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Page 1: Wave climate off northern Norway

Wave climate off northern Norway

S V E R R E H A V E R

Statoil, Stavanger, Norway

The wave climate off northern Norway is considered and the investigation is based on wave measure- ments made at Troms~bflaket by means o f a waverider buoy during the years 1977-81. Data quality of waverider measurements is briefly commented upon; however, more emphasis is given to an evaluation of the long-term representativity of the actual measuring period and to a procedure accounting approximately for a lack of representativity. The wave climate is presented in terms of a smoothed joint probability density function of the significant wave height, H s, and the spectral peak period, Tp. Based on this distribution a consistent design curve in the Hs, Tp space is established.

INTRODUCTION

An important topic in ocean engineering is the wave climate at a given location or, for example, a given pipeline route. For such studies the existence of a sufficient amount of wave measurements becomes crucial. With respect to the Norwegian continental shelf most wave data series have a duration in the order of some few years, but such dura- tion is considerably shorter than the time period usually adopted as the return period for the design conditions. Accordingly, an evaluation of the long-term representativity of the measuring period is a very important part of the climate consideration. Herein hindcast wave data will be adopted for this purpose and in this way a lack of repre- sentativity can be accounted for quantitatively, provided the hindcast data represent the relative severity from year to year reasonably well.

Subsequently the wave climate off northern Norway will be considered. The investigation is based on wave measure- ments from Troms~bflaket and hindcast data from grid point (18, 20), which is assumed to be representative for the Troms¢flaket area. The measurements are achieved during the years 1977-81, while the hindcast values cover the years 1955-81.

WAVE DATA AND DATA QUALITY

The wave measurements used herein are achieved by means of a surface following waverider buoy, manufactured by Datawell NV, the Netherlands. An accelerometer on board the buoy measures the vertical acceleration, and by double integration an estimate of the sea surface elevation is obtained. On board a weathership located within some few kilometres of the buoy, the signal thus obtained is sampled at a frequency of 2 Hz and stored on a magnetic tape. The aim of the recording procedure is a time series of approxi- mately 20 minutes every third hour. Finally the time series are analysed both in the t i m e - and frequency domain using the NEPTUN system, Torsethaugen and Krogstad, I which includes a detailed data control concerning format errors, non-physicalities, and so on.

It should be noted that the time series actually analysed are only estimates of the sea surface elevation and not the

Accepted September 1984. Discussion closes June 1985.

0141-1187/85/020085-08 $2.00 © 1985 CML Publications

actual sea surface elevation. Use of a floating buoy as the sensoring unit will most probably introduce errors of both a systematic and random nature. Additionally, some noise will most probably interfere with the measured signal on the way through the recording procedure as shown schema- tically in Fig. 1. The errors possibly introduced by the Waverider are expected to be rather small; however, they will not be detected by the aforementioned data control and will therefore be present in the data. A detailed con- sideration of the accuracy associated with the present time series is out of scope of this investigation, but some comments pertaining to this topic are given below.

As far as sea state characteristics such as the significant wave height and the spectral peak period are considered, the actual time series will most probably be of sufficient accuracy. Their adequacy becomes more questionable as characteristics of the individual waves are to be investigated, especially if the waves are nonlinear. It is occasionally observed that the buoy is passed by sharp wave crests in a more or less submerged condition. Furthermore, if a wave crest is of a three-dimensional form, which can be the case in short crested seas, the buoy, even if floating exactly on the wave crest course, will not experience the highest part of the wave crest. Such waves will induce a three-dimensional motion of the buoy and the correspond- ing crest height will be somewhat underestimated.

Another possible source of bias is the orbital motion of the waverider buoy. A consequence of this is that the buoy stays too long at a wave crest and too short at the wave trough, resulting in a slight distortion of the wave profile. Accordingly, steepnes,~ and asymmetry parameters might be somewhat biased, while the zero upcrossing wave period should not be significantly affected.

BLACK BOX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

GENERATING ~ RECORDING PROCESS PROCEDURE

RECEIVED k DATA PROCESS RANK

~(ti)

Figure 1. An illustration of the process of estimating the sea surface elevation, ~(t)

Applied Ocean Research, 1985, Vol. 7, No. 2 85

Page 2: Wave climate off northern Norway

Wave climate o f f northern Norway: S. Hayer

In passing on this topic it may be concluded that regarding a description of the wave climate as expressed in reasonable sea state characteristics, the present time series are considered to be of reasonable accuracy. The time series can also be used for predicting extreme individual wave heights and corresponding wave periods, but when applying these results one should bear in mind that the wave heights might be slightly underestimated, especially if they are associated with sharp crests or three-dimensional crests. With respect to an investigation of nonlinear waves, non-Gaussmn processes, and, for example, the spectral behaviour in the equilibrium range, the actual time series will most probably be inadequate.

