Proceedings of the Eleventh (2001) International Offshore and Polar Engineering Conference Stavanger, Norway, June 17-22, 2001 Copyright © 2001 by The International Society of Offshore and Polar Engineers ISBN 1-880653-51-6 (Set); ISBN 1-880653-54-O(VoL I11); ISSN 1098-6189 (SeO

Joint Distribution for Wind and Waves in the Northern North Sea

Kenneth Johannessen, Trond Stokka Meling and Sverre Hayer Statoil

Stavanger, Norway

ABSTRACT For design purposes, it has been common to estimate the 100-year response by exposing the structure to the simultaneous action of 100-year wind, 100-year wave and 10-year current. Present design codes, see e.g. NORSOK N-003, recommend a less conservative approach by stating that the aimed response extremes can be predicted accounting for the actual correlation between the environmental processes. This requires a joint probabilistic model for the weather parameters of interest for the problem under consideration. In this paper a joint probabilistic model of mean wind speed, significant wave height and spectral peak period will be presented. Such a model will be needed if a long-term response prediction of motions and anchor lines for a floater is carried out.

Simultaneous wind and wave measurements covering the years 1973 - 1999 from the Northern North Sea are used as a database. The wind speed is chosen as the primary parameter since the wind is assumed to have the strongest influence on the loads on the mooring lines of a semi-submersible structure. The significant wave height is assumed to have second most influence and the spectral peak period is assumed to have least influence on the loads.

The joint model is used to establish a contour surface, giving combinations of the weather parameters for which the exceedance corresponds to a return period of 100 years. The paper is closed by briefly indicating the application of the joint model to the mooring line loads.

KEY W O R D S Waves, wind, contour plots, line tension

P R E D I C T I O N OF L O N G T E R M E X T R E M E S For design purposes, the 100-year response has often been estimated by combining 100-year wind, 100-year wave and 10-year current. A more consistent approach is to estimate the long-term responses by a full long term analysis, i.e.:

F x ( x ) = f IIFxlwH°.or, (x[w,h,t)-fWH.oV ~ (w,h , t ) .dhdwdt (1)

Fxrvm,,or ° (x [ w, h, t) is the distribution function of the 3-hour extreme value given the weather parameters and

fwH,,0r, (w,h,t) is the joint probability density function of the weather characteristics of interest for the problem under consideration. The choice of a 3-hour duration of the short-term condition is of course somewhat arbitrary. A consistent estimate for the 100-year extreme value, Xjoo, is now obtained by solving:

1 Fx(Xloo) = 1 - - -

N10o (2)

where N~oo is the number of 3-hour conditions in 100 years.

The challenge related to Eq. (1) will often be to establish the short-term distribution of X in the case of complicated response problems, e.g. the horizontal motions of a cantenery moored semi submersible. A possible approach is to assume that the conditional distribution of X can be modelled by a Gumbel distribution, i.e.:

(3)

19

For a given combination of weather characteristics, wi, hi, tk, the distribution parameters can be estimated by fitting the model to a simulated sample of 3-hour maxima. The simulated response is obtained by exposing the floater to the simultaneous action of a wind spectrum characterised by wj and a wave spectrum characterised by hj and tk. A 3-hour time history of the response quantity under consideration is simulated and a realisation of the 3-hour maximum is identified. For a case where the Gumbel model is expected to be the "correct" asymptotic model, a reasonable number of 3-hour simulations could be 20-30. Provided that point estimates for the distribution parameters for a sufficient number of combinations o f weather characteristics are carried out, response surfaces can be fitted to the point estimates, i.e. #(w, h, t) and [3(w, h, t) are estimated by continuous functions and Eq. (1) is convenient solved numerically.

