Proceedings
ProceedingsofofOMAE’01
OMAE'01:
th
20
International
Conference
on
Offshore
Mechanics
Engineering
20 th International Conference on Offshore Mechanics and
and Artic
Arctic
Engineering
Rio
de Janeiro,
June
3-8, 2001
June
3-8, 2001,Brazil,
Rio de
Janeiro,
Brazil
OMAE2001/S&R-2178
OMAE'01-S&R-2178
PROBABILITY DISTRIBUTIONS OF WAVE HEIGHTS AND PERIODS
IN MEASURED TwO-PEAKED SPECTRA FROM
THE PORTUGUESE COAST
C. Guedes Soares, A. N. Carvalho
Unit of Marine Technology and Engineering, Technical University of Lisbon
Instituto Superior T~cnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Email: [email protected], utl.pt
Therefore, it is very useful to assess the ability of the present
probabilistic models to model full-scale data so as to verify
the adequacy of the conclusions obtained previously on the
basis of numerically simulated data.
ABSTRACT
An analysis is made of measured two-peaked sea spectra
from a deep-water location at the Portuguese Coast. The
spectra are organised in different groups according to the
relative nature of their two component wave systems. The
probability distributions of wave height and period are
determined and compared with several theoretical models.
This paper deals with the analysis of buoy data measured by
a directional waverider buoy at a location of the Portuguese
coast.
From the data of one year, the two-peaked spectra have been
identified and have been classified into 9 classes, according to
the relative value of the energy and peak frequency of each of
the two wave systems.
The results of firing various
probabilistic models to the data in each class are reported
here.
INTRODUCTION
Most work on developing probabilistic models of short-term
sea state characteristics has concentrated on wind waves in
single wave systems, which are appropriately described by
single peaked spectra. However, the importance of combined
sea states of sea and swell (Guedes Soares, 1984) is becoming
generally recognized and thus the interest of verifying whether
the available probabilistic models are applicable in these
situations.
E X P E R I M E N T A L DATA
The experimental data used in this study has been collected
by a Datawell waverider buoy located off the port of Sines in
Portugal. The buoy is located at a water depth of 97 m.
There has been a series of studies by Rodriguez and Guedes
Soares, which were based on numerical simulations of twopeaked spectra and considered the marginal distributions of
wave height (Rodriguez et. al, 1999) and wave period
(Rodriguez and Guedes Soares, 2000), joint distribution of
height and period (Rodriguez and Guedes Soares, 1999), as
well as the group structure (Rodriguez et. al., 2000) in these
sea states. The main conclusions were that for some of the
combined sea states the probabilistic models developed for
single wave systems were still applicable. However, for the
sea states with larger differences in the peak period of the two
wave systems some discrepancies were identified.
The measured time series have 30 minutes duration and the
scalar spectra have been determined by usual spectral analysis
procedures. The sea surface elevation was sampled at a rate of
0.64 Hz during 30 minutes every three hours. The spectral
estimators were obtained using the Welch method with cosene
window and 25% overlap. Records consisting of 2304 data
points were segmented in 18 partitions of 128 points.
The wave data used in this study has been collected during
the year 2000 and corresponds to a total of 2272 spectra. The
whole data set was scrutinized to identify the existence of two
peaked spectra, which were identified according to the criteria
described in Guedes Soares and Nolasco (1992). From these,
149 spectra were identified as being two peaked, which
corresponds to about 7%, a percentage that is lower that the
ones reported in Guedes Soares (1991) and in Guedes Soares
and Nolasco (1992). Their distribution as a function of Hs is
given in Table 1.
The limitation that could be attributed to those studies is that
they were based on numerical simulations and while that type
of simulations are well established for single peaked spectra, it
is not clear how the interaction between the two wave systems
is being properly taken into account in those numerical
simulations.
1
Copyright ©2001 by ASME
0<Hs<1
1<Hs<2
2<Hs<3
3<Hs<4
4<Hs<5
5<Hs<6
6<Hs<7
7<Hs<8
TOTAL
Table
All spectra
96
1121
812
129
54
47
7
6
2272
Soares (1984), in which the other two parameters are the
significant wave height and the mean period o f the sea state.
