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Stochastic model for joint wave and wind loadson offshore structures

Ove Ditlevsen

Technical University of Denmark, Lyngby, Denmark

Abstract

The stochastic wave load environment of offshore structures is of such a complicated nature that any

engineering analysis requires extensive simplifications. This concerns both the transformation of the wave

field velocities and accelerations to forces on the structure and the probabilistic description of the wave

field itself. In this paper the last issue is in focus. The modeling follows the traditional structure of sub-

dividing the time development of the wind driven wave process into sea states within each of which the

wave process is modeled as a stationary process. The wave process of each sea state is modeled as an affi-

nity in height and time of a Gaussian process defined by a normalized dimensionless spectrum of Pierson-

Moskowitz type. The affinity factors are the so-called significant wave height Hs and the characteristic zero

upcrossing time Tz. Based on measured data of (Hs, Tz) from the North Sea a well fitting joint distributionof (Hs,Tz) is obtained as a so-called Nataf model. Since the wave field is wind driven, there is a correlation

between the time averaged wind velocity pressure Q and the characteristic wave height in the stationary

situation. Using the Poisson process model to concentrate on those load events that are of importance for

the evaluation of the safety of the structure, that is, events with Q larger than some threshold q0, available

information about the wind velocity pressure distribution in high wind situations can be used to formulate

a Nataf model for the joint conditional distribution of (Hs, Tz, Q) given that Q > q0. The distribution of the

largest wave height during a sea state is of interest for designing the free space between the sea level and the

top side. An approximation to this distribution is well known for a Gaussian process and by integration

over all sea states given Q > q0, the distribution is obtained that is relevant for the free space design.

However, for the forces on the members of the structure also the wave period is essential. Within the linear

wave theory (Airy waves) the drag term in the Morison force formula increases by the square of the ratio

between the wave height and the wave length, and the mass force term increases proportional to the ratioof the wave height and the square of the period. For a strongly narrow band Gaussian process Longuet-

Higgins has derived a joint distribution of the height and the period. However, simulations show that the

PiersonMoskowitz spectrum (or any other standard spectrum for wind driven sea waves of similar

bandwidth such as the JONSWAP spectrum) does not provide a sufficiently narrow banded process for the

distribution of Longuet-Higgins to make a good fit. Surprisingly it turns out that the random time

L between two consecutive 0-upcrossings and the random wave height H observed between the two

0167-4730/02/$ - see front matter # 2002 Published by Elsevier Science Ltd.P I I : S 0 1 6 7 - 4 7 3 0 ( 0 2 ) 0 0 0 2 2 - X

Structural Safety 24 (2002) 139163 www.elsevier.com/locate/strusafe

E-mail address: [email protected] (O. Ditlevsen).

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0-upcrossings behave such that L and the ratio H/L are practically uncorrelated and both normally dis-

tributed except for clipping the negative tails. This result is of global nature and is therefore very difficult if

not impossible to obtain by analytical mathematical reasoning. Finally, by combining all the derived dis-

tributional models into a Rosenblatt transformation, a first order reliability analysis of a tubular offshoreplatform can be made with respect to static pushover. Correction for non-linear wave theory can be taken into

account crudely by using the deterministic 5th order Stokes wave in the limit state formulation. A dynamic

analysis will be more complicated, of course, but the provided distributional information and the demon-

strated modeling principles are judged as generally applicable.# 2002 Published by Elsevier Science Ltd.

Keywords: Extreme wind driven sea waves; Local maxima and period properties of Gaussian process; Nataf model for

wave and wind data; Offshore structure loads; Sea wave stochastics during wind storm; Wave and wind loads

1. Introduction

Loads on offshore structures are dominated by natural forces that are only describable by theirstatistical properties. Due to random changes of the wind velocity and its direction the typical

wave heights and periods changes randomly by time. However, these changes are sufficiently slowthat the concept of sea state makes sense. A sea state is characterized as a wave situation that isapproximately constant during some time interval of duration 12 h, say, in the sense that a

moving time window average of the wave height and the wave period as defined below are approxi-mately constant during the time interval. The time window for averaging could be 10 min, say.

A sea state is traditionally characterized by the so-called significant wave height Hs and thecharacteristic zero-crossing period Tz. The definition of Hs is based on a calibration to the visual

classification of the waves by experienced ship captains. This characterization has turned out to

fit reasonably well with the definition of the significant wave height Hs to be the average height ofthe upper third of the size ordered sample of all wave heights within the time interval of theconsidered sea state. The concept of characteristic zero-crossing period Tz is a generalization of

the period of a pure sinusoidal wave. The real sea surface may be modeled as made up of asuperposition of a large number of sinusoidal waves with random heights and periods. At anygiven position the resulting surface necessarily moves up above and down below the level of the

undisturbed surface (denoted as the zero level). This movement is not periodic for example in thesense that there is a constant time interval between an upcrossing and the next upcrossing of thezero level. The time intervals vary randomly around an average value denoted Tz and measured

during the considered time chosen for a sea state duration. For technical evaluations the sub-

division into sea states is convenient even though the real situation is a gradual and continuouschange of the character of the sea surface geometry.

Table A1 in the Appendix shows 1 year of observations of ( Hs, Tz) allocated to the center of 0.3m length intervals and 0.5 s time intervals. The observations are made at a position in the

Northern North Sea. These data will be used in the following as an example data base for for-mulating probability distribution models aimed at reliability evaluations of offshore structures.The data are for a given location, and data from other locations will in general differ from these

data. However, the probability models may be more generally applicable in the sense thatapplications at other locations may only require a calibration of the distribution parameters, and

not a formulation of an entirely different distribution model.

