo

lve

d to

−2

00

m

Orig

ina

lL

inX

−2

00

mze

tab

lze

tab

n

Sim

ulationoftheD

raupner‘1995

New

Yearw

ave’time

series‘evolved’atdifferentdistances

(seetitlefor

distance)fromoriginalobservation

site.The

originalDraupnertim

eseriesis

the

thickred

line.The

dottedline(‘Linx’)

shows

resultsfromthe

linearmethodofM

agnusson,

Donelan&

Drennan

1999.The

blacklines

areresultsprovided

byK

arstenTrulsen,w

herethe

fulllines

(‘zetabn’)showthe

resultsfromnonlinearcalculations,and

dottedlines(‘zetabl’)

showthe

resultsfromthe

samecalculationsw

ithoutnonlinearterms.

-8 -6 -4 -2 0 2 4 6 8

01

00

02

00

03

00

04

00

05

00

06

00

07

00

08

00

0

time

se

ries_

hz1

19

98

-10

-25

15

:40

:00

+0

00

0 - 1

99

8-1

0-2

5 1

7:4

0:0

0+

00

00

Ekofisk

Waveridertim

eseries,1998

October

25,15:40–17:40UT

C.

-8 -6 -4 -2 0 2 4 6 8

10

12

01

00

02

00

03

00

04

00

05

00

06

00

07

00

08

00

0

time

se

ries_

hz2

19

98

-10

-25

15

:40

:00

+0

00

0 - 1

99

8-1

0-2

5 1

7:4

0:0

0+

00

00

Ekofisk

lasertime

series,Flare

North,1998

October25,15:40–17:40U

TC

.

-8 -6 -4 -2 0 2 4 6 8

10

12

01

00

02

00

03

00

04

00

05

00

06

00

07

00

08

00

0

m

time

/ se

co

nd

s

time

se

ries_

hz3

19

98

-10

-25

15

:40

:00

+0

00

0 - 1

99

8-1

0-2

5 1

7:4

0:0

0+

00

00

Ekofisk

lasertime

series,Flare

South,1998

October25,15:40–17:40U

TC

.

-8 -6 -4 -2 0 2 4 6 8

10

122

50

02

51

02

52

02

53

02

54

02

55

02

56

02

57

02

58

02

59

02

60

02

61

0

metres�

time

/ s

time

se

ries_

hz2

19

98

-10

-25

15

:40

:00

+0

00

0 - 1

99

8-1

0-2

5 1

7:4

0:0

0+

00

00

Ekofisk

lasertime

series,Flare

North,1998

October25,15:40–17:40U

TC

,partoftime

series.

-8 -6 -4 -2 0 2 4 6 8

10

122

58

02

58

52

59

02

59

52

60

02

60

5

metres�

time

/ s

time

se

ries_

hz2

19

98

-10

-25

15

:40

:00

+0

00

0 - 1

99

8-1

0-2

5 1

7:4

0:0

0+

00

00

Ekofisk

lasertime

series,Flare

North,1998

October25,15:40–17:40U

TC

,partoftime

series.

-8 -6 -4 -2 0 2 4 6 8

10

121

90

01

92

01

94

01

96

01

98

02

00

02

02

02

04

02

06

02

08

02

10

02

12

0

metres�

time

/ s

time

se

ries_

hz3

19

98

-10

-25

15

:40

:00

+0

00

0 - 1

99

8-1

0-2

5 1

7:4

0:0

0+

00

00

Ekofisk

lasertime

series,Flare

South,1998

October25,15:40–17:40U

TC

,partoftime

series.

-8 -6 -4 -2 0 2 4 6 8

10

122

03

22

03

42

03

62

03

82

04

02

04

22

04

42

04

62

04

82

05

02

05

2

metres�

time

/ s

time

se

ries_

hz3

19

98

-10

-25

15

:40

:00

+0

00

0 - 1

99

8-1

0-2

5 1

7:4

0:0

0+

00

00

Ekofisk

lasertime

series,Flare

South,1998

October25,15:40–17:40U

TC

,partoftime

series.

