EFFECT OF THRESHOLD VALUES ON THE PREDICTED …EFFECT OF THRESHOLD VALUES ON THE PREDICTED EXTREME WAVES-A CASE STUDY IN UAE TERRITORIAL WATERS ... Burj Al Arab, Mina Siyahi, Jebel

Emirates Journal for Engineering Research, 12 (3), 77-93 (2007) (Regular Paper)

77

EFFECT OF THRESHOLD VALUES ON THE PREDICTED EXTREME WAVES-A CASE STUDY IN UAE TERRITORIAL WATERS

S. Neelamani

Environment and Urban Development Division, Kuwait Institute for Scientific Research, 13109 Safat, Kuwait Email: [email protected]

(Received April 2007 and accepted September 2007)

محطة داخل المياه اإلقليمية 25تم دراسة تأثير تغيير قيمة العتبة على ارتفاع الموجة الھام المتطرق وذلك عند تم استخدام طريقة ويبل وطريقة جمبل لتحليل األمواج . 4.5mإلى 0.75mقيمة العتبة تتراوح من . لإلمارات

قيمة المؤشر لطريقة ويبل تتراوح من . المحطات المتطرفة وقد بينت الدراسة أن طريقة ويبل ھي األفضل لجميعسوف يتم استخدام المحطة المطابقة . واختيار أفضل مؤشر مستند على المعامل األعلى لقيمة االرتداد 1.3إلى 0.8

توقع ارتفاع الموجة عندال يوجد اتجاه مؤكد . ومقياس المؤشر لتوقع الموجات الھامة المتطرفة لفترات عودة مختلفةفي بعض المحطات يوجد . لمحطات مختلفة 4.5mإلى 0.75mام المتطرف عندما تتراوح قيمة العتبة من الھ

تخفيض في قيمة ارتفاع الموجة الھام المتطرف المتوقع مع تزايد قيمة العتبة وفي محطات أخرى قيمة ارتفاع االختالف بين الحد األعلى . 4.5mإلى 0.75mالموجة الھام المتطرف المتوقع منقلبة عندما تتغير قيمة العتبة من

نتائج ھذه الدراسة ستكون مفيدة في . 1.6mسنة عودة ھو 100والحد األدنى لقيمة ارتفاع الموجة الھام لفترة .التصميم المثالي للعديد من المشاريع المستمرة والمخططة للمستقبل القريب في المنطقة الساحلية لإلمارات

The effect of varying the threshold value on the predicted extreme significant wave height is investigated. 25 different locations in the UAE territorial water is used for this case study. Threshold value is varied from 0.75 m to 4.5 m with increment of 0.25 m. Gumbel and Weibull extreme value distributions are used for the extreme wave analysis and Weibull distribution is found to be better for all the locations. The value of shape parameter of the Weibull distribution is varied from 0.8 to 1.3 with increment of 0.05 and the best shape parameter is selected (based on the highest coefficient of regression value) and the corresponding location and scale parameter are used for predicting the extreme significant waves of different return periods. It is found that there is no definite trend in the predicted extreme significant wave height, when the threshold value is varied from 0.75 m to 4.5 m, for different selected locations. For some locations, the predicted extreme significant wave height is found to reduce with increased threshold value and for some other location, the predicted extreme wave height value oscillates when the threshold value is changed from 0.75 to 4.5 m. It is found that the difference between the maximum and minimum predicted 100 year return period significant wave height value is about 1.6 m for some locations in the UAE territorial waters and for few other locations, the difference is as small as 0.20 m. It is hence recommended to the user/owner of the project to select a design value in between the minimum and maximum predicted significant wave height for different return periods by keeping the risk and project cost in mind. A large number of marine projects are in progress and many new projects are planned for the near future in the UAE territorial waters. The results of the present study will be highly useful for optimal design of these marine structures. Keywords: Extreme waves, UAE territorial waters, Threshold value, Gumbel and Weibull

distribution, Significant wave height, Return periods, Hindcasted wave data

1. INTRODUCTION Establishing the design wave condition for different types of marine structures (Seawater intake structures, Breakwaters, Port and harbor structures, Shore protection structures, Submarine pipelines, Open sea loading/unloading terminals, Oil terminals and offshore platforms etc.) are very essential both from safety and economic point of view. A lack of the

correct information on the design wave condition will result either in an unsafe structure or with an over designed (and hence uneconomical) structure. For example the weight of armor unit of a breakwater depends on the design significant wave height to the power of 3. Hence selection of 3 m or 4 m significant wave height results in an armor unit of weight in the ratio of 27:64. Hence it is important to predict the