140 O. Ditlevsen / Structural Safety 24 (2002) 139163

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There are two quite different wave loading situations of importance for the safety of an offshorestructure. For the ultimate failure situation the severe weather situation with large wind pressures

and large waves are of primary interest. Another important situation is the slowly increasing metalfatigue and the appearance of growing cracks due to the constant exposure to wave forces of everyday size. This deteoriation process may finally lead to an ultimate failure if the process is not keptunder control by inspection, maintenance and repair. The probabilistic tools to deal with the two

situations are quite different. The reliability analysis relevant in the ultimate loading situation is by

and large based on the theory of vectors of random variables and their joint probability distribu-tions, while the long term wave force exposure requires a random process description of the waveload. In this paper, the theory for treating the ultimate failure situation will be described.

It is not intended to make comparisons with current models and practice used in the offshoreindustry, but solely to provide some partly new ideas, tools and results that may be of interest foruniversity people and reliability engineers working with offshore structure problems.

2. Sea state distribution

By summing the numbers in the rows of Table A1, the sample ofHs is obtained. Fig. 1 (left) showsthe cumulative distribution of the Hs-data together with the fitted distribution function model

FHs x 2x

1; x > 1

0:7 m; 2:5 m 2

that is, a truncation of the normal distribution function x

with truncation point at its mean

=.

Fig. 1. Fitted truncated normal distribution functions (1) and (3) of significant wave height Hs and characteristic zero-

crossing period Tz compared to the corresponding empirical distribution functions of one year of data from the

Northern North Sea [1]. The data pairs of (Hs,Tz) are given in Table A1 and are shown in the form of a 3D histogram

in Fig. 2 together with the corresponding fitted two-dimensional density (Nataf density type).

O. Ditlevsen / Structural Safety 24 (2002) 139163 141

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By summing the numbers in the columns of Table A1, the sample of Tz is obtained. Fig. 1(right) shows the cumulative distribution of the Tz-data together with the fitted distribution

function model

FTz t log t

3

log 6:8 s ; 0:17 4that is, a lognormal distribution function with mean and standard deviation of log Tz. It isseen that both the marginal distribution models of (Hs, Tz) fit surprisingly well to the empirical

distributions of the data. Next step is to formulate a model for the joint distribution of the tworandom variables Hs and Tz. The straight forward way to obtain such a model is to set up the

marginal transformations by which each of the marginal distributions of the data transform into

Fig. 2. The data pairs of (Hs,Tz) given in Table A1 shown in the form of a 3D histogram together with the corre-

sponding fitted two-dimensional density (Nataf density type) (9).


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normal distributions, and then apply these marginal transformations to the data. The pairs of

transformed data with marginal normal distributions are next assumed to have a bivariate nor-mal distribution with correlation coefficient estimated from the transformed data. Finally back-

transformation of the obtained bivariate normal density function gives a bivariate density modelfor the original data pairs. This is the so-called Nataf model for formulating joint density of vec-tor data on the basis of given marginal distributions of the single components of the random

vector [2]. Clearly, this is only one of infinitely many possibilities of formulating a joint distribu-tion with given marginal distributions. However, the experience shows that the Nataf modelbesides being easy to implement is often giving a reasonably well fitting joint distribution model.

In the present case the transformations are

X 1 2 max 0:01; Hsf g

1

!5

Y logTz

6

where a small correction from Hs to max{ +0.01, Hs} is introduced to secure that the observa-tions from the first row in Table A1 are taken into account without causing the transformation fromHs to Xto be undefined. Each of the random variables Xand Yare standard normal. The correlationcoefficient =Cov[X, Y] is estimated as % 1

n20i0

29j0nijxiyj % 0:70, with n=20i0 29j0ij 975,

where (xi, yj) are the transformed data points, and nij is the number of observations of (xi, yj).Then the conditional distribution of Y given X=x is normal with mean E[YjX x]=x andstandard deviation ffiffiffiffiffiffiffiffiffiffiffiffiffi1

2p . ThusFY y

X x y xffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

p

7

or

FTz tjHs h logt

1 2 h

1

!ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p

0BB@

1CCA 8

so that the joint density of (Hs, Tz) becomes

fHs;Tz h; t fTz tjHs h fHs h

2

tffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p

logt

1 2 h

1

!ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p

0BB@

1CCA h

; h > ; t > 0

9

This density function is shown in Fig. 2 together with histogram plots of the data.


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3. Wind pressure distribution

It is consistent with the indications in the Danish Code: Loads for the Design of Structures

[3] to adopt a marked Poisson process model for the occurrence and size of extreme wind velocitypressures in open sea. In fact, it is assumed that the occurrence of storm events can be modeled asevents in a homogeneous Poisson process of intensity =0.02 y1. A wind storm event is defined

as the occurrence of a 10 min wind speed average of more than 27 m/s in the height 10 m abovethe ground in open landscape of so-called class 0.05 (farm country). By calibration to the condi-tions at open sea, the storm event may be defined as the occurrence of a 1 min velocity pressure

average of more than

q0 z 5log 100z 2N=m2; z in m 10

at height z above the sea level. The velocity pressure q0(10 m) corresponds to a 1 min velocity

average of 43.2 m/s, or a 10 min velocity average of about 43.2/1.11=38.9 m/s.Let N(t) denote a homogeneous Poisson process of intensity . Since P[N(t)=0]=exp(t),

the waiting time to the first occurrence of the event is exponentially distributed with para-

meter . Then it follows from the well known way of construction of the Poisson processthat the time distance between two consecutive events is exponentially distributed with para-meter .

If the Poisson process events either have or do not have a given property A, and if this property

is assigned to any of the events independently from event to event with a given probability P(A),

then an event with the property A obviously occurs within the time interval from t to t+t withthe probability P(A)t+o(t). Those points of the Poisson process that are marked with theproperty A thus make up an outcome of a homogeneous Poisson process with intensity P(A).

This is expressed by saying that the Poisson process of marked points is obtained by thinning ofthe original Poisson process with the thinning probability P(A).

The marks on the points in a Poisson process may be independent outcomes of a random

variable or random vector. If the points are wind storm events, then the marks may be the largestwind pressure at a given location during the wind storm, or it may be the largest wave height atthe location, etc.