The hindcast data included are generated by means of the Norwegian wave model, NOWAMO, operated by the Norwegian Meteorological Institute. Experience seems to indicate that the hindcast values do not fit too well to existing observations, Haug and Guddal. 2 On average the hindcast model overestimates both the relative frequency of very low values of the significant wave height, Hs, and the relative frequency of very large wave heights. This is clearly indicated by Fig. 2, showing the estimated cumula- tive distribution functions for H s. However, although the hindcast values in general do not fit too well to the true values, they are expected to reflect the relative severity of the various years reasonably well.

REPRESENTATIVITY OF WAVE MEASUREMENTS

Data recovery is considered in some detail in Tryggestad et al. 3 It is found that regarding the various months, the percentage data recovery (relative to all three-hour periods for each month) varies between 74 and 84%, except for September where the data recovery is only 64%. On average the data recovery is close to 80%. Furthermore, the sample is also found to be reasonably well balanced between the winter, summer, and spring/autumn seasons.

Missing observations might be a possible source of bias errors, depending on the cause for being missed. By considering the winter seasons of the actual data sample, it is found that about 60% of the missing observations are caused by the monthly change of crew on the weathership

f I

i.. - . 0.. .... o. . . . . . t t H .

WAVE MEASUREMENTS ; 1977 - 1981 WAVE HINDCAST ; 1977 - 1981

---WAVE HINDCAST ; 1955 - 1981 i

H~ [Y,]

Figure 2. Es t imated cumulat iw, distribution func t ions j a r the significant waw, height

AMI, which is operating the waverider buo> .~ These missing observations are not expected to introduce any bias errors. The rest of the missing observations, which are assunled to be absent due to failures of the recording procedure, are grouped with respect to the significant wave height by guessing the value of H s in the missing observation. It is found that the percentage of absent observations slightly increases with increasing value of H s but for the actual site, the estimated cumulative distribution function for H s is not significantly affected by this bias. 4 It is therefore concluded that the actual sample is representative for the wave climate within the measuring period.

If reliable extreme values are to be obtained from the present sample, it is necessary that the actual five-year period is representative for the long-term wave climate at the actual site. Subsequently this will be investigated by means of hindcast wave data and too much attention should not be given to the absolute values. At best the hindcast data can be expected to reflect the relative severity of the wave climate (in terms of Hs) from year to year reasonably well.

Hindcast data for the years 1977-81 and for the years 1955-81 are considered herein. For these time periods the estimated cumulative distribution functions for H s are shown in Fig. 2. It is seen that the measuring period seems to be slightly less severe than the long-term average wave climate and this has to be accounted for if reliable extreme values are to be predicted. It is therefore of some interest to establish a procedure for a consistent combination of the information included in the measured sample with that of the hindcast sample.

It is suggested that the distribution function for H s estimated from the measurements is reasonably close to the underlying true distribution for the measuring period. The actual hindcast procedure gives the sea state charac- teristics once every sixth hour. In spite of this, it is suggested that the hindcast sample corresponding to the measuring period is a representative sample regarding all three-hour periods within this period, not in terms of H s, directly, but rather in terms of a transformed variable I:Is =g(Hs) . The transformation will most probably be of a case dependent nature. However, the on average form of

:g(Hs) can be found from tire following equation,

Ffl s [t:I s = g(Hs) l = FHs(H s) ( 1 )

where FHs(Hs) is the cumulative distribution for H s obtained from the measured sample and Ffils(fls) is the distribution function for fr s obtained from the corresponding hindcast sample.

Based on the distributions corresponding to the measur- ing period, Fig. 2, a smoothed transformation approxi- mately satisfying equation (1) is found to be: a

/ts ~ 2-23Hs ° s -- 2.58; U s > 2.0 m (2)

Assuming equation (2) to be valid also with respect to the period 1955-81, the cumulative distribution for /4s, obtained from the hindcast data can be transformed to an estimated distribution of H s. This transformed distribution is compared to that obtained from five years of measure- ments in Fig. 3. By fitting Weibull distributions to the tail regions, it is indicated that the measuring period corresponds to extremes being 5-10% less than those predicted from the modified data sample.

