nc

Shan40, C

lcuedarmeies aainomiine Te prl formulas.

ve cresuationwaveits forcturese of off

of crest height distri-ate crest distributions(Tayfun, 1980, 1986)3), Tung and Huang

Coastal Engineering 78 (2013) 112

Contents lists available at SciVerse ScienceDirect

Coastal Eng

j ourna l homepage: www.e lsev(Chakrabarti, 1987; Longuet-Higgins, 1952; Ochi, 1998). In the idealGaussian sea model the individual cosine wave trains superimpose

(1985), Dawson et al. (1993), Kriebel and Dawson (1993), Tayfunand Al-Humoud (2002), Tayfun (2004), and others. Forristall (2000)used second-order nonlinear simulations of a set of unimodalThe wave eld is not Gaussian even in innitely deep water, butapproaches a Gaussian eld in the limit when the wave steepnesstends to zero. The wave crest distributions in an ideal Gaussian ran-dom sea are generally regarded to obey the Rayleigh probability law

exist many theoretical and/or empirical modelsbutions of nonlinear random waves. Approximbased on the narrow-band model of sea wavesinclude those described by Huang et al. (198tain an adequate air gap so that the impact of the highest wave crestson the underside of the deck structures can be prevented. Finally, fora ship in the ocean, the occurrence of green water on deck, the waveslamming on the bow are, and the extreme vessel roll motion are alldependent on the extreme wave crests.

and other more suitable methods should be applied to predict the dis-tribution of wave crest heights for the nonlinear randommodel of thesea elevation.

The crest distributions of nonlinear random waves have been aresearch topic for more than three decades. In the literature, there Corresponding author at: Department of Naval ArchiShanghai Jiaotong University, Shanghai 200240, Chifax: +86 21 34206701.

E-mail address: wyg110@sjtu.edu.cn (Y. Wang).

0378-3839/$ see front matter 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.coastaleng.2013.03.002shore structures such asplatform or a semisub-sually designed to main-

lead to underestimation of wave crests which increases in severityas the wave energy increases. In this case the application of theRayleigh distribution to the wave crests becomes nonconservative,a xed platform, a jack-up rig, a tension legmersible platform, their deck elevations are uMonte Carlo simulation

1. Introduction

The probability distributions of waportance to the design and safety evalshore structures and ships. Firstly, theincluding specifying sound safety limimportant for all kinds of coastal struand breakwaters. Secondly, in the cast height are of vital im-of coastal structures, off-crest height assessment,overtopping hazards, issuch as seawalls, dikes

linearly (add) without interaction, and therefore, the model is alsocalled the linear sea model.

Waves in the real world are nonlinear. Real waves show a smallbut easily noticed departure from a Gaussian surface. The crests arehigher and sharper than expected from a summation of sinusoidalwaves with random phase, and the troughs are shallower and atter(Forristall, 2000). Consequently, the linear Gaussian sea model canFinite water depth 2013 Elsevier B.V. All rights reserved.Transformed Rayleigh methodWave steepness

using the method with thosmethod and some empiricaCalculating nonlinear wave crest exceedaTransformed Rayleigh method

Yingguang Wang a,b,, Yiqing Xia a

a Department of Naval Architecture and Ocean Engineering, Shanghai Jiaotong University,b State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 2002

a b s t r a c ta r t i c l e i n f o

Article history:Received 22 May 2012Received in revised form 4 March 2013Accepted 6 March 2013Available online 9 April 2013

Keywords:Wave crest height exceedance probabilitiesNonlinear mixed sea states

This paper concerns the camixed sea states. The excewave model into a Transfocrest exceedance probabilitthe need for long time-domed sea state, two wind sea denergy. The wave steepnessaccuracy and efciency of thtecture and Ocean Engineering,na. Tel.: +86 21 34206514;

rights reserved.e probabilities using a

ghai 200240, Chinahina

lation of the wave crest height exceedance probabilities in fully nonlinearnce probabilities have been calculated by incorporating a fully nonlineard Rayleigh method. This is an efcient approach to the calculation of wavend, as all of the calculations are performed in the probability domain, avoidssimulations. The nonlinear mixed sea states studied include a swell dominat-nated sea states, and two states of mixed wind sea and swell with comparableuence and the nite water depth effects are also considered in the study. Theransformed Rayleigh method are validated by comparing the results predictededicted by using the Monte Carlo simulation method, the theoretical Rayleigh

ineering

i e r .com/ locate /coasta lengJONSWAP spectra to obtain parametric wave crest distributions.Prevosto et al. (2000) and Prevosto and Forristall (2004) also devel-oped a perturbated narrowband model for probability distributionsof nonlinear wave crests. Fedele and Arena (2005) have derived ana-lytical expressions for the probabilities of exceeding crest height in anon-Gaussian sea state, and the proposed distributions consider

and the other covers the lower frequency components. Each modi-ed PiersonMoskovitz spectrum is expressed in terms of three pa-rameters and the total spectrum is written as a linear combinationof the two:

S 14

X2j1

4j14

4mj

j j H2sj

4j1exp

4j 14

mj

4 1

where Hs1, m1 and 1 are the signicant wave height, modal fre-quency and spectral shape parameters for the lower frequency com-ponents of the sea while Hs2, m2 and 2 correspond to the higherfrequency components of the sea.

2 Y. Wang, Y. Xia / Coastal Engineering 78 (2013) 112second order nonlinearities. Their proposed analytical probabilitiesare validated by performing Monte Carlo simulations of nonlinearsea states with rectangular and unimodal JONSWAP spectra. Fedeleand Arena (2005) also validated their proposed model against thedata of the wave elevation measured at the Draupner eld in thecentral North Sea.

