Jan 1, 2018 - and tests different methods of analysis of existing wave data. .... fonned crests) as they passed a pole located about ... significant wave period in tile following way. .... the following expression was obtained: ..... (2300 km) and a wind speed of 100 knots. ..... The authors wish to express their gralitute to Mr. D.
ISRAEL
JOURNAL
OF EARTH-SCIENCES
Val.
30 1982
pp.
120-134
Evaluation of the Wave Characteristics at the Mediterranean Coast of Israel
Dav S. Rosen and Eliezer Kit Israel Coastal and Marine Engineering Research Institute, Technion - Israel Institute of Technology, Haifa, Israel
Abstract
Nomenclature
Different methods of evaluation of long-term and extreme wave conditions were tested using wave data gathered over an 18 yr period at Ashdod, on the Mediterranean coast of Israel. Long-term joint density probability of wave heights and periods estimated for Ashdod is used to evaluate confidence domains for different confidence coefficients. Extreme valUes
1\, all, 0,
n, "
C. ('2,C e. exp fl
H, fl,
Hi, H:
are shown to be conservatively estimated by a Gumbell distribution, according to which the 100 yr average-recurrence deep-water significant wave height is about 8.70 m. f-/,
H*
H
conslilnts constants conslants 2.7IR2R ... variilble vilriahles i-th villue in the silmple of deepwater significilnt wave heights scale factor (constant) l-l. variable lower limil of (Weibull distribution) upper limiting bound of the variable H, (Asymplote III) modal value of the log-normal density mean value of Ihe sample of H, values
K
index, indicates rank of a vilriable in a silmple of values correlation coefficient between
I-
data III a sample (Gumbel, Asymptote)) economic lifetime of "a structure
In 11
11
(X, )
natural number
logarithm of lrials
_...
number
of trials
al level
continuous /I
(H,)
Received January.
process
variale
x, of a X
average number of chances 10 occur per year which ha~ a sea state H .•
1982
•
MEDITERRANEAN
P(X ~ x) P(X
~
P(H,
Xi)
(H" T),
p(H:, p (r, 0)
-
~ H;) -
p(H;) p(T') p
-
-
Ti)
-
q (Xi)
-
0",
-
standard deviation of the population of In values
O"y
-
standard deviation of the population of In T values
cumulative probability of occurrence of deep-water significant wave heights smaller th3n H; probability density function of H; probability density function of T' joint probability density function of H, and T values joint probability 6 values
6
of r 3nd
density
the exponent expression distribution
radial coordinate of a point in the r, 6 plane standard deviation of the sample of H, values
variable,
standard deviation of T values
radial
f
coordi-
of the sample
X
In H, value of a variable in a sample mean value of the natural
y y
of the
H,
values
y
Itl H.
'
Jr;T', similar to X reduced variate III the method constant
Gumbel
y
confidence
4J(H, ~ H:) -
extreme function
A
level (value) of the joint probability density function
n
-
coefficient cumulative of H,
3.14159 ...
probability
variable,
angular
coordi-
collected at Ashdod from 1958 by the Coast Study Division of the Israel Ports Au thori ty. This is the largest and most extensive set of wave data gathered to date on the Medi terranean coast of Israel. Hence, this set of data has been used as an almost unique data basis for the design of coastal and marine structu res in Israel.
modal value of the log-normal probability density distribution function of the T values
logarithms In T
random nate
and tests different methods of analysis of existing wave data. The basic set of wave data used here were
T*
X
H
1. Introduction
variable in the sample of wave periods corresponding to the sample of H, values variable wave period corresponding to the H; value mean value of the sample of T values
T'
values in a sample of wave data correlation coefficients of X, Y values
The present study describes the evaluation of wave characteristics on the Mediterranean coast of Israel
of the of the function
ri
T
and
p....
of nonin I trial
r3ndom nate
-
H,
cumulative exceedance
r
511
of
T
(see Ochi, 1978a, p. 65) return period risk
r
coefficient
PII.T
power of asymptotic cumulative
R
correlation
cumulative probability of occurrence of X values smaller than x probability of Xi value
-
121
COAST WAVES
)
The safety of any coastal or marine structure, as well as its optimal utilization, primarily depend on the nature and reliability of the data used for its design. Of particular importance is· information on wave characteristics at the site as its analysis allows: a) the evaluation of the combined and marginal probability of occurrence of wave heights, periods and directions, which are used in the optimization of the. functions of a structure (e.g., yearly average number of days which a ship can safely moor or unload at a certain berth), and b) the evaluation of the probabilitY of occurrence of extreme sea states, which is required for structural design, Le. for the detennination the structure.