S. Neelamani

78 Emirates Journal for Engineering Research, Vol. 12, No.3, 2007

design wave heights for different return periods. In UAE territorial waters, though a lot of coastal and offshore projects are in progress, sufficient investigation on extreme wave height prediction was not carried out so far covering its territorial waters. The procedure for the prediction of extreme wave height involves selection of a threshold value as an important input during the data preparation stage. The effect of selecting the threshold value also needs to be studied in order to know its influence on the predicted extreme wave heights for different return periods. This is also not known as on today for the UAE territorial waters. These are the major motivations for the present work. Work on establishing the wave heights for UAE territorial waters and Kuwaiti territorial waters with threshold wave height of 1.0 m were carried out by Neelamani et al.[1,2]. The present paper is a continuation of this work and reveals the effect of threshold value on the predicted wave heights at 25 different points in the UAE territorial waters.

A marine structure designed for the design sea state of North Atlantic Ocean (the design wave climate are severe due to possible fully developed sea conditions) or Bay of Bengal (cyclone effects are frequent) need not be adopted for the UAE's marine condition, which is basically fetch limited. The Arabian Gulf (Figure 1), a marginal sea in a typical arid zone, is an arm of the Indian Ocean. It lies between the latitude of 24-30 degree N. The gulf covers an area of 226,000 square km. It is 990 km long and its width ranges from 56 to 338 km. It has a total volume of 7000 to 8400 km3 of seawater (Emery [3], Purser and Seibold [4], El-Gindy and Hegazi [5]). The entire basin lies upon the continental shelf. The average water depth of the Arabian Gulf is about 35.0 m. But depths more than 107 m occur in some places. The gulf's water depth increases in the south east direction. The Arabian Gulf is connected to the Gulf of Oman and the Arabian Sea through the Strait of Hormuz, which is 56 km wide and with an average water depth of 107 m and allows water exchange between the Arabian Gulf and Arabian Sea. Further details of the Oceanographic Atlas of Arabian Gulf can be obtained from Al-Yamani et al [6]. In the Arabian Gulf, in general the dominant wind direction is northwesterly (Elshorbagy et al.[7]). Arabian Gulf is one of the very active marine areas on the earth. Most of the oil produced in the Gulf countries is transported through the Arabian Gulf waters. It is also strategically important area. Most of the countries around the Arabian Gulf rely on the seawater for desalination and for cooling purposes of power plants.

The United Arab Emirates is a federation of the seven Persian Gulf States of Abu Dhabi, Ajman, Dubai, Fujairah, Ras al-Khaimah, Sharjah, and Umm-al-Qaiwain and are located in the eastern part of the Arabian Peninsula. It is situated in between 22o 50' and 26o North latitude and between 51o and 56o 25' East longitude. UAE is surrounded by Qatar on the

northwest, Saudi Arabia on the west, south and southeast, Oman on the southeast and northeast and with 1318 km of Arabian Gulf coast on the North and Northeast (Figure 2). The total area of UAE is about 77,700 square km. It is in a strategic location along the southern approaches to the Strait of Hormuz, a vital transit point for world crude oil. Over the last few years the emirates, especially Dubai has attracted international acclaim for its dynamic and innovative approach to development in the coastal zone. Projects at Jumeirah Beach, Burj Al Arab, Mina Siyahi, Jebel Ali and the Palm Island developments are all testimony to this reputation

There are many offshore platforms in operation at present and many more are planned for future. Design of these marine structures requires estimate of design wave height for different return periods. An attempt was made by Neelamani et al.[1] to report the extreme waves in the UAE territorial waters for different return periods. In that work, the threshold value for extreme wave analysis was kept constant with 1.0 m. However, it is felt that the predicted extreme wave height value vary when the threshold value is changed. Hence, further investigation is carried out to report the effect of varying the threshold value on the predicted extreme significant wave heights for different return periods and is presented in this paper in detail.

