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Design sea state assessment using genetic algorithm approach
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Ocean Engineering 34 (2007) 148–156 www.elsevier.com/locate/oceaneng
Design sea state assessment using genetic algorithm approach Jasna Prpic´-Orsˇ ic´, Roko Dejhalla, Anton Turk Department of Naval Architecture and Ocean Engineering, Faculty of Engineering, University of Rijeka, Vukovarska 58, HR-51000 Rijeka, Croatia Received 25 May 2005; accepted 16 November 2005 Available online 17 April 2006
Abstract Two major statistical issues can be distinguished in the procedure of wave extreme prediction. The first issue is that predicted extreme values must be based on data collected in a relatively short time. The second issue is extrapolation of the observed data into its extreme region, typically lying well beyond from even the most extreme available observation. The process of extrapolation plays a fundamental role in this area of analysis and therefore it is essential to fit empirically a convenient probability distribution that describes the available data as closely as possible. Determination of extreme values probability distribution parameters by genetic algorithm is applied to improve the methodology of extreme sea state prediction. Illustrative applications of the method are given for a North Atlantic sea environment. The results are presented as crest height maximum values occurring with a given probability or in a design storm that has a specified return period. r 2006 Elsevier Ltd. All rights reserved. Keywords: Long-term prediction; Wave extremes; Probability density function; Genetic algorithm
1. Introduction An important step often encountered in a design is estimation of an extreme design wave on the basis of recorded or hindcast data (Soares et al., 1996). This generally involves selecting and fitting a suitable probability distribution to wave height data, and extrapolating this to locate a suitable design wave, such as the so-called ‘‘50-year wave’’. This is characteristically large wave height that might be expected with a certain small probability during the lifetime of the structure. In the procedure of wave extreme prediction, two major statistical issues can be distinguished. The first issue is that predicted extreme values, for let us say 100 years, must be based on data collected in a relatively short time, for example 10 years. It is understandable that reliable data collected over 100 years will certainly not be available. The second issue is extrapolation of the observed data into its extreme region, typically lying well beyond even the most extreme of the available observation. Extremes are scarce by definition and the main difficulty in dealing with any extreme value Corresponding author. Tel.: +385 51 651 452; fax: +385 51 675-818.
E-mail address: [email protected] (J. Prpic´-Orsˇ ic´). 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.11.016
analysis is the lack of needed data. Asymptotic theory suggests that such maxima are well modeled by generalized extreme value distributions that were originally developed by Fisher and Tippet and later systematized by Gumbel (Chakrabarti, 2001). An important requirement to an adequate analysis method is that the short-term (conditional) exceedance probabilities are consistently accumulated into a resulting long-term (marginal) exceedance probability, so it is important to find the distribution that covers the data best. 2. Wave environment prediction 2.1. Extreme wave height of an extreme sea state—design sea state method In its general form, the All Sea State (ASS) method requires the complete probability density function of the response for each specific condition, that is, each combination of speed, heading, wave spectrum and sea state. The short-term probability density function has to be calculated for all wave direction intervals and sea state intervals defined by the significant wave height and the wave period. The prediction of the characteristics of long-term extreme
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values deals with the occurrence of rare events as opposed to the short-term statistics which determine the normal deviations. The accuracy in estimating the characteristics of small or moderate sea states, or in calculating the ship response at these levels, could introduce an appreciable error into the estimation. Therefore, for this reason as well as for savings in computation, for the majority of ship design application, the Design Sea State (DSS) method is more appropriate. The local storm conditions are described by the return periods associated with a particular environmental condition. The return period TR is defined as the average interval of time, normally in years, in which the condition is exceeded (Mansour and Priston, 1995). According to DSS method, adequate significant wave height and zero crossing period define an extreme sea state. The most common approach is to model long-term marginal distribution of significant wave height H1/3 and the same for characteristic periods. The first of these is the most important one. The probabilities for the significant wave height are known from the wave scatter diagram, and the total number of observations n during return period can be obtained from the total number of observations during observation time. The most probable extreme value expected to occur in n observations, xn, can be evaluated from the initial cumulative distribution function F(xn): F ðxn Þ 1
1 N 50 years
(1)
and the given return period (for example 50 years), as the extreme value that is most likely to occur during return period. This value is typically lying well beyond even the most extreme of the available observation. The problem is thus one of extrapolation of the observed distribution of data (collected over a relatively short period of, say, 10 years) into its extreme region. In offshore calculations, an extreme sea state with a return period of 50 years, lasting 3 h is generally considered. The resulting extreme wave amplitude in this sea state can be calculated from Rayleigh distribution: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi za;maxð50 years stormÞ ¼ 2m0z lnðNÞ, (2) where m0z is the wave energy zero moment or variance of the process. The number of cycles N during 3 h storm can be determined by using an average wave period belonging to the extreme significant wave height as presented by Ochi (1978, 1998) and Hughes (1983). However, the probability that the largest peak value may exceed the probable extreme value is quite large (E0.632), and hence it is not appropriate to use this value for engineering design. For the purposes of structural design, an extreme value for which the probability of being exceeded is some acceptably small value must be obtained. This small probability value, a, is called risk parameter. Although the decision regarding which probability value should be taken for design purposes depends on the designer’s own judgment; a risk parameter value of 0.01–0.05 is often considered as a standard design practice. If S is the storm duration in
149