It is well known that not all sea states have unimodal wave spectraand narrow (or nite) spectral bandwidth. Frequently, sea states aredue to the coexistence of various wave systems. In particular, localwind waves often develop in the presence of some background lowfrequency swell coming from distant storms, and the resultingmixed sea states will have bimodal wave spectra (Guedes Soares,1984). Although validations of the existing probabilistic wave crestmodels are done basically for sea states with unimodal spectra,there also exist studies on wave crest statistics in bimodal sea states,and a description of recent results in this eld is given as follows:Toffoli et al. (2006) study the effect of the angle of spread betweentwo coexisting wave systems on the statistics of second-orderwaves in unimodal and bimodal seas.

Arena and Guedes Soares (2009a) validate the model of Fedeleand Arena (2005) against second-order Monte Carlo simulations forfour bimodal wave spectra in deepwater recorded in the North AtlanticOcean and in the North Sea. Arena and Guedes Soares (2009b) investi-gated the nonlinear structure of high wave groups in bimodal sea statesand the results in their paper are validated by means of Monte Carlosimulations of nonlinear sea waves. Petrova and Guedes Soares(2009) estimated the probability distributions of wave heights in bi-modal seas and compared with the Rayleigh model and with othermodels that take into account either the effect of spectral bandwidthor the effect of wave nonlinearities. Petrova et al. (2011) investigatethe effect of angle of spread between two crossing wave systems (char-acterized by bimodal spectra) on the nonlinearity of wavesmeasured ina deep-water basin. Petrova and Guedes Soares (2011) presentedresults for the distribution of wave heights from laboratory generatedbimodal sea states. In their study, data collected at the DHI offshorebasin were analyzed and compared with results based on wave recordsfrom the MARINTEK offshore basin. Finally for shallow water Chernevaet al. (2005) investigated the probability distributions of peaks, troughsand heights of wind waves measured in the coastal zone of theBulgarian part of the Black Sea. In their study various theories fornon-Gaussian random process are applied.

To move a step further, in this paper, the probabilistic structureof the wave crest height distributions in nonlinear mixed sea stateswill be systematically investigated by utilizing a Transformed Ray-leigh method. The nonlinear mixed sea states studied will includea swell dominated sea state, two wind sea dominated sea states,and two states of mixed wind sea and swell with comparable ener-gy. Finite water depth effects (e.g. in the coastal regions) will alsobe considered in the study. The accuracy and efciency of theTransformed Rayleigh method for calculating the crest height ex-ceedance probabilities will be validated by comparing the resultspredicted using the method with those predicted by using the MonteCarlo simulation method, the theoretical Rayleigh method and someempirical formulas.

2. The nonlinear mixed sea states

2.1. The bimodal wave spectra for mixed sea states

In order to derive the wave crest distributions in the nonlinearmixed sea states, the bimodal structure of the wave spectra shouldbe studied rst.

To describe the mixed sea states, Ochi (1998) developed a six-parameter spectrum model by a superposition of two modiedPiersonMoskovitz spectra. One of the modied PiersonMoskovitz

spectra is for the higher frequency components of the wave energyRodriguez et al. (2004) (also in Rodriguez and Guedes Soares, 1999,2001; Rodriguez et al., 2002) utilized the above bimodal OchiHubblespectrum with nine different parameterizations to represent threetypes of sea state categories (please note that in these papers thesea states were all numerically simulated; however, full scale evi-dence of the situation can be found in Guedes Soares and Carvalho,2003, 2012):

I Swell dominated sea states: The most important part of the energyis concentrated on the low frequency spectral part but with a sig-nicant contribution from high frequency components.

II Wind sea dominated sea states: The main part of the wave eldenergy is associated with the high frequency spectral peak butsignicantly inuenced by the swell.

III Mixed wind sea and swell with comparable energy: The wave eldenergy is comparably distributed over the high and low frequencyranges.

Each category is represented by three different inter-modaldistances between the wind sea and the swell spectral components.These three subgroups are denoted in Table 1 by a, b, and c re-spectively. The exact values of the six parameters are given in Table 1.

In Fig. 1, a bimodal OchiHubble spectrum is plotted with a Matlabtoolbox (Brodtkorb et al., 2000) for a swell dominated sea state with in-nite water depth (sea state typeI and sea state group b in Table 1).In the following sections, this spectrum will be called Spectrum 1.

Similar plots have been made for Spectrum 2 and Spectrum 3 inFigs. 2 and 3 respectively, and these two spectra are for two windsea dominated sea states with innite water depth. Similar plotshave also been made for Spectrum 4 and Spectrum 6 in Figs. 4 and 5respectively, and these two spectra are respectively for two mixedswell and wind sea states with comparable energy in an innitelydeep sea.

In order to study the effects of nite water depth, Spectrum 5(a bimodal OchiHubble model for the shallow water case) has alsobeen plotted. This spectrum follows the original Spectrum 4 but in-cludes a correction parameter for a nite water depth of 30 m, i.e. itis obtained by multiplying the original Spectrum 4 by a function

Table 1Target spectrum parameters for mixed sea states (cf. Rodriguez et al., 2004).

Sea state type Sea state group Hs1 Hs2 m1 m2 1 2 S1

I a 5.5 3.5 0.440 0.691 3.0 6.5 0.0314b 6.5 2.0 0.440 0.942 3.5 4.0 0.0293c 5.5 3.5 0.283 0.974 3.0 6.0 0.0274

II a 2.0 6.5 0.440 0.691 3.0 6.0 0.0533b 2.0 6.5 0.440 0.942 4.0 3.5 0.1016c 2.0 6.5 0.283 0.974 2.0 7.0 0.0988

III a 4.1 5.0 0.440 0.691 2.1 2.5 0.0446b 4.1 5.0 0.440 0.942 2.1 2.5 0.0700c 4.1 5.0 0.283 0.974 2.1 2.5 0.0617

0.4 0.6 0.8 1 1.20

5

10

15

Frequency [rad/s]

S(w)

[m2

s / r

ad]

Fig. 1. Spectrum 1 for a swell dominated sea state (type: I; group: b) as a function of

0.4 0.6 0.8 1 1.2 1.4 1.60

2

4

6

8

Frequency [rad/s]

S(w)

[m2

s / r

ad]

fp1 = 0.94 [rad/s]fp2 = 0.45 [rad/s]

Fig. 3. Spectrum 3 for a wind sea dominated sea state (type: II; group: b) as a functionof radian frequency.