It should be noted that the average relation between Hs and /ts given in equation (2) is at best valid as far as the yearly wave climate is considered. The investigation carried

86 Appl ied Ocean Research, 1985, Vol. 7, No. 2

Page 3: Wave climate off northern Norway

Wave climate off northern Norway: S. Hayer

O ~

O . I

0.7C

0,.~

0.2(

O.OE 0.30

Figure 3.

,3

/, /,

/, ,/,,"

/ /

HINDCAST DATA (1955-1981)

H s [M]

Cumulative distribution function for H s

- - MEASUREMENTS (1977-1981)

MEASUREMENTS (1977-1981) AND

O.NWN

O.IOm~

O .NN

O.lll~

0.70

~ m

0.3O

0.20

0.06 0.3O

Figure 4.

J _ ~ . I 1 l WAVE MEASUREMENTS ; 1977 - 1981 WAVE HINDCAST ; 1977 - 1981

--- WAVE HINDCAST ; 1955 - 1981 ~i

• /

I # / '

/"

/ !

1

~ 2 3 ¢ O O 7 I I I O 3O ~

H~ [M]

Cumulative distribution function for Hs, March

100

.10

m

g

L

ioo lO

z

out herein indicates that the relation will vary over the year. The cumulative distribution functions for Hs and Hs are shown for four months in Figs. 4-7. It is seen that applying equation (1) to the distributions of these figures will result in slightly different relations. Accordingly, an average relation for each month must be established if updated monthly distributions are to be obtained

Herein the monthly climate is considered in terms of the respective extremes of H s and for such purpose the effects of a possible lack of representativity can be approxi- mately accounted for by the following approach. Straight lines are fitted to the tail regions of the estimated monthly distributions of Hs and Hs and extrapolated into the range of extremes. The actual return periods are shown in Figs. 4-7. It should be noted that the return periods as shown in these figures are consistent with a duration of three hours for each occurence. The predicted extremes corres- ponding to a return period of n years are denoted by

OJ~3oq

0.119M

O.n~

0 .~

0.TO ~ m

o.5o

0,3O

0.20

0.06 0.30

Figure 5.

O . ~ O . I o.mml

0 . ~

0.TO

~-~ o ~

0.3O

0.29

0.06 0.3O

Figure 6. September O.BDDDN

O . ~ OJm9

O.~

~ . o.7O

0.3O

0.3O

O~D

0.08 0.3O

Figure 7. November

I I

/""' I J J

.-C/" i /I I

I i

i

WAVE MEASUREMENTS 1977 - 1981 WAVE HINDCAST ; 1977 1981

--- WAVE HINDCAST ; 1955 1981

1 2 3 4 O O 7 ee lo m

H S [M]

Cumulative distribution function for H s, July

100

10

1

I

1

Cumulative

/"

~/I. f / " /

.//~¢/"

L

WAVE MEASUREMENTS ; 1977 - 1981 WAVE HINDCAST ; 1977 - 1981 WAVE HINDCAST ; 1955 - 1981

I 2 3 4 $ O 7 O I lO

H s l M ]

distribution function for

ioo

io

1

g

: u

I I ~] WAVE MEASUREMENTS ; 1977 - 1981 WAVE HINDCAST ; 1977 - 1981

--- WAVE HINDCAST ; 1955 - 1981

i 7

/'i '"i"

30

_ _ m

ioo

IO

1

2 3 O 6 7 eg lo 3o 3o

H~ IM]

Cumulative distribution function for Hs,

Applied Ocean Research, 1985, Vol. 7, No. 2 87

Page 4: Wave climate off northern Norway

Wave climate o f f northern Norway." S. Hayer

H(sn,~,/~s, n), and/2/(s,n)-/, respectively. The hat indicates that the extremes are obtained from hindcast data. Assuming that the ratio between the extremes predicted from measurements and the corresponding extremes predicted from hindcast data is independent of the length of the time periods considered, corrected extremes read;

/~(n) H ( n ) _ 14(n) s,2"7

- " ' , , s /~(n; (3) 8,S

H (n) combines the information of the actual wave climate included in the measured sample with the information of long-term representativity included in the hindcast data.