hours, the maximum amplitude, with probability of exceedance a, can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi N 3600 S ¼ 2m0z ln . za;maxð50 years stormÞ ¼ 2m0z ln a aT z (3) There are various approaches of estimating the extreme wave period associated with the design wave height which has been obtained (Sarpkaya and Isaacson, 1981). The first is to repeat the entire procedure using wave periods instead of wave heights as statistics. An alternative approach involves using the predicted wave height to set a lower limit TL to the wave period (due to wave breaking), sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32pH s , (4) T zL ¼ g where g is gravity acceleration, and Hs is significant wave height, and then using a series of values of wave period above this lower limit to find the worst possible effect on the structure. 2.2. Extreme value probability distribution The long-term probability could be obtained by the above method of order statistics. This method, however, is limited by the lack of long-term wave data. Often, one takes the help of a theoretical probability distribution function. The process of extrapolation plays a fundamental role in this area of analysis and therefore it is essential to empirically fit a convenient probability distribution that describes the available data as closely possible. There are several such formulas available (Soares et al., 1996). These include the lognormal distribution and the Extremal Types I, II and III known also as Gumbel (Fisher–Tippett type I), Frechet (Fisher–Tippett type II) and Weibull (Fisher–Tippett type III) probability distribution. Although these all have a theoretical base, they are used here essentially as an empirical fit to the data. The lognormal and Weibull distributions seem to fit best the empirical wave data. The lognormal distribution has been fitted to many wave data with a varying degree of success. The cumulative probability distribution is given as ln H s mln H s F ðH s Þ ¼ F ¼ FðuÞ, (5) sln H s where gamma function is given as Z x 1 x2 exp FðxÞ ¼ pffiffiffiffiffiffi dx, 2 2p 1
(6)
mln H s and sln H s are, respectively, the mean value and the standard deviation of significant wave heights natural logarithm. The line u can be expressed as u¼
1 sln H s
ln H s
mln H s , sln H s
(7)
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where mln H s =sln H s is the intercept and 1=sln H s is the slope. The 3-parameter Weibull probability distribution can be expressed as Hs e a F ðH s Þ ¼ 1 exp , (8) y where e represents location parameter, y is scaling parameter and a is shape parameter. The accuracy of extreme value prediction is significantly affected by the choice of these parameters. If the above expression is rearranged, it may be written as a linear equation: lnð lnð1 F ðH s ÞÞÞ ¼ a lnðH s eÞ a ln y,
(9)
where a ln y is the intercept and a is the slope. One way to represents data points precisely is to express the cumulative distribution with 4-parameter function developed by Ochi (1998) and Mansour (1990) in the form F ðxÞ ¼ 1 eqðxÞ k
qðxÞ ¼ axm epx ,
(10) (11)
where a, m, p and k are distribution parameters which have to be estimated according to empirical data. 2.3. Probability distribution parameters estimation Having selected one distribution as a likely model, it remains to estimate the parameter values that will provide the best empirical fit between the distribution and the data. The most straightforward approach is to plot the individual data points on the selected probability paper and then draw a straight line through these by eye. The paper is usually constructed by taking the logarithm of the cumulative distribution function twice. Therefore, a small difference between data and cumulative distribution function drawn on the probability paper may result in a substantial difference between the histogram and probability density function. Another way to plot the data is to take logarithm of the return period. In the case of Weibull distributions, this would apply only if a or e were chosen in advance. Since Weibull distribution includes all three empirical coefficients, this increases the adaptability of this probability distribution to fitting empirical data. But it also increases the effort required to fit the data, as one of the three parameters must be estimated before the data can be plotted (Mansour, 1990; Mansour and Priston, 1995; Parunov and Senjanovic´, 2000). One of the possibilities is to presume location parameter e ¼ 0 which corresponds to 2-parameter Weibull distribution and find a best fit line by one of the three possible methods: the method of moments, the method of least squares and the method of maximum likelihood (Oliver and Ochi, 1981; Ochi, 1998). The other way is to assume the value of shape parameter a. This parameter is then reestimated until the best fit is obtained (for example a ¼ 0.75, 1.0, 1.4 and 2.0). All of these methods presume that one parameter is chosen in advance
and the other two are then estimated in a way that obtained cumulative distribution displays the reasonably fit to majority of the data. The parameters involved in 4-parameter Ochi distribution can be determined numerically by a nonlinear minimization procedure. The extreme value can then be estimated based on this probability function. 2.4. Genetic algorithm application Another way to search for the fittest distribution is a genetic algorithm (GA) (Coit and Smith, 2002; Strelen, 2003). The objective is to minimize the discrepancies between the empirical probability distribution and the probability distribution with randomly generated parameters. The GAs are evolutionary problem-solving techniques based on the Darwinian principle of ‘‘survival of the fittest’’ to obtain efficient solutions. The basic idea behind GAs is to extract features from a population of individuals, subjected to the existing environment, which can yield better performance. GAs are extremely flexible optimization tools because they are able to avoid local optima as they start from multiple points. They can handle nonlinearities and constraints and provide an efficient search of large spaces using only a suitable representation of the problem and a ‘‘black-box’’ evaluation mechanism. The simple GA works as follows. Initially, a population of individuals of a fixed size is randomly generated. Each individual is characterized by a string of genes and represents one possible solution to the problem that has been tackled. Strings can then be evaluated, giving a measure of how good the solutions are. This is known as the fitness value. Using the fitness value, the population undergoes the natural selection. The strings that are better suited have more chance of surviving than the other weaker ones. The surviving individuals are then paired up randomly and subjected to crossover with probability pc to form new offspring strings. The idea is that the good properties from the two parents will combine to form an even better offspring. This population is then mutated with probability pm where small random changes are made to the offspring in order to maintain diversity. The newly generated strings are then reevaluated and given a fitness score. The process repeats until stopped, usually after a fixed number of generations. The best strings found can then be used as near-optimal solutions to the problem. The GA used in this study fundamentally followed the simple GA, described by Goldberg (1989). The three most important aspects of using GA are:
definition of the objective function, definition and implementation of the genetic representation, definition and implementation of the genetic operators.
A noninteractive procedure has been developed to minimize a measure for the accuracy, which has been
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chosen as the objective function: X ZðH S Þ ¼ ½F ðH s Þ F^ ðH s Þ2 .
The representation or coding of the variables has a considerable impact on the search performance, as the optimization is performed on this representation of the variables. The two most common representations, binary and real number coding, differ mainly in how the recombination and mutation operators are performed. Having experience with both coding approaches (Dejhalla et al., 2002), for the presented problem, we have chosen the real coding. Because the mutation operator is the only one that creates new variable values in the real coding approach, the mutation probability pm must be taken as a much greater value than in the binary coding. To transform a minimization problem to a maximization problem needed for the GA procedure, it is necessary to map the objective function to a fitness function form (socalled ‘‘raw fitness’’) through one or more mappings. The raw fitness must be scaled in order to help the GA maintain diversity between very similar individuals.