2 peaks
31
70
3l
5
5
7
0
0
149
However, recently Henriques and Guedes Soares (1998) have
concluded that using peak period instead o f the mean period
improved the quality o f the fit.
Sea state
category
- Distribution of the number of spectra with Hs
a)
b)
c)
Swell-Dominated Sea States. The main part o f the wave
field energy is associated with the low-frequency
spectral peak, but significantly influenced by wind-sea.
One o f these is the ratio o f energies associated to each wave
system, or Sea-Swell Energy Ratio (SSER), defined as the ratio
o f the wind-sea frequency band zero order spectral moment to
the zero order moment o f the frequency spectrum
corresponding to the swell wave field
(2)
0
and the subscripts sw and ws stand for swell and wind-sea
parameters, respectively.
Those wave fields with SSER value smaller than one
represent swell dominated sea states and those with SSRE
value greater than one correspond to the wind-sea dominated
category. If the SSRE value is close to one they are included
in the category o f sea states with two spectral peaks with
comparable energy content.
0.202 0.473
2
0.319 0.575
17
III
0.518 0.573
4
Wind-sea
I
0.192 5.033
6
Dominated
II
III
0.321 6.434
0.485 5.745
8
2
state
Mixed wind-sea
I
0.165 1.852
2
and swell systemsof
II
0.328 1.595
9
Comparable energy
Ill
0.514 1.552
3
The identified two-peak spectra have been grouped into nine
categories, in terms o f the two dimensionless parameters.
These three categories have been denoted by the letters a, b
and c, respectively, as indicated in column 1 o f Table 2.
The other parameter is the frequency separation between the
spectral frequency peaks, f p , corresponding to the swell and
wind-wave systems, named as Inter-modal Distance (ID) and
expressed as
p.. - f p ~
fp. + f p .
I
II
Thus, the combination o f the above described characteristics
o f wave fields in terms o f ID and SSER, leads to nine classes
o f sea states which are representative o f a wide part o f the
bimodal spectra that can be observed at sea. After identifying
the two-peaked spectra and classifying them in the mentioned
nine classes, they have been normalized and then all data was
pooled in order to obtain the mean spectra shown in figure 1.
The frequency was normalized by the peak frequency o f the
individual spectral component with highest energy. The
spectral ordinates were normalized by the largest value. The
spectra o f these nine kinds o f sea states are shown in figure 1
and the corresponding values o f ID and SSER are given in
table 2.
where m. is the n th order spectral moment, given by
IO=
spectra
Those wave fields with ID value close to zero correspond to
sea states with swell and wind-sea spectral peaks very near to
each other. Those with ID value nearer to one represent sea
states with swell and wind-sea systems located in very
different frequency zones, that is to say, the swell and windsea spectral peaks are considerably separated. Finally, those
wave fields with intermediate values o f l D are included in the
group o f two-peaked sea states whose modal spectral
frequencies are moderately separated. These three groups are
denoted by I, II and III, as indicated in column 2 o f Table 2.
These three categories o f sea states can be characterized by
means o f two a nondimensional parameters.
" n = 0, 1, 2,...
Nr. o f
Table 2: Shape parameters of the two-peaked spectra, mean spectral
bandwidth values and number of spectra of the wave records in each
spectrum type.
Sea States with Comparable Influence o f Wind-Sea and
Swell. The wave field energy is comparably distributed
over the low and high frequency ranges.
m,, = I f " S O C ) d f
SSER
Swell
sea state
Wind-Sea-Dominated Sea States. The most important
part of the energy is concentrated on the high-frequency
spectral part, but with a significant contribution from
low-frequency components.
11)
Dominated
sea
Sea states with two peak spectra can be classified in the
following main categories:
Sea
state
group
Initially only the spectra with H s >__2m were chosen for the
present study. This led to a total o f 48 spectra but classes IIIb
and IIIc did not have any spectra. Therefore, in order to have
all classes represented, 5 additional spectra were chosen with
H s < 2m leading to a total o f 53 spectra distributed as
indicated in Table 2.