An important design problem is to determine the distance between the still water surface and

the topside of an offshore platform such that there is a given probability 1pN that all wave topspass under the topside within an N year time period.With reference to the exponential waiting time distribution the probability p that a wave hits

the topside in a single storm is obviously related to pN by

p 1N

log 1 pN %pN

N11

where the last approximation is valid for pN 1.


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4. Wind pressure and significant wave height distribution

Severe sea states with large waves are generally accompanied by heavy wind. The waves are

caused by the wind over some sea area that may be remote from the location of an offshoreplatform. Therefore there may be severe waves without much wind at the location or there maybe heavy winds before the sea has raised to severe waves. Nevertheless, under stationary condi-

tions there is correlation between the significant wave height and the wind pressure at the samelocation. For the reliability analysis of the structure it is important to have a model for the jointdistribution of the random mean wind pressure Q at a standard reference height of z=10 m, and

the sea state random variables (Hs,Tz) during a wind storm taken to be of duration as the seastate. Detailed information on this issue has not been available to the author at the time of writ-ing this text (and is possibly not available in general). Therefore an anticipatory modeling is used

to illustrate the technique of model formulation in this type of problem under consideration of

available information. It is consistent with the information given in the wind load code part of(DS410 1982) to assume that the conditional distribution of the velocity pressure Q given that Q5q0 is exponential asymptotically as q0 ! 1:

P Q4 qQ5 q0 1 el qq0 ; q5 q0 12

with l=0.0075 m2/N that together with (10) give lq0=9.0 for z=10 m. Using the probabilityP(Q5 q

Q 5q0) as thinning probability on the wind storm Poisson process of intensity , it isdirectly seen that the maximal wind velocity pressure during a time period of N years has the

distribution function

P

maxNyears

Q4 qjQ5 q0

exp Nel qq0 ; q5 q0 13This double exponential distribution function is of a type called the Gumbel distribution type

and is the distribution for yearly extreme wind pressures given in the wind load code [3].

The directional properties of the wind velocities are not treated on the modeling level chosen inthis text. For offshore structures the wind and the waves may be assumed to have a coincidingdirection that is most severe for the structure. If more detailed information is available for the

actual site of the structure, the modeling may be refined with respect to the directional informa-

tion by extending the set of random variables, but otherwise just following straight forward gen-eralizations of the probabilistic methods explained herein.

The Nataf distribution modeling technique applied to obtain the probability density (9) of (Hs, Tz)can be applied to obtain a probability density model for the triple (Hs, Tz, Q) such that the dis-

tribution of (Hs, Tz) is preserved, and such that the marginal distribution of Q is consistent withthe conditional distribution (12). Since the distribution of (Hs, Tz, Q) is not conditional on theevent Q >q0, it is necessary to know the unconditional distribution of Q to apply the Nataf

technique. It is obviously consistent with (12) to adopt the unconditional exponential distributionP(Q > q)=elq, q > 0. This distribution is not necessarily a good fit to data for the low windpressures, but for the structural reliability assessments it is essentially the upper tail that matters


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under the assumption that the larger waves come together with the larger wind pressures. The

transformation formulas (5) and (6) hereafter extend to the three marginal transformations

X 1 2 Hs

1 !; Hs > 14Y logTz

; Tz > 0 15

Z 1 1 elQ ; Q > 0 16where the correction from Hs to max{ +0.01, Hs} in the transformation (5) has been removed as

irrelevant because no measurement data for Hs will be used here as input to the transformation.

The estimated marginal distribution parameter values are =0.7 m, =2.5 m, =log (6.8 s), =0.17, and l=0.0075 m2/N. Moreover the correlation coefficient between the image variables X

and Y is estimated to XY=0.70.

According to the Nataf model the joint distribution of the three image variables X,Y,Z is takento be normal. For complete determination of this normal distribution the correlation coefficientsXZ and YZ need to be obtained. Data information may exist in the offshore literature for esti-

mation of the value ofXZ, but for illustration of its influence the correlation coefficient =XZ iskept as a free parameter in this paper.

While there are obvious reasons to have dependency between Q and Hs, it is less obvious whe-ther there is dependency between Q and Tz. For example, it is not physically obvious whether Tzshould have a tendency to increase or to decrease as Q increases. It might be conjectured that the

random variation of Tz is caused by directional effects of the wind combined with the distantgeographic topography of the land as seen from the actual site. Possibly also the actual characterof the wind flow with respect to vorticity and turbulence has an influence on the wave period.

Since all the possible causes are mixed together at the present modeling level, it is hardly unrea-sonable to assume that YZ=0.

The conditional distribution function of Hs given Tz=t and Q=q then is

FHs hjTz t; Q q FX xjY y; Z z

x

E X

jY

y; Z

z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar X E XjY y; Z z p x

XYy

XZzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2XY 2XZp 17

x 1 2 h

1

!; h > 18

y logt

; t > 0 19

z 1 1 elq ; q > 0 20


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from which the conditional density fHs hjTz t; Q q is obtained by differentiation:

fHs hjTz t; Q q x XYy XZzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2XY 2XZ

p dx

dhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2XY 2XZ

p 21

dx

dh

2

h

x ; h > ; q > 0; t > 0 22

The conditional density

fX x

jY

y; Z > z0

1

z0 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2XYp x XYyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2XYp

XZ x XYy 1 2XY

z0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2XY

1 2XY 2XZ q

0B@

1CA 23

is obtained by elementary but lengthy manipulations within the mathematics of the normal dis-tribution. Thereafter

fHs hjTz

t; Q > q0

fX x

jY

y; Z > z0

dx

dh; h > ; t > 0

24

Some contour curves for the conditional joint density fHs;Tz h; tjQ > q0 fHs hjTz t; Q > q0 fTz t are shown in Fig. 3 for XZ=0.0, 0.4, 0.6, 0.7.