of the safety level of
The wave climate at a site is determined by monitoring the waves there for a long period. At present, the monitoring usually -consists of wave measurements performed using sophisticated instruments. The "instrumental measurements'~-consist of 10- to 20-min intervals taken every 3 or 6 h or only dally, ancl they usually extend over a period of, at most, a few years. In the past, when sophisticated wave measu ring instrumen ts were not available, the monitoring was carried out using simple m-ethods. The simplest one consisted of "visual observations" and -the quality of the data obtained depended on the skills of the observers and their experience. A more
•
122
D.S. ROSEN AND E. KIT
complex method consisted of "visual wave measurements", by which the wave parameters were visually determined by means of a graduated stick, binoculars and a stopwatch. At Ashdod, the "visual wave measurement" method was employed until 1975 and.the "instrumen tal wave measu remen t" me thod from 1977. As visual monitoring could be performed only during dayJigh t, the "visual measurements" were performed less frequently Ulan the "instrumental measurements" (only once to UHee times per day). AI though such data seem to be very sparse, the reliable evaluation of the long-term com bined and marginal probabili ty distribu tions of the wave heights, periods and directions is feasible because of the large number of data. The reliable evaluation of extreme sea states (i.e. significant wave heights witil large return periods) is also feasible because it covers relatively many years and a reasonably large sanlple of yearly sea-state maxima is available. A previous study conducted by Rosen and (1978) indicated that tile wave climate in deep at Ashdod fairly well represents tile deep-water climate on the Mediterranean coast of Israel. study
was
based
on "visual
Vajda water wave That
wave measurements"
perfonned simultaneously for 25 monUls at Ashdod (souUlern section of the coast) and at Hadera (about 100 km nortil of Ashdod). Therefore, to evaluate the wave characteristics on the Mediterranean coast of Israel, the raw data gatllered at Ashdod were initially processed in such a way as to form two homogeneous samples of data, one of daily maximum deep-water significant wave heights, tJle oUler of yearly maximum deep-water significant wave heights. Further, Ule Ashdod wave data are compared here wi Ul wave data gaUlered and published by Ule U.S. Army Naval Weatller Service Command (1970). In Ule present study tile significant wave height refers to the deep-water unless oUlerwise specified.
Hs
2. Description Present Study Raw Data
significant
wave height
represented ilie height of Ule highest breaking wave during 5 to 10 min of continuous monitoring of the elevations of the breakers' crests above MSL datum. These elevations were continuously determined by a shore-based observer, using a graduated stick, who aligned each breaker crest witJl the horizon (see Fig. 1). The estimated height of tJle crest was later corrected by removing Ule influence of tJle tide on the sea level. The tide was recorded at Ashdod hourly, using a mareograph which was installed, for the period 1958-1965, inside Ule cooling basin of tJle Eshkol electric power station and, since 1966, inside Ashdod Port. FurUler, ilie maximum breaker height was detemlined by assuming Ule highest crest to be 75% of the total breaker heigh t. The use of this assumption, made by the Coast Study Division of the Israel Ports Auiliority, leads to somewhat conservative values (higher waves). This has been indicated by published data~ (Coastal Engineering Research Center, 1977, Vol. I1, fig. 7-45, p. 7-82; Vol. Ill, fig. C-6, p. C-35). The corresponding' main wave direction at a water deptJl of about 30 m was visually detemlined from a high point on the shore using binoculars and a compass. The wave period was obtained from visual observation by timing 10 consecutive waves (11 wellfonned crests) as they passed a pole located about 1 km offshore. Assuming tJlat ilie smaller waves were probably not counted, Ule visual period is assumed to be closer to ilie significant period, and is considered as such in ilie following text. This assumption was supported also by tile hindcasted values for one of tJle most severe stomis (Stiassnie, 1978). Group 11 (1973-1975). Waves were visually measured at Ule sanle times as in Group I, but tileir heights were determined relative to a pole placed at 12 m water depth. Wave directions and periods were measured in Ule same way as for Group I. It must be stressed Ulat tilese two bodies of data are incomplete since consecutive observations from one or more
of the Sets of Wave Data Used in the
The raw data available for Ule present study can be divided into four groups. These groups were derived according to tile me UlOds of measu remen t and Ule periods of data acquisition. Group I (1958-1971). Three daily visual wave measurements were carried ou t usually at 0600,0900 and 1200 GMT. The visually measured
wave height
- •••-. __
._
....L~
~t!.!!!.1
,...~1I!!!!!'