Belgian

locations:

010

2030

4050

6070

8090

0

100

200

300

400

Hs (cm)

010

2030

4050

6070

8090

0

0.5 1

1.5 2

2.5 3

1/10/1997−31/12/1997 GB

1 station AK

Z

Hmax / HsWave

recordatthe

off-shorestationAkkaert(A

KZ

)

010

2030

4050

6070

8090

0 50

100

150

200

250

300

350

Hs (cm)

010

2030

4050

6070

8090

0

0.5 1

1.5 2

2.5 3

1/10/1997−31/12/1997 GB

1 station A2B

Hmax / Hs

Wave

recordatthe

near-shorestationA

2B

0100

200300

400500

600700

800900

−2.5

−2

−1.5

−1

−0.5 0

0.5 1

1.5 2

2.5

time [s]

elevation [m]

Typicalrecord(10

Hz)from

aW

averiderbuoy

closetoZ

eebruggeharbour.

262264

266268

270272

274276

−2

−1.5

−1

−0.5 0

0.5 1

1.5 2

2.5

Wave 49

Tim

e (sec)

Highestw

avein

theprevious

record(Z

eebruggeharbour).

Ekofisk:

cumulative

andextrem

evalue

statistics

Spectralanalysis

andanalysis

ofindividualwaves

Analysedusing

proceduresdescribedby

Rijksw

aterstaat,with

slightmodifications.

Rejectionofpoor-quality

datapoints.

This

couldnotbe

donejustbyrejecting

pointswhich

deviatedby

more

thanfourtim

esthe

standarddeviation

ofthesea-surfacedisplacem

entfromthe

meanw

aterlevel,sincethe

purposeofMA

XW

AV

Eis

toinvestigate

extremew

aves.

Itw

asfoundnecessary

toperform

visualqualitycontrolon

every20-m

inutedatarecord,to

remove

dataerrors.

Sincethe

purposeoftheM

AX

WA

VE

projectisto

investigateextrem

ew

aves,computer

processingfordataquality

controlisregarded

asinadequate,sincetherewould

bea

riskofrem

ovinggenuineextrem

eeventsin

additionto

data‘spikes’.

The

time-consum

ingnatureofthevisualinspectionhasm

eantthatonlyspecifiedstorm

periodshavebeencom

pletelychecked.

Subtractionofthe

meanw

aterlevel

Spectralanalysisinvolving

performingfastFouriertransform

son200-secondsub-series

ofthedata,w

ithcosinetaperingon

theouter10

percentofthesub-series.

Splitting

upthe

dataintoindividualw

avesusingthe

zerodow

n-crossingcriterion.Short

wavesw

itha

periodoflessthan

onesecondare

addedtothe

previousw

aveorto

thenext

wave.

The

dataarestoredin

tablesinan

SQ

Ldatabase

usingfreely

availabledatabasesoftw

are

(PostgreS

QL:http://w

ww.postgresql.org/).

—T

ime

stampin

ISO

format(Y

YY

Y-M

M-D

DH

H:M

M:S

S.S

S+

hhmm

)

—R

ecordinginterval

—N

o.ofsamples(usually2395)

—Table

ofindividualwave

heights

—Table

ofindividualwave

periods

—Totalnum

berofwaves

Hm

axH

eightofthehighestw

ave

H1� 50

Averageheightofthe

highest1/50ofthe

waves

H1� 10

Averageheightofthe

highest1/10ofthe

waves

H1� 3

Significantw

aveheight(I):

averageheightofthehighest1/3

ofthew

aves

Tm

axP

eriodofthelongestw

ave

T1� 3

Averageofthe

periodofthe

longest1/3ofthe

periods

TH

1 3A

verageoftheperiod

ofthehighest1/3

ofthew

aves

HC

MM

aximum

crestheight

HT

DLeveloflow

esttroughdepth

10 Czz

Energy

(variance)densityspectrum

at0.01Hz

intervals

Hm

0S

ignificantwave

height(II)from

spectrum(4

timesthe

standarddeviation)