LONG-TERM DESCRIPTION OF WAVE CLIMATE

Provided that the sea surface elevation at a fixed location for short time periods can be accurately modelled by a stationary Gaussian process, a short-term sea is in a statistical sense, completely characterised by the frequency spectrum, s( f ) . In engineering applications an analytical expression involving some few characteristic parameters is most often adopted for s(f) .

Herein the main characteristics are chosen to be Hs as defined from the spectrum, and the spectral peak period, Tp. The reason for choosing the spectral peak period instead of other characteristic periods is that this period is less correlated to the significant wave height than the other periods, s Together with Hs it will therefore include somewhat more information about a general sea state.

Furthermore, it will often be of main interest to predict a reasonable location of the spectral peak.

The joint frequency table for H s and Tp for the measur- ing period is given in Table 1. An inspection of the frequency table indicates that some observations, underlined in Table 1, are of questionable accuracy. For such observa- tions the underlying time series and corresponding spectral density functions should be carefully examined in order to detect errors possibly introduced through the recording and analysing procedure. This is not done and too much weight will not be given to these observations subsequently.

When characterising a short-term sea state by only two parameters, the corresponding spectral shape will in general not be completely defined. A variability of a fundamental nature in the underlying spectral shape is likely to occur even if H s and Tp are constant. These variabilities will mainly be reflected in a variable peakedness of the wind generated part of the wave spectrum and in the spectral densities of the low frequency range due to a more or less random occurence of swell components. Accordingly, with respect to a more complete description of the wave climate, at least two more parameters should be included, one being sensitive to the spectral peakedness and the other to the amount of swell present.

A spectral form for combined seas is proposed by Ochi and Hubble. 6 This spectrum is a superposition of two single peaked spectra and is characterised by six parameters, making it inconvenient for practical applications.

Assuming that a short-term sea state is approximately characterised by H s and T v, the long-term variation in the

H~

[M]

.0- .5

.5- 1.0

1.0- 1.5

1.5- 2.0

2.0- 2.5

2.5- 3.0

3.0- 3.5

3.5- 4.0

4.0- 4.5

4.5- 5.0

5.0- 5.5

5.5- 6.0

6.0- 6.5

6.5- 7.0

7.0- 7.5

7.5- 8.0

8.0- 8.5

8.5- 9.0

9.O- 9.5

9.5-10.0

10.0 - 10.5

10.5-11.0

11.0 - 11.5

11.5 - 12,0

12.0 - 12.5 Sum

2 I 6

2 7 101

54

7

13 2 I 8 5 3

238 418 157 74 75 43

441 780 450 175 119 79

254 662 593 2581 121 94

"72 512 613 3241 130 78

I 264 498 299 140 79

3 85 349 253 121 63

24 188 218 119 59

4 85 135 94 54

28 102 87 57

11 51 82 38

I 15 48 31

8 32 22

4 15 19

2 4 16

Ii. 2 7

21 0 7

2

I

SUM

I 42

6 8 5 I I 1136

38 16 10 2 3 0 I 2168

66 50 22 9 I 2137

42 37 16 9 I 1834

34 23 14 I I 1364

30 12 8 924

18 9 5 640

17 12 2 403

18 2 0 0 295

10 4 2 198

10 I 0 I 107

5 I 68

10 5 53

7 I 31

5 2 ! 18

5 0 15

2 3 7

8 I 10

I I 2

0

I I

I I

0

I I

88 Applied Ocean Research, 1985, Vol. 7, No. 2

Page 5: Wave climate off northern Norway

wave climate can be described by the joint probability distribution of H s and Tp. The sample probability density function for H s and Tp, fHsTv(Hs, Tp), can easily be calculated from Table 1. The sample density function can be considered as a first estimate for the underlying, unknown probability density function, fHT (Hs T,), but its validity is restricted to the range of observations. Regard- ing an extrapolation to more rare sea states, a smoothed probability density function is needed. For this purpose the joint probability density function is conveniently written;

fHsTp(H s, Tp) = fTp IHs(Tp IHs) fHs(Hs) (4)

where fHs(Hs) is the marginal probability density function for H s and fT_lHs(TplHs) is the conditional probability density functio[a for Tp given H s.