3Y. Wang, Y. Xia / Coastal Engineering 78 (2013) 112(d) that ranges between 0 and 1 according to the similarity law ofBuows et al. (1985):

S5 S4 d S4 k ; d 3 k ; d k ; 3 k ;

" # S4

kd 3 kdk 3 k

" #: 2

In the above formula d is a dimensionless frequency dened by

d=g

pand kd is the wave number associated with the linear disper-

sion relation:

2 gkd tanh kdd 3

where g and d are the acceleration of gravity and water depth, respec-tively. In formula (2), S5() represents the nite water depth dimen-sional spectrum, and S4() represents the innite water depthdimensional spectrum. Fig. 6 shows the correction parameter (d)calculated according to Eq. (2) with d = 30 m and in Fig. 7 thesolid blue line represents the obtained Spectrum 5 in our study.

2.2. The second order non-linear wave model for mixed sea states

The second order nonlinear wave model for mixed sea states canbe obtained by adding to the linear Gaussian sea model quadraticterms allowing for interactions between the elementary cosinewaves. Here, the Gaussian sea model is obtained as a rst orderapproximation of the solutions to differential equations based onlinear hydrodynamic theory of gravity waves. In this paper, we onlyconsider a long-crested and unidirectional sea where all the wavestravel along the x-axis with positive velocity. The rst order wave sur-face elevation l can then be approximated by the following Fourierseries based on the model rst proposed by Rice (1944, 1945)

l x; t ReXNnN

An2ei ntknx 4

radian frequency.as N tends to innity. In Eq. (4), Re denotes the real part of the com-plex number, x stands for the distance along the x-axis, t denotes

0.4 0.6 0.8 10

5

10

15

Frequency [rad/s]

S(w)

[m2

s / r

ad]

Fig. 2. Spectrum 2 for a wind sea dominated sea state (type: II; group: a) as a functionof radian frequency.time, and for each elementary sinusoidal wave An denotes its complexvalued amplitude (An is complex Gaussian), n the angular frequency,and kn the wave number. Because l should be a real valued eld, weneed to assume that j = j, kj = kj. If l is assumed to bestationary and Gaussian, then the complex amplitudes An are alsoGaussian distributed, that is, An = n(Un iVn), where Un and Vnare independent zero mean and variance one Gaussian random vari-ables, and n

2 is the energy of waves with angular frequencies nand n.

The mean square amplitudes are related to the mixed sea statewave spectrum S() in Eq. (1) by:

E Anj j2h i

2S nj j 5

where = c/N and c is the upper cut-off frequency beyondwhich the power spectral density function S() may be assumed tobe zero for either mathematical or physical reasons. The value ofupper cut-off frequencyc can be established using the following for-mula (Shinozuka and Deodatis, 1991):

c0 S d 1 0 S d 6

with chosen to be a very small positive number (0 b 1,e.g. = 0.00001, = 0.0001). At this point it should be noted thatwhen generating sample functions of the simulated stochastic pro-cess according to Eq. (4), the time step t separating the generatedvalues of l(x,t) in the time domain has to obey the Nyquist frequencycondition (Shinozuka and Deodatis, 1991):

t2= 2c : 7

The above condition is necessary in order to avoid aliasingaccording to the sampling theorem (Shinozuka and Deodatis, 1991).

In Eq. (4), the individual frequencies,n and wave numbers, kn arerelated through the linear dispersion relation:

n2 gkn tanh knd 80.5 1 1.50

2

4

6

fp1 = 0.68 [rad/s]fp2 = 0.45 [rad/s]

Frequency [rad/s]

S(w)

[m2

s / r

ad]

Fig. 4. Spectrum 4 for a sea state of mixed wind sea and swell with comparable energy(type: III; group: a) as a function of radian frequency.

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

fp1 = 0.45 [rad/s]fp2 = 0.94 [rad/s]

Frequency [rad/s]

S(w)

[m2

s / r

ad]

Fig. 5. Spectrum 6 for a sea state of mixed wind sea and swell with comparable energy(type: III; group: b) as a function of radian frequency.

4 Y. Wang, Y. Xia / Coastal Engineering 78 (2013) 112where g and d are the acceleration of gravity and water depth, respec-tively. For deep water, Eq. (8) simplies to:

n2 gkn: 9

Real wave data does not follow the linear Gaussian model. The lin-ear Gaussian seamodel can be corrected by including quadratic terms.Following Langley (1987) the quadratic correction q is given by

q x; t ReXNnN

XNmN

AnAm4

E n;m ei ntknx ei mtkmx 10

where the quadratic transfer function E(n,m) is given by:

E i;j

gkikjij

12g

i2 j2 ij

g2

ikj2 jki2

ij i j

1g

ki kji j 2 tanh ki kj

d

gkikj

2ij 12g

i2 j2 ij

:

11

For deep water waves the quadratic transfer function simplies to:

E i;j

12g

i2 j2

; E i;j

12g

i2j

2 12

where i and j are positive and satisfy the same relation as in thelinear model. Finally, by combining Eqs. (4) and (10) the wave surfaceelevations for the nonlinear mixed sea states can be written as:

x; t l x; t q x; t : 13For nonlinear random waves in a mixed sea state, the wave crestswill become higher and steeper, and the troughs of the nonlinear

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

d

(d)

Fig. 6. Correction factor (d) as a function of the dimensionless frequency d.waves will become shallower and atter. Obviously, the Rayleigh dis-tribution which is good for predicting the crests of linear Gaussianwaves will underestimate the crests of nonlinear random waves inthe mixed sea state. In the existing literature some empirical and heu-ristic distribution functions for wave crest heights have been pro-posed, and in Section 3 of this paper we will briey review severalof these empirical distributions. In Section 4 of this article the theoret-ical background of a Transformed Rayleigh method proposed for cal-culating the crest distributions of nonlinear randomwaves in a mixedsea state will be elucidated. Finally in Section 5 some calculation ex-amples utilizing the Transformed Rayleigh method will be given.

