-
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: [email protected] (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
Arena, F., Guedes Soares, C., 2009a. Nonlinear crest, trough and
wave height distribu-tions in sea states with double-peaked
spectra. Journal of Offshore Mechanicsand Arctic Engineering 131
(Paper 041105 (8 pages)).
Arena, F., Guedes Soares, C., 2009b. Nonlinear high wave groups
in bimodal sea states.Journal of Waterway, Port, Coastal, and Ocean
Engineering (ASCE) 135, 6979.
Brodtkorb, P.A., Johannesson, P., Lindgren, G., Rychlik, I.,
Rydn, J., Sj, E., 2000. WAFO a Matlab toolbox for analysis of
random waves and loads. Proceedings of the 10thInternational
Offshore and Polar Engineering Conference, 3, pp. 343350.
Buows, E., Gunther, H., Rosenthal, W., Vincent, C.L., 1985.
Similarity of the wind wavespectrum in nite depth water: 1 spectral
form. Journal of Geophysical Research90 (C1), 975986.
Chakrabarti, S.K., 1987. Hydrodynamics of Offshore Structures.
Computational MechanicsPublications Inc.
Cherneva, Z., Petrova, P.G., Andreeva, N., Guedes Soares, C.,
2005. Probability distribu-tions of peaks, troughs and heights of
wind wavesmeasured in the Black Sea coastalzone. Coastal
Engineering 52 (7), 599615.
Cramr, H., 1954. Mathematical Methods of Statistics. Princeton
University Press118120.
Dawson, T.H., Kriebel, D.L., Wallendorf, L.A., 1993. Breaking
waves in laboratory-generated JONSWAP seas. Applied Ocean Research
15, 8593.
Fedele, F., Arena, F., 2005. Weakly nonlinear statistics of high
random waves. Physics ofFluids 17 (2), 110.
Forristall, G.Z., 2000. Wave crest distributions: observations
and second-order theory.Journal of Physical Oceanography 30 (8),
19311943.
Grime, A.J., Langley, R.S., 2003. On the efciency of crossing
rate prediction methodsused to determine extreme motions of moored
offshore structures. AppliedOcean Research 25, 127135.
-
Guedes Soares, C., 1984. Representation of double-peaked sea
wave spectra. OceanEngineering 11 (2), 185207.
Guedes Soares, C., Carvalho, A.N., 2003. Probability
distributions of wave heights andperiods in measured combined sea
states from the Portuguese coast. Journal of Off-shore Mechanics
and Arctic Engineering 125 (3), 198204.
Guedes Soares, C., Carvalho, A.N., 2012. Probability
distributions of wave heights and periodsin combined sea-states
measured off the Spanish coast. Ocean Engineering 52, 1321.
Huang, N.E., Long, S.R., Tung, C.C., Yuen, Y., Bliven, L.F.,
1983. A non-Gaussian statisticalmodel for surface elevation of
nonlinear random wave elds. Journal of GeophysicalResearch American
Geophysical Union 88, 75977606.
Huang, N.E., Bliven, L.F., Long, S.R., Tung, C.C., 1986. An
analytical model for oceanicwhitecap coverage. Journal of Physical
Oceanography 16, 15971604.
Kriebel, D.L., Dawson, T.H., 1993. Distribution of crest
amplitudes in severe seas withbreaking. Journal of Offshore
Mechanics and Arctic Engineering 115, 915.
Langley, R.S., 1987. A statistical analysis of non-linear random
waves. Ocean Engineering14 (5), 389407.
Longuet-Higgins, M.S., 1952. On the statistical distribution of
the heights of sea waves.Journal of Marine Research 11, 245266.
Machado, U., Rychlik, I., 2003. Wave statistics in non-linear
random sea. Extremes 6(2), 125146.
Ochi, M.K., 1998. Ocean Waves: The Stochastic Approach.
Cambridge University Press 24.Petrova, P.G., Guedes Soares, C.,
2009. Probability distributions of wave heights in bi-
modal seas in an offshore basin. Applied Ocean Research 31 (2),
90100.Petrova, P.G., Guedes Soares, C., 2011. Wave height
distributions in bimodal sea states
from offshore basins. Ocean Engineering 38 (4), 658672.Petrova,
P., Tayfun, M.A., Guedes Soares, C., 2011. The effect of
third-order nonlinearities
on the statistical distributions of wave heights, crests and
troughs in bimodal crossingseas. Proceedings of the 30th
International Conference on Ocean, Offshore and ArcticEngineering
(OMAE 2011), Rotterdam, The Netherlands, Paper OMAE2011-50313.
Prevosto, M., Forristall, G.Z., 2004. Statistics of wave crests
from models vs. measure-ments. Journal of Offshore Mechanics and
Arctic Engineering 126 (1), 4350.
Prevosto, M., Krogstad, H.E., Robin, A., 2000. Probability
distributions for maximumwave and crest heights. Coastal
Engineering 40 (4), 329360.
Rice, S.O., 1944. Mathematical analysis of random noise. Bell
System Technical Journal23, 282332.
Rice, S.O., 1945. Mathematical analysis of random noise. Bell
System Technical Journal24, 46156.
Rodriguez, G.R., Guedes Soares, C., 1999. The bivariate
distribution of wave heights andperiods in mixed sea states.
Journal of Offshore Mechanics and Arctic Engineering121,
102108.
Rodriguez, G.R., Guedes Soares, C., 2001. Correlation between
successive wave heightsand periods in mixed sea states. Ocean
Engineering 28 (8), 10091030.
Rodriguez, G.R., Guedes Soares, C., Pacheco, M.B., Prez-Martell,
E., 2002. Wave heightdistribution in mixed sea states. Journal of
Offshore Mechanics and Arctic Engi-neering 124 (1), 3440.
Rodriguez, G.R., Guedes Soares, C., Pacheco, M.B., 2004. Wave
period distribution inmixed sea states. Journal of Offshore
Mechanics and Arctic Engineering 126,105112.
Rychlik, I., Leadbetter, M.R., 1997. Analysis of ocean waves by
crossing- and oscillation-intensities. Proceedings of the
International Offshore and Polar EngineeringConference, 3, pp.
206213.
Rychlik, I., Johannesson, P., Leadbetter, M.R., 1997. Modelling
and statistical analysis ofocean-wave data using transformed
Gaussian processes. Marine Structures 10,1347.
Shinozuka, M., Deodatis, G., 1991. Simulation of stochastic
processes by spectral repre-sentation. Applied Mechanics Reviews
ASME 44 (4), 191204.
Tayfun, M.A., 1980. Narrow-band nonlinear sea waves. Journal of
Geophysical Research85, 15481552.
Tayfun, A., 1986. On narrow-band representation of ocean waves.
Journal of GeophysicalResearch 91, 77437752.
Tayfun, M.A., 2004. Statistics of wave crests in storms. Journal
of Waterway, Port,Coastal, and Ocean Engineering (ASCE) 130 (4),
155161.
Tayfun, M.A., Al-Humoud, J., 2002. Least upper bound
distribution for nonlinear wavecrests. Journal of Waterway, Port,
Coastal, and Ocean Engineering (ASCE) 128(4), 144151.
Toffoli, A., Onorato, M., Monbaliu, J., 2006. Wave statistics in
unimodal and bimodalseas from a second-order model. European
Journal of Mechanics B/Fluids 25,649661.
Tung, C.C., Huang, N.E., 1985. Peak and trough distributions of
nonlinear waves. OceanEngineering 12 (3), 201209.
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