Although one mainly need accurate distribution parameters in a rather limited subspace o f the total weather space, it is quite obvious that a rather large number of 3-hour simulations will be necessary. For a complicated response problem, a 3-hour simulation may well take 3 hour or more in computational time. For such a problem a full long term analysis will be impractical and it would be more convenient to estimate proper long-term extremes by exposing the structure to a short-term sea state. This can be done utilising the principle o f contour surfaces, see e.g. Meling et al. (2000). Based on the joint model,

fWH,,oT° (w,h,t) , one can estimate "sphere" where each point at the surface corresponds to an exceedance probability of 100

years. In practise this is done by transforming fwn,,oL (w, h, t) into a non-physical space consisting o f independent standard Gaussian variables, Uj (reflecting the marginal variability in W), U2 (reflecting the conditional variability of H~,o given W) and Us (reflecting the conditional variability of Tp given W and Hmo), utilising e.g. a Rosenblatt transformation scheme, Madsen et al (1986). In this space we know that all 100-year combinations (u~, u2, us) will be located on a sphere with a radius, r, given by

1 ~b(r) = 1 - - -

NI0o . With Nloo = 292000, this yields r = 4.5. Transforming this sphere back to the physical parameter space yields a "sphere" consisting of "100-year combinations" o f W, H,o, and T~

If the short term distribution could have been modelled by a delta function, i.e. no inherent variability in the 3-hour extreme value, an accurate estimate for the 100-year response could have been found as the most probable 3-hour maximum value in the most unfavourable combination on the "sphere" surface. This would be valid for all response problems, but of course the unfavourable combination (i.e. the point on the surface) would in principle be different from problem to problem. In practise one can not neglect the inherent variability of the 3-hour maximum. This can be accounted for approximately by selecting a somewhat higher fractile of the 3-hour extreme value distribution as the characteristic short-term value. For a response problem characterised by two slowly varying

parameters, e.g. H,,,o and Tp, a fractile in the order of 85-90% is recommended when the 100-year value is to be estimated in Haver et al. (1998). I f the problems involve 3 slowly varying weather characteristics, a fractile in the order of 65% is recommended as 100-year values are to be predicted (Meling et al., 2000). In the end of the paper this will be illustrated.

It is seen from this brief discussion that in order to predict consistent estimates for response extremes corresponding to a prescribed return period, one will in principle need a joint probabilistic model o f the slowly varying environmental parameters of importance for the problem under consideration.

Characteristic parameters The weather is described by the following 3 parameters:

• 1-hour mean wind, W

• Significant wave height, Hmo

• Spectral peak period, Tp

Simultaneous description of wind and waves We seek a joint density distribution of the characteristic parameters, W, H,,o and Tp. In this analysis the response will probably be dominated by the variability of the wind and therefore, W is chosen as the primary parameter. Based on this the following joint density function seems reasonable:

L',,'~,,,~'i:,, (w, 17: t) = j;~.(w) • f,.,-,,,o:,;, (hi w)..l:,:..~.; ,,,:.,(t ih, w) (4)

20

Marginal distribution for the wind, W We will assume that the marginal distribution of the l-hour mean wind speed at 10 m can be described by the 2-parameter Weibull distribution:

r ~,[- ,.t

(5)

where a and 13 is the shape and scale parameters, respectively.

Based on measurements from the Northern North Sea in the period 1973-99, and the method of moments, the values a = 1.708 and fl = 8.426 were determined for these parameters. These parameters seem to give a good description of the wind speed distribution and correspond to a 100 years extreme wind of 39.0 m/s. The distribution based on the measurements is plotted together with the fitted Weibull distribution in Fig. 1.

The data basis from the Northern North Sea consists of composite measurements from the fields Brent, Troll, Statfjord, Gullfaks and the weather ship Stevenson. For periods where measured data are missing, model data from the Norwegian hindcast archive (WINCH, gridpoint 1415) have been filled in, thus a 20 year long continuous time-series has been used.

8 "5 >,

.Q

I I ~ ] r ~ WeibuJl Model 1

f]e

J . / / , , -

/

z 3 4 s

Fig. 1.