3. Some empirical wave crest distributions

For a linear sea in the narrow band limit, the wave crest height ex-ceedance probabilities can be calculated according to the followingRayleigh law (Chakrabarti, 1987; Longuet-Higgins, 1952):

P Ac > h exp 8hHs

2 14

where h is the crest height, and Hs is the signicant wave height.Tayfun (1980) and Huang et al. (1986) produced crest height

distributions from the Stokes model. There is some disagreement be-tween these authors on the exact form of the resulting distribution.Eq. (15) below is taken from the review by Forristall (2000) withthe original wave steepness R = kHs replaced by an effective wavesteepness R*:

P Ac > h exp 8R

2

1 2Rh

Hs

s1

" #22435 15

0.5 1 1.50

2

4

6

Frequency [rad/s]

S(w)

[m2

s / r

ad]

Fig. 7. Finite depth (d = 30 m) of Spectrum 5 (solid blue line) corresponding to in-nite water depth wave of Spectrum 4.where the wave effective steepness R* is given by:

R kHsf 2 kd kHscoshkd 2 cosh2kd

2 sinh3kd 1

sinh2kd

16

where k is the wave number and d is the water depth.In a seminal paper Forristall (2000) developed a two parameter

Weibull distribution for the wave crest heights based on secondorder simulations:

p Ac > h exp hHs

: 17

The parameters and are given in terms of S1, which is ameasure of steepness and the Ursel number Ur, which is a measure

5Engineering 78 (2013) 112of the impact of water depth on the non-linearity of waves. Thesequantities read:

S1 2g

HsT1

2 18

Ur Hs

k12d3

19

where T1 is the mean wave period calculated from the ratio of therst two moments of the wave spectrum, k1 is the wave number fora frequency of 1/T1, and Hs is the signicant wave height. In thecase of a second order long-crested sea (the 2D case):

0:3536 0:2892S1 0:1060Ur 20

22:1597S1 0:0968 Ur 2: 21

In the case of a second order short-crested sea (the 3D case), theparameters and are given as:

0:3536 0:2568S1 0:0800Ur 22

21:7912S10:5302Ur 0:284 Ur 2: 23

At present the above Forristall distribution is considered to be themost convenient surface model being available for routine work.

4. Transformed Rayleigh method for wave crest distributions

4.1. Theoretical background of the Transformed Rayleigh method

From Section 2.2 we know that (0,t) (corresponding to x = 0 inEq. (13)) is the height of the sea level at a xed point as a functionof time t. In order to simplify the notation, we shall write (t) for(0,t). For the denition of a wave, one often uses the so-calledmean down crossing wave, where the wave is considered as a partof a function between the consecutive down crossings of the meansea level.

Assume (t) crosses a mean sea level u* (here also the levelmost frequently crossed by ) nite many times. Denote by ti,0 b t1 b t2 b , the times of down crossings of u*. The crest Mi, say,of the ith wave is the global maximum of (t) during the intervalti b t b ti + 1.

For a specic nonlinear mixed sea state, (t) will be a non-Gaussianstochastic process. Here we will use a simple (but widely effective)model for a non-Gaussian sea where (t) is expressed as a functionof a stationary zero mean Gaussian process t with variance one( V 0 1), i.e. (Rychlik and Leadbetter, 1997; Rychlik et al.,1997)

t G t ; dGd

> 0;G 0 0: 24

Note that, once the distributions of crests in t are computed,then the corresponding wave crest distributions in (t) are obtainedby simple transformations involving only the inverse of G. Therefore,in order to use the model it is necessary to estimate the transforma-tion G. It is well known that, for a zero mean Gaussian process t ,the level up-crossing rate u of the level u by t is given by thecelebrated Rice's formula:

u 12

exp u2

2 2

! 1

2

exp

u2

2

!25

Y. Wang, Y. Xia / Coastalwhere 2 and 0

2 are the variances of t and the derivative t ,respectively. We notice that

2 1.We turn now to the level up-crossing rate (u) of the non-Gaussian

process (t). For all G-functions satisfying the properties in Eq. (24), thefollowing relationship exists:

u G1 u

12

exp

G1 u 2

2

0B@

1CA: 26

The following equation then holds (Rychlik and Leadbetter, 1997):

u 0 exp

G1 u 2

2

0B@

1CA: 27

If the level up-crossing rate (u) of the non-Gaussian process (t)is known, then the above equation can be used to obtain the inversetransformation G1(u) (Rychlik and Leadbetter, 1997):

G1 u

2 ln u

0 s

if u0

2 ln u

0 s

if ub0

:

8>>>>>>>:

28

Next, we turn to the relation between the level up-crossing rate(u) and the probability distribution of the crest height FM u . Asshown in Rychlik and Leadbetter (1997), we have that:

1FM u min0zu

z 0 : 29

Combining Eqs. (27) and (29) we obtain the following relationship:

FM u 1 exp G1 u 2

2

0B@

1CA 30

for all u 0, and G1(u) is dened by Eq. (28). Eq. (30) can also bewritten in a form of:

FM u 1 exp G1 u 2

2m0

0B@

1CA 31

where m0 is the zero-order spectral moment of the stationary zeromean and variance one Gaussian process t . m0 2 1.We notice that the exponent in the right part of Eq. (31) is theRayleigh probability that the linear crest exceeds the predenedthreshold u. That is why this model is called a TransformedRayleigh method.