(12)
Hs
The measure for the accuracy is based on selected distribution function F(Hs) and empirical distribution function F^ ðH s Þ for parameter tuple (significant wave height range) in order to fit empirical data best using values within parameter space. Table 1 GA settings
Chromosome length Population size Number of generations Number of evaluations
Weibull model
Ochi model
27 binary bits 120 individuals 120 14 400
36 binary bits 120 individuals 120 14 400
151
Table 2 Significant wave height distribution according to SHIPREL data Total
0.87
12.4
62.6
166.1
254.3
242.7
155.7
71.03
25.4
7.86
0.96
1000
11–12 10–11 9–10 8–9 7–8 6–7 5–6 4–5 3–4 2–3 1–2 0–1 Hs, m
— — — — — — — — — 0.088 0.17 0.62 o4
— — — — — — 0.17 0.29 0.72 1.5 3.1 6.6 4–5
— — — — 0.17 0.35 0.68 1.9 4.5 10.0 20.0 25.0 5–6
— — 0.17 0.22 0.43 1.4 3.0 6.9 17.0 37.0 63.0 37.0 6–7
— — 0.17 0.76 1.5 3.5 8.4 20.0 42.0 74.0 82.0 22.0 7–8
— — 0.76 1.68 3.4 7.0 15.0 31.0 54.0 72.0 51.0 6.9 8–9
— 0.77 1.08 1.95 4.4 8.3 16.0 27.0 39.0 38.0 18.0 1.2 9–10
0.5 0.63 1.02 1.88 3.4 5.8 9.8 14.0 17.0 13.0 4.0 — 10–11
— 0.43 0.67 1.2 1.8 2.7 4.0 5.3 4.9 3.3 1.1 — 11–12
— — 0.3 0.44 0.72 1.1 1.5 1.5 1.4 0.9 — — 12–13
— — — — 0.16 0.32 0.32 0.16 — — — — 13–14 Tz, s
0.5 1.83 4.17 8.132 15.98 30.47 58.87 108.05 180.52 249.79 242.37 99.32 Total
Cumulative distribution function CFD SHIPREL 1.00
P (Hs)
0.75
0.50
0.25
0.00 0.0
Empirical data Weibull distribution Lognormal distribution 2.5
5.0
7.5
10.0 Hs, m
12.5
15.0
17.5
20.0
Fig. 1. Cumulative probability distribution of significant wave height (lognormal and Weibull distributions).
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The results presented in this study have been carried out by means of GA (Goodman, 1996) employing
3. Predicted extreme values in the North Atlantic sea The application of the computational methods is given for the SHIPREL (reliability methods for ship structural design) wave scatter diagram that describes the North Atlantic sea environment given in Table 2 (Hogben et al., 1986, Soares et al., 1996). This scatter diagram, recommended by a consortium of maritime classification societies and university research groups, describes the North Atlantic sea environment combining information from zones 8, 9, 11, 15, 16 and 17 which are weighted proportionally to their size so as to reflect the percentage of time that ships crossing the Atlantic would spend in each one. Data in the table represents the collected
the linear scaling of the raw fitness, the stochastic uniform sampling as selection operator, the two-point crossover as crossover operator, the multi-bit mutation as mutation operator, crossover probability pc ¼ 0.5, mutation probability pm ¼ 0.3.
a,max,m
In the procedure, the GA settings given in Table 1 have been used.
28.0 27.0 26.0 25.0 24.0 23.0 22.0 21.0 20.0 19.0 18.0 17.0 16.0 15.0 14.0 13.0 12.0 11.0 10.0
Extreme wave amplitude (3 hour storm) SHIPREL =0.01 =0.05 =0.1 =1.0
=0.01 =0.05 =0.1 =1.0 Weibull model Lognormal model
0
10
20
30
40 70 50 60 Return period, years
80
90
100
Fig. 2. Extreme wave amplitude (lognormal and Weibull distributions).
0.4
0.6 0.8 1
Hs -, m 2 4
6
8 10
20 1
3.00
Weibull distribution adapted to empirical data
2.00 1.00 0.1
-1.00
P (Hs )
ln (-ln(1-P(Hs)))
0.00
-2.00 -3.00 0.01 -4.00 Empirical data -5.00
GA method of data fitting
-6.00
Aproximative method
-7.00 -1.0
0.001 0.0
1.0 ln (Hs -), m
2.0
3.0
Fig. 3. Weibull probability distribution of significant wave height.
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return period and risk parameter. The similar analyses have been done for the 4-parameter Ochi distribution. Comparison has been made between two approaches of obtaining extreme wave amplitudes: 3-parameter Weibull model and 4-parameter Ochi model. In both cases, parameters are obtained by GA approach. In Fig. 7 and 8, the resulting CFD curves obtained by Ochi and Weibull models are compared with empirical data. The extreme wave amplitudes of 3-h storm, for different return periods and different risk parameters, obtained by Weibull and Ochi models are compared in Fig. 9.
observations distribution of the significant wave height and zero crossing periods. The CFD curves obtained by lognormal and Weibull models are compared with empirical data in Fig. 1. The extreme wave amplitudes of 3-h storm obtained by lognormal and Weibull models are compared in Fig. 2. The line fitting and the resulting CFD curves for Weibull model derived from GA compared with traditional Weibull model line and empirical data are presented in Figs. 3 and 4. The extreme wave amplitudes of 3-h storm obtained by Weibull model derived from GA and traditional Weibull model are compared in Fig. 5. The amplitudes at extreme sea state are computed for different return periods and different risk parameters. The extreme wave amplitudes obtained by ASS and DSS methods using Weibull distribution with GA optimized parameters are shown in Fig. 6 as a function of
4. Conclusions The difference in extreme wave amplitudes obtained by two different methods, All Sea State and Design Sea State
Cumulative distribution function CDF SHIPREL 1.00
P (Hs )
0.75
0.50
Empirical data Aproximative method GA method of data fitting
0.25
0.00 0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
Hs, m Fig. 4. Weibull cumulative probability distribution of significant wave height.
20.0
Extreme wave amplitude (3 hour storm) SHIPREL
19.0 18.0 17.0 a,max,m
16.0
=0.01
=0.01
=0.05 =0.1
=0.05 =0.1
=1.0
=1.0
15.0 14.0 13.0 Traditional method for Weibull distribution Genetic algorithm model
12.0 11.0 10.0 0
10
20
30
40
50
60
Return period, years Fig. 5. Extreme wave amplitude.
70
80
90
100
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20.0 Extreme wave amplitude SHIPREL
19.0 18.0 =0.01
a,max,m
17.0 16.0
=0.05
15.0
=0.1
=0.01 =0.05 =0.1 =1.0
14.0 =1.0
13.0 12.0
All Sea State method Design Sea State method (3 hours storm)
11.0 10.0 0
10
20
30
50 60 40 Return period, years
70
80
90
100
Fig. 6. Extreme wave amplitude (ASS and DSS methods).