5. Extreme wave heights

During a sea state characterized by its duration , and by a given significant wave height Hs=h

and characteristic zero-crossing period Tz=t, the conditional distribution function of the max-

imal wave height Hmax can be modeled as

FHmax xjHs h; Tz t exp

texp 2 x

h

2 !& '; x5 0; h > 0; t > 0 25

This model is based on Gaussian process theory and is derived in Section 6. The assumption

that the sea elevation is a Gaussian process raises a question about the validity of (25) for seastates with large significant wave heights. However, since (25) is asymptotically valid for

decreasing significant wave height, the distribution may not be seriously wrong, except for


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Fig. 3. Contour curves for the conditional joint density ofHs and Tz given a storm defined by the wind pressure con-

dition Q > q0 for different correlation coefficients XZ between the Gaussian random variables X=1[FHs Hs

] and

Z=1[FQ(Q)].

Fig. 4. Upper tail of complementary distribution function of maximal wave height Hmax during a wind storm of

duration =2 h.


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extremely large significant wave heights where the physics sets a limit to how high a wave can be.

Here it will be assumed that (25) is applicable for engineering analysis.Given a wind storm defined by the event Q > q0, the complementary distribution function of

Hmax then becomes

1 FHmax xjQ > q0

1

10 1 exp

texp 2 x

h

2 !& ' fHs hjTz t; Q > q0 fTz t dtdh

26

Fig. 4 shows the upper tail of the complementary distribution function (26) for the sea stateduration =2 h, and the correlation coefficients XZ=0.0, 0.2, 0.4, 0.6, 0.7.

By the thinned Poisson process argument the probability that the maximal wave height during

N years is larger than x is then

1 exp N 1 FHmax xjQ > q0 % N 1 FHmax xjQ > q0 27

where the approximation is valid if N[1FHmax xjQ > q0 ] 1.For large wave heights the linear theory of Airy waves becomes considerably in error with

respect to the wave shape. A comparison between an Airy wave and a 5th order Stokes wave ofthe same height and wave length is made in Fig. 5. It is noted that the wave top is considerablyhigher up and, consequently, that the wave bottom is also higher up and more flat for the Stokeswave than for the Airy wave. For the example of the figure the height above the zero level is about 2/

3 of the total wave height. Obviously the wave top position has decisive importance for the design of

the free space between the sea surface and the top side of an offshore structure. It is also a generalproperty that the Stokes wave has a shorter period than the corresponding Airy wave.

Fig. 5. Wave profiles for Stokes and Airy wave.


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Clearly it is a problem how the non-linearity of the large waves should be taken into account,

and how it influences the probability distribution of the wave height when deviating from Gaus-sianity of the wave elevation process. The ad hoc assumption made here is that the height dis-

tribution is not affected seriously within the regime of relevant wave heights, but that theasymmetry with respect to through and crest of the Stokes wave should be taken into account bynot dividing the wave height in two equal heights above and below the zero level. The fact that

there is a physical limit of the wave steepness before the wave breaks gives reason to conjecturethat the real extreme wave heights during sufficiently long time are smaller than those occurringin a Gaussian process, because in the long run any level will be crossed by a Gaussian process.

Thus the ad hoc assumption is likely to be conservative rather than the opposite, but a proof ofthis conjecture is not known to the author. There might be a range of intermediate levels wherethe ad hoc assumption is to the unconservative side.

Remark. Besides the essential contribution from the wave elevation of the water surface also theelevation due to tides and storm surges should be taken into account if these phenomena occur to an

essential degree at the actual location. Obviously this adds a couple of extra random variables to the

problem. It is on the safe side to add this contribution independently to the free height, but in a more

detailed model it could be taken into account that high tides do not necessarily occur during violent

sea states while storm surges may have tendency to occur together with violent sea states.

Another contribution to add to the free height is the possible future random settlement of thesea bottom occurring because of the decrease of the oil pressure in the oil carrying geological

structures beneath the platform.

6. Crossing and local maxima theory

To relate the characteristic zero-crossing period Tz of a Gaussian sea surface elevation processX(t) to the covariance characteristics or, equivalently, the spectral characteristics of the process,but also of other reasons that will become clear shortly, the mean upcrossing rate of the Gaussian

process X(t) through the level u is considered. The random number Nofu-upcrossings in the timeinterval [0, 1] has the mean

E N ffiffiffiffiffiffiffiffiffiffi

l2

2l0s uffiffiffiffiffi

l0p 28This is the well known Rice formula [4,5] for the mean number ofu-upcrossings per time unit by

a stationary Gaussian process. It is seen that if l2=1, that is, if the process is not differentiable,then the mean number ofu-upcrossings is infinite. Due to the stationarity of the process the meannumber of u-downcrossings per time unit is equal to the mean number of u-upcrossings per time

unit, denoted as the u-upcrossing rate.In particular the zero-upcrossing rate is (0)=

ffiffiffiffiffiffiffiffiffiffiffil2=l0

p=2. The characteristic zero-crossing period

Tz is then defined as


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Tz 1 0 2

ffiffiffiffiffil0

l2

s29

and from the relation !0Tz=2, the characteristic angular frequency is defined as !0=ffiffiffiffiffiffiffiffiffiffiffil2=l0

p. It

should be noted, however, that the mean distance between to consecutive zero-upcrossings is notexactly equal to Tz. In fact, E[1/N] 6 1/E[N]. If the point process ofu-upcrossings should happento be a Poisson process, in which case the consecutive distances have independent exponentialdistributions, then E[1/N]=1/E[N]. This property is generally not valid for a stationary Gaussianprocess except asymptotically as u! 1.