"),. d,' {,-z, Fig. 1. Sketch of wave measurement method.
•
MEDITERRANEAN
days are often missing. Moreover, the three daily maximum measurements cover only a quarter of a day. Group III (1977, May 1978-March 1979). The wave data included in this group refer to instrumental wave measurements performed offshore from Ashdod, at a depth of approximately 19 m using a Datawell waverider. The waves were measured continuously for 10 min at 3-h intervals and were recorded on paper only. Evaluation of the significant wave heights from tilese paper records was not performed until recently. In order to preserve homogeneity of the sample of wave data used, tile data of Group III were, in general, not included in tJle statistical evaluation. However, tile wave records of yearly maxima were processed manually. The significant wave heights evaluated from them allowed us to increase the size of the sample of yearly maxima of signif1cant wave heights. The wave periods were detennined also from the paper records, while tJle wave directions were obtained, for this Group, at tJle same time and in the same way as in Groups I and 11. Group IV. The wave data included in this group refer to wave measurements performed WitJI the same wave rider buoy as in Group III bu t the data were recorded on magnetic tape and processed on a mini computer (Nova 3) at tJle Israel Coastal and Marine Engineering Research Institute, Technion - Israel Institu te of Technology, Haifa, Israel. The data consists of measurements performed at Hadera during January-April 1978 and at Ashdod from April 1979. The waves were measured also at 3-h intervals as in Group Ill, but for a duration of about 20 min. These measuremen ts were used only to obtain the yearly maxima of significant wave heights in order to have homogeneity of tJle data, as was explained previously. The wave direction was determined at tJle same hours and in the same manner as in Groups I, II and Ill. Processing of the Raw Wave Data
123
COAST WA VES
each one an equal weigllt; the duration of an observation was taken as 5 min. The choice of only tile maximum daily wave heigllt was meant to limit tile dependence on consecutive measurements, as the otJler two values were obtained in tile same quarter of tJle day. The evaluation of the deep-water significant values was performed in the following ways: a) The wave heigllts of Group I" which consisted, as mentioned in Sect. 2, of maximum breaker heights during 5-min periods, were transformed into deepwa ter significant wave heigll ts preserving tile same significant wave period in tile following way. First, for each maximum breaker height, Hbr m ax' the deep-water maximum wave' height, Ho: nlaX' during 5 min (300 s) was determined. Since the breaking depth was unknown, use was made of tile empirical formula of Komar and Gauglla.rl (1972), also taking in to accoun t the refraction coefficien t and tile shore slope (which is about 1:80). Then this formula became (in metric units):
Ho,••", = (H~;~~•••/O.53 T~~)· (cos a~./cos
ao)12S
(1)
where Ts '" significant wave period""" T; T '" wave period (in seconds), visually determined by dividing the total time during which a number of waves passed a fixed point to the number of waves (10) counted during tJlis period; ao '" angle between the deep-water wave crest and the shore line (in degrees); ab '" angle between line.
the breaker
line and the shore
Secondly, assuming tilat the short-term distribution of tlle wave heigllts behaves according to the Rayleigh distribution, the deep-water significant wave heights were calculated. The deep-water maximum wave heigllt obtained was assumed to represent the most probable highest deep-water wave during the 5-min period. Thus, using tile relationship between tile most probable higllest Wave and the significant wave heigh t according to Longuet-Higgins (1952, formula 2), tJle deep-water significant wave heigllt was obtained:
The raw wave data used were processed in such a way as to obtain a homogeneous sample of deepwater significant waves. Therefore, the following (2) Ho,,,,", = [lln(N)]l. H, constraints were introduced: (a) only tile maximum where N is the number of waves during the observadaily wave height was selected from the three daily tion period: N'" 300/T. visual measurements available in Groups I and Il, Finally, hindcast analysis performed for one of the together with its respective period and direction, and (b) differences among observers and among - heaviest recorded storms during the period covered estimates of durations was neglected, so as to give by Group I confirmed, in anotJler way, the deep-
•
124
D.S. ROSEN AND E. KIT
water values obtained
from the visual wave measure-
ments. The results of the hindcast analysis have shown consistently similar wave values with those obtained from tlle measured breakers (Stiassnie, 1978). b) The evaluation of the deep·water significant wave heights of the other groups of data was performed again using the values of tlle maximum waves, assuming parallel straigh t con tour lines and using refraction and shoaling coefficients presented by Coastal Engineering Research Cen ter (1977, Vo\. H, fig. 7-45, p. 7-82; Vo\. III, fig. C·6, p. C-35). Afterwards, tl1e transformation from deep-water maximum wave height to deep water significant wave height was performed as for Group I.
