Marc Prevosto , Raymond Nerzic Abstract Standard models for the ...

Wave Climate Parameterization of the Distribution
of Maximum Wave Heights and Crest Elevations
Marc Prevosto 1 , Raymond Nerzic 2
Abstract
Standard models for the distribution of wave heights and crest elevations are
useful for computing maximum wave and crest heights, on the basis of significant
wave heights and mean wave periods, for the purpose of designing offshore structures.
However, there can be differences of 10-20% in the results obtained with different
models, which are unsuitable in terms of design criteria. The discrepancies could be
explained by the problems of accurately representing wave properties in a given sea
state, particularly for strong and steep waves, because of the non-linearity in the wave
kinematics which is not suitably taken into account in the common models.
A Corrected Weibull-Stokes (CWS) model was thus proposed by Nerzic and
Prevosto (1997) on the basis of standard Weibull models, modified using a third order
Stokes expansion. This model was validated with North Sea wave data, but the analysis
shown that the model might be site dependent, in relation with water depth and
sea state properties such as directionality, and spectral shape.
The present paper describes a procedure which allowed to derive the parameters
of the CWS model from satellite based directional wave spectra, using a second
order non-linear wave model. The procedure was validated for the same North Sea
area and the developments indicated that the methodology could be an effective tool
to parameterize the distribution of maximum wave and crest heights for design purposes
in any oceanic area.
Introduction
The Corrected Weibull-Stokes (CWS) model proposed by Nerzic and Prevosto
(1997) for the distribution of wave and crest heights is based on standard Rayleigh
and Weibull models, modified using a third order Stokes expansion of the wave en-
1.IFREMER, DITI/GO/COM, BP 70, 29280 PLOUZANE, FRANCE
2.OPTIMER, 3 rue Jean Monnet, District de Montpellier, 34830 CLAPIERS, FRANCE
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OCEAN WAVE KINEMATICS, DYNAMICS AND LOADS ON STRUCTURES 397
velope.
In the model, the Stokes expansion allows to take into account the non-linearity
in the wave kinematics, in relation with wave steepness. The effect of the steepness
on wave crest elevation is particularly clear from site measurements, as shown
on figure 1 from wave data at Frigg field in the North Sea. It can be observed also
that the effect of steepness on wave height is small. And this is consistent with the
Stokes expansion model.
However, other wave properties can influence the distribution of wave heights
and crest elevations, for example the directionality and the spectral shape as well as
the water depth or the wave length-to-depth ratio. And this tends to indicate that the
distributions of wave heights and crest elevations might be site dependent, in relation
with wave climate and water depth. Unfortunately, there are only few oceanic sites
with a wave data base that can allow analysing the wave distribution. And this particularly
because of the limited accuracy of measurement of maximum wave crests
by wave buoys.
Wave observations from satellite give access to second order statistical information:
the significant wave heights from the altimeter and the wave directionality
and the wave spectral shape from the synthetic aperture radar (SAR). These data do
not permit to describe the non-linear characteristics of the wave field but they provide
information about the crestedness and the frequency peak narrowness of the sea state.
1
1.65
C max /H 1/3
0.95
0.9
0.85
0.8
0 0.02 0.04 0.06 0.08
mean steepness s
z
H max /H 1/3
1.55
1.45
1.4
0 0.02 0.04 0.06 0.08
mean steepness s
z
Figure 1 . C max /H 1/3 mode (left) and H max /H 1/3 mode (right) vs. Steepness – Frigg field data
Second order non-linear wave models allow computing time series of non-linear
waves from directional spectra. This permits to supplement the spectral information
given by satellites in order to generate data bases of wave elevation time series
on a specific site and then the maximum wave heights and crest elevations. So, we
can avoid the difficult theoretical transformation of the statistics of the wave spectral
data set to the wave statistics through the non-linear transfer of the wave kinematics.
Then, the wave data base can be processed as actual data and thus, they can provide
the fitted parameters of the distribution of the maxima in any oceanic area.
This methodology was applied to directional wave spectra from satellite observations
around Frigg field and the results were compared to the parameters derived
from the analysis of site measurements. The analysis allowed stating the ability of
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OCEAN WAVE KINEMATICS, DYNAMICS AND LOADS ON STRUCTURES
the proposed methodology to replace the processing of in-situ wave data.