Subsequently fHs(Hs) and fTpIHs(T p IH s) are fitted to the observations separately. Thereafter numerical values for the joint distribution are obtained by means of equation (4). Hereinfn~(Hs) is modelled by a log normal distribution for Hs <~ r7 and by a Weibull distribution for Hs > r~, i.e.:

1 { (lnHs--O)Z} __ e x p - - - Hs <~ rl

x/2n~H s 2K 2

f~(H~) = ( 5 )

H ~- 1

where 0 = the mean and K 2 = the variance of ln(/-/$) and where additionally, continuity is required for fHs(I-Is)and FH,(Hs) at H s = '7. Accordingly this mixed model herein referred to as the lonowe mode l s should be considered as a three parameter model.

A consistent estimation procedure of the lonowe para- meters becomes rather complicated and an approximate approach is therefore adopted. 0 and K 2 are estimated as if a log normal model was to be applied, ~/-+ oo, thereafter r/is varied until a reasonable fit to the tail region is obtained. It should be noted that this is a reasonable procedure only as far as r~ is sufficiently large. For the actual case a reason- ably good fit is obtained for ~ = 4.5 m, 0 = 0.701, g2 = 0.307, ~ = 2.165, and/3 = 1.311, see Fig. 8.

The conditional distribution of Tp given H s is herein modelled by the log normal distribution, i.e.:

1 exp { fT,~ m~-IHs(TpIHs)=VL n a Tp 2a a I

where U --- E[lnTp] and o 2 = Var [lnTp] (6)

The log-normal distribution is found to be a reasonable choice as far as more severe sea states are concerned. It is seen from Fig. 9 that the estimated distribution function for lnTp appears as a straight line on normal probability paper, indicating that Tp is reasonably well modelled by a log-normal distribution.

For each class of H a including a sufficient number of observations, p and a e are estimated according to the moment principle. The estimates are shown versus H s in Fig. 10. In order to obtain reasonable parameter estimates for large values o f Ha, continuous functions are fitted to the estimates and extrapolated into the range covering the most extreme sea states.

The adequacy of the fitted conditional distribution is shown in Fig. 11. It is seen that as far as H s < 9 - 1 0 m , a reasonable fit seems to take place regarding the central region. With respect to the more severe sea states, the

Wave climate off northern Norway." S. Hayer

0 . ~

0 . ~

0 . ~

0 . ~

0 , ~

O70 %

u - I 0 . ~

o.~'Q

0.116 0 3 0

Figure 8.

J

I 2 3

H s

4 S 6 7 1110 2~1

(M]

Cumulative distribution function for Hs

~-~® i 10

, t ' I

1- I

t ° =s

,9999

,$99

,99

,95

,8O a-

® ,50

c~

,20

Figure 9. Hs

,05

.GI

.001

2.2 2.4 2.6 2.8 3.0

LN Tp

Cumulative distribution function for Tp given

adequacy of the conditional distribution can not be finally verified before more data become available. It is expected, however, that the central position is reasonably well located, while larger uncertainties are associated with the scatter around this position, see Fig. 10.

The adequacy of the fitted simultaneous distribution can be indicated by considering the marginal distribution of Tp and the conditional mean of Hs given Tp. The marginal distribution for Tp is shown in Fig. 12. A reasonably good fit between the model and the empirical distribution is observed, bearing in mind that for low and moderate sea states emphasis was given to the central part of the distribu- tion. The conditional mean of H s given Tp is displayed in Fig. 13. The fit is seen to be very good for periods less than 12-13 s, while for larger periods a more questionable fit

Applied Ocean Research, 1985, Vol. 7, No. 2 89

Page 6: Wave climate off northern Norway

Wave climate o f f northern Norway: S. Hayer

2.5 - -

2.0

3.0 , ! •

+I" ~/0 4(Hs+l) -2 +0.5 LN(Hs+I) +0,5 + SAMPLE VALUES

14 ~6 8 16 12 14

H s [M]

~2

0.4

0.3

0.2

0.1

Figure ] O. Tp given H s

I

: 1 1 0,065Hs - 0 ' 8 -0 ,01H S +0,05; Hs<5M

(~2= 0,065Hs_0, 8 ; Hs>5M ,

I~ 4- SAMPLE VALUES

2 4 6 8 10 12 14

H s [M]

~rameters in the conditional distribut~n of

18

16

14

T p 12

10 [sl

8

6

4

2

" / ' " / 0 75 e"L~ 5o • r /0 i0 L/'.:2"~%"

\.i . / . , , : . z 212"

~ " / / / / Z /

;4:" /

FROM SMOOTHED MODEL + EMPIRICAL 10 AND 90% POINTS

o EMPIRICAL 25 AND 75%POINTS

EMPIRICAL MEDIAN

THREE LAGEST OBSERVATIONS

i i i i

2 4 6 8 10 12 14

H~ [M-B

Percentage points for the conditional distribu- Figure 11. tion o f Tp given H s

is observed. However, this is most probably caused by a poor description of the swell dominated sea states. Regarding extreme sea states the joint model is therefore expected to be of a reasonable accuracy.