(3)
It should be noted that ID and SSER are closely related to
two o f the four parameters used in the model o f Guedes
2
Copyright ©2001 by ASME
[
[-
m
1
1
olI
I 0.
laIa
0. 8
0. 6
Lo.6 i
00 .. 6 6
I
~08
I
iIo.4
0. 4
8
I
i
I
II a
Ila
0. 8
'0.8
0. 6
i0.6
0. 4
0
i
I
I
0
0
-
0
1
--
,
,
2
7
3
3
4
5
4
L
1
2
2
o
3
fOi
3
0
4
4
5
5
00
Ib
0. 8
0. 8
Ib
o A
0. 6
0. 4
0. 4
0.4
0. 2
0. 2
i0"20
0
0
0
0
1
1
2
2
3
3
4
5
4
5
I 0
.
,
0
0
I
.
1
i
.
2
2
1
Ic
Ic
3
3
4
4
0
i o
5
0
0
1,
0. 4
0.4
0. 4
i6.4
0. 2
0. 2
0.2
.
2
3
3
4
4
5
3
3
2
4
,
4
5
51
0. 6
0. 2
o.~
i
0
0
0
1
2
IIIc
0. 4
0
1
io.8
!06 i
0
1
0. 8
0. 6
0.6
!i
55
1
II c
0. 8
0.8
0. 6
4 4
0. 2
0.2
1
1
1
0. 8
.
33
IIIb
0. 6
0.6
0. 4
22
0. 8
IIII bb
06
0. 6
1
1
1
08
I
I
0. 2
16.2
----..._._.,
0
1
-
1o.4
°ii
0.2
IIIa
Ilia
0. 4
0. 2
0. 2
1
0
1 1
2
2
3
3
4
4
5 5
o0
I
1
2 2
33
44
5 5
Figure 1: Shape of the normalized mean spectra in each class
iF erug :1
S epah fo
t h e ron m a l i z de m nae
s p e c t r a i n ae c h c l s a
=~
H
(5)
WAVE H E I G H T D I S T R I B U T I O N M O D E L S
VariousHEIGHT
theoretical
and empirical models
have been proposed
WAVE
DISTRIBUTION
MODELS
to characterize the wave height probability distribution. The
Various
theoretical
empirical
have been
more
relevant
modelsand
dealing
with models
zero-crossing
waveproposed
heights
were
discussed in
et.al.,
1999) anddistribution.
their usefulness
to
characterize
the(Rodriguez
wave height
probability
The
to
predict
the
wave
height
distribution
in
mixed
sea
states
was
more relevant models dealing with zero-crossing wave heights
examined. Only the most relevant ones will be considered here.
were discussed in (Rodriguez et.al., 1999) and their usefulness
Longuet-Higgins (1952) established that in a stationary,
to
predictand
the extremely
wave height
distribution
in mixedtheseawave
states
was
gaussian
narrow
banded process
height
may
be
regarded
as
twice
the
envelope
amplitude
and
that
these
examined. Only the most relevant ones will be considered here.
are distributed according a Rayleigh probability distribution
Longuet-Higgins
(1952) established that in a stationary,
given
by
gaussian and extremely narrow banded process the wave height
may be regarded as twice the envelope amplitude and that these
are distributed according a Rayleigh probability distribution
P(~ by
> ~ o) = exp given
(4)
where wave heights were normalized as:
The spectral bandwidth of the process is defined in terms of the
parameter
v
=(non2
_ 111/2
~, m2
(6)
However, even for a narrow band process (v ~ 0) the
assumption of the wave height as twice the amplitude of the
envelope amplitude is not totally exact due to the modulated
structure of a narrow band process and the time lag between a
crest and the adjacent trough. Various authors have suggested
that wave heights fits more adequately to other probability laws,
such as the Weibull distribution, which for the normalized wave
height may be written as
/
(7)
where (~ and 13 are parameters to be determined by using some
3
Copyright 02001 by ASME
4
Copyright 2001 by ASME
Naess (1985) derived an expression for the crest-to-trough wave
heigh in a stationary Gaussian narrow banded wave train given
by:
procedure for fitting empirical data. Forristall (1978) used data
recorded during hurricanes in the Gulf of Mexico and obtained a
good agreement between the Weibull distribution and the
observed data for (~=2.126 and 13=8.42.