!1! W i n d speed (m/s)

0.999999 0.99999 0.9999

0.999

0.99

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0,05

Cumulative frequency distribution of 1-hour mean wind speed for the Northern North Sea.

Conditional distribution of H,,o for given W The 2-parameter Weibull distribution was suggested as the conditional distribution of significant wave height for given wind speed. Based on the measurements from the Northern North Sea, the distribution of wave heights within wind groups with a bin size of 1.5 m/s was found. Next, the adequacy of the conditional 2-parameter Weibull distribution was considered by plotting the cumulative distributions on Weibull scale. The 2-parameter Weibull distribution would be considered suitable if these curves were approximately linear.

For each wind class, the Weibutl scale and shape parameters a and 13 were estimated from regression analysis (the least squares method), including all the wave classes where measurements were available. To evaluate the goodness of the estimated Weibull parameters, the corresponding cumulative distribution was also plotted in Fig. 2. The corresponding density function is plotted in Fig. 3.

3.0<=Wind<4.5 (m/s) 5 4 3 / f

In( 2

in( 0 ¢ A /? 1- -1

~ -2 f

-3 . ' ' ' /

-S

-1 0 1 2

In(Sign. wave height)

0 Measurements - - Weibull

4.5<=wind<6.0 (m/s) 4 3

In( 1 8:,-

in Y~" ( - 1 <~ 1- -2 f

-3 ..- -4

-5 --,¢. -6

-1 0 1 2

In(Sign. wave height)

<> Measurements

- - Weibull

1 6 . 5 < = W < 1 8 . 0 (m/s )

4

2

A 0

,~-.2 c . . t w

-¢ 4 /'0

-8 0 0.5 1 1.5 2

In(Sign. wave height)

<> Measurements - - Weibull

2.S

Fig. 2.

2

1

O

-3

2 4 . 0 < = W i n d < 2 5 . 5 ( m / s )

1 1 1 1 t f I I I I I I I I ~ I I I I I I I I I I I ( I l i l t I I I I I l l I

1.2 1.4 1.6 1.8 2 2,2 2.4 2.6

In(Sign, wave height)

Measurements Weibull

Cumulative distributions for significant wave height within different classes of wind speed Based on measurements and a fitted 2-parameter Weibull distribution.

21

The graphs in Figs. 2 and 3 indicate that the adapted probability distribution could be improved for the lower classes, of wind speed. For higher classes, the 2-parameter Weibull distribution seems to follow the measurements better.

3.0<=W<4.5 4.5<=W<6.0

o . , ,~ , , , , , , , , ' ' . . . . . ' I ' , I o.g , i ,u • ~i i i i } i i J 0.4 . 1~ I ~ I ) I I --Mea~lll~rnent$

. . . . . . . . . . . . . 11 II I ~eibllll ~ 0.3 X . . . . . . . . . .

o,} bAI . . . . . . II ,., i J-~. k l i i i i \

11.1 ) i I Xk~L I I I 0.1

0 1 2 3 4 5 ~ 7 0 o ~ 2 3 4 s

Wave height Wave height

16.S<=W<tS.0 24,0<=W<25.5 o.3~ o.,I J I ! !.U,[ [ t [ :;:, .... I I I ' i~

0'15 I I / I ~ i F / /

o 2 4 g lO o s lO Wave helghl Wave height

Fig. 3. The probability density function for significant wave heights within different wind classes. Based on measurements (the "un-smooth " curve) and a f i t ted 2-parameter Weibull distribution (the smooth curve).

Further, a and fl were plotted versus wind speed. Based on these plots the following parameterisations were proposed for c~ and

Ctfit = a l + a2 . IJV

and

(6)

8

7

6

sp 5 e

4

3

2

/ v

/

/ <

/

/ . o ~ ~ ,0< ~ '0. MeasuRrnen~

- - Fit

0 10 2O 3O 4O 50

W,~ weed (rrVs)

Fig. 5. The Weibull shape parameter, a, estimated from the measurements versus a f i t ted estimate.