Weibull-Stokes model
Weibull model with Stokes expansion. The Weibull-Stokes model is based on
the Weibull (or Rayleigh) distribution and on the assumption that the sea surface process
is a distortion of a first-order Gaussian process due to non-linear effects. The amplitudes
of the first order process are distributed as a Weibull distribution (or a
Rayleigh distribution for narrow banded process) and the non-linear effects can be
introduced with a second- or third-order Stokes expansion.
Thus, after Vinje (1989), the following relations between crest and wave
heights (C and H) and wave amplitude (a) of the first order process can be considered
at infinite water depth:
and (1)
where b2 = 1/2 and b3 = 3/8 are the parameters of the third-order Stokes expansion,
in infinite water depth and km is a mean wave number, related to the mean wave period
Tm , defined either from the spectral moments (Tm = (m0 /m2 ) 1/2 H 2a 1 b3( kma )
) or as the mean
zero down-crossing period Tz .
Then, the maximum wave heights and crest elevations are asymptotically distributed
as a Gumbel distribution with parameters derived from the parameters of
the Gumbel distribution associated to the first order process. For example, with aN the mode parameter of the maxima distribution of the first order process, it gives for
the mode parameter of the maximum crest elevations:
(2)
Corrected Weibull-Stokes model. After the analysis of wave data from the Frigg
field in the North Sea (cf. Nerzic and Prevosto (1997)), a Corrected Weibull-Stokes
(CWF) model was proposed, with two correction factors. The first correction factor,
αk, was for the mean wave steepness or wave number and the second factor, αa , for
the wave energy term or the wave amplitude. This resulted in a simple asymptotic
parameterized Gumbel law for the non-normalized maxima of wave height and crest
elevation with the mode and the scale parameters given by relations like (2).
The parameters θ, β (resp. scale and shape parameter of the Weibull law) and
αk were as follows (Table 1), with the two different definitions of the couple (Hs ,Tm ),
first with (Hm0 ,T02 ) and then with (H1/3 ,Tz ). The correction factor αk applies to the
wave number km given by Tm transformed by the dispersion relation and the correction
factor αa is entered in θ.
2
= ( + ) C a 1 b2( kma) b3( kma) 2
= ( + + )
aCN aN 1 b2( kma N)
b3( kma N)
2
=
( + + )
Table 1 . Parameters of the Weibull-Stokes model – Frigg field data
Weibull-Stokes Model Hmax / Hs Cmax / Hs θ β αk θ β αk Hs = Hm0 / Tm = T02 0.73 2.38 0.7 0.71 2.08 0.6
Hs = H1/3 / Tm = Tz 0.77 2.38 0.7 0.78 2.18 0.7
The differences resulting from the selected definition of significant wave

OCEAN WAVE KINEMATICS, DYNAMICS AND LOADS ON STRUCTURES 399
height and mean wave period are mainly on the scale parameter, about 5% to 10%
larger for H 1/3 than for H m0 , reflecting the bias between the two significant wave
height parameters. The main results are presented in figure 2, with the data-to-model
ratios from the mode parameters, for both H max /H m0 and C max /H m0 . The bias are
nil and the standard deviations are less than 2 or 3%.
C max /H 1/3
1.15
1.1
1.05
1
0.95
0.9
H max /H 1/3
1.15
1.05
0.95
0 0.02 0.04 0.06
mean steepness s
z
0.08 0 0.02 0.04 0.06
mean steepness s
z
0.08
Figure 2 . Cmax/H1/3 (left) and Hmax /H1/3 (right) Model-Data mode ratio vs. Steepness – Frigg
field data / Weibull-Stokes model
PARAMETERIZATION OF THE WEIBULL-STOKES MODEL
If the CWS model of distribution of maximum wave and crest heights is a
very general model, the parameters of the model are site dependent because the wave
properties vary from one oceanic area to an other. The main site characteristics which
can influence the shape of maximum waves are the water depth, because it is clear
that offshore waves are distorted when propagating in shallow waters, and also the
meteorological and oceanographic features which act on the wave directionality and
on the wave spectral shape. The directional and spectral properties of waves act on
the steepness of waves and on the wave pattern, and can lead to very different sea
states, from long unidirectional swell to mixed wind seas. Thus, the site dependent
wave climate has a main influence on the distribution of maximum wave and crest
heights.