0.4
0.6 0.8 1
2
Hs , m 4
6
8 10
20 1
3.00
Weibull and Ochi distribution adapted to empirical data
2.00
0.1
0.00 -1.00
P (Hs)
ln (-ln(1-P(Hs )))
1.00
-2.00 -3.00
0.01
-4.00
Epirical data
-5.00
GA Weibull distribution
-6.00
GA Ochi distribution
-7.00 -1.0
0.001 0.0
1.0 ln (Hs), m
2.0
3.0
Fig. 7. Four-parameter probability distribution of significant wave height.
methods, is notable especially for higher risk parameters. For SHIPREL, scatter diagram difference between extreme wave amplitudes for 100-year return period and for risk parameter value of 0.01 is more then 1 m. A ship spends most of the time on small or moderate seas and therefore accuracy in estimating the characteristics of small or moderate sea states, or in calculating the ship response at these levels, could introduce an appreciable error into the estimation when using All Sea State method. For these reasons, as well as for savings in computation, the Design Sea State is more appropriate for the majority of ship design application. Also, from the safety point of view, the Design Sea State method can be recommended because of higher extreme values estimation.
The extreme wave amplitudes obtained by lognormal distribution are significantly different from those obtained by Weibull distribution. For example for return period of 50 years, the difference in extreme wave amplitude during 3-h storm is more than 5 m for risk parameter 1.0. The discrepancies increase as the return period increases and as risk parameter decreases. So, for the return period of 100 years and for risk parameter value of 0.01, the difference becomes even 9 m. However, the Weibull distribution seems to cover the empirical data better, which is understandable because of possibility of 3-parameter tuning. The extreme wave heights obtained by application of genetic algorithm (GA) procedure of estimating Weibull parameters are slightly lower than results obtained by
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Cumulative distribution function CDF SHIPREL 1.00
P (Hs )
0.75
0.50
Empirical data GA Ochi model GA Weibull model
0.25
0.00 0.0
2.5
7.5
5.0
10.0
12.5
15.0
17.5
20.0
Hs, m Fig. 8. Four-parameter probability distribution of significant wave height.
20.0
Extreme wave amplitude (3 hour storm) SHIPREL =0.01 =0.01 =0.05
19.0 18.0
a,max ,m
17.0
=0.05
=0.1
16.0
=0.1
15.0 =1.0
14.0
=1.0
13.0 12.0 GA Ochi distribution GA Weibull distribution
11.0 10.0 0
10
20
30
40 50 60 70 Return period, years
80
90
100
Fig. 9. Extreme wave amplitude obtained by 4-parameter Ochi distribution.
least-squares and maximum likelihood approach. The difference in extreme wave amplitude during 3-h storm is nearly 0.5 m for all risk parameters and for all return periods longer than 5 years. The discrepancies slightly increase as the return period increases and as risk parameter decreases. So, for the return period of 100 years and for risk parameter value of 0.01, the difference becomes 0.6 m. The Weibull distribution obtained by GA approach cover the empirical data better because of simultaneous tuning of all parameters, whereas for traditional approach one of the parameters is predefined. Ochi 4-parameter distribution, which has the possibility to set four parameters in order to obtain the best fit curve, covers the empirical data even better. For the return period of 100 years and for risk parameter value of 0.01, the extreme
wave amplitude obtained by Weibull distribution is 17.6 m while the same obtained by Ochi distribution is 19.0 m. The use of GA optimization procedures for estimating parameters of distribution in order to predict an extreme design wave on the basis of recorded or hindcast data is well suited and can be generalized to consider a broader range of problems. GA is uniform for all theoretical distributions; hence, it is applicable for multi-mode and mixed distributions as well as for the selection of a theoretical distribution that covers the given data best. For the estimation of extreme values, it is highly desirable to represent the data precisely by a certain probability distribution over the entire range of the cumulative distribution and, generally speaking, that can be better obtained if the distribution is multi-parameter. The main
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advantage of GA optimization procedures application in estimation of distribution parameters is that the procedure remains the same if the number of parameters is greater, only the range of new parameters must be estimated. Acknowledgment The research presented in this paper was made possible by the financial support of the Ministry of Science of the Republic of Croatia, under the Project no. 0069-007. References Chakrabarti, S.K., 2001. Hydrodynamics of Offshore Structures. Computational Mechanics Publications, WIT Press, Southampton. Coit, D.W., Smith, A.E., 2002. Genetic algorithm to maximize a lowerbound for system time-to-failure with uncertain component Weibull parameters. Computers & Industrial Engineering 41. Dejhalla, R., Mrsˇ a, Z., Vukovic´, S., 2002. A genetic algorithm approach to the problem of minimum ship wave resistance. Marine Technology 39 (3), 187–195. Goldberg, E.D., 1989. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley Longman, Reading, MA. Goodman, E.D., 1996. An Introduction to GALLOPS—the ‘‘Genetic Algorithm Optimized for Portability and Parallelism’’ system. Release
3.2, Technical Report #96-07-01, Michigan State University, East Lansing. Hogben, N., Dacunha, N.M.C., Olliver, G.F., 1986. Global Wave Statistics. British Maritime Technology Ltd., Feltham. Hughes, O.F., 1983. Ship Structural Design—A Rationally Based, Computer Aided Optimization Approach. Wiley, New York. Mansour, A.E., 1990. An Introduction to Structural Reliability Theory. Ship Structure Committee SSC-351, Washington. Mansour, A.E., Priston, D.B., 1995. Return periods and encounter probability. Applied Ocean Research 17, 127–136. Ochi, M.K., 1978. Wave statistics for the design of ships and ocean structures. SNAME Transactions 86, 47–76. Ochi, M.K., 1998. Ocean Waves—The Stochastic Approach. Cambridge Ocean Technology Series 6, Cambridge. Oliver, J.C., Ochi, M.K., 1981. Evaluation of SL-7 Scratch Gauge Data. Ship Structure Committee, SSC-311, Washington. Parunov, J., Senjanovic´, I., 2000. Methods for long term prediction of extreme sea states. Journal of Naval Architecture and Shipbuilding Industry (Journal Brodogradnja) 48 (2) (in Croatian). Sarpkaya, T., Isaacson, M., 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold Company Inc., New York. Soares, C.G., Dogliani, M., Ostergaard, O., Permentier, G., Pedersen, P.T., 1996. Reliability based ship structural design. SNAME Transactions 104, 357–389. Strelen, J.Ch., 2003. The genetic algorithm is useful to fitting input probability distributions for simulation models. In: Proceedings of the Business and Industry Symposium—ASTC 2003, Society for Computer Simulation, San Diego.