The smooth parameters a and fl (Eqs. 6 and 7) were further used to calculate the conditional mean and standard deviation for significant wave height given wind speed as given by Eqs (8 and 9). The mean value and the standard deviation based on the measurements and on the parameterisation are plotted in the Figs. 6 and 7.

E(H,,,0) : [~-F(+ + 1)

and

STD(tt,,,0) = /3 - [~(~ + 11)- r~ (+ + 1)] °5

~: t5 / ~ : b i + b2 - t~ 'b:' (7)

From regression analysis (least square method) the parameters ~ '° a~, a:, b~, b:, b3 were estimated. All the points were included in ~ .,,~' 5

the analysis. The Figs. 4 and 5 show the fitted Weibull ,~ parameters versus the Weibull parameters based on the o *** _**.,.x*"

o measurements.

20

15 so /

10 / " 1> Measurement= ~ - . '

(s)

0 10 20 30

W i n d s p e e d [m /s )

40 50

(9)

,, .~a-a

l

10 20 30

Wind speed (m/s)

. . . t -

• From rneasuren

Fig. 6. Expectation value for significant wave height versus wind speed according to measurements and a smooth Weibull representation.

Fig. 4. The Weibull scale parameter, ,8,, estimated from the measurements versus a fitted estimate.

22

2

1.5

1

0 .5

0

,I(" •

, ~ "" - Parameterised L~k. & : ~ " ~ • From measurements

10 20 30 40 SO

W i n d s p e e d (m/s)

Fig. 7. Standard deviation for significant wave height versus wind speed according to measurements and a smooth Weibull representation.

C O N D I T I O N A L D I S T R I B U T I O N OF T~ F O R G I V E N H~o AND W

FINDING A SUITABLE D I S T R I B U T I O N F O R T~ The wind speed data used in this analysis were sorted into classes with a bin size of 1.5 m/s in the range 0 - 36 m/s, that is 24 classes. Subsequently, the data for significant wave height in each wind speed class were sorted in wave classes with a bin size of 0.5 m in the range 0 - 12.5 m, that is 25 classes. This lead to 600 combinations of wind speed and wave height and with a limited number of data for each combination.

In order to find a suitable distribution for the peak period, Tp, for given wave height and wind, the distribution of Tp within each wind-wave class was plotted. Due to the somewhat limited number of data within each class, the plots were not particular smooth. Still they seemed to indicate that a log-normal distribution would be suitable for the distribution of Tp. The log-normal distribution is given by:

r ] f" In(t)--'Uln(rP) "~ 2 1 fr , , ( t) / (10)

where ~o,,crp) and o),crp) are the expectation value and standard deviation of ln(TA, respectively. The mean value and standard deviation of ~, were calculated for each combination of H,,o and W and used for the calculation of/-0,,crp~ and crz, crp), i.e. the mean value and standard deviation in the log-normal distribution, according to the relationships:

f p ~;., ] /21 O).) = 113.

(11)

and

= In[o~), + 1] (12) O-!n(7),)

pz), and CrVp are the mean value and standard deviation of T~, from the measurements for each of the 600 wave-wind classes. Fig. 8 shows a selection of the ~,-distributions for random combinations of wind and wave magnitudes.

33 me~sulement~ 21 rr~os~emontl

o~ a35

o,o~ & - I o.os I - A ~ J ! X 2 L _ o

o 5 10 15 20 o 5 10 15

- W~.75.HsB.7 s L c t ~ t m a l

971 rne~osurements 137 m e c ~ u r e m e n t s

~3 I/V

°.,,', ! ! ! I ............ i °' U o ~ o ~ a l * Uog~o,mal

0.t ' 0,1

0 S ~0 IS 20 0 ~ ~0 15 20

Fig. 8. A selection of the Tp-distributions for random combinations of wind and wave magnitudes. The dotted lines show the calculated log-normal distribution while the other lines show the distribution found directly from the measurements.