In a latter paper Longuet-Higgins (1980) showed that the data
examined by Forristal (1978) could be adequately fitted by
introducing the finite spectral bandwidth effect in the
relationship between the amplitude mean root square and mo. In
this way, the exceedance probability distribution takes the form
P(~ > ~o) = exp
-
734v 2
- 4(1_ p ~(~9})
P(~ >~o) =exp
(9)
where p(x/2) represents the value of the normalized
autocorrelation function of the sea surface elevation at the time
when it attains its first minimum:
(8)
R(O)
This distribution improves the predictions given by equation
(8).
Ila
la
o
Ilia
~4 f
e.
ft.
"""
"",,, ~ ,,x,
",
i°I
,g
• ", "",'x.
lo ~
m
lib
Ib
~1~ •
2
°
IIIb
'%i
-\
i>1
I
2
3
~
$
7
S
10",
S
i
i
;
1
;
;
ff
5
10'
1
2
lie
IIIc
"-.
4
3
5
i
7
Ic
. - Nt~s
" >x
1
e
i
lO'
11
3
*
S
~10'
o
X\N'"
~
2
3
'~
5
S
I~
r
tO
~10,
]!
1
2
3
4
5
Figure 2: Empirical and Theoretical Distribution of Wave Height in Different Spectral Groups.
4
Copyright @2001 by A S M E
S
distributions. Increasing ID an overprediction of the Rayleigh
distribution and an improvement of the fit for the Weibull
distribution are observed. It is possible to see that there is an
underestimation of the Naess model in all cases. When ID takes
large values the Weibull distribution is able to predict the
observations over the main part of wave heights.
Results and Discussion
The models described in the previous section have been
applied to the nine mean spectra and Table 3 shows the value of
these parameters. It is also shown the mean value of the spectral
bandwidth parameters e and v, corresponding to each group of
the kind of sea state considered. It should be noted that the
spectral bandwidth increases with the inter-modal distance but
there is not a direct relationship between both parameters.
Sea-Swell energy equivalent sea states
For small and large ID values there is an overprediction of all
models for higher height waves (~>5). It is for intermediate ID
values that the best fitting is found for all range of wave heights.
In this case the Weibull distribution gives good results, while the
Rayleigh and Naess models produce overprediction and
underestimation, respectively. Once again a similar behaviour is
observed for Rayleigh and Weibull for small inter modal
distance.
The mean values of p corresponding to each group of wave
spectra records for each kind of sea state are given in table 3.
Note that in the limit, when the spectral bandwidth approaches
zero this distribution converges to a Rayleigh distribution.
Sea state
category
~
Sea state
group
Swell
o
p
¢t
[1
I
0.869
0.620
-0.627
1.849
5.714
II
0.848
0.815
-0.405
1.960
6.877
sea state
III
0.823
0.701
-0.492
2.043
7.328
Wind-sea
I
0.787
0.453
-0.856
2.032
7.519
Dominated
Dominated
A
B
seastate
Mixedwind-sea
And swell systems with C
Comparable energy
II
0.780
0.490
-0.489
2.007
7.449
III
0.700
0.450
-0.439
2.149
8.899
I
0.823
0.559
-0.608
2.058
7.581
I1
0.785
0.540
-0.397
2.181
8.886
III
0.730
0.580
-0.438
2.272
8.714
WAVE PERIOD DISTRIBUTION MODELS
The study of the probability density function of wave periods
has received less attention than that of wave heights. This is due
to the intrinsic difficulty to determine the distributions for wave
periods, even under the assumptions that waves are linear and
have narrow-band frequency spectrum.
A common procedure to obtain the probability density of wave
periods has been to derive it as a marginal distribution from the
joint distribution of wave heights and periods. The earlier
method was presented by Longuet-Higgins (1975), based on the
assumption of a narrow banded spectral density function. The
expression of the marginal distribution of wave periods given by
this author is
Table 3: Parameters of the fitted distribution of wave height.