It is worth noting that the amplitude ax of the pure harmonic vibration x(t)=ax cos !0t, andthe amplitude ax: of the corresponding velocity x

:t =ax!0sin!0t are related as ax:=!0ax. It is

then natural to define the characteristic angular frequency !0 of a general stationary random

process X(t) by the relation D[X:

(t)]=!0D[X(t)], or !0= ffiffiffiffiffiffiffiffiffiffiffil2=l0p , which coincides with the defini-tion given above for a stationary Gaussian process.Considering the time interval [t, t+h] with h#0 and X=X(t), the conditional density of thederivative X

:=X

:(t) given an upcrossing of level u at time t is obtained from

fX: x

: X < u; hX: > u X / fX: x

: P u hx: < X < uX: x:

1ffiffiffiffiffil2

p x:ffiffiffiffiffil2

p

uffiffiffiffiffil0

p

u hx:ffiffiffiffiffi

l0p

!

1ffiffiffiffiffil2p

x:ffiffiffiffiffil2p

uffiffiffiffiffil0p

hx:ffiffiffiffiffil0p o h

! / x:

l2 exp x: 2

2l2

; x

:

> 0

30

This is the well known result that the upcrossing velocity X:

(t) given the event Cu,t of a u-upcrossing at time t has a Rayleigh distribution of mean E[X

:t jCu;t]=

ffiffiffiffiffiffiffiffiffiffiffiffil2=2

p, that is, X

:(t)/

ffiffiffiffiffil2

pgiven Cu,t is distributed with standard Rayleigh density. Note that the conditioning event Cu,t has

zero probability. Note also that X:

(t) given that X:

(t) > 0 has the truncated normal density2(x

:=ffiffiffiffiffil2

p), x

:> 0. Thus the conditioning on a u-upcrossing at t changes the distribution drastically

by introducing the factor x:

on the truncated normal density.

The spectrum SX: (!) of X

:(t) is obtained from the spectrum SX(!) of X(t) by multiplication by

!

2

. Thus the upper tail of the spectrum is amplified by differentiation of the process and the highfrequency components become more dominant in the appearance of the sample functions of X: (t)

than in the appearance of the sample functions of X(t). This may be expressed by saying that X:

(t)has a less degree of regularity than X(t). Fig. 6 shows a piece of a sample function of a Gaussian

process of zero mean. The points of local minima are the 0-upcrossings of the derivative processX:

(t). The mean number of 0-upcrossings of the process X(t) itself and of its derivative processX:

(t) areffiffiffiffiffiffiffiffiffiffiffil2=l0

p=2 and


p=2, respectively, and the last is at least as large as the first. For a

highly regular process the two numbers are close to each other, while for a highly irregular pro-cess the last number is much larger than the first. This leads to the definition of the regularity

factor as the ratio of the two numbers:


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l2ffiffiffiffiffiffiffiffiffil0l4

p ; 044 1 31

The following elegant derivation of the probability distribution of the local maxima of a sta-tionary Gaussian process is based on an idea by the Swedish mathematician Jack de Mare , [6].

The stationary Gaussian process X(t) is obviously the sum of the conditional mean (linear

regression) E[X t X: (0), X (0)] and the corresponding non-stationary Gaussian residual process(t), which according to linear regression theory is stochastically independent of E[X t

X: (0), X (0)].Listing that E[X(t)]=0, Cov[X(t), X(0)]=l0(t), Cov[X(t), X

:(0)]=l0: (t), Cov[X(t), X (0)]= l0(t),

Var[X:

(0)]=l2, Var[X (0)]=l4, Cov[X:

(0), X (0)]=0, it is

X t E X t X: 0 ; X 0 t l0 : t t l2 0

0 l4

!1X:

0 X 0

" # t

: t l0l2

X:

0 t l0l4

X 0 t 32

with the residual variance

Var t l0 l20 : t t l2 0

0 l4

!1 : t t

!

l0 1 : t 2l0l2

t 2l0l4

! 33

In particular, :

(0)=0 and (0)=l2/l0, and therefore

Fig. 6. Illustration of regularity factor interpretation %N0/Nmin. The shown sample curve has Nmin=8 and N0=6within the considered time window.


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X 0 0 l0l4

X 0 0 l2l4

X 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var 0 p

U 34

where U is a standard Gaussian random variable independent of the Gaussian random variableX (0), and Var[(0)]=l0l22=l4. This expression is valid independent of what might be the ran-dom value of X

:(0). However, as it has been shown above, the distribution of X (0)/

ffiffiffiffiffil4

pchanges

from the Gaussian distribution to the standard Rayleigh distribution, if it is given that X:

(0)=0(an event of probability zero) and X (0) > 0 corresponding to a 0-downcrossing of the first deri-vative process X

:(t). Then X(0) is a local maximum of X(t). Consequently

Y X 0 ffiffiffiffiffil0

p Zffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p

U 35

where Z is standard Rayleigh, U is standard normal, and Z and U are mutually independent.Convolution integration hereafter gives the probability density of Y as

fY y ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p

yffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p

y ey2=2 yffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p

36

which is the Rice density function for the local extremes in a differentiable stationary Gaussianprocess of zero mean and unit variance [4,5]. Consistent with the expression (35) the standardRayleigh density is obtained for the regularity factor ! 1, while the standard normal distribu-tion is obtained for ! 0. Plots of the density function for a set of values of are shown in Fig. 7

Fig. 7. Left: Rice density functions for local maxima in stationary Gaussian process of zero mean, unit variance, and

regularity factor . Right: comparison between a simulated empirical distribution function and the Rice distribution

function. The process is defined by the PiersonMoskowitz spectrum (39) corresponding to Hs=4 m, Tz=1 s, and

truncation point at the angular frequency !=10p s1. The truncation causes the regularity factor to be %0.65 in steadof =0 for the untruncated spectrum. The truncated spectrum is normalized to have l0=1 corresponding to a process

of variance 1.


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(left). Fig. 7 (right) shows comparisons with simulation results.