P A R A M E T E R I Z A T I O N OF T H E MEAN VALUE IN T H E L O G - N O R M A L D I S T R I B U T I O N

To get an idea of how Tp vary with H,,o and W, the mean value of Tp was plotted in a 3D diagram shown in Fig. 9. The figure shows that for a constant value of H,,o, the period decreases with increasing wind speed and for a constant Wthe period increases with increasing wave height. To describe this behaviour the following function was proposed:

- - - [ '.'-~(h)') >" ] (14) r , , (w, h) = r ( h ) . [ ] + o . j

where T(h) is the conditional mean peak period for a given

value of the wave height. Correspondingly, w(tO is the conditional mean wind speed for a given value of the wave

height. The term ill + 0 . \ ~(h~ y J will then adjust the expected peak period according to whether the actual wind speed is above or below the expected wind speed for the particular wave height. The coefficient's 0 and 7 will take into account how the expected peak period will vary with wind speed for different wave heights.

where

~;" (13)

23

.... : ....... : ........ ~i....!.... "i' " - . . . . .

. '"" "'''i ' "" 1.1.1

o . : i i "

3 o ~ . " ...... :"- ' " " ....... : 12

t,'.'J'-'s' 0 0 W i n d Hs [m]

Fig. 9. The conditional mean peak period as a fimction o f significant wave height and wind speed.

The mean peak period, ~'(h), was plotted as a function of wave height and the following parameterisation was proposed:

7"(h) = ct + cz • h ¢:+ 05)

The parameter's ci, c2, c3 were estimated using regression analysis (least squares method). All points from the measurements were included in the regression and the result is shown in Fig. 10.

20

15

,,, / 10 ,t if

/

- - Fit: E [ T p ] = 4 . 8 8 3 + 2 . 6 8 * h ^ 0 . 5 2 ¢

0 2 4 6 8 10 12 14 16

S i g n i f i c a n t w a v e h e i g h t (m)

Fig. 10. The conditional mean peak period, =i,: ~1~t0, as a function of significant wave height according to measurements

from the Northern North Sea and a smooth parameterization.

Correspondingly, the mean wind speed was plotted as a function of significant wave height and the following parameterisation

was proposed for ~:(h) "

~ ( h ) = dt + d2 "h c!s 06)

As for the period, a regression analysis was performed and the result is shown in Fig. 1 1.

35

30

25

20

10

5

0

/

/ /

/ /

/ / - - M e a s u r e d

¢ / - - - - F i t : F : ( W ) = l . 7 6 4 + 3 , 4 2 6 * h ^ 0 . 7 8

0 5 10 15 2 0

H s [m]

Fig. 11. The conditional mean wind speed as a function o f significant wave height according to measurements f rom the Northern North Sea and a smooth parameterisation.

To give reasonable values for the two coefficient's 0 and y, a relation between wind speed and expected period had to be found. Eq. 14 was rewritten as:

_ ( ,+-----+c+> 't +' ( ]7> 7~,,c,%,~>.-.?:~,(+) _ 0 . k . ~ y Tp(/O

I +%,,x,...__]](:, ) 1 For given wave height, the normalised period ~(/+) : ) w-..i~(h i

was plotted as a function of the nonnalised wind \ , . Fig. 12 shows a selection of these plots for different wave heights.

2.0<=Hm0<2.5 4 .0<=Hm0<4.5

0.3 0.3 ~ ~

0.2 ~ , ~ ~ 0,2 "7 ~

~ o +-+. ~ o "-_. ++ +., ,, +-. ~. +

..o+2 - " .a,.+ z -o.2 ~--j%

.o,3 -I .o.s o o + I . i s -i++ +1 ++s o o . s I 1.+

mo~limd wind mofmal;sed wind

6.0<=Hm0<6.5 8.0<=Hm0<8.5 0.25

o.2 •

.... " - 1+ .... o+1 + ~ +

I -1+.+I++ -0.05 -1.s -1 .o.s o o.s 1 ~o+6 .o.4 ~.2 o 0.2 o.4 o.6

Nor~li~d w;nd No~l i~d wind

Fig. I2. The normalised expected peak period as a function o f normalised wind speed. The data points are from measurements while the solid lines are based on regression analyses (least squares).