Samples of Wave Data The data processed, as explained above, were divided in two wave samples: the first one, with a relatively large number (5168) of daily maxima of deep-water significant waves, the second witl1 relatively few values (18) containing yearly maxima of deep-water significant wave heights and associated periods. A third sample of wave data acquired by the U.S. Army Naval Weather Service Command (1970) in the eastern Mediterranean by visual observation from ships was used for comparison with the former one. These samples formed the basis for the statistical analyses perfonned in the present study. 3. Evaluation of the Marginal Probability Distribution of Significant Wave Heights and Wave Periods In order to evaluate tl1e long-ternl cumulative probability distribution of the significant wave heights at Ashdod, use was made of a relatively large sample (5168 values) of daily maxima of deep-water significant wave heights. Different authors used different
distributions
to describe
tl1e cumulative
probability
distribu tion of significan t wave heigh ts and periods. Such distributions include tlle exponential, Weibull, normal and 10g-norn1a1 distribu tions. These are the most commonly used ones and are presented below.
Exponential Distribution This
distribution
has
been used
and advocated
by many autllOrs for tile evaluation of tile long-ternl cumulative probability distribution of significant wave heights. Copeiro (1978) reaches tlle conclusion that: "the exponential function could then describe statistically tile growth of the sea caused by wind fields reaching tl1e observation site, originated by differential pressure centers". The technique is to plot log (1 - P(H s ~ /-Ps)f' versus H~ tl1rough tile higher values of HL assuming them to be exponentially distributed. Here: P(Hs ~ HD = cumulative probability of occurrence of deepwater significant wave heights smaller than H~. H~ is the i-th value in a sample of deep-water significant wave heights. Accor'dingly, the sample of data was plotted on semi-log paper in Fig. 2. Following Bretschneider and Rocheleau (1978), a straight line was drawn through tl1e higher Hs values, using leastsquares regression, in order to obtain the exponential distribu tion of the deep-water significan t wave heights. The function obtained may be expressed as: P (l-J., ~ H:)
P(H,
~ H;)
= 1 - 12.08[10-o('RI/:]
(l)
= 1-12.08exp[
(2)
-1.565H;]
Weibull Distribution This distribution, proposed originally by Weibull, was suggested for application to coastal engineering by Bretschneider (1964). In fact it is the general case
'0
0_
Ashdod-
,,-\).5.
(19'0-1971)
Novy-CMedileffon.on)
Bul fil line
====-:-,
I o '.
100
10
I PIH.
Fig, 2. Exponential probability
it)
EXCEEDING
distribution
O' FREQUENCY
IC)'
C'(.)
of deep-water significant wave heights,
.'.
MEDITERRANEAN
of Ule exponen tial distribu tion, the exponen t being raised to _._.'-1some power, while in the exponential f-li~'1.33 I distribution the--lpower of the exponent is l. The general fonn of the Weibull distribution is given by: P (X;;;; x)
= 1-
/
/
I--
exp ( - (x - a)! bY
(3)
For ilie distribu tion of significan t waves ilie following fonn is generally used:
P(H. ~ H;)
= 1- exp[ -«H;-
Ho)/(He - He,)t] (4)
where a , HO = ilie lower limit of the variable, b, Hc = the scale factor and e, 'Y= the shape factor. If a = 0 then ilie distribu tion is known as the Frechet distribu tion. The use of iliis distribu lion has been advocated by many authors (Battjes, 1972; Houmb & Overvik, 1977), ilie most prominent being the International Commission for Ule Reception of Large Ships (1979). When the data are plotted on Weibull paper it is expected that they will fi t a straight line. However, to do so, a good choice of the Ho, i.e. the lower limit of the variable, has to be made. Thi.svalue of Ifs is found when I -P(Hs -
... ~
0.60
l!l ~ O.~ ::;
iii ~
o
0140
0::
... 0.30
020
0.10
01 H.,.
Z" DEEP WATER
04,.56 SIGNIFlCANT
WNE
HEIGHT
(m
J
Fig. 6. Deep-water significant wave height offshore Ashdod: histogram and density probability distributions.