We can see that in order to calculate the wave crest distributions(i.e. FM u ) of the non-Gaussian process (t), the critical task is to cal-culate the level up-crossing rate (u) of (t). In the next subsection,the principles of a numerical procedure for calculating this (u) willbe elucidated.

4.2. Saddle Point Approximation of the level up-crossing rates (u)

We rst rewrite the dening equation for the process (0,t)(i.e. (t)). Assume that the angular frequencies j and average

energies j, j = 1,2,3,N, are chosen. Denote by the column vector

6 Y. Wang, Y. Xia / Coastal Engineering 78 (2013) 112containing j while the column vector of j. Dene (Machado andRychlik, 2003):

Z t U1iV1 ei1t UniVn einth iT X t iY t 32

where X(t) and Y(t) are real, and

Q qmn ; qmn E m;n E m;n mn; 33

R rmn ; rmn E m;n E m;n mn; 34

W wmn ;wmm m and wmn 0 if mn 35

S QWWR; 36

where m, n = 1, 2, 3 N, and

E m;n gkmknmn

12g

m2 n2 mn

g2

mkn2 nkm2

mn m n

1g km knm n 2

tanh km kn d

g kmkn2mn

12g

m2 n2 mn

37

E m;n g

kmknmn

12g

m2 n2mn

g

2

mkn2nkn2

mn mn

1g km knmn 2

tanh km kn d

g kmkn2mn

12g

m2 n2mn

:

38

Then (Machado and Rychlik, 2003):

t TX t 12X t TQX t 1

2Y t TRY t 39

_ t TWY t 12X t TSY t 1

2Y t TSTX t : 40

Please note that the relations in Eqs. (39) and (40) are only validfor second order unidirectional random waves. We turn now to thederivation of the formula for the characteristic function of ((0),_ 0 ), denoted by:

M 1; 2 E exp i 1 0 2 _ 0 f g : 41

We notice that the variables X(0) and Y(0) are independentstandard Gaussian and their joint probability density function f(Z)is given by:

f Z 12

p n exp 12ZT IZ

42

where I is a (2N, 2N)-dimensional identity matrix. Therefore, thecharacteristic function of ((0), _ 0 ) is given by:

M 1; 2 E exp i 1 0 2 _ 0 f g

12

p n exp itTZ12ZT IA Z

dZ1dZn 43

where the matrix A = A(i1,i2) and the vector t = t(i1,i2) aredened as follows:

A i1; i2 i1Q i2Si2S

T i1R

; t i1; i2 i1i2W

: 44We utilize the following famous result from Cramr (1954)

exp it

TZ12ZTAZ

dZ1dZn

2

p ndet A p exp

12tTA1t

:

45

By combining Eqs. (43) and (45) we can now obtain the explicitformula for the characteristic function of the vector ((0), _ 0 ):

M 1; 2 1

det IA p exp 1

2tT IA 1t

: 46

The joint probability density function of ((0), _ 0 ) can then becomputed by means of the inverse Fourier transform:

f u; y 12 2

exp i 1u 2y f gM 1; 2 d1d2: 47

Assume that (t) is non-Gaussian. For a xed level u, let (u) bethe expected number of times, in the interval [0, 1], the process (t)crosses the u level in the upward direction. We then have the follow-ing extension of Rice's formula (Machado and Rychlik, 2003):

u a:a:u0 zf 0 _ 0 u; z dz 48

where a:a:u means that the equality is valid for almost all u. We seethat the computation of (u) requires the estimation of the joint den-sity of (0) and _ 0 . Therefore, the level up-crossing rate (u) of (t)can be calculated by combining Eqs. (47) and (48):

u 12 2

0

y exp i 1u 2y f gM 1; 2 d1d2dy: 49

Because typically the matrix A is very large (it may have dimen-sions (500, 500) and more), and because each evaluation of the char-acteristic function requires the evaluation of the inverse (I A)1,the numerical integration in the above equation becomes extremelyslow. In order to improve the computational efciency, we will usea Saddle Point method to approximate the joint density of ((0),_ 0 ) in Eq. (47). The cumulant generating function, K(s1,s2) =ln M( is1, is2) of ((0), _ 0 ) by Eq. (46) becomes:

K s1; s2 12ln det IA 1

2tT IA 1t 50

where the matrix A = A(s1,s2) and the vector t = t(s1,s2) are calcu-lated as follows:

A s1; s2 s1Q s2Ss2S

T s1R

; t s1; s2 s1s2W

: 51

The Saddle Point Approximation f^ u; y of the density f 0 ; _ 0 u; y is dened by (Machado and Rychlik, 2003):

f^ u; y 12

K s^1; s^2 1=2 exp K s^1; s^2 s^1us^2yf g: 52

In the above equation, the so-called Saddle Point, s^1; s^2 is theunique solution of the following system of equations (Machado andRychlik, 2003):

K1 s1; s2 K s1; s2 s1

u

K2 s1; s2 K s1; s2 s2

y:

8>>>: 53

Furthermore K is theHessianmatrix of the cumulant generating func-tion K. By numerically implementing the procedures from Eqs. (49) to(53), the level up-crossing rate (u) of (t) can then be calculated.

When applying the Transformed Rayleigh method and utilizingthe Saddle Point Approximation of the level up-crossing rates, someapproximations of the cumulant generating function of ((0), _ 0 )can further be made in order to increase the computational efciency(for theory, see Appendix A where eigendecompositions have beenapplied).