Remark. The decomposition of the random local maxima into a sum of two independent random

variables, a Gaussian variable and a Rayleigh variable, was earlier discovered by Krenk using theknown fact that the local maxima have Rice distribution [7]. Krenk defines an upper and a lower

envelope to X(t) as X0A(t) where X0(t)=X(t)+X (t)/!2 (interpreted as an instantaneousequilibrium position) and A(t)=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX:

t =!h i2

X t =!2h i2r

(interpreted as an instantaneous amplitude

of a locally harmonic oscillation), with the frequency ! defined as the value that minimizes the var-

iance of the process X0(t), a choice that gives the value ! ffiffiffiffiffiffiffiffiffiffiffil4=l2

p[=2 times the mean number

of local maxima of X(t) per time unit]. Elementary calculations reveal that X0(t) and A(t) are

mutually independent with Gaussian and Rayleigh distribution, respectively. Thus Krenks upper

envelope has the same marginal distribution as the local maxima. From Krenks derivation it then

follows that the local maxima can be represented by (35).

The distribution function (25) can now be justified as a version of an approximate distribution

of the extreme of the stationary Gaussian process X(t) over the time derived from the Ricedistribution under the assumption of the occurrence of a large number of local maxima duringthe time .

It is seen that the upper tail of the Rice distribution (36) is of the form y ey2=2 asymptotically

as y!1. Thus the complementary distribution function ofYis asymptotically 1-FY(y)= ey2=2.The distribution function of Y represents the cumulative distribution of the sample of localmaxima of a single sample function from the Gaussian process X(t)/

ffiffiffiffiffil0

passuming that the pro-

cess is ergodic. Those sample values that are placed in the upper tail above a high level y are

spread along the sample function as rare points that reasonably can be assumed to occur aspoints in a homogeneous Poisson process. In fact, it can be shown that the y-upcrossings occurasymptotically as the points in a homogeneous Poisson process, and that each such upcrossing is

followed by a single local maximum [8,9]. The intensity of the Poisson process is obviously that

fraction 1FY(y) of the mean rateffiffiffiffiffiffiffiffiffiffiffil4=l2

p2

of local maxima occurrences that are local maximaabove level y. Thus the intensity is

1 FY y 12

ffiffiffiffiffil4

l2

s 1

2

ffiffiffiffiffil4

l2

sey

2=2 12

ffiffiffiffiffil2

l0

sey

2=2 37

asymptotically as y!1, that is, the Poisson intensity of local extremes of X(t)/ ffiffiffiffiffil0p above level yis asymptotically the 0-upcrossing rate (0) of X(t) thinned by the probability ey

2=2 correspond-ing to the standard Rayleigh distribution. The regularity factor is without influence on thisasymptotic result.

The distribution function (25) hereafter follows from the usual argument that the time to theoccurrence of the first event in the Poisson process has exponential distribution with parameter

equal to the intensity of the Poisson process.


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The random maximal wave height during time is Hmax=2Yffiffiffiffiffil0

pand

ffiffiffiffiffil0

p=Hs/4. Moreover,

Tz=2


p. Thus

FHmax xjHs h; Tz t exp 12ffiffiffiffiffil2l0

sey2=2

( ) 38from which the right side of (25) is obtained after substitution of


p2

1=t, and y=2x/h.

7. Joint distribution of period and wave height

It is also of considerable interest to know the joint distribution of the random time distance

between consecutive zero upcrossings, here called the random zero upcrossing period, and the

random wave height defined as the difference between the largest local maximum and the smallestlocal minimum of the process between the two upcrossing points. An exact mathematicalexpression for this distribution is not known, so recourse has to be taken to simulation.

The Gaussian wave elevation process considered in this paper is defined from an upper trun-cation of the so-called PiersonMoskowitz spectrum. The PiersonMoskowitz spectrum is basedpartly on physical considerations and partly on empirical observations of wave processes. The

basic dimensionless form of the spectrum is

S ! 22

2

!

5exp 1

2

!

4" #39

which defines a stationary Gaussian process of standard deviation 1 [i.e. l0=10 S ! d! 1] and

mean number of zero level upcrossings per time unit equal to 1 [i.e. l2=1

0!2S ! d! 2 2].

This basic process is mapped into the process of the sea state defined by ( Hs, Tz) simply by letting

the dimensionless time be t/Tz and the dimensionless process be X/ffiffiffiffiffil0

p=4X/Hs. Thus it is suffi-

cient to make simulation studies with the basic Gaussian process with the spectrum (39). It isnoted that l4 1. Therefore truncation at an upper frequency !u makes the fourth spectralmoment much dependent on the truncation frequency, implying that the regularity factor isstrongly dependent on the truncation frequency. However, the effect of truncating at a high fre-quency where the residual contribution is

1!u

S ! d! ( 1, is just that some additive fast fluctu-ating residual process of small variance is neglected as being without essential engineering

importance. In the simulations referred to herein !u =10p

is used giving1!u S ! d! % 0:005. Theresulting regularity factor is about 0.65. Obviously the regularity factor should not play an

essential role in any engineering application.The sample functions are approximated by the trigonometric polynomial

X t Xnk1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS !k !

pXkcos !kt Yksin !kt 40

where Xk and Yk for k=1,. . . n are mutually independent standard normal variables, and !k+1=!k+!, !=/100, !n=!u =10, and n=994 giving !1 %0.2.


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Fig. 8 (left) shows the empirical marginal distribution of the random zero upcrossing period L.It is seen that the normal distribution provides a good fit. Fig. 8 (right) shows the correspondingempirical distribution of the wave height H. It is seen that a Rayleigh distribution provides a

good fit. The estimates E[L]

%0.96 (seen to be close to Tz=1), D[L]

%0.36, and E[H] ffiffiffiffiffiffiffiffi2=p % 1:78are obtained.

Remark. By regression technique an approximate joint distribution of the time distance T~ between a

local maximum and the following local minimum and the vertical height H~ from the minimum to the

maximum is derived in [10] and referred in detail in [11]. In a collection of examples the calculated

joint distributions have a bimodal appearance with respect to T~. This appearance obviously reflects

that the regularity factors of the considered processes are less than one. There is no simple relation

between the complicated distribution of (T~, H~) and the joint distribution of the zero crossing period

L and the wave height H investigated by simulation herein.