In addition to applying the models to the complete data set
corresponding to the normalized mean spectra, they have also
been applied to each individual sea state and it was observed
that despite some variability, there was the same trend as the
one of the mean spectra.
Swell dominated sea states
V 2
It can be observed that the case (Ia) is the only one in which
the three distributions produce an adequate fit to the main range
of wave heights, but not for the highest value which are
overpredicted. It can also be observed that in this case the
Rayleigh distribution gives the best fit to the empirical
probability while the Weibull distribution shows a small
deviation which increases with wave height. With the increase
of ID, the best and worse fittings are the Naess model, for the
highest heights values (~>6) and the rest of the range,
respectively. In contrast, the Rayleigh and Weibull distribution
gives the opposite behaviour. In all cases it is possible to see an
overprediction of the observed probabilities of the Naess model,
especially for intermediate and large values of ID, where this
deviation is significant. Another observation, in these cases, is
the similar behaviour of the Rayleigh and Weibull distributions.
(11)
where the wave periods were normalized as:
x-
Tm~
(12)
m o
Another theoretical expression for the wave period distribution
was derived by Cavanie et al. (1976). These authors used the
wave crest distribution to derive the joint distribution of wave
heights and periods as a starting point. The theoretical marginal
distribution of wave periods given by them takes the form
0t 3 [32x
p(x ) =
2 _(/2
(13)
.~_Ot 4 [~ 2
Wind-sea dominated sea states
When the SSER is large and ID is small none of the models is
able to characterise the observed probabilities over the entire
range of wave heights, in particular for the highest value which
is underestimated. For small and intermediate values of ID there
is a very similar behaviour of the Rayleigh and Weibull
5
where
oc=l/~2II+ 1-x/~-S-e2);
13=/~l_eZ
(14)
and
Copyright ©2001 by ASME
=
upcrossing period, which implies no correlation between
individual wave heights and periods. However, the joint
distribution of wave heights and periods of wave records with
finite bandwidth displays a clear asymmetry about this period,
mainly for low heights. To remove this inconsistency LonguetHiggins (1983) revised his model and presented an alternative
approach, from which the marginal distribution of wave periods
adopts the following expression
(15)
mom4 J
is a spectral bandwidth parameter. It should be noted that the
dimensionless period is given by
"~ = ~'r = ~
Tin,
(16)
mo
where ~- is a function of e that remains close to 1 one for values
P(x)=t.
of g from 0 to 0.95. Then, ~- = 1 is used here.
It should be noted that the model proposed by LonguetHiggins (1975) shows symmetry of periods about the mean zero
la
,) ,,=j
4
(17)
which, according to Shum and Melville (1984), seems to give
good results.
Ib
le
14 r
~L
! ",\
/
"\
'l
i
?'
o6~
/:'
O4
~i
\
1
~2
{Y
• s
o
F
,
o~
x
i
lib
Ila
IIc
~~
14.
... L~..,.J l
12
/,,\
I
/,<~
~
1
o
o5
~
I
':\,I
~
!i
1-7
~ /
i,
i tx-
-
-o
2S
05
1
is
2
25
x
IIIb
Ilia
IIIc
~
f,
-!!
!
I,
,r
o~
%
,1
i
- :: .....
ms
o
o~
1
Is
2
i
2s
I
3
1~
2
22
....
3
Figure 3: Empirical and Theoretical Distribution of W a v e Period in Different Spectral Groups.
6
Copyright ©2001 by ASME
Results and Discussion
mixed sea states IIc, IIIc and Illb. However, in the last two
cases the models give rise to a systematic overestimation
around the mean period. The largest deviations are when the
observed probabilities tend to bimodal, like in the cases Ia,
Ilia and Ib, where none of the models can be fit.
The general trend of the results of of the same nature as the
obtained earlier with numerically simulated data.
Figure 3 shows the histogram of the normalised period data
in all classes. Also shown are the probability density function
resulting from fitting the distribution of Longett-Higgins and
of Cavani6 et al to the data.