5. Calculation examples and discussions

In this section, we demonstrate the accuracy and efciency of theTransformed Rayleigh method by some example calculations. Thenonlinear mixed sea states with Spectrum 1, Spectrum 2, Spectrum3, Spectrum 4, Spectrum 5 and Spectrum 6 specied in Section 2.1are chosen for our calculations. Fig. 8a shows our calculationresults for the wave crest height exceedance probabilities for a swell

dominated sea state with Spectrum 1. In Fig. 8a, the blue solid linerepresents the results of the wave crest height exceedance probabili-ties calculated using the theoretical Rayleigh probability distribution.The continuous green line represents the Monte Carlo simulationresults of the wave crest height exceedance probabilities of thenonlinear mixed sea state. In this nonlinear simulation, 2,000,000wave elevation points (200 repetitions of a simulation of 10,000wave elevation points) were generated in the simulated time seriesin order to reduce the variance of the estimate. A wave elevationpoint has two coordinates (The rst coordinate is time. The secondcoordinate is the value of the wave elevation above the mean waterlevel.). The randomized second order non-linear waves were simulat-ed from Spectrum 1 by summation of sinus functions with randomphase angles uniformly distributed in the range of [0, 2] while thesampling interval is dened by the Nyquist frequency (i.e. by usingEq. (7) in Section 2.2). The wave crest height time series were thenextracted from these 2,000,000 wave elevation points. Spline interpo-lation was rst done before extracting wave crest height time series.

0 1 2 3 4 5 6 7

10-5

10-4

10-3

10-2

10-1

100The probability of X exceeding x

1-F(

x)

x (m)

Monte Carlo simulationTransformed Rayleigh methodTheoretical Rayleigh method

100

3

4

me

Surface elevation from mean water level (MWL).

)

a

b

. Non

7Y. Wang, Y. Xia / Coastal Engineering 78 (2013) 11260 70 80 90

-4

-3

-2

-1

0

1

2

Ti

Dis

tanc

e fro

m M

WL.

(m

Fig. 8. a. Wave crest height exceedance probabilities for a sea state with Spectrum 1. b

state from t = 50 s to t = 150 s together with the linear simulation results.110 120 130 140 150 (sec)

linear simulationnonlinear simulationmean water level

linear simulated time series of wave surface elevations of Spectrum 1 for a mixed sea

8 Y. Wang, Y. Xia / Coastal Engineering 78 (2013) 112After extracting these time series, exact kernel density estimates (byusing the Epanechnikov kernel density function) were performed inorder to obtain probability density function of the wave crest heights.Then cumulative trapezoidal numerical integration was carried outon the above probability density function for getting the probabilitydistribution (F) of the wave crest height. Finally, the wave crestheight exceedance probabilities were calculated based on the proba-bility distribution by using the formula P = 1 F. The above MonteCarlo simulation results are utilized as the standards against whichthe accuracy of the results from the theoretical Rayleigh methodand the Transformed Rayleigh method is checked. We notice fromthe upper, left tails in Fig. 8a that in a small region (about [0, 3 m])of the wave crest heights the theoretical Rayleigh method still givespredictions that t the simulation results well. However, as soon asthe wave crest is higher than about 3 m, the theoretical Rayleighmethod will predict overly nonconservative exceedance probabilitiesof the wave crest heights in the nonlinear mixed sea states. In Fig. 8a,the red dashed line represents the results of the wave crest height ex-ceedance probabilities obtained from the Transformed Rayleighmethod.When applying the Transformed Rayleigh method and utiliz-ing the Saddle Point Approximation of the level up-crossing rates,some approximations of the cumulant generating function of ((0),_ 0 ) were made in order to speed up the computations (for theory,see Appendix A). We see from the gure that the TransformedRayleigh method gives more close predictions of the wave crestheight exceedance probabilities to the simulation results than theoriginal theoretical Rayleigh method does. The reason for the goodts between the simulation results and the results from theTransformed Rayleigh model is that the level up-crossing ratesobtained from the Saddle Point Approximation are quite accurate.Grime and Langley (2003) also demonstrated that the Saddle PointApproximation can predict very accurate crossing rates for determin-ing extreme motions of moored offshore structures (see Fig. 1 inGrime and Langley (2003)). The proposed Transformed Rayleighmodel whose accuracy is mainly affected by the Saddle Point Approx-imation can thus predict quite accurate results.

Fig. 8b shows our nonlinear simulated time series of wave surfaceelevations of Spectrum 1 in a time range from t = 50 s to t = 150 s.In order to exemplify its nonlinear characteristics, in Fig. 8b we havealso added a red wave linearly simulated from Spectrum 1 for com-parison purpose. We notice that the crests of the nonlinear randomwaves in the mixed sea state are higher and steeper, and the troughsof the nonlinear waves are shallower and atter.

Fig. 9 showsour calculation results for thewave crest height exceed-ance probabilities for a wind sea dominated sea state with Spectrum 2.In thegure, theMonte Carlo simulation results of thewave crest heightexceedance probabilities of the nonlinear mixed sea state are againrepresented by the continuous green line. 2,000,000 wave elevationpoints were generated in the simulated time series in order to reducethe variance of the estimate, and the time series simulation and thepost statistical processing took about 42 s on a Dell OptiPlex 360 desk-top computer. In the same gure, the red dashed line again representsthe calculation results of thewave crest height exceedance probabilitiesof themixed sea state from utilizing the Transformed Rayleigh method,and in this example the calculation using the Transformed Rayleighmethod took only about 8 s. We can clearly see that the results fromthe TransformedRayleighmethod aremuchbetter than those predictedby the theoretical Rayleigh method, which are represented by a solidblue line in the gure. Besides the high accuracy and efciency of theTransformed Rayleigh method, we can also notice from the gure thatthe method gave slightly conservative predictions in the region ofabout [3.5 m, 7 m], and the design based on these predictionswill resultinmore safemarine structures.We have carried out similar calculationsfor awind sea dominated sea statewith Spectrum 3, and our calculationresults are summarized in Fig. 10. We can see that the tendencies of the