A similar distribution of period and amplitude has been derived in [12,13] under an assumptionthat the spectrum gives a narrow band process with slowly varying amplitude process A(t) and

phase shift process (t). The obtained joint density does not depend on the regularity factor

(and thus on l4), but is derived from the representation X=Acos[!" t + ] where !" =l1/l2=point

of gravity of the spectrum. The details of the derivation is given in [14], p. 312. The Longuet-Higgins density of (R, T) and the corresponding marginal densities of R and T are

fR;T r; t 4ffiffiffi

2p

1 ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2p r2 er2 r

ffiffiffi2

pffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

p 1 1t

" #1

t2; r5 0; t5 0 41

Fig. 8. Left: empirical distribution function of the random zero upcrossing period L fitted by the normal distribution

of mean 0.96 and standard deviation 0.36. Right: empirical distribution functions of the wave height H defined as the

difference between the largest local maximum and the smallest local minimum of the sample curve between the two

upcrossing points. The distribution is fitted by the Rayleigh distribution of mean 1.78ffiffiffiffiffiffiffiffi

=2p % 2:23.


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fR r 41 r e

r2 ffiffiffi

2pffiffiffiffiffiffiffiffiffiffiffiffiffi1 2p r

; r5 0 42

fT t 1 t 1 2

t2 2 1 t 2 3=2; t5 0 43where the spectral width parameter =l1/

ffiffiffiffiffiffiffiffiffil0l2

pis 0.92 for the PiersonMoskowitz spectrum.

Contour curves for the density (41) are shown in Fig. 9 together with the simulated observations.The Longuet-Higgins distribution of T given by (43) deviates from the empirical distribution as

shown in Fig. 9 (Top right), in particular in the upper tail. As mentioned the Longuet-Higgins

Fig. 9. Top left: contour curves for the Longuet-Higgins density (41) placed on top of the scatter plot of 310 simulated

points (L, H). Top right: the marginal Longuet-Higgins distribution function of the period (right curve) compared to

the normal distribution function of the random zero upcrossing period L fitted to the simulated data as shown in

Fig. 8. Bottom: the marginal Longuet-Higgins distribution function of the half wave height (right curve) compared to

the Rayleigh distribution function of the random half wave height H/2 per period fitted to the simulated data as

shown in Fig. 8.


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density is based on an assumption of slowly varying amplitude and phase. This assumption islikely to give less good predictions of the tails of the distribution of the period.

The angular shape with vertex at the origin of the cloud of observations in Fig. 9 suggest that asimpler two-dimensional distribution may be obtained for the pair (S, L)=(H/L, L), whereS=H/L can be interpreted as the steepness (in the time domain) of the local wave. Indeed, thescatter plot of (S, L) turns out as in Fig. 10. The two random variables S and L are practically

uncorrelated, and the distribution ofS is as the distribution L very close to be normal (of course,

both clipped at zero). On basis of this observation it is particularly convenient if the bivariatedistribution can be modeled with reasonable accuracy as a bivariate normal distribution. Gra-phical tests for bivariate normality are shown in Fig. 11. The top right diagram shows the

empirical distributions of qUffiffiffiffiffiffiffiffiffiffiffiffiffi

1 q2p

W, q=0, 0.2, 0.4,. . .,1, where U=(LE[L])/D[L] andW=(SE[S])/D[S]. All empirical distribution functions seem to fit reasonably with the standardnormal distribution. The bottom diagram shows the empirical distribution of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU2 W2

pcom-

pared to the standard Rayleigh distribution. These graphical tests are indicators of bivariatenormality and independence of S and L. However, the actual bivariate sample of size 310 is notsufficiently large to state that for all applications the distribution is sufficiently well modeled by a

bivariate normal distribution. The tendency of clustering of the observation points in the lower

left corner and the upper left corner may indicate non-normality and statistical dependencebetween Sand L. This may not exclude the application of the bivariate normal distribution in the

range of large values ofS. In fact, the drag force term in Morisons formula is proportional to thesquared water particle velocity, and within the linear wave theory the particle velocities are

directly proportional to S. Therefore it is the large values of S that are relevant in pushoverreliability analysis of tubular offshore platforms. On the other hand, the mass force term inMorisons formula is proportional to the water particle accelerations that, in turn, are propor-

tional to S/L. Therefore small values of L combined with large values of S are relevant, that is,values in the lower right corner of the scatter plot shown in Fig. 12. These heuristic arguments

support that the possible deviations from normality are not critical for the reliability analysis.

Fig. 10. Left: scatter plot of 310 simulated outcomes of (H/L, L). Right: normal distribution fit to the empirical dis-

tribution of H/L. The mean is estimated to 2.25 and the standard deviation to 0.89.


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Thus, by the same argument as used for the derivation of the distribution of Hmax, the condi-

tional distribution of the maximal value of S during a given sea state becomes

FSmax xjHs h; Tz t exp

t V1S 1

4t

hSx

!& '44

Moreover

FL ljHs h; Tz t V1Ll

t 1

!45

Fig. 11. Top left: scatter plot of the normalized data shown in Fig. 10. Top right: empirical distributions of linear

combinations of the coordinates of the points in the scatter plot compared to the standard normal distribution func-

tion. Bottom: empirical distribution function of the distance from the origin in the scatter plot to the observation

points compared to the standard Rayleigh distribution.


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The distributions (44) and (45) are suggested to be used in the pushover reliability analysis even

though such an application may be outside the range of applicability of the Gaussian processtheory. However, as discussed at the end of Section 5, this may not be critical. At least the dis-tribution shape is correct asymptotically as the wave height decreases. Possibly some parametervalues may be corrected in the presence of data for the wave height distribution during sea states

with large significant wave heights. In any case, the reliability analysis, to be practicable, must be

simplified by not considering the detailed random variation of the surface elevation from zerolevel upcrossing to zero level upcrossing, but by replacing this variation by the deterministicshape of the 5th order Stokes wave, say.