of
•
MEDITERRANEAN
comparison probability accentuate
-
I-..I-
Marginal Probability
Distribution
Y = In Ti, Ti is the visually determined
period
corresponding
Equation
(10)
Ht
Y = 1.67, and 0y =
was plotted
together
with
As mentioned before, Ochi (l978a) found that lognormal probability density functions describe the histograms of Ule significant wave heights and wave periods reasonably well. Hence, he suggested expressing Ule joint density probability function of the significant wave heights and wave periods in the following manner:
0-.o-y27T •V
. exp{ - (1/2X1- p~y). - 2pxy
[((x
. (( J( - x)/ 0-. ) . ((y
005
o
I
2
3
4
5
6
1
8
9
10
11
12
13
14
r;---z 1PXY
-'7;)/0-.)2 -Y)/o-y)
-1-
Fig. 7. Wave period at Ashdod. density probability distribution.
histogram
and log-normal
where X = In Ifs. Y = In T, p Xy = correlation cient of X, Y values.
coeffi.
PXY
= In(p~IT·V(e"!-1)(e"~-1)-1- 1)
(12)
UX,(Ty
where PUT = correlation coefficient Since similar results were found data as presented used in Ulis work probability density wave periods. Thus,
of lis, T values. for the Ashdod
above, Ule same expression was to describe the long-term joint distribution of wave heights and confidence domains for different
r
confidence coefficients were calculated using a numerical integration method and the results were plotted in Fig. 8. The confidence domain is defined here as Ule area enclosed by the contour curve. This is Ule intersection 'of -the joint probability density function with a plane parallel to the (H, T) plane at a certain level A of Ule joint probability density. The confidence coefficient is defined as the cumulative
r
probability of the corresponding domain. In order to find the confidence domains
1
= H, . T·
ten tZ u; iii ::: >-
the
4. Joint Probability Distribution of Deep-Water Significant Wave Heights and Wave Periods, Confidence Domains and Confidence Coefficients
.
0.15 0.10
VI
>-
wave
histogram in Fig. 7. The comparison shows a good fit of the log-normal distribution with the histogram. This indicates that a log-normal distribution may be used to describe the long-term probability density distribu lion of the wave periods offshore of Ashdod.
p(H" T)
, :;..~~0
H(ln T' - I.G7)/O,JOSn
where
to
020
WAVE PERIOD (,I
(10)
0.305.
L::~ LO ;2
7
(2300 km) and a wind speed of 100 knots. The resulting value obtained was a wave height of 36 m. In order to examine the sensitivity of the results for extreme sea states to Ule limiting values, other limiting values of I? m, 20 m and 1000 m were also used to find the best-fit line to the data using leastsquares regression. The resul ting distribu tions were as follows:
•..6
In(15 - H;) = 0.0872 ·In[ -In[(H:. ~ H;)]J + 2.3113
Fig. 11. Log-normal distribution significant wave heights.
.7~ _
5iL ~ Z
Cl
•
;;; '"
'" ...
~ ..
'"
l'l 2 CUMULATIVE
(26)
1nl
In(20 - H;) = 0.057,
1~(H,
~ H:)J] + 2.7144 (27)
In(36 - H;) = 0.0271 'In[ -In[(H, In(1000
-1I;)
~ H;)]J + 3.4373
(28)
= 0.000833 (29)
'In[ - In[(H, ~ H;)]J + 6.90286
Since each dist.ribu lion requires a different plotting scale in order to be plotted on Weibull paper, they were not plotted together on Weibull paper. However, the results for different return periods [1/ using Asymptote I and oUler methods. Log-normal.
The use of this distribution
evaluation of extreme by Weiss (1957). It is reasons with oUler distribution obtained
for the
values was originally proposed presented here for comparative distributions. The log-normal (also using Eq. (8» is shown
together WitJl the actual expression is given by:
data
in Fig.