However, there are very few site data to analyse the influence of wave climate
on the distribution of maximum wave and crest heights. There are few extensive wave
surveys available for such analysis, excepted for the North Sea and the Gulf of Mexico.
And often the available wave data come from buoy measurements, therefore with
limited accuracy of wave crest data.
But the developments of second-order non-linear wave simulation models allow
to generate wave elevation time series from directional wave spectra. This allows
to generate synthetic data bases of wave statistics from a climatic description of wave
conditions in any oceanic area, provided that the requested sea state statistics are available,
that means statistics of significant wave height, directionality and spectral shape.
And the only extended source for such wave statistics are from satellite observations.
Wave statistics from satellite observations. Satellite observations provide sta-
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OCEAN WAVE KINEMATICS, DYNAMICS AND LOADS ON STRUCTURES
tistics of offshore sea states in two different ways: significant wave heights from altimetric
data and directional spectra from SAR images. The main advantage of
satellite observations is their availability for any oceanic area, with only a limitation
at distances less than 5 to 10 km from shore.
The satellite data have some limitations, particularly the SAR directional
spectra are limited to long waves, with a cut-off frequency about 0.14 Hz. But a procedure
described in (Hajji, 1993) allows to complement the spectra in the high frequency
band with wind wave spectra generated from scatterometer data.
Thus, the directional wave spectra derived from satellite observations are provided
with a resolution of 24 angular bands (15° angle) and 22 frequency bands (from
1/25 Hz to 1/4 Hz).The frequency resolution is quite low, with a cut-off frequency
about 0.25 Hz, and the angular resolution is 15°. This can lead to some limitations
in the use of the spectra, which are discussed in Bitner-Gregersen & al. (1996). The
limitations affect mainly the accuracy of wave period parameters (peak frequency,
mean period, etc. and therefore the steepness parameters) rather than the accuracy
of wave height parameters, because the wave energy outside the frequency band remains
very low. Thus, the results of second order wave simulations are only slightly
affected, with mainly some bias on wave period parameters.
Second-order wave simulation. The wave simulation model applied for the generation
of non-linear irregular waves is a second-order non-linear wave model for
multidirectional waves developed by Prevosto (1998) on the basis of the model described
by Ding & al. (1994). Note that 25 seconds were necessary to simulate a wave
time series of 4096 time step by a Personal Computer with a 133 MHz processor
and 32 Mb RAM.
Procedure for parameterization. The basis for the parameterization of the CWS
model is simply to generate a data set of maximum wave and crest heights, H max
and C max , from a data base of directional wave spectra derived from satellite observations,
using the second-order wave simulation model.
To be adequate for the purpose of deriving the distribution of maximum wave
and crest heights at a given oceanic site, the wave spectra data base must be representative
of the wave climate at the site. In particular, wave spectra for storm conditions
must be present to analyse the distribution of extreme wave and crest heights.
Thus, several wave time series have to be simulated to get a sufficient data set of H max
and C max to allow the analysis of their distributions.
Then, the H max and C max data have to be grouped by class intervals of H s
and T m and the H max /H s and C max /H s ratio distributions in each class interval have
to be fitted to Gumbel distributions. The mode and scale parameters of the fitted distributions
can then be analysed in relation with the proposed Corrected
Weibull-Stokes model, in order to determine the parameters θ, β and α k .
This procedure can be applied for any oceanic area, provided that satellite
based directional wave spectra are available. And with the procedure, a model of distribution
of maximum wave and crest can therefore be derived for any oceanic area,
but the procedure requests that a representative data base of wave spectra be built.

OCEAN WAVE KINEMATICS, DYNAMICS AND LOADS ON STRUCTURES 401
Site case analysis
The methodology was applied to a wave spectra data base over 3 years from
satellite observations in an area of 200kmx200km centred on the Frigg field in the
North Sea. The data base consisted of 118 directional wave spectra. The wave height
– period and the wave height – steepness scatter diagrams of this data base are presented
at figure 3, with Hs = Hm0 , the significant wave height, Tm = T02 , the spectral
km mean period, sm = ----- HS , the mean steepness and km , the mean wave number, re-
2π
lated to Tm with the dispersion relation.