Tm

02S

pectrallyaveraged

wave

period

fpF

requencyw

here10 C

zzhasits

maxim

um

The

cumulative

distribution(C

DF

)oftheindividualw

aveheights,crestheightsand

troughdepthsfor

specificperiodswere

alsodeterm

inedby

sortingthem

intoincreasing

order

12

3N

.Ifiis

theposition

inthis

ordering,thecorresponding

valueofthe

estimated

CD

Fis

givenby

Fi

N1

.The

CD

Fvaluesare

fittedto

Rayleighdistributions

FR

x1

expx

22β

2(1)

byestim

atingtheparam

eterβto

be∑

i x2i

2N

11� 2.

The

datawere

alsofitted

toto

Weibulldistributions

FW

x1

expx

αγ

(2)

byleast-squaresregressionoflog

log1

Fagainstlog

x,usingthe

upperhalfofthedata

sample(i.e.F

05).

Alternative

methodsinclude

them

aximum

likelihood

methodand

them

ethodofmom

ents,but

theleast-squaresfitting

hastheadvantage

ofsimplicity,and

theupperhalfofthe

distribution

waschosenasw

eare

mainly

lookingfor

theupperextrem

evalues.

Fitting

thew

holesam

plebyleastsquares

regressionisnotso

good,asthesm

allestvaluesof

wave

heightetc.were

foundto

causeanoticeable

adverse(negative)biasin

theparam

eterγ.

0 1 2 3 4 5 6 7

01

23

45

67

8

-log(1-F(x))

Cre

st h

eig

ht / m

time

se

ries_

hz1

19

98

-10

-25

15

:40

:00

+0

00

0 - 1

99

8-1

0-2

5 1

7:4

0:0

0+

00

00

Ra

yle

igh

dis

t.

Cum

ulativedistribution,E

kofiskw

averider,1998-10-2515:40–17:40

0 2 4 6 8

10

12

02

46

81

01

2

-log(1-F(x))

Cre

st h

eig

ht / m

time

se

ries_

hz2

19

98

-10

-25

15

:40

:00

+0

00

0 - 1

99

8-1

0-2

5 1

7:4

0:0

0+

00

00

Ra

yle

igh

dis

t.

Cum

ulativedistribution,E

kofisklaserF

lareN

orth,1998-10-2515:40–17:40

0.1 1

110

-log(1-F(x))

metres

timeseries_hz1 1998-10-25 15:00:00+

0000-1998-10-25 17:00:00+0000 (769 w

aves)

Crests: . . . . . . . observed

Weibull, alpha =

2.961 m, gam

ma =

1.827R

ayleigh, parameter =

2.129T

roughs: . . . . . . . . observedW

eibull, alpha = 2.958 m

, gamm

a = 1.978

Rayleigh, param

eter = 2.081

Wave heights: . . . . . . . observed

Weibull, alpha =

5.835 m, gam

ma =

1.977R

ayleigh, parameter =

4.123

Ekofisk

waverider,1998

October25,15:00–17:00U

TC

,cumulative

distributions.

0.1 1

110

-log(1-F(x))

metres

timeseries_hz2 1998-10-25 15:00:00+

0000-1998-10-25 17:00:00+0000 (754 w

aves)

Crests: . . . . . . . observed

Weibull, alpha =

3.493 m, gam

ma =

1.804R

ayleigh, parameter =

2.541T

roughs: . . . . . . . . observedW

eibull, alpha = 3.214 m

, gamm

a = 2.157

Rayleigh, param

eter = 2.220

Wave heights: . . . . . . . observed

Weibull, alpha =

6.593 m, gam

ma =

2.026R

ayleigh, parameter =

4.663

Ekofisk

laserFlare

North,1998

October25,15:00–17:00U

TC

,cumulative

distributions.