EXTREME SEA STATES

From the marginal distribution of H s, extreme values corresponding to return periods of 1, 10, and 100 years. are found to be:

H (D = 10.4 m S , 5

Hs °) = 12.8 m (7) ,5

H}lOO) = 15.0 m ,5

,9999 !

,999

,99

,95

,30

~'~, 50 k-

Ch

kL

,20

,05

,01

,001

Figure 12. T~

/ /

/ ) /

f v

/ .

! I

1- r

i FROM JOINT MODEL

• FROM MEASUREMENTS

I r

i

E B 6 9 ,2 ~ 16 ~i ~ , 2~

T v [S]

Marginal cumulative distribution/unction Jor

6

F~

u:, 4 :12

LD

4- 8

z' / ¢

SAMPLE VALUES SMOOTHED MODEL

I I i

Y" + + X

6 1 2 1 8 2 4

tp iS]

L i t

4 I

i i

- - @ . . . . . . . L - - q

t i i

31]

Figure 13. Conditional mean of H s given Tp. The empirical value given within parentheses is due to one observation which is o f a auestionable accuracy. (Underlined in Fig. 6)

90 Applied Ocean Research, 1985, Vol. 7, No. 2

Page 7: Wave climate off northern Norway

The corresponding expected spectral peak period can be taken from Fig. 14, i.e.

E[TpIH s = 10.4 m] = 15.3 s

E[TplH s = 12.8 m] = 16.8 s (8)

E[TplH s = 15.0 m] = 18.0 s

Interval estimates for the spectral peak period can be obtained from Fig. 11.

The adequacy of the extremes given in equation (7) rests to a large extent on the long-term representativity of the data sample. It is previously concluded that with respect to waves the actual time period is slightly less severe than the long-term average climate. Updated predictions accounting approximately for this effect can be obtained by means of equation (3) and Fig. 2. The extremes thus obtained read;

Hs(1) = 11.1 m

n (1°) = 13.9 m (9)

H O°°) = 16.2 m

With respect to the corresponding spectral peak periods reference is made to Figs. 11 and 14.

For some offshore activities monthly extremes are of some interest. Extremes accounting approximately for the possible lack of long-term representation for the various months, can be obtained by means of probability plots like those shown in Figs. 4-7 and the scaling formula given by equation (3). The monthly extremes obtained by this procedure are given in Fig. 15.

So far a commonly adopted design procedure is as follows. At first the marginal extreme of Hs correspond- ing to a specified return period is estimated. Thereafter a reasonable interval for the corresponding spectral peak period is determined and the worst case within this interval is chosen as the design case.

Suggest now that the period interval of interest is restricted by Tpl = 12 s and Tpu = 24 s, respectively. Based on the five years o f measurements the design curve will be the line between 12 s and 24 s located at Hs = 15 m. This curve is shown by a broken line in Fig. 16. The adequacy of this design curve can be questioned since the probability of exceeding it will vary very much over the period interval. In order to obtain a design curve of constant probability of exceedence over the period interval, the significant wave height must be allowed to vary over the interval. Denoting

Wave climate off northern Norway: S. Hayer

12

H s

[M] s

f

. 1 o '

. . . . X 1

\- ,X\ / i /

" ~ V MONTHLY EXTREMES

MARGINAL EXTREME~

1 2 3 4 5 6 7 8 9 10 11 12

MONTH NO.

Figure 15. Monthly extremes for the significant wave height

H~

[M]

l e I

! 14 i i

[

!

! I

I TpI

6

ORIGINAL DESIGN CURVE

I

_ _

[

/i ! : - t

4 8 12 16 20 24 28

Tp ~S]

Figure 16. Design curves corresponding to return pertou~ of 1 O0 years

2O

16

03

12

LU 4

,¢.,-,

2 4 14

. ~ ~,,,. I ' 'P I * , , , " U -

+ SAMPLE VALUES

SMOOTHED MODEL

6 8 10 12

H S [M]

J

Figure 14. Conditional expectation for the spectral peak period given the significant wave height

the modified design curve by H(n)(tp), it is uniquely deter- mined by;

fHsTp(Hs, Tp) dH s dTp = q(n) 7"pl ~q(~" )

o o

P

ATp J fFrsTp(Hs, Tp) dHs = constant (10)

. } " ) ( r p )

where q(n) is equal to the inverse of the number of occur- rences in n years.