Caires and Sterl [8] have estimated the 100 year return significant wave height value from the ERA-40 data for the whole oceans of the earth. The wind data used in their study is obtained from grid of 1.5o x 1.5o. Unfortunately this course grid cannot provide much information for marine areas off UAE since the width of the Arabian Gulf itself is of the order of 1.5o only. Hence we have purchased wind data for finer grid size of 0.5o x 0.5o and the wind speeds are linearly interpolated for grid size of 0.1o x 0.1o for running the WAM model for hind casting the significant wave height and for further extreme analysis of waves.

2. LITERATURE REVIEW There are a large number of works being carried out around the world on extreme value prediction of winds and waves. Gumbel [9] is the first who has developed a statistical method for predicting the extreme values of natural random events like wind speed. Recorded annual maximum wind speed for as many years as possible, is the input for this method. Gumbel's extreme value distribution is widely used by the wind engineering community around the world, since the method is simple and robust. St. Denis[10,11] has discussed Gumbel distribution in the context of extreme wave prediction. Information related to the collection of data samples for extreme value analysis can be found from Nolte[12], Cardone et al.[13], Petrauskas and Aagaard [14] and Jahns and Wheeler [15]. Details regarding the plotting formula used for the extreme wave predictions are available in Kimball [16],

Effect of Threshold Values on the Predicted Extreme Waves-A Case Study in UAE Territorial Waters

Emirates Journal for Engineering Research, Vol. 12, No.3, 2007 79

Figure 1. Map of the Arabian Gulf and UAE Shoreline

Figure 2. UAE map along with boundaries

S. Neelamani


Gringorten[17] and Petrauskas and Aagaard [14]. The procedure to extreme wave height predictions are explained in Sarpkaya and Isaacson[18] and in Kamphuis[19]. Extreme value analysis for waves is discussed in detail in Mathiesen et al.[20], Goda et al.[21] and Goda[22]. Coles[23] has provided the statistical details of extreme value prediction based on the annual maximum data points and Peak Over Threshold (POT) method. Additional information on POT and its application is provided in Ferreira and Guedes Soares[24] and Leadbetter[25]. All these literatures provide the information and knowledge for carrying out a detailed extreme value analysis and are used for the present work.

3. INPUT DATA GENERATION For the present work, the wave data is hindcasted using WAM model for a total period of 12 years, starting from 1st Jan 1993 to 31st December 2004. The output from the WAM model is the significant wave height and the mean wave period for every one hour. The data is hindcasted for the whole Arabian Gulf waters with a grid size of 0.1o x 0.1o. The model was validated using measured data as provided in Al-Salem et al. [26]. The extreme wave analysis is carried out for a total of 25 different locations off the UAE territorial waters as shown in Figure 3. Each location has a total of 105,192 raw data points. The longitude, latitude and the water depth of each location is given in Table 1.

The maximum and average significant wave heights for these 25 locations based on the 12 year hindcasted data is provided in Figure 4. The highest maximum significant wave height is hindcasted at location 21 (Hs = 5.2 m) and the lowest maximum significant wave height is hindcasted at location 1 (Hs = 2.73 m). Similarly the maximum average wave height for 12 year has occurred at location 21 with Hs = 0.71 m and the minimum average wave height has occurred at location 1 with Hs = 0.38 m.

3. METHODOLOGY The Gumbel and Weibull distribution is used for the extreme wave height prediction. The input data point selection is done carefully. The statistics of long term prediction of wave requires that the individual data points used in the statistical analysis be statistically independent. Hence any hourly wave height depends very much on the wave height of the previous hours and hence the theoretical condition of statistical independence is not met. Hence, in order to produce independent data points, only storms should be considered. The commonly used method to separate wave heights into storms is called Peak Over Threshold (POT) analysis (Coles[23]). Mathiesen et. al.[20] recommends that the minimum time interval between local maxima be somewhat longer than time lag for which the auto-correlation function is 0.3-0.5.