Ocean Engineering 34 (2007) 148–156 www.elsevier.com/locate/oceaneng
Design sea state assessment using genetic algorithm approach Jasna Prpic´-Orsˇ ic´, Roko Dejhalla, Anton Turk Department of Naval Architecture and Ocean Engineering, Faculty of Engineering, University of Rijeka, Vukovarska 58, HR-51000 Rijeka, Croatia Received 25 May 2005; accepted 16 November 2005 Available online 17 April 2006
Abstract Two major statistical issues can be distinguished in the procedure of wave extreme prediction. The first issue is that predicted extreme values must be based on data collected in a relatively short time. The second issue is extrapolation of the observed data into its extreme region, typically lying well beyond from even the most extreme available observation. The process of extrapolation plays a fundamental role in this area of analysis and therefore it is essential to fit empirically a convenient probability distribution that describes the available data as closely as possible. Determination of extreme values probability distribution parameters by genetic algorithm is applied to improve the methodology of extreme sea state prediction. Illustrative applications of the method are given for a North Atlantic sea environment. The results are presented as crest height maximum values occurring with a given probability or in a design storm that has a specified return period. r 2006 Elsevier Ltd. All rights reserved. Keywords: Long-term prediction; Wave extremes; Probability density function; Genetic algorithm
1. Introduction An important step often encountered in a design is estimation of an extreme design wave on the basis of recorded or hindcast data (Soares et al., 1996). This generally involves selecting and fitting a suitable probability distribution to wave height data, and extrapolating this to locate a suitable design wave, such as the so-called ‘‘50-year wave’’. This is characteristically large wave height that might be expected with a certain small probability during the lifetime of the structure. In the procedure of wave extreme prediction, two major statistical issues can be distinguished. The first issue is that predicted extreme values, for let us say 100 years, must be based on data collected in a relatively short time, for example 10 years. It is understandable that reliable data collected over 100 years will certainly not be available. The second issue is extrapolation of the observed data into its extreme region, typically lying well beyond even the most extreme of the available observation. Extremes are scarce by definition and the main difficulty in dealing with any extreme value Corresponding author. Tel.: +385 51 651 452; fax: +385 51 675-818.
E-mail address: [email protected] (J. Prpic´-Orsˇ ic´). 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.11.016
analysis is the lack of needed data. Asymptotic theory suggests that such maxima are well modeled by generalized extreme value distributions that were originally developed by Fisher and Tippet and later systematized by Gumbel (Chakrabarti, 2001). An important requirement to an adequate analysis method is that the short-term (conditional) exceedance probabilities are consistently accumulated into a resulting long-term (marginal) exceedance probability, so it is important to find the distribution that covers the data best. 2. Wave environment prediction 2.1. Extreme wave height of an extreme sea state—design sea state method In its general form, the All Sea State (ASS) method requires the complete probability density function of the response for each specific condition, that is, each combination of speed, heading, wave spectrum and sea state. The short-term probability density function has to be calculated for all wave direction intervals and sea state intervals defined by the significant wave height and the wave period. The prediction of the characteristics of long-term extreme
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values deals with the occurrence of rare events as opposed to the short-term statistics which determine the normal deviations. The accuracy in estimating the characteristics of small or moderate sea states, or in calculating the ship response at these levels, could introduce an appreciable error into the estimation. Therefore, for this reason as well as for savings in computation, for the majority of ship design application, the Design Sea State (DSS) method is more appropriate. The local storm conditions are described by the return periods associated with a particular environmental condition. The return period TR is defined as the average interval of time, normally in years, in which the condition is exceeded (Mansour and Priston, 1995). According to DSS method, adequate significant wave height and zero crossing period define an extreme sea state. The most common approach is to model long-term marginal distribution of significant wave height H1/3 and the same for characteristic periods. The first of these is the most important one. The probabilities for the significant wave height are known from the wave scatter diagram, and the total number of observations n during return period can be obtained from the total number of observations during observation time. The most probable extreme value expected to occur in n observations, xn, can be evaluated from the initial cumulative distribution function F(xn): F ðxn Þ 1
1 N 50 years
(1)
and the given return period (for example 50 years), as the extreme value that is most likely to occur during return period. This value is typically lying well beyond even the most extreme of the available observation. The problem is thus one of extrapolation of the observed distribution of data (collected over a relatively short period of, say, 10 years) into its extreme region. In offshore calculations, an extreme sea state with a return period of 50 years, lasting 3 h is generally considered. The resulting extreme wave amplitude in this sea state can be calculated from Rayleigh distribution: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi za;maxð50 years stormÞ ¼ 2m0z lnðNÞ, (2) where m0z is the wave energy zero moment or variance of the process. The number of cycles N during 3 h storm can be determined by using an average wave period belonging to the extreme significant wave height as presented by Ochi (1978, 1998) and Hughes (1983). However, the probability that the largest peak value may exceed the probable extreme value is quite large (E0.632), and hence it is not appropriate to use this value for engineering design. For the purposes of structural design, an extreme value for which the probability of being exceeded is some acceptably small value must be obtained. This small probability value, a, is called risk parameter. Although the decision regarding which probability value should be taken for design purposes depends on the designer’s own judgment; a risk parameter value of 0.01–0.05 is often considered as a standard design practice. If S is the storm duration in
149