12
0.05
T 02 (s)
10
8
6
4
0 2 4 6
H (m)
m0
mean steepness s m
0.04
0.03
0.02
0.01
0 2 4 6
H (m)
m0
Figure 3 . Wave height – period (left) and wave height – steepness (right) scatter diagrams
– Satellite data base
The data base was limited to 118 spectra, with a maximum significant wave
height of 5.6 m and a maximum steepness of 0.05. This can be compared to the maximum
significant wave height of 12.4 m and to the maximum steepness of 0.08, in
the Frigg wave data base used for the validation of the Weibull-Stokes model. However,
the data base was sufficient to test the methodology, because several synthetic
wave time series can be generated from each directional wave spectra. 100 wave time
series were simulated for each wave spectra, i.e. a total of 11800 wave time series.
The Hmax and Cmax data were grouped by class intervals of Hs and Tm , with
bandwidth of resp. 0.5 m and 1.0 s. Then, the Hmax /Hs and Cmax /Hs ratio distributions
in each class interval were fitted to Gumbel distributions, using the maximum likelihood
method. And the parameters of the fitted distributions were analysed in relation
with the proposed Corrected Weibull-Stokes model.
The procedure was applied for the two different definitions of the couple
(Hs ,Tm ) considered in the study. The parameters θ, β and αk derived from the analysis
are presented hereafter:
Table 2 . Parameters of the Weibull-Stokes model – Satellite data
Weibull-Stokes
Hmax / Hs Cmax / Hs Model
θ β αk θ β αk Hs = Hm0 / Tm = T02 0.67 2.11 0.6 0.70 2.00 0.5
Hs = H1/3 / Tm = Tz 0.80 2.48 0.6 0.82 2.32 0.5

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OCEAN WAVE KINEMATICS, DYNAMICS AND LOADS ON STRUCTURES
The main results are presented in figure 4, with the data-to-model ratios of
the mode parameters, for both H max /H s and C max /H s . They show a good agreement
between the model and the data. The bias are nil and the standard deviations are less
than 2 to 4%.
Figure 4 . C max /H s (left) and H max /H s (right) Model-Data mode ratio vs. Steepness – Satellite
data / Weibull-Stokes model
There are some differences between the parameters θ and β calculated for
(H m0 ,T 02 ) and for (H 1/3 ,T z ), which cannot be explained only by the bias between
H m0 and H 1/3 . In fact, several pairs of parameters θ and β would be suitable for the
model, that means with low bias and standard deviations between the mode and the
scale parameters from the models and from the data.
The parameters θ and β show also some differences with the parameters derived
from field measurements and presented in table 1, particularly the parameters
for H m0 and T 02 . However, the predictions of actual maximum wave and crest heights
by the present models are good, as indicated by the ratios between the field data and
the predictions with the model based on satellite data. The data-to-model ratios of
the mode parameters are presented in figure 5 and they show a good agreement. The
bias are less than 1% and the standard deviations are less than 3%.
Figure 5 . C max /H s (left) and H max /H s (right) Model-Data mode ratio vs. Steepness – Frigg
field data / Weibull-Stokes model from Satellite data

OCEAN WAVE KINEMATICS, DYNAMICS AND LOADS ON STRUCTURES 403
Conclusions
The methodology developed for the parameterization of the distribution of
maximum wave and crest heights from satellite based directional wave spectra, using
second-order wave simulations, has proven its efficiency for the tested site case. This
gives a complementary evidence of the suitability of the Weibull-Stokes model to
represent the distribution of maximum wave and crest heights and thus to derive the
extreme wave height and crest elevation parameters for design purposes.
The results indicated also the ability of the procedure to derive the distribution
of maximum wave and crest heights from sea observations by satellites, without the
help of site measurements, even if site data can provide a better accuracy. This important
result came probably for two reasons. First, the satellite observations allow
deriving the directional wave spectra with sufficient accuracy for the purpose, in spite
of the low accuracy and resolution of the data. Secondly, the second-order wave simulation
models are adequate to represent the non-linear characteristics of ocean waves
which concern the wave heights and crest elevations.
The procedure was tested for a site of the North Sea in 100 metres of water,
in almost deep-water conditions. In shallower waters, the non-linearities are affected
also by the water depth and mainly by the wavelength-to-depth ratio. The ability of
the proposed procedure to parameterize the distribution of maximum wave and crest
heights in shallow waters still has to be demonstrated.Acknowledgments
The authors are grateful to Elf for permission of using Frigg Field data, and
Météomer for providing the data set of inverted SAR spectra
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