2 4

As can be seen from the plots, the trend is nearly linear indicating that y is close to 1. The slope however, seem to vary for the different wave groups. To investigate this, 0 from all the wave groups were plotted as a function of the significant wave height. This plot is shown in Fig. 13.

-0.05

-o.1 1

i ~_ -0.15

e

-0,2

-0.25 ] o 1

i i i

ie

2 3 4 5 6 7 8 9 10 Hm0 (m)

Fig. 13. The slope, O, Eq. (17) between normalized period and normalised wind speed as a function of the wave height.

It is not straightforward to see any simple relationship between the wave height and 0. Further, with intention on finding a general trend for the slope, 0, as a function of significant wave height, the wind classes with a very limited number of data were excluded from the analysis. The peak period versus wind speed when the outlayers were excluded from the data is shown in Fig. 14. Fig. 15 shows the slope, 0, versus significant wave height when the outlayers are excluded.

2 . 0 < H m 0 < = 2 , 5 4 . 0 < H m 0 < = 4 . 5

o.s o,3

+os ~ .o.2 "~

N~ ~ri=e¢~ ~n~ Sl~=d N ~ a I~'a ed q'~nd ~Peed

6 . 0 < H m 0 < = 6 . 5 8 . 0 < H m 0 < = 8 . 5

°"'"'"~'J"'""*""lttlttlI~ll~lllllli :2, " I l l l ] l l l l l l l l l i l l ~ - + 4 1 ~ , / ] l l l l I I I t 111111 I I I I 1 =~~,'

~.$ ~.4 -o3 .0.2 .0.1 0 0,1 0.2 0.3 0.4 0.5 .0,2 .0.1 0 0,1 0,2 0.3 Norm=lised ~n~ speed Nonm~liscd ~+~nd Sl:~md

Fig. 14. The normalized expected peak period as a function of normalised wind speed. Outlayers are excluded from the data.

-0.16 - 0 . 1 7

- 0 , 1 8

- 0 . 1 9

-0.2

- 0 . 2 1

- 0 . 2 2

- 0 . 2 3

- 0 . 2 4

0 1 2 3 4 S 6 7 8 9 1 0

S i g n i f i c a n t w a v e h e i g h t ( m )

Fig. 15. The slope, 0,, Eq. (17)between normalized period and normalised wind speed as a function of the wave height. Outlayers are excluded from the data.

By considering the Figs. 13 and 15 no obvious expression for 0 did occur and thus based on best fit by eye the constant value 0 = - 0.19 was proposed for the slope. From the above analysis all the parameters in Eq. (14) have been estimated. To evaluate the goodness o f the proposed model, the periods were calculated for all 600 wind/wave combinations. The difference between measurements and estimate was investigated and the result can be seen in Fig. 16. Table 1 gives the estimated peak period for a number of combinations of wind speed and significant wave height.

4 1 ........ °'2

o. ~ ° .

2. h_ 0-

-4 : 4O

• . . ' " " i . . . . . . . . . " ' " " " i " " " ! " . . i ~ .

" ' ' " : ' " i : ' " " . . . . . . . . . • , . ' . , .

. . . . . . . . . ' " : ".. . ." . . . . . . . :~'.1 . . . . . . . ~ ' 7 ~ , . i

2° . . . . . . . . . . . . . . . . . 10 ~ 5

W i n d [ m / s ] 0 0 Hma I m l

Fig. 16. Difference in peak period between measurements and calculations for all combinations of wind speed and significant wave height for 0 = -0.19. A negative sign means that the estimated peak period is lower than the measured peak period.