11 and its
PROOA8I.UTY
(%)
of extreme
statistically dependent events of a physical continuous process are transfonned in a sample of discrete events with statistical independence, and showed tJle application of the method for the evaluation of extreme significant wave heights. According to it, each sea state (variate Hs) is assigned an average duration per year and the total average number of occurrences of a sea state (Hs) thus becomes a function of Ule sea state itself. Therefore - while for a sample of discrete events tJle probability 1>(x ';;;x/) that an event will not be exceeded in 11 trials is given by (x ';;;x/) = [P(x ';;;x/)ln, where P(x ';;;x/) is tJle probability of non-exceedance in one trial - for a continuous evolution. process the number of trials n is a function of x itself, Le.n = n(x) and tJlen
(31 ) For significant wave heights, Copeiro found that tJle n(Hs) function was satisfied by the empirical linear rela tionship:
n(H,) = A (H;- B) (H;, and B are in meters)
(30)
Quite surprisingly, the fit to Ule actual data is quite good. Values for representative return periods are presented in Table 1 and in Fig. 13 WiUl oUler distribu lions. b) Copeiro Method Copeiro
(1978)
proposed
a me thod by ~hich
the
deep-water
(32)
Further, according to his findings, the most suitable distribution for P(Hs';;; m) is tJle exponential distribution, while the WeibuU distribution is less suitable and the log-normal is unsuitable due to its poor fit to extreme values. According to tJle method described above, an evaluation of the n(Jfs) function was done on the basis of the available data concerning tJle yearly average durations of sea states at Ashdod. The data was plotted in FIg. 12 and from it the function obtained, assuming linear dependence, was 11
(H,)=
69H, +34
(H, is in meters)
(33)
•
MEDITERRANEAN
TABLE 1. Extreme Values of Deep-Water Significant Wave Heights Offshore of Ashdod 85 a Function of Return Period and Method of Evaluation ... LO'C.,tlr. ~\ Gu",lI.11 '11'", W."vll rthHfl .T9 No."'.' '11I','''''11 Ie",tt"'d 1.'0 1.11 ,I '.01 10.11 10.7" 4.111 .1' '.1 1.lr''''' il'\"r "abloi"ltI ... "'Ill"". .1 [.,., "~,I •T., probability '.84 10.01 1.71 •• S,Z4 •. '.J'4 •.et 1.;'0 1.4 '.01 & •.e. t.l5 '.T& .... 9.••• 10. •10.21. '.liT S.l' "T ".05 1.n •.... ol. .11 T.1l ','9 10 •.•• 41_tr. '.1' '.18 '.20 8.14 •".20 1.'1 1".1 '.10 41.". ,I' 6.1 '.!., .34 •tS,11 I.e, '" 10.0" 9.96 ".H 0& I." •_hOM •. ~ ••• 10.00 S. T.ZI ,•. 41,,,. ".11 It' '.45 6.11 & '25 8.U •.•• ,rt t.15 ., '.8' .1. ..... .19 .••0 .70 '1",. 1[.tropolalloll I'1.01 1t1 ...• .,6" ~5 '.10 1.'0 !.•.•SfuPlC'\OfI' ,.. '.14 '.55 ...,.,• III IIorlll.1 40 1,01 Il,. praboblli')' 10.2:1 [,po","'1I1 T ••T,5.2:0 I.'" 1.44 TU '.22 8.11 '.12: 1.01 &.70 •. 1.IU 1.2:1 '.17 I.' .10 .1' '.00 1.'9 511 eIU.f,. !I,,'10' e7 "lu ••fUIlCllo".1 )IT .,.21 "rift"_,,"pt&ll Lo, pfMPI'1 'IT 6." ,dol' ••'" .1 dr 'rt",fOf"""9 da)'. ':110 AI,""pl0" Ht. '58", ~IOOOIJl Hl·2O ••
Afterwards, using the exponential and Weibull distributions previously obtained in Sec. 4, extreme values were evaluated on the basis of the Copeiro ,oflod "".rav· p.rlod 'I'urll dl.I,. ptriod '00 "'lIfOl (,.or 1000 1000 10 400 .00 •.0 '0 .J• .0 400 100 lOO 10 lI'Iuho4 lOO '0 method, i.e. .00 -
H:) l= [1-
,
..