Figure 3. Locations in the UAE territorial waters for extreme wave

analysis

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28Location

Hs

(m)

Maximum HsAverage Hs

Figure 4. Maximum and average value of the hindcasted significant

wave height for UAE territorial waters based on the hindcasted waves from 1.1.1993 to 31.12.2004.

Table 1. Longitude, Latitude and local water depth at 25 different locations in the UAE territorial waters

Location Longitude (oE)

Latitude (oN)

Water depth (m)

1 51.8 24.5 20 2 52.1 24.5 41 3 52.1 24.2 19 4 52.4 24.4 19 5 52.7 24.6 16 6 52.9 24.7 14 7 53.5 24.6 14 8 53.8 24.5 16 9 54.2 24.6 15 10 54.4 24.8 19 11 54.6 25.1 20 12 54.9 25.2 21 13 55.1 25.5 21 14 55.4 25.7 29 15 55.7 25.9 39 16 52.1 25.3 16 17 52.6 25.2 19 18 53 25.2 19 19 53.5 25.2 30 20 53.9 25.4 29 21 54.3 25.5 38 22 54.7 25.6 45 23 55 25.8 48 24 55.3 26 79 25 55.7 26.2 81

UAE



They also recommend that generally a time interval of two to four days suffices. For the present work with 12 years of data, one can hence expect about 1095 to 2190 data points for each location, if the time interval between the data is considered as 4 days and 2 days respectively. It means that for each location, one can expect about 91 and 182 data points per year if the time interval between data is 4 days and 2 days respectively. The No. of storm events/year with threshold wave height of 1.0 m for different locations is provided in Figure 5.

The number of events is less than 91 (Which is the expected number of data, if the average data interval is 4 days). Hence threshold value of 1.0 m or more will definitely increase the degree of randomness of the data for the extreme value analysis. From Figure 5, it can be seen that there are 21 locations which has more than 50 No. of storm events/year (with threshold significant wave height of 1.0 m). It is also seen that there are only 4 locations which has 40 to 50 storm events/year. There are no locations amongst the selected 25 locations with less than 40 storm events/year. This important information is vital for marine operations around these locations. In the present work the threshold values of 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.25, 3.5, 3.75, 4.0, 4.25 and 4.5 m were used and the extreme value analysis is repeated for each selected threshold values. The data points for each location are arranged in the descending order. The probability of exceedence, Q is calculated using the formula Q = (i-c1)/(N+c2) (1) where; i: Rank and N: Total number of data points c1=0.44 and c2 = 0.12 for Gumbel distribution and c1=0.20+ (0.27/α) and c2 = 0.20 + (0.23/α) for Weibull distribution, where α is the shape parameter. The value of α is varied from 0.8 to 1.3 with an increment of 0.05 and the value of α, which gives best fit for the data set is selected.

The detailed description of Gumbel and Weibull distribution can be found from many sources (For example, see Kamphuis [19]). For the purpose of quick reference and use for the readers, the salient details of these distributions are provided in Annexure A.

3.1 No. of events/year for different threshold value at different locations in the UAE territorial waters

It is essential to know how the no. of events changes when the threshold value is changed. The number of events/year at location 1 for different threshold value is given in Figure 6. From this figure, it can be seen that significant wave heights of the order of 2.0 m may occur, but very rarely in a year in location 1. Similar plot, for location 5 is given in Figure 7. The trend of the plot is similar to Figure 6, though the number of events/year is slightly different. Similar plot for location 10 is given in Figure 8.

01020304050607080

0 5 10 15 20 25 30

Location

No.

of s

torm

eve

nts/

year

w

ith th

resh

old

Hs

of 1

.0 m

Figure 5. No. of storm events/year with threshold significant wave

height of 1.0 m in the UAE territorial waters

0

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20

30

40

50

60

70

0 1 2 3 4 5

Threshold wave height (m)

No.

of e

vent

s/ye

ar

Figure 6. The number of events/year for different threshold value in

location 1

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Figure 7. The number of events/year for different threshold values

in location 5

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in location 10

One can see in this figure that the no. of events/year for any threshold value is different, when compared to Figures 6 and 7. Similar plots for location 15, 20 and 25 are given in Figures 9, 10 and 11 respectively. All these figures provide vital statistical information, which are essential for any type of marine works (constructions, operations and maintenance) in the UAE territorial waters.

