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
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Figure 2: Empirical and Theoretical Distribution of Wave Height in Different Spectral Groups.
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Copyright @2001 by A S M E
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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
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13=/~l_eZ
(14)
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=
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
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