Herein ATp -- 1.5 s is adopted, and H(sX°°)(Tp) obtained from equation (10) and the smoothed joint model is shown

Applied Ocean Research, 1985, Vol. 7, No. 2 91

Page 8: Wave climate off northern Norway

Wave climate o f f northern Norway. S. Hayer

in Fig. 16. It is seen that the significant wave height corresponding to the most probable peak period is some- what larger than the marginal extreme, while the significant wave height corresponding to the more rare periods is significantly reduced.

It should be noted that the design curves presented in Fig. 16 are established from wave measurements, In order to account for the indicated lack of long-term representa- tivity, the design curve should be somewhat increased.

CONCLUSIONS

The results presented herein indicate that:

The measuring procedure generally adopted for the Norwegian continental shelf is of a sufficient accuracy as far as the wave climate is to be predicted in terms of the most commonly adopted sea state characteristics.

Although the hindcast values do not fit too well to the observations, the hindcast wave data seem to be a con- venient mean with respect to an evaluation of the long-term representativity of the measuring period, especially if this is to be accounted for quantitatively.

A reasonable empirical model for the joint distribution of the significant wave height and the spectral peak period is obtained by adopting a lonowe model for the significant wave height (i.e. a log normal distribution for low values and a Weibull distribution otherwise) and a log normal distribution for the spectral peak period.

A consistent design curve with respect to the choice of simultaneous values of the significant wave height and spectral peak period can easily be established when joint distribution of the two characteristics is known. Typical for this new design curve is that the significant wave height corresponding to the most probable peak period is some- what increased compared to the marginal extreme, while the significant wave height corresponding to more rare periods is significantly reduced.

It should be noted that the most probable largest response values will vary for sea states located along the design curve. A specified return period for a response quantity can only be achieved if a long term response analysis is carried out.

ACKNOWLEDGEMENTS

This paper is to a large extent based upon results of projects carried out while the author was at the Norwegian Hydro- dynamic Laboratories. The projects were financially supported by NTNF, Esso Exploration and Production, Norway, Norwegian Contractors, Statoil, and the Norwegian

Petroleum Directorate. Their support and interest for these projects are deeply acknowledged.

NOMENCLATURE

E [ ] expectation operator fx(x) probability density function for x .fxl),(xly) conditional probability density function tot

x given y Jxy(X,Y) joint probability density function for x a n d y Fx(x ) cumulative distribution function for x H s significant wave height, herein defined as four

times the square root of the spectral moment of order zero

/~s significant wave height produced by the hind- cast model

H (n) extreme values of the significant wave height s , m

corresponding to a return period of n years and obtained from measurements achieved during tn years

/~(n) extreme value of the hindcast significant wave Sg/gl

height corresponding to a return period of n years and obtained from m years of hindcast data

H (n) extreme value of the significant wave height based on a joint consideration of measurements and hindcast data

Tp spectral peak period, herein defined as the inverse of the spectral peak frequency

Var[ ] variance operator

REFERENCES

1 Torsethaugen, K. and Krogstad, H. Neptun-A computer program for the analysis of ocean wave records, Continental Shelf Institute, Report P218/1/79, Trondheim, 1979

2 Haug, O. and Guddal, J. Hindcasting the wind and wave climate of Norwegian waters, The Norwegian Meteorological Institute, Oslo/Bergen, 1981

3 Tryggestad, S. et al., Environmental Conditions at Troms~flaket, ISBN82-7257-126-9, The Norwegian Petroleum Directorate, Stavanger, 1983

4 Hayer, S. On the prediction of extreme sea states, NHL-report 183222, Norwegian Hydrodynamic Laboratories, Trondheim, 1983

5 Hayer, S. Analysis of uncertainties related to the stochastic modelling of ocean waves, UR-80-09, Department of Marine Technology, The Norwegian Institute of Technology, Trondheim, 1980

6 Ochi, M. K. and Hubble, E. N. Six parameter wave spectra, Proceedings of the 15th Coastal Engineering Conference, Honolulu, 1976

92 Applied Ocean Research, 1985, VoL 7, No. 2