,. "" "" "", ____ ~. ., _. 11 ••••••
"
I"., ,I
A ••••• oO·
(~(H, ~
13'
COAST WA VES
''
.•
11
(~•• 0''elfllpl. la",
l
•... ,..
exp( - O.692H:)I2')J(~QII:+Jd) (34)
using the Weibull function and
~(H, ~ H:) = [l-12.08cxp(
-1.565H:)](fiQ//:+3d) (35)
using the exponen tial function. It should be mentioned here Ulat the return period is already expressed in years and not in days, due to the way n (H s) was obtained. The results are presented in Table J for comparison wiUl the other methods of evaluation of extreme values. They are also presented in Fig. 13.
"-Bosed00
Ashdod data
100 YEARLY
~oo
200
AVERAGE
N'J, 2 m). For smaller waves it is not clear if this applies as well, bu t using the same assumption servative values are obtained.
con-
The data used in the present study were composed of the whole sample of visual wave measurements taken at constant short time intervals (daily maxima
in our case). The plots of the data sample on Weibull, semiJog (exponential), normal and log-normal paper did not pennit fitting a straight line through all the data. A reasonably well-fitted straight line could be drawn through the data on wave heigh ts larger than 4 m when Weibull or exponential distributions were assumed, using Ule method of least-squares regression. The plots of the distribu tion using normal and log-normal functions do not fit well the data for larger wave heights (which seem to lie on a straight line) because they were built using all the data. The evaluation of extreme deep-water significant wave heights was tested by employing three methods of calculation. The use of the sample of yearly maxima of deep-water significant wave heights which are assumed independent is somewhat handicapped by the relatively small size of the sample. As pointed ou t by Borgman (1975), extrapolations beyond twice the sample size are in fact of low reliability. The use of Asymptote III seems more reasonable for the description of a physical process which may be assumed to be limited 'by an upper bound. As seen from Table J and Fig. 13, the results for different limi ting values of the upper wave-heigh t bound indicate clearly that the Asymptote I is in fact the limiting case of AsymptQte Ill. The average return p'eriod R (years) represents the average time intervaJ between the occurrences of deep-water wave heights larger thati lis. When designing a structure to last a lifetime L (years) we must be concerned with the occurrence of wave heights, which have only a small chance of occurrence, defined as a small risk value r of being exceeded during the lifetime L of the structure. Thus, tile extreme wave height to be used for the design of a structure is given by: 1 R(years)
= l-(l-r)'l'
(36)
For a more comprehensive explanation the reader is referre'd to ~n article by Borgman (1963). 7. Conclusions It was found in regard to the long-term wave characteristics in deep water at the Mediterranean coast of Israel that the lorig-tenn joint and marginal density probability fUllctions of the significant wave heights and associated periods cart be reasonably evaluated assuming log-nomlal distribu tions. However, rare values of significant wave height are estimated better assuming a Weibull distribution.
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MEDITERRANEAN
For the evaluation of extreme sea slales, the relatively simple method proposed by Gumbel for the use of Asymptote I gives il an advantage over lhe other ones. Furthermore, the Gumbel melhod gives the largest values and in fact is lhe upper bound for the extreme-values distribu lions. However, its use is possible only for samples WiUl sizes larger than 10. When such samples are not available, and Ule wave data cover only a few years, the method proposed by Copeiro, using the Weibull probability funclion, seems to give good estima tes of ex treme wave he igb Is. The confidence domains of joint probability of occurrence of deep·water significant wave heighls and periods can be used to find lhe cri lical wave period values associa ted wi tb a given design signi ficanl wave height and confined by a certain confidence inlerval. Acknowledgement The authors wish to express their gralitute to Mr. D. Divon, Head of Ule Coasts Survey Division of lhe Israel Ports Au thorily, for providing most of the raw data used in Ule present study. Thanks are also expressed to Prof. M. Diskin, Director of the Israel Coastal and Marine Engineering Research Insitute, for providing financial support, and for his useful remarks concerning the manuscript. Also, we would like 10 express our gratitude to Prof. M.L. Vajda, Or. M. Sliassnie and Or. E. Naheer, Israel Coastal and Marine Engineering Research Institute, for their comments and for reviewing the manuscript.
valuable
References B~ttjes, 1.A. 1972. Long term w~ve height distributions at seven stations ~round the British Isles. Deutch. Hydrogr. Z.99. Borgman, E.L. 1963. Risk critcria. Proc. WW3: 1-35.