Fig. 13 shows the correlation functions of the random sequence of zero crossing periods L andthe corresponding random sequence of wave heights H per period as they are observed con-secutively along the time axis. The appearance of the sequences as almost white noise supports

the Poisson process assumption on which the distributions (25) and (44) are based.

8. Pushover reliability analysis

The information embedded in the conditional distribution functions modeled above is the least

information needed on the load side for making an approximate reliability analysis with respectto pushover failure of an offshore platform subjected to wind and waves. For example, consider a

horizontal cut at the sea bottom with a coordinate system of origin vertically below the point ofgravity of the self weight masses of the structure and the one axis parallel to the waves. Then themoment M about this axis of all the static and dynamic forces transmitted through the cut is a

function M=(R, Smax, L) of the three random variables R=Q/q0, Smax, and L (neglectingmoments caused by live loads on the platform). For the design of the foundation or a reliability

analysis of a give foundation design, the upper tail of the probability distribution of the random

Fig. 12. Scatter plot of 310 simulated observations of (H/L2, L).


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moment M (or a similar response quantity) is of obvious interest. Given a wind storm defined by

the event Q > q0, the probability that M> m for any given m is

P R; Smax; L > mR > 1

r;s;l >m1

h1

t0fL ljx; h; t fSmax sjh; t fHs hjt; q fTz t fR rjR > 1 dh dt

dsdr46

Calculation of this integral by direct numerical integration requires large computer efforts. Anapproximate evaluation of the integral can be obtained by the well known First Order Reliability

Method (FORM). If needed, a FORM calculation may be supplemented by a Second Order

Fig. 13. Top left: sequence of zero crossing periods L (bottom curve) and the corresponding sequence of wave heights

H per period (top curve) as they are observed consecutively along the time axis. Top right and bottom: estimated

correlation coefficient functions for the two sequences after transformation to Gaussian sequences. The correlation

functions show that the sequences are close to pure white noise.


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Reliability Analysis (SORM) or by suitably designed simulation calculations [2]. The Rosenblatt

transformation based on the formulated distributions is

y1 FQ qQ > q0 y2 FTz t y3 FHs hjTz t; Q q y4 FSmax sjHs h; Tz t

47

while the limit state is defined by the equation g(r,t,h,s,l)=m(r,s,l)=0. By a suitable numericalalgorithm the geometric reliability index G(m) (HasoferLind) is calculated and the FORMapproximation to the probability P[(R,Smax,L) > mjR > 1] then becomes [G(m)].

Table A1

One year of 975 observations of (Hs, Tz) at a position in the Northern North Sea [1]

Tz (s) 4.25 5.25 6.25 7.25 8.25 9.25 10.25 11.25 12.25 13.25

0.45 0 0 0 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.75 1 8 4 3 5 5 3 0 0 0 0 0 0 0 0 0 0 0 0 0

1.05 1 12 10 20 12 12 9 2 0 0 0 0 0 0 0 0 0 0 0 0

1.35 0 1 12 15 33 18 10 5 0 0 0 0 0 0 0 0 0 0 0 0

1.65 0 0 5 13 32 19 14 3 5 1 0 0 0 0 0 0 0 0 0 0

1.95 0 0 4 14 18 15 7 14 3 2 0 0 0 0 0 0 0 0 0 0

2.25 0 0 0 6 20 18 16 9 7 0 1 0 0 0 0 0 0 0 0 0

2.55 0 0 0 1 15 25 10 6 5 2 0 0 0 0 0 0 0 0 0 0

2.85 0 0 0 1 11 25 10 7 5 4 0 0 0 0 0 0 0 0 0 0

3.15 0 0 1 3 8 19 15 9 9 4 0 0 0 0 0 0 0 0 0 0

3.45 0 0 0 3 2 4 15 18 7 1 1 0 0 0 0 0 0 0 0 0

3.75 0 0 0 0 1 14 16 21 10 10 5 0 0 0 0 0 0 0 0 0

4.05 0 0 0 0 0 2 8 10 12 14 0 0 0 0 0 0 0 0 0 0

4.35 0 0 0 0 1 3 0 6 7 10 0 3 1 0 0 0 0 0 0 0

4.65 0 0 0 0 0 0 3 5 6 5 6 0 0 1 1 0 0 0 0 0

4.95 0 0 0 0 0 2 1 5 6 1 4 3 1 0 0 0 0 0 0 0

5.25 0 0 0 0 1 2 4 2 3 1 5 1 0 0 0 0 0 0 0 0

5.55 0 0 0 0 0 0 2 3 0 0 0 0 2 0 0 0 0 0 0 0

5.85 0 0 0 0 0 0 2 1 0 2 3 2 0 1 0 0 0 0 0 06.15 0 0 0 0 0 0 0 1 2 0 4 0 3 0 0 0 2 0 0 2

6.45 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6.75 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0

7.05 0 0 0 0 0 0 0 1 0 2 0 4 2 0 0 0 0 0 0 0

7.35 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0

7.65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7.95 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0

8.25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8.55 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0

8.85 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0

Hs (m)

Appendix 1


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References

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[7] Krenk S. A double envelope for stochastic processes, DCAMM report, 134. Danish Center for Applied Mathe-

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[8] Crame r H, Leadbetter MR. Stationary and related stochastic processes. New York: Wiley; 1967.

[9] Leadbetter MR, Lindgren G, Rootze n H. Extremes and related properties of random sequences and processes.

New York: Springer-Verlag; 1983.

[10] Rychlik I. Regression approximations of wavelength and amplitude distributions. Advances in Applied Prob-

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[11] Lindgren G. Wave analysis by Slepian models. Probabilistic Engineering Mechanic 2000;15:4957.

[12] Longuet-Higgins MS. On the joint distribution of the period and amplitude in sea waves. Journal of Geophysical

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