25

Table 1. Estimated peak period for combinations of wind speed and significant wave hei~ ht.

H~0 (m) W Estimated Tp

12

15

15

15

18

( m / s ) ( s ) 30 14.4

20 17.1

30 16.1

40 15.1

30 17.7

From Fig. 16 two conclusions can be drawn:

The estimates and measurements were consistent in the area containing the majority o f the measurements.

For more rare wind speed and wave height combinations, the estimated peak period differ from the measured peak period. The difference for large wind speed and wave heights is of the order up to 2 seconds.

To improve the estimate in the area o f interest, which is for large waves and wind speed, two separate courses were tried:

1. An iteration was made where 0 was the parameter to be optimised. The value giving the smallest RMS value, i.e. the sum of the difference squared for all the 600 possible combinations of H,,o and W. would be considered as the best fit. In addition the mean value of the difference between the measurements and the estimate should be as small as possible.

2. By assuming the relationship between wind speed and peak period not to be linear as originally was proposed, the following equation was assumed to describe the relationship better:

LS(h) - ( l ) ( ,~ . . .~ ( ;0) ) k v,,{h) )

where 2 is a constant.

08)

Eq. (18) is based on the idea that the outlayers in Fig. 12 indicate some curvature.

None of these two methods gave significant improvements to the estimated wave periods, therefore it is recommended that Eq. (17) is used with 0 = - 0.19. Further work in optimising the parameter, 0 and possibly A, combined with omitting the low wave classes could still lead an improvement in the model used for calculations of the peak period.

P A R A M E T E R I S A T I O N OF THE STANDARD DEVIATION IN THE L O G - N O R M A L DISTRIBUTION

Eq. 14 gives an estimate on the mean value recommended to be used in the log-normal distribution for the period. As for the standard deviation, Eq. (19) is proposed based on experiences on the relationship between standard deviation and mean value from the measurements:

aTi~ = [ - 1 . 7 - 10 .3 + 0 . 2 5 9 . e×p(-0 .113 .h) ] .fz~ (19)

R E C O M M E N D A T I O N S The joint distribution for wind and waves can be expressed by inserting the parameters given in Table 2 in Eq. (4). Based on these parameters, a contour body expressing different combinations o f wind speed, significant wave height and peak period was found for the 100-year level. This is shown in the Figs. 17 - 19.

Table 2. Parameters recommended for use in the joint wind and wave distribution (Eq. 4)for Veslefrikk.

Wind distribution

Conditional distribution, Significant wave height

Conditional distribution, Peak period

Shape = c~ = 1.708

Scale = [3 = 8.426

Shape: a = 2.0 ~ 0. t 35 • w

Scale: fi= 1.8 ~ 0. ll)0. w ~.:<"

Mean value:

u r . , . a = l n ,-=-~:-i

Standard deviation '

where

, , 7 ) ,= (4 ,883+2 .68 .h ~ ..... ) .Ll ......0,19.i, t.7(,4<~4:?e,;~,~7~

and

o-7)~ = [ - l . 7 • 10 ....3 + 0.259. exp(-0.1 I3. h)] .,u :,)

w and h is the wind speed and significant wave height

26

0 5 1D 15 Significant waveheight He [nl]

Fig. 17. Contour surface o f the jo in t distribution f o r wind and waves - 100 year return period. 2D - Wind speed versus significant wave height.

0 5 lope ak period Tp Is] 15 ~

Fig. 18. Contour surface o f the jo in t distribution f o r wind and waves - 100 year return per iod 2D - Significant wave height versus peak per iod

35. , " "

30. '

D> . ' "

1

Signil@c~n D O

........ i " . . ._., ......... ' i ......:., . .

Fig. 19. Contour surface o f the jo in t distribution f o r wind and waves - 100 year return period.