hours, the maximum amplitude, with probability of exceedance a, can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi N 3600 S ¼ 2m0z ln . za;maxð50 years stormÞ ¼ 2m0z ln a aT z (3) There are various approaches of estimating the extreme wave period associated with the design wave height which has been obtained (Sarpkaya and Isaacson, 1981). The first is to repeat the entire procedure using wave periods instead of wave heights as statistics. An alternative approach involves using the predicted wave height to set a lower limit TL to the wave period (due to wave breaking), sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32pH s , (4) T zL ¼ g where g is gravity acceleration, and Hs is significant wave height, and then using a series of values of wave period above this lower limit to find the worst possible effect on the structure. 2.2. Extreme value probability distribution The long-term probability could be obtained by the above method of order statistics. This method, however, is limited by the lack of long-term wave data. Often, one takes the help of a theoretical probability distribution function. The process of extrapolation plays a fundamental role in this area of analysis and therefore it is essential to empirically fit a convenient probability distribution that describes the available data as closely possible. There are several such formulas available (Soares et al., 1996). These include the lognormal distribution and the Extremal Types I, II and III known also as Gumbel (Fisher–Tippett type I), Frechet (Fisher–Tippett type II) and Weibull (Fisher–Tippett type III) probability distribution. Although these all have a theoretical base, they are used here essentially as an empirical fit to the data. The lognormal and Weibull distributions seem to fit best the empirical wave data. The lognormal distribution has been fitted to many wave data with a varying degree of success. The cumulative probability distribution is given as ln H s mln H s F ðH s Þ ¼ F ¼ FðuÞ, (5) sln H s where gamma function is given as Z x 1 x2 exp FðxÞ ¼ pffiffiffiffiffiffi dx, 2 2p 1
(6)
mln H s and sln H s are, respectively, the mean value and the standard deviation of significant wave heights natural logarithm. The line u can be expressed as u¼
1 sln H s
ln H s
mln H s , sln H s
(7)
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where mln H s =sln H s is the intercept and 1=sln H s is the slope. The 3-parameter Weibull probability distribution can be expressed as Hs e a F ðH s Þ ¼ 1 exp , (8) y where e represents location parameter, y is scaling parameter and a is shape parameter. The accuracy of extreme value prediction is significantly affected by the choice of these parameters. If the above expression is rearranged, it may be written as a linear equation: lnð lnð1 F ðH s ÞÞÞ ¼ a lnðH s eÞ a ln y,
(9)
where a ln y is the intercept and a is the slope. One way to represents data points precisely is to express the cumulative distribution with 4-parameter function developed by Ochi (1998) and Mansour (1990) in the form F ðxÞ ¼ 1 eqðxÞ k
qðxÞ ¼ axm epx ,
(10) (11)
where a, m, p and k are distribution parameters which have to be estimated according to empirical data. 2.3. Probability distribution parameters estimation Having selected one distribution as a likely model, it remains to estimate the parameter values that will provide the best empirical fit between the distribution and the data. The most straightforward approach is to plot the individual data points on the selected probability paper and then draw a straight line through these by eye. The paper is usually constructed by taking the logarithm of the cumulative distribution function twice. Therefore, a small difference between data and cumulative distribution function drawn on the probability paper may result in a substantial difference between the histogram and probability density function. Another way to plot the data is to take logarithm of the return period. In the case of Weibull distributions, this would apply only if a or e were chosen in advance. Since Weibull distribution includes all three empirical coefficients, this increases the adaptability of this probability distribution to fitting empirical data. But it also increases the effort required to fit the data, as one of the three parameters must be estimated before the data can be plotted (Mansour, 1990; Mansour and Priston, 1995; Parunov and Senjanovic´, 2000). One of the possibilities is to presume location parameter e ¼ 0 which corresponds to 2-parameter Weibull distribution and find a best fit line by one of the three possible methods: the method of moments, the method of least squares and the method of maximum likelihood (Oliver and Ochi, 1981; Ochi, 1998). The other way is to assume the value of shape parameter a. This parameter is then reestimated until the best fit is obtained (for example a ¼ 0.75, 1.0, 1.4 and 2.0). All of these methods presume that one parameter is chosen in advance
and the other two are then estimated in a way that obtained cumulative distribution displays the reasonably fit to majority of the data. The parameters involved in 4-parameter Ochi distribution can be determined numerically by a nonlinear minimization procedure. The extreme value can then be estimated based on this probability function. 2.4. Genetic algorithm application Another way to search for the fittest distribution is a genetic algorithm (GA) (Coit and Smith, 2002; Strelen, 2003). The objective is to minimize the discrepancies between the empirical probability distribution and the probability distribution with randomly generated parameters. The GAs are evolutionary problem-solving techniques based on the Darwinian principle of ‘‘survival of the fittest’’ to obtain efficient solutions. The basic idea behind GAs is to extract features from a population of individuals, subjected to the existing environment, which can yield better performance. GAs are extremely flexible optimization tools because they are able to avoid local optima as they start from multiple points. They can handle nonlinearities and constraints and provide an efficient search of large spaces using only a suitable representation of the problem and a ‘‘black-box’’ evaluation mechanism. The simple GA works as follows. Initially, a population of individuals of a fixed size is randomly generated. Each individual is characterized by a string of genes and represents one possible solution to the problem that has been tackled. Strings can then be evaluated, giving a measure of how good the solutions are. This is known as the fitness value. Using the fitness value, the population undergoes the natural selection. The strings that are better suited have more chance of surviving than the other weaker ones. The surviving individuals are then paired up randomly and subjected to crossover with probability pc to form new offspring strings. The idea is that the good properties from the two parents will combine to form an even better offspring. This population is then mutated with probability pm where small random changes are made to the offspring in order to maintain diversity. The newly generated strings are then reevaluated and given a fitness score. The process repeats until stopped, usually after a fixed number of generations. The best strings found can then be used as near-optimal solutions to the problem. The GA used in this study fundamentally followed the simple GA, described by Goldberg (1989). The three most important aspects of using GA are:
definition of the objective function, definition and implementation of the genetic representation, definition and implementation of the genetic operators.
A noninteractive procedure has been developed to minimize a measure for the accuracy, which has been
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chosen as the objective function: X ZðH S Þ ¼ ½F ðH s Þ F^ ðH s Þ2 .
The representation or coding of the variables has a considerable impact on the search performance, as the optimization is performed on this representation of the variables. The two most common representations, binary and real number coding, differ mainly in how the recombination and mutation operators are performed. Having experience with both coding approaches (Dejhalla et al., 2002), for the presented problem, we have chosen the real coding. Because the mutation operator is the only one that creates new variable values in the real coding approach, the mutation probability pm must be taken as a much greater value than in the binary coding. To transform a minimization problem to a maximization problem needed for the GA procedure, it is necessary to map the objective function to a fitness function form (socalled ‘‘raw fitness’’) through one or more mappings. The raw fitness must be scaled in order to help the GA maintain diversity between very similar individuals.