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Borglll~n, E.L. 1975. Extremal statistics in ocean engineering. Proc. Civil Engineering in the Oceans Ill. Vol. 1. 1'1'.117-133. ASCE. Bretschneider, C.L. 1964. Investigation of the statistics of wave heights. J. Waterw. Harbour. Div. Am. Soc. Civil Eng. 90, No. WWl: 153-166. Uretschneider, C.L. and R.E. Rocheleau. 1978. An evaluation of extreme wave climate at Keahole Point, Hawaii. Proc. 16th Coastal Eng. Conf. Vol. 1. Chap. 7. p. 152173. ASCE. Urown, L.P., M. Nekhom and M.L. Patch. 1975. A MonteCarlo determination of extreme ocean waves. Proc. Offshore Technology Conference. Paper no. OTC 2194. Vol. 1. Texas, USA. Cartwright, D.E. and M.S. Longuet·Higgins. 1956. The statistical distribution of maxima of a random function. Proc. R. Soc. London, Ser. A. 237: 212-232.
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COAST WAVES
Coastal Engineering Research Center. 1977. Shore Protection Manual. US Army. Collins, 1.1. 1974. Wave statistics and rubble mound structures. lnt. Symp. on Ocean Wave Measurement and Analysis. Vol. 1. p. 486-499. ASCE. Copeiro, Enrique. 1978. Extremal prediction of significant wave height. Proc. 16th Coastal Eng. Conf. Vol. I. Chap. 14. p. 2R5-304. ASCE. Darbyshire, 1. 196J. Prediction of wave characteristic over North Atlantic. 1. lost. Navig. 14: 339-347. Draper. L. 1963. Derivation of a "design wave" frol11 instrumenIal records of sea waves. Proc. Inst. Civ. Eng. (London) 26: 296-304. Fisher, R.A. and L.n.C. Tippet. 1928. Limiting forms of the frequcncy distribution of the largest or smallest member of a sample. I'roc. Cambridge Philos. Soc. 24: 180-190. GlImbel, EJ. 1958. Statistics of Extremes. Columbia Univ. Press, New York. I-Ioffman. D. and 0.1. Karst. 1975. The theory of the Rayleigh distribution and some of its applications. 1. Ship Res. 19: 172-191. lIollmb. O.G. 1976. Wave Statistics for Design, in Port Engineering. 2nd ed. P. Uruun, ed. Golf I'ubl. Co. lIollston, Texas. p. 136-140. I-Iollmb, a.G. and I. Overvik. 1977. On the duration of sea state. Univ. of Trondheim, Div. of I'ort and Ocean Engineering, Norway. International Commission for the Reception of Large Ships. 1979. Methods for analY7.ing wind, wave and swell data to estima te on an annual basis the number of days, and the maximum duration of periods during which port and ship operations will be impeded by these elements. Permanent International Association of Navigation Congresses. lardine, T.P. 1979. The reliability of visllally observed wave hei!!h ts. J. Coastal Eng. Vol. 3: 33 -38. Khanna, 1. and P. Andru. 1974. Lifetime wave height curve for Saint John Deep, Canada. Ins!. Symposium on Ocean Wave Measurements and Analysis. Vol. 1. pp. 301-319. ASCE. Linsley, K., M. Kohler and 1. Paulhus. 1958. Hydrology for Engineers. Chap. 11. Civil Eng. Series. McGraw Hill. p.245--259. Longuet-l-Iiggins, M.S. 1952. On the statistical distribution of the heights of sea waves. J. Mar. Res. 11: 245 - 266. Ochi. M.K. 1973. On prediction of extreme values. 1. Ship Res. 17: 29-37. Ocld,
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coastal waves. Proc. 16th_.Co.astal Eng. Conf. Vol. 1. Chap. 2. p. 59-75. ASCE. Ochi, M.K. 1978b. Probabilistic extreme values and their implication for offshore structure design,'" Proc. O.T.C. 1978.Pap~no.3161.p.978-992. Ochi, M.K. and J.E. Whalen. 1979. Estimation of extreme waves critical for the safety of offshore stmctures. I'roc. OT.C. 1979. Paper no. 3596. p. 2085-2090. Resio, DT. 1975. Extreme wave heights for Cleveland lIarbour. Proc. Civil Engineering in the Oceans Ill. Vol. I. p. 62-78. ASCE. Rosen, D. and L. Vajda. 1978. Hadera offshore coal unloading terminal. Progress Report No. 5. IIadera wind and
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