Example of application of the joint model The joint model is utilised for predicting extreme mooring line loads for semi submersible with 12 mooring lines. The application is presented into some detail in Meling et al. (2000), and only some few results will be reviewed herein. Solving Eq. (1), the following extremes are obtained for the line loads in the intact case:

x~o0 = 3523kN (20)

x 1o000 = 6_~.~4kb (2 I)

Utilising methods from the field of structural reliability, one can identify the most likely combination of environmental characteristics as these values occur. The combinations are:

xt00 ( 7 8 % ) " w = 35.7m/s , h~,0 = 14.1m, t ; = 14.3s (22)

xt0000 (91%) :w = 42.1m/s, h,,0 = 16.9m, tp = 14.8s (23)

The percentages given in parenthesis, are the most likely fractiles of the short-term extreme value occurring in combinations with the given environmental conditions.

I f instead of using a full long term analysis one will use the contour surface principle, one will first of all have to identify the most unfavourable parameter combination on the 100- and 10000-year surfaces for the problem under consideration. For the mooring line problem, the following environmental conditions are identified:

"* /

1 0 0 - y e a r " w = .36rms, hmo = 14.4m, tp = 13.3s (24)

l O 0 0 0 - y e a r ' w = 42.3m/s, hmo = 17.9m, tl~ = 14.8s (25)

For these sea states we have estimated the median (50%) and 90% fractile of the 3-hour extreme value. We have also estimated the fractile we have to select in order to equal Eqs. (20) or (21), respectively. The results are:

1 O0 - y e a r : .~.~481c~\; (~0,4), 3945kN (90%), 3523 (66%) (26)

10000 -), 'ear : 5820;~:N(50%), 6865kN (90%), 6334(76%) (27)

The advantage by using the contour surface principle is that good estimates for the n-year response may be achieved by use of a short-term analysis. This saves time and resources at an early stage in design. However, before final design a long-term analysis must be performed. For more details on the response example reference is made to Meling et al. (2000).

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Conclusions - - ~ r (" " ~ ] w - ~ ( h ) 7

The equation Tz'(w'h):z"h)'[l+O't.-~;i--JJ (Eq. 14) with 0 = -0.19 and y = 1.0 describes the peak period as a function of significant wave height and wind speed.

Attempts to improve Eq. (14) have failed. For large classes of wind speed and significant wave height the difference between measured and estimated peak period is of the order up to 2 seconds.

The wind speed and significant wave height can be described by a two-parameter Weibull distribution. The peak period can be described by a log-normal distribution.

Simultaneous wind and waves can be described by the joint density function:

/~,~o .,,~, ( w, h, 1) = f~4w) . J}.~,.o~,,.( hl w ) . j~',,i~i,.o'~.( t[ ;¢~, w ) (Eq. 4).

An efficient method to predict a specified n-year response is to apply the contour line/surface approach. However, this method requires that the proper fractile of the extreme value distribution of the response is known. When the short-term variability of both wind and waves are included, a fractile around 66% is found to be adequate in order to predict the 100-year response. For the 10000-year cases, the proper fractile seem to be approximately 76%. Further studies should be performed in order to validate the proposed fractile levels. In addition, different structures should be tested to check if the fractile levels are case dependent.

R e f e r e n c e s

Haver, S., Gran, T. M. and Sagli, G. (1998): Long-Term Response Analysis of Fixed and Floating Structures, Proceedings of Ocean Wave Kinematics, Dynamics and Loads on Structures, ASCE, Houston, Texas, April 30 - May 1, 1998, pp. 240-248.

Madsen, H.O., Krenk, S. and Lind, N.C. (1986): Methods of Structural Safety, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1986.

Meling, T. S., Johannessen, K., Haver, S. and Larsen, K. (2000): Mooring Analysis of a Semi-Submersible by use of IFORM and Contour Surfaces, Proceedings of ETCE/OMAE2000 Joint Conference for the New Millennium, no. OMAE2000/osu oft-4141, February 14-17, 2000, New Orleans, LA, USA.

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