three lines in Fig. 10 are similar to those of the three lines in Fig. 9.If we look more closely at Figs. 8a, 9 and 10 and compare them, wecan nd that in Fig. 8a the differences between the results from the the-oretical Rayleighmethod and the Transformed Rayleighmethod are thesmallest, while in Fig. 10 the corresponding differences are the biggest.Spectrum 1 represents a swell dominated sea state whose steepnessparameter value is 0.0293 calculated using Eq. (18). The steepness pa-rameters for the two wind sea dominated sea states Spectrum 2 andSpectrum 3 are 0.0533 and 0.1016, respectively (these values are alsolisted in the last column of Table 1). Steepness is a parameter that char-acterizes the degree of nonlinearity of thewaves. The larger the value ofa steepness parameter is, the more nonlinear the corresponding wavesare. Therefore, we can see that it is more advantageous to apply theTransformed Rayleighmethod to predict the wave crest height exceed-ance probabilities of the more nonlinearmixed sea states. Finally weshould point out that the results in Figs. 8a, 9 and 10 are suitable to becompared because Spectrum 1, Spectrum 2 and Spectrum 3 all havean equivalent signicant wave height value of 6.8007 m which isequal to four times the standard deviation of each spectrum.

The high efciency of the Transformed Rayleigh method can bemore clearly demonstrated by applying it to the prediction of thecrest height exceedance probabilities of waves in nite water depth.Fig. 11 shows the calculation results for the wave crest height exceed-ance probabilities for Spectrum 5 with a water depth of 30 m. In thiscase, the time series simulation with 2,000,000 wave elevation pointsand the post statistical processing took about 5.5 min on a DellOptiPlex 360 desktop computer (the simulation results are represent-ed by the continuous green line in Fig. 11). By comparison, it took lessthan 13 s to obtain the results of the wave crest height exceedanceprobabilities represented by the red dashed line in Fig. 11 by applyingthe Transformed Rayleigh method. In Fig. 11, the solid pink line rep-resents the results predicted by the theoretical Rayleigh method,and the small black dots represent the wave crest height exceed-ance probabilities calculated using Forristall's 3D formulas (i.e. usingEqs. ((17)(19)) and Eqs. ((22)(23))). In Fig. 11, the solid blue linerepresents the wave crest height exceedance probabilities calculatedusing Forristall's 2D formulas (i.e. using Eqs. ((17)(21))). We cansee that Forristall's 3D formulas predicted slightly higher wave creststhan Forristall's 2D formulas did. However, the two Forristall modelsand the Transformed Rayleigh method all gave better predictions ofthe wave crest height exceedance probabilities than the theoreticalRayleigh method did. In Fig. 11, the small blue + represents awave crest height exceedance probability predicted by using theTayfun model (i.e. by using Eqs. ((15)(16))). We can notice that theTayfun model gave poorer predictions in the wave crest height regionof about [3 m, 5.5 m] than the Transformed Rayleigh method did.

As another example, the Transformed Rayleigh method was uti-lized to calculate the wave crest height exceedance probabilities fora mixed swell and wind sea state with comparable energy (Spectrum4 with an innite water depth), and the calculation results are shownin Fig. 12 by the red dashed line. We spent only about 12 s to obtainthis red dashed line on a Dell OptiPlex 360 desktop computer. In thesame gure, the continuous green line again represents the MonteCarlo simulation results of the wave crest height exceedance proba-bilities of the mixed sea state, and the simulation with 2,000,000wave elevation points spent about 43 s this time. We can clearly seefrom the gure that the results from the Transformed Rayleigh meth-od are fairly good in this case, while the results obtained from usingthe theoretical Rayleigh method (represented by the blue solid line)are overly unconservative.

Finally, the Transformed Rayleigh method was applied to calculatethe wave crest height exceedance probabilities for an innite deepnonlinear mixed sea state with Spectrum 6, and the calculation re-sults are shown in Fig. 13 by the red dashed line. We spent onlyabout 12 s to obtain this red dashed line on a Dell OptiPlex 360 desk-top computer. In the same gure, the continuous green line again

represents the Monte Carlo simulation results of the wave crest

0 1 2 3 4 5 6 7

10-5

10-4

10-3

10-2

10-1

100The probability of X exceeding x

1-F(

x)

x (m)

Monte Carlo simulationTransformed Rayleigh methodTheoretical Rayleigh method

Fig. 9. Wave crest height exceedance probabilities for a wind sea dominated sea state with Spectrum 2.

9Y. Wang, Y. Xia / Coastal Engineering 78 (2013) 112height exceedance probabilities of the mixed sea state, and the simu-lation with 2,000,000 wave elevation points spent about 46 s thistime. In Fig. 13, the small blue dots represent the wave crest heightexceedance probabilities calculated using Forristall's 3D formulas,and the small blue + represents a wave crest height exceedanceprobability calculated using Forristall's 2D formulas. We can see thatin the innite deep water Forristall's 3D formulas predicted lowerwave crests than Forristall's 2D formulas did, and the results calculatedby using Forristall's 3D model t more closely with the simulation re-sults. However, the two Forristall models and the Transformed Rayleighmethod all gave much better predictions of the wave crest height ex-ceedance probabilities than the theoretical Rayleigh method did.