Swell dominated sea states
In the swell dominated case, especially for small and large
intermodal distance, the distribution tends to bimodal. For
intermediate intermodal distance the shape of the distribution
is close to symmetric. This is due the increase of intermediate
periods while for small and large intermodal distance the
number of small and large periods increases. Except for
intermediate intermodal distance where there is a slight
deviation, none of the models are able to characterise the
empirical distribution of wave periods. It can be observed that
both theoretical models underestimate the probability of
intermediate periods and overestimate the probabilities of
small and long wave periods.
Acknowledgements
The data analyzed in the present paper was supplied by the
Port of Sines within the scope of the project "Radar
Monitoring Network of the Sea State in the Coastal Waters of
the Iberian Peninsula" (RADSEANET). The analysis was
made within the scope of the project "Reliability Based
Structural Design of FPSO Systems" (REBASDO), which is
partially funded by the European Commission under the
contact ENK6-2000-00107). The authors are grateful to Dr. Z.
Cherneva for several helpful discussions.
Wind-sea dominated sea states
For wind-sea dominated spectra the shape of distribution is
more regular and the models are able to characterise the wave
period distribution for intermediate and large intermodal
distance. The two models are very similar, with a small
deviation for small and large wave periods.
REFERENCES
Cavani6, A., Arhan, M. and Ezraty, R., 1976, "A Statistical
Relationship between Individual Heights and Periods of Storm
Waves", Proc. Conf. on Behaviour of Offshore Structures, Vol.
2, pp. 354-360.
Forristall, G. Z., 1978, "On the Statistical Distribution of
Wave Heights in a Storm", J. Geophys. Res., 83:2553-2558
Sea-Swell energy equivalent sea states
When the energy is comparable the shape of the distribution
is close to symmetric, except for the case of small intermodal
distance, which tends to bimodal. Thus, the models are able to
predict the distribution of wave periods except for the last
case. However, there is an overestimation of both models
around the mean period.
Guedes Soares, C., 1984, "Representation of double-peaked
sea wave spectra", Ocean Engineering, Vol. 11, pp. 185-207.
Guedes Soares, C., 1991, "On the Occurrence of Double
Peaked Wave Spectra", Ocean Engineering, Vol. 18, N ° 1/2,
pp. 167-171
Guedes Soares, C. and Nolasco, M.C., 1992 "Spectral
Modelling of Sea States with Multiple Wave Systems",
Journal of Offshore Mechanics and Arctic Eng., Vol.l14,
pp.278-284.
Henriques, A. C. and Guedes Soares, C., 1998, "Fitting a
double-peak spectral model to measured wave spectra",
Proceedings of the 17th International Conference on Offshore
Mechanics and Arctic Engineering, C. Guedes Soares (Eds.),
ASME, New York, Vol. II.
Longuet-Higgins, M.S., 1975, "On the Joint Distribution of
Wave Periods and Amplitudes of Sea Waves", J. Geophys.
Res., Vol. 80, pp. 2688-2694.
Longuet-Higgins, M. S., 1980, "On the Distribution of
Heights of Sea Waves: Some Effects of Nonlinearity and
Finite Bandwidth", J. Geophysical Research, 85(C3): 15191523.
Longuet-Higgins, M.S., 1983, "On the Joint Distribution of
Wave Periods and Amplitudes in a Random Wave Field",
Proc. Roy. Soc. of London, Vol., 389(A), pp. 241-258.
CONCLUSIONS
In relation to the relative usefulness of the different models
to predict the observed exceedance probability, the Rayleigh
distribution gives rise to a systematic overprediction of the
observed wave heights. This model is able to reproduce the
observed probability only in the case of swell dominated sea
states with low intermodal distance.
In the cases of wind-sea dominated sea states with large
intermodal distance and bimodal sea state with similar energy
and intermodal distance not too large, the best overall fits
between observed and predicted probabilities is given by
Weibull distribution.
The Naess model only gives good results for higher waves
heights in swell dominated sea states with intermediate and
large intermodal distance.
The theoretical models, when compared with the observed
probabilities seem to be useful to predict the obtained wave
period distribution, at least approximately, only in the cases of
7
Copyright ©2001 by ASME
Naess, A., 1985, "On the Statistical Distribution of Crest to
Trough Wave Heights", Ocean Engineering, 12:221-234
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