0.1 1

110

-log(1-F(x))

metres

timeseries_hz3 1998-10-25 15:00:00+

0000-1998-10-25 17:00:00+0000 (773 w

aves)

Crests: . . . . . . . observed

Weibull, alpha =

3.100 m, gam

ma =

1.759R

ayleigh, parameter =

2.256T

roughs: . . . . . . . . observedW

eibull, alpha = 2.802 m

, gamm

a = 2.003

Rayleigh, param

eter = 1.966

Wave heights: . . . . . . . observed

Weibull, alpha =

5.886 m, gam

ma =

2.035R

ayleigh, parameter =

4.125

Ekofisk

laserFlare

South,1998

October25,15:00–17:00U

TC

,cumulative

distributions.

-6 -4 -2 0 2 4 6 8

01

00

02

00

03

00

04

00

05

00

06

00

07

00

08

00

0

m�

time

/ se

co

nd

s

time

se

ries_

hz3

19

98

-10

-25

11

:00

:00

+0

00

0-1

99

8-1

0-2

5 1

3:0

0:0

0+

00

00

(90

5 w

ave

s)

Ekofisk

laser,Flare

South,1998

October25,11:00–13:00U

TC

.

Ekofisk

laser,Flare

South,1998

October25,11:00–13:00U

TC

.Data

atlocationofsensor.

Ekofisk

laser,Flare

South,1998

October

25,11:00–13:00UT

C.Linearpropagation

to

450mat5

mintervals.

0.1 1

11

0

-log(1-F(x))

me

tres

time

se

ries_

dra

up

ne

r 19

95

-01

-01

15

:20

:00

+0

00

0-1

99

5-0

1-0

1 1

5:4

0:0

0+

00

00

(84

wa

ve

s)

Cre

sts

: . . . . . . . ob

se

rve

dW

eib

ull, a

lph

a =

4.1

90

m, g

am

ma

= 1

.54

9R

ayle

igh

, pa

ram

ete

r = 3

.25

0T

rou

gh

s: . . . . . . . . o

bse

rve

dW

eib

ull, a

lph

a =

3.7

47

m, g

am

ma

= 1

.93

2R

ayle

igh

, pa

ram

ete

r = 2

.59

0W

ave

he

igh

ts: . . . . . . . o

bse

rve

dW

eib

ull, a

lph

a =

7.9

40

m, g

am

ma

= 1

.89

3R

ayle

igh

, pa

ram

ete

r = 5

.66

4

Draupner,1995

January1storm

,15:20U

TC

,cumulative

distributionofw

aveheight,crest

heightandtrough

depth.

Conclusions

1.T

heextrem

ewavesobserved

donothave

anunusualshape:they

tendto

havesharpcrestsand

roundtroughs,w

itha

pronouncedcrest:trough

asymm

etry.

2.T

hestatistics

ofindividualwave

height,crestheight,andtrough

depth,aregenerallyin

agreementw

ithR

ayleighorWeibulldistributions,although

particularlyextremeeventssuchasthe

Draupner1995

New

Yearw

aveare

highlyunusualaccordingto

thenorm

ally-acceptedGaussian

andnon-G

aussianstatisticalm

odels.

3.T

hew

aveletanalysismethodcan

beusefulfor

detecting‘groupiness’andindividuallarge

waves.

4.A

nalysisofthe

useofthequasi-Lagrangiancorrectionm

ethodonW

averidertime

seriesforthe

Stenfjellcaseindicatesthatthis

methodis

insufficienttotransform

observationaldataintotim

eseriesw

hoseextrem

evaluesofcrestheightand

troughdepthare

equivalenttoobservations

usingE

ulerianmeasurem

enttechniques.

5.‘P

ropagation’ofwave

observationsto

hypotheticallocationsdifferent

fromw

herethew

avem

easurementsare

made,in

ordertoevaluatethe

riskofextrem

ewaveslargerthan

thoseactuallyobserved,may

bepossible,but

furtherstudyisrequiredto

determinethe

statisticalreliabilityofthe

linearand

nonlinearmethodsproposed.