6. Concluding remarks

The detailed mathematical procedures of a Transformed Rayleighmethod for calculating wave crest height exceedance probabilities0 1 2 3

10-5

10-4

10-3

10-2

10-1

100The probabili

1-F(

x)

x

Monte Carlo simulationTansformed Rayleigh methoTheoretical Rayleigh metho

Fig. 10. Wave crest height exceedance probabilities forin nonlinear mixed sea states are outlined in this article, and themethod has been applied in predicting the exceedance probabilitiesof wave crest heights in six nonlinear mixed sea states. For the venonlinear mixed sea states with innite water depth, the predictedwave crest height exceedance probabilities are compared with thosecalculated by using the Monte Carlo simulationmethod, and the accu-racy and efciency of the Transformed Rayleigh method are convinc-ingly validated. The high efciency of the Transformed Rayleighmethod is further demonstrated by applying it to calculate the wavecrest height exceedance probabilities of a nonlinear mixed sea statewith a nite water depth of 30 m. In all cases studied, the TransformedRayleigh method gave slightly conservative predictions of the wavecrest height exceedance probabilities, and the designs based on thesepredictions will result in more safe marine structures. Finally, it is no-ticed that for all the nonlinear mixed sea states the theoretical Rayleighmethod gave overly unconservative predictions of the wave crestheight exceedance probabilities.4 5 6 7

ty of X exceeding x

(m)

dd

a wind sea dominated sea state with Spectrum 3.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

10-5

10-4

10-3

10-2

10-1

100The probability of X exceeding x

1-F(

x)

x (m)

Monte Carlo simulationTransformed Rayleigh methodForristall's 2D formulaForristall's 3D formulaTheoretical Rayleigh methodTayfun model

Fig. 11. Wave crest exceedance probabilities for Spectrum 5 with water depth of 30 m.

10 Y. Wang, Y. Xia / Coastal Engineering 78 (2013) 112Acknowledgment

This research work is supported by the funding of an independentresearch project from the Chinese State Key Laboratory of OceanEngineering (grant no. GKZD010038). We thank the two anonymousreviewers for their useful comments that signicantly improved thequality of this paper.

Appendix A

We consider the following eigenvalue problems:

Q ;R A 1

where is the (N, N) matrix whose rows are the eigenvectors of Q or-dered according to increasing absolute value of their eigenvalues i,i = 1, ,N, and is a diagonal matrix with i as its elements. is the (N, N) matrix whose rows are the eigenvectors of R orderedaccording to increasing absolute value of their eigenvalues i,i = 1,0 1 2 3

10-5

10-4

10-3

10-2

10-1

100The probabil

1-F(

x)

x

Monte Carlo simulatioTransformed RayleighTheoretical Rayleigh m

Fig. 12. Wave crest height exceedance proba ,N, and is a diagonal matrix with i as its elements. Because Qand R are symmetric their eigenvalues are real. Therefore, we havethe following eigendecompositions:

Q T;R T: A 2

The sea surface and its derivative in Eqs. ((39)(40)) can now bewritten as:

t TX t 12X t TX t 1

2Y t TY t A 3

_ t TWTY t 12X t TSY t 1

2Y t TSTX t : A 4

By means of matrix algebra, the cumulant generating functionK(s1,s2) in Eq. (50) can be re-expressed as:

K s1; s2 12ln det IA 1

2tT IA 1t4 5 6 7

ity of X exceeding x

(m)

n methodethod

bilities for a sea state with Spectrum 4.

bili

x

tionigh ulaulah m

roba

11Engineering 78 (2013) 112where now

A s1; s2 s1 s2 W

TWT

s2 WTWT

Ts1

24

35A 5

t s1; s2 s1s2W

: A 6

For most of sea spectra, a considerable number of eigenvalues iand i are very close to zero. We propose to replace the m smallesteigenvalues i and i by zeros and use Eqs. (A-3) and (A-4) to deneap(t):

ap t TX t 12X t T X t 1

2Y t T Y t A 7

0 1 2 3

10-5

10-4

10-3

10-2

10-1

100The proba

1-F(

x) Monte Carlo simulaTransformed RayleForristall's 2D formForristall's 3D formTheoretical Rayleig

Fig. 13. Wave crest height exceedance p

Y. Wang, Y. Xia / Coastal_ap t TWTY t X t T SY t A 8

where and are the matrices and with the rstm rows replacedby zeros, and S WTWT .

The positive integer m is decided by using the following relation-ship of the ratio of variances:

V 0 V ap 0

V 0 0:00001 A 9

i.e. select m as the largest m such that

12 m2 12 m2

12 m2 N2 12 m2 N2

0:00001:

A 10Now dene Kap(s1,s2) to be the cumulant generating function of

ap 0 ; _ap 0

, then

Kap s1; s2 12ln det I A

12tT I A 1

twhere now

A s1; s2 s1 s2 W

TWT

s2 WTWT

Ts1

24

35:

A 11

The matrix A s1; s2 is (2N, 2N)-dimensional but it contains blocksof zeros. Therefore the computation of the cumulant generating func-tion can be speeded up.

Here we give a calculation example regarding the dimensionlesswave spectrum 1. By applying the criterion in Eq. (A-10) we can re-placem = 244 (of 257) eigenvalues i in by zeros. Similarly, we canreplacem = 244 (of 257) eigenvalues i in by zeros. By doing thesethe relative error of the ap-variance is less than 0.00001. The aboveprocedures reduced the dimension of the matrices, which has to beinverted, from (514, 514) to (26, 26) without noticeably affectingthe accuracy of the Saddle Point method.

Therefore, the computational efciency can be increased.

4 5 6 7

ty of X exceeding x

(m)

method

ethod

bilities for a sea state with Spectrum 6.References

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12 Y. Wang, Y. Xia / Coastal Engineering 78 (2013) 112

Calculating nonlinear wave crest exceedance probabilities using a Transformed Rayleigh method1. Introduction2. The nonlinear mixed sea states2.1. The bimodal wave spectra for mixed sea states2.2. The second order non-linear wave model for mixed sea states

3. Some empirical wave crest distributions4. Transformed Rayleigh method for wave crest distributions4.1. Theoretical background of the Transformed Rayleigh method4.2. Saddle Point Approximation of the level up-crossing rates (u)

5. Calculation examples and discussions6. Concluding remarksAcknowledgmentAppendix AReferences