(12)
Hs
The measure for the accuracy is based on selected distribution function F(Hs) and empirical distribution function F^ ðH s Þ for parameter tuple (significant wave height range) in order to fit empirical data best using values within parameter space. Table 1 GA settings
Chromosome length Population size Number of generations Number of evaluations
Weibull model
Ochi model
27 binary bits 120 individuals 120 14 400
36 binary bits 120 individuals 120 14 400
151
Table 2 Significant wave height distribution according to SHIPREL data Total
0.87
12.4
62.6
166.1
254.3
242.7
155.7
71.03
25.4
7.86
0.96
1000
11–12 10–11 9–10 8–9 7–8 6–7 5–6 4–5 3–4 2–3 1–2 0–1 Hs, m
— — — — — — — — — 0.088 0.17 0.62 o4
— — — — — — 0.17 0.29 0.72 1.5 3.1 6.6 4–5
— — — — 0.17 0.35 0.68 1.9 4.5 10.0 20.0 25.0 5–6
— — 0.17 0.22 0.43 1.4 3.0 6.9 17.0 37.0 63.0 37.0 6–7
— — 0.17 0.76 1.5 3.5 8.4 20.0 42.0 74.0 82.0 22.0 7–8
— — 0.76 1.68 3.4 7.0 15.0 31.0 54.0 72.0 51.0 6.9 8–9
— 0.77 1.08 1.95 4.4 8.3 16.0 27.0 39.0 38.0 18.0 1.2 9–10
0.5 0.63 1.02 1.88 3.4 5.8 9.8 14.0 17.0 13.0 4.0 — 10–11
— 0.43 0.67 1.2 1.8 2.7 4.0 5.3 4.9 3.3 1.1 — 11–12
— — 0.3 0.44 0.72 1.1 1.5 1.5 1.4 0.9 — — 12–13
— — — — 0.16 0.32 0.32 0.16 — — — — 13–14 Tz, s
0.5 1.83 4.17 8.132 15.98 30.47 58.87 108.05 180.52 249.79 242.37 99.32 Total
Cumulative distribution function CFD SHIPREL 1.00
P (Hs)
0.75
0.50
0.25
0.00 0.0
Empirical data Weibull distribution Lognormal distribution 2.5
5.0
7.5
10.0 Hs, m
12.5
15.0
17.5
20.0
Fig. 1. Cumulative probability distribution of significant wave height (lognormal and Weibull distributions).
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The results presented in this study have been carried out by means of GA (Goodman, 1996) employing
3. Predicted extreme values in the North Atlantic sea The application of the computational methods is given for the SHIPREL (reliability methods for ship structural design) wave scatter diagram that describes the North Atlantic sea environment given in Table 2 (Hogben et al., 1986, Soares et al., 1996). This scatter diagram, recommended by a consortium of maritime classification societies and university research groups, describes the North Atlantic sea environment combining information from zones 8, 9, 11, 15, 16 and 17 which are weighted proportionally to their size so as to reflect the percentage of time that ships crossing the Atlantic would spend in each one. Data in the table represents the collected
the linear scaling of the raw fitness, the stochastic uniform sampling as selection operator, the two-point crossover as crossover operator, the multi-bit mutation as mutation operator, crossover probability pc ¼ 0.5, mutation probability pm ¼ 0.3.
a,max,m
In the procedure, the GA settings given in Table 1 have been used.
28.0 27.0 26.0 25.0 24.0 23.0 22.0 21.0 20.0 19.0 18.0 17.0 16.0 15.0 14.0 13.0 12.0 11.0 10.0
Extreme wave amplitude (3 hour storm) SHIPREL =0.01 =0.05 =0.1 =1.0
=0.01 =0.05 =0.1 =1.0 Weibull model Lognormal model
0
10
20
30
40 70 50 60 Return period, years
80
90
100
Fig. 2. Extreme wave amplitude (lognormal and Weibull distributions).
0.4
0.6 0.8 1
Hs -, m 2 4
6
8 10
20 1
3.00
Weibull distribution adapted to empirical data
2.00 1.00 0.1
-1.00
P (Hs )
ln (-ln(1-P(Hs)))
0.00
-2.00 -3.00 0.01 -4.00 Empirical data -5.00
GA method of data fitting
-6.00
Aproximative method
-7.00 -1.0
0.001 0.0
1.0 ln (Hs -), m
2.0
3.0
Fig. 3. Weibull probability distribution of significant wave height.
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return period and risk parameter. The similar analyses have been done for the 4-parameter Ochi distribution. Comparison has been made between two approaches of obtaining extreme wave amplitudes: 3-parameter Weibull model and 4-parameter Ochi model. In both cases, parameters are obtained by GA approach. In Fig. 7 and 8, the resulting CFD curves obtained by Ochi and Weibull models are compared with empirical data. The extreme wave amplitudes of 3-h storm, for different return periods and different risk parameters, obtained by Weibull and Ochi models are compared in Fig. 9.
observations distribution of the significant wave height and zero crossing periods. The CFD curves obtained by lognormal and Weibull models are compared with empirical data in Fig. 1. The extreme wave amplitudes of 3-h storm obtained by lognormal and Weibull models are compared in Fig. 2. The line fitting and the resulting CFD curves for Weibull model derived from GA compared with traditional Weibull model line and empirical data are presented in Figs. 3 and 4. The extreme wave amplitudes of 3-h storm obtained by Weibull model derived from GA and traditional Weibull model are compared in Fig. 5. The amplitudes at extreme sea state are computed for different return periods and different risk parameters. The extreme wave amplitudes obtained by ASS and DSS methods using Weibull distribution with GA optimized parameters are shown in Fig. 6 as a function of
4. Conclusions The difference in extreme wave amplitudes obtained by two different methods, All Sea State and Design Sea State
Cumulative distribution function CDF SHIPREL 1.00
P (Hs )
0.75
0.50
Empirical data Aproximative method GA method of data fitting
0.25
0.00 0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
Hs, m Fig. 4. Weibull cumulative probability distribution of significant wave height.
20.0
Extreme wave amplitude (3 hour storm) SHIPREL
19.0 18.0 17.0 a,max,m
16.0
=0.01
=0.01
=0.05 =0.1
=0.05 =0.1
=1.0
=1.0
15.0 14.0 13.0 Traditional method for Weibull distribution Genetic algorithm model
12.0 11.0 10.0 0
10
20
30
40
50
60
Return period, years Fig. 5. Extreme wave amplitude.
70
80
90
100
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20.0 Extreme wave amplitude SHIPREL
19.0 18.0 =0.01
a,max,m
17.0 16.0
=0.05
15.0
=0.1
=0.01 =0.05 =0.1 =1.0
14.0 =1.0
13.0 12.0
All Sea State method Design Sea State method (3 hours storm)
11.0 10.0 0
10
20
30
50 60 40 Return period, years
70
80
90
100
Fig. 6. Extreme wave amplitude (ASS and DSS methods).
0.4
0.6 0.8 1
2
Hs , m 4
6
8 10
20 1
3.00
Weibull and Ochi distribution adapted to empirical data
2.00
0.1
0.00 -1.00
P (Hs)
ln (-ln(1-P(Hs )))
1.00
-2.00 -3.00
0.01
-4.00
Epirical data
-5.00
GA Weibull distribution
-6.00
GA Ochi distribution
-7.00 -1.0
0.001 0.0
1.0 ln (Hs), m
2.0
3.0
Fig. 7. Four-parameter probability distribution of significant wave height.
methods, is notable especially for higher risk parameters. For SHIPREL, scatter diagram difference between extreme wave amplitudes for 100-year return period and for risk parameter value of 0.01 is more then 1 m. A ship spends most of the time on small or moderate seas and therefore accuracy in estimating the characteristics of small or moderate sea states, or in calculating the ship response at these levels, could introduce an appreciable error into the estimation when using All Sea State method. For these reasons, as well as for savings in computation, the Design Sea State is more appropriate for the majority of ship design application. Also, from the safety point of view, the Design Sea State method can be recommended because of higher extreme values estimation.
The extreme wave amplitudes obtained by lognormal distribution are significantly different from those obtained by Weibull distribution. For example for return period of 50 years, the difference in extreme wave amplitude during 3-h storm is more than 5 m for risk parameter 1.0. The discrepancies increase as the return period increases and as risk parameter decreases. So, for the return period of 100 years and for risk parameter value of 0.01, the difference becomes even 9 m. However, the Weibull distribution seems to cover the empirical data better, which is understandable because of possibility of 3-parameter tuning. The extreme wave heights obtained by application of genetic algorithm (GA) procedure of estimating Weibull parameters are slightly lower than results obtained by
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Cumulative distribution function CDF SHIPREL 1.00
P (Hs )
0.75
0.50
Empirical data GA Ochi model GA Weibull model
0.25
0.00 0.0
2.5
7.5
5.0
10.0
12.5
15.0
17.5
20.0
Hs, m Fig. 8. Four-parameter probability distribution of significant wave height.
20.0
Extreme wave amplitude (3 hour storm) SHIPREL =0.01 =0.01 =0.05
19.0 18.0
a,max ,m
17.0
=0.05
=0.1
16.0
=0.1
15.0 =1.0
14.0
=1.0
13.0 12.0 GA Ochi distribution GA Weibull distribution
11.0 10.0 0
10
20
30
40 50 60 70 Return period, years
80
90
100
Fig. 9. Extreme wave amplitude obtained by 4-parameter Ochi distribution.
least-squares and maximum likelihood approach. The difference in extreme wave amplitude during 3-h storm is nearly 0.5 m for all risk parameters and for all return periods longer than 5 years. The discrepancies slightly increase as the return period increases and as risk parameter decreases. So, for the return period of 100 years and for risk parameter value of 0.01, the difference becomes 0.6 m. The Weibull distribution obtained by GA approach cover the empirical data better because of simultaneous tuning of all parameters, whereas for traditional approach one of the parameters is predefined. Ochi 4-parameter distribution, which has the possibility to set four parameters in order to obtain the best fit curve, covers the empirical data even better. For the return period of 100 years and for risk parameter value of 0.01, the extreme
wave amplitude obtained by Weibull distribution is 17.6 m while the same obtained by Ochi distribution is 19.0 m. The use of GA optimization procedures for estimating parameters of distribution in order to predict an extreme design wave on the basis of recorded or hindcast data is well suited and can be generalized to consider a broader range of problems. GA is uniform for all theoretical distributions; hence, it is applicable for multi-mode and mixed distributions as well as for the selection of a theoretical distribution that covers the given data best. For the estimation of extreme values, it is highly desirable to represent the data precisely by a certain probability distribution over the entire range of the cumulative distribution and, generally speaking, that can be better obtained if the distribution is multi-parameter. The main
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advantage of GA optimization procedures application in estimation of distribution parameters is that the procedure remains the same if the number of parameters is greater, only the range of new parameters must be estimated. Acknowledgment The research presented in this paper was made possible by the financial support of the Ministry of Science of the Republic of Croatia, under the Project no. 0069-007. References Chakrabarti, S.K., 2001. Hydrodynamics of Offshore Structures. Computational Mechanics Publications, WIT Press, Southampton. Coit, D.W., Smith, A.E., 2002. Genetic algorithm to maximize a lowerbound for system time-to-failure with uncertain component Weibull parameters. Computers & Industrial Engineering 41. Dejhalla, R., Mrsˇ a, Z., Vukovic´, S., 2002. A genetic algorithm approach to the problem of minimum ship wave resistance. Marine Technology 39 (3), 187–195. Goldberg, E.D., 1989. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley Longman, Reading, MA. Goodman, E.D., 1996. An Introduction to GALLOPS—the ‘‘Genetic Algorithm Optimized for Portability and Parallelism’’ system. Release
3.2, Technical Report #96-07-01, Michigan State University, East Lansing. Hogben, N., Dacunha, N.M.C., Olliver, G.F., 1986. Global Wave Statistics. British Maritime Technology Ltd., Feltham. Hughes, O.F., 1983. Ship Structural Design—A Rationally Based, Computer Aided Optimization Approach. Wiley, New York. Mansour, A.E., 1990. An Introduction to Structural Reliability Theory. Ship Structure Committee SSC-351, Washington. Mansour, A.E., Priston, D.B., 1995. Return periods and encounter probability. Applied Ocean Research 17, 127–136. Ochi, M.K., 1978. Wave statistics for the design of ships and ocean structures. SNAME Transactions 86, 47–76. Ochi, M.K., 1998. Ocean Waves—The Stochastic Approach. Cambridge Ocean Technology Series 6, Cambridge. Oliver, J.C., Ochi, M.K., 1981. Evaluation of SL-7 Scratch Gauge Data. Ship Structure Committee, SSC-311, Washington. Parunov, J., Senjanovic´, I., 2000. Methods for long term prediction of extreme sea states. Journal of Naval Architecture and Shipbuilding Industry (Journal Brodogradnja) 48 (2) (in Croatian). Sarpkaya, T., Isaacson, M., 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold Company Inc., New York. Soares, C.G., Dogliani, M., Ostergaard, O., Permentier, G., Pedersen, P.T., 1996. Reliability based ship structural design. SNAME Transactions 104, 357–389. Strelen, J.Ch., 2003. The genetic algorithm is useful to fitting input probability distributions for simulation models. In: Proceedings of the Business and Industry Symposium—ASTC 2003, Society for Computer Simulation, San Diego.