S. Neelamani


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in location 15

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3.2 Prediction of the wave height for the selected return period

The return period, TR and the probability of exceedence are linked by the following expression:-

Q = 1 / (l TR) (2)

where 'l' is the number of event/year. For the present problem, we know the total number of storm events exceeding different threshold value for each location in UAE territorial waters. Since the data is for a total duration of 12 years, the value of l can be calculated immediately (Refer Figure 5 for threshold value of 1.0 m for all 25 locations and Figures 6, 7, 8, 9, 10 and 11 for location 1, 5, 10, 15, 20 and 25 for different threshold values respectively). Similar information is available for all the other locations. Now according to Gumbel distribution, the wave height expected for a selected return period HTR can be estimated as follows:

HTR = g -b In[In(1/P)] (3) i.e. HTR = g -b In[In{(lTR)/(lTR-1)}] (4)

According to the Weibull distribution, the wave height expected for a selected return period HTR can be estimated from the following formula:

HTR = g +b [In(1/Q)] 1/α (5) i.e. HTR = g +b [In(lTR)] 1/α] (6)

Now it is possible to obtain the extreme wave height for any selected return period and for any selected threshold wave height, if the value of the parameters of these distributions are known.

4. RESULTS AND DISCUSSION 4.1 Procedure for the extreme significant wave

height prediction

The following are the steps used for the long-term prediction of waves:- - The data set for each location is obtained based on

peak over threshold value from 0.75 m to 4.5 m with an interval of 0.25 m and for all 25 locations from the hindcasted data for the period from 1.1.1993 to 31.12.2004. Steps 'b' to 'f' is carried out for each data sets pertaining to a threshold wave height and repeated for all the 16 different threshold wave heights selected.

- The wave heights obtained at each location is arranged in descending order.

- The plotting formula, as discussed in eqn.1 is used to reduce the wave height data to a set of points describing the probability of exceedence of wave height, Q.

- The wave height is then plotted against the reduced variate of Gumbel distribution (-In [In (1/P)]) and Weibull distribution ([In (1/Q)] 1/α).

- A straight line is fitted by using least square techniques through the points to represent a trend. The slope and intercept is obtained. From this, the parameters of the probability distribution are obtained.

- Eqn. 4 and 6 is used for predicting wave heights for chosen return period (12 year, 25 year, 50 year, 100 year, 200 year etc.) for Gumbel and Weibull distribution respectively.

It is found that the Weibull distribution is better than Gumbel distribution and hence all the extreme wave prediction is carried out by using Weibull distribution. A typical Weibull distribution plot for location 21 and for threshold wave height of 1.0 m is provided in Figure 12. The equation of the best line fit and the correlation coefficient are provided. Similar plots are prepared for all the 25 locations and for different threshold values.

4.2 Weibull distribution parameters for different locations and different threshold values

The Weibull distribution parameters (Shape parameter, Scale parameter and Location parameter) for different locations and for different threshold value are estimated. Typical plots of the shape parameters



obtained for all the 25 locations based on the Weibull distribution for threshold value of 1.0 m, 2.0 m, 3.0 m and 4.0 m are given in Figures 13, 14, 15 and 16 respectively. It can be seen that the shape parameter is closer to 1.3 for all 25 locations, when the threshold value is 1.0 m. However, the shape parameter varies from 0.8 to 1.3, when the threshold value is increased to 2.0 m to 4.0 m.

Similarly typical plots of the scale parameter for all the 25 locations based on the Weibull distribution for threshold value of 1.0 m, 2.0 m, 3.0 m and 4.0 m are given in Figures 17, 18, 19 and 20 respectively. It can be seen from these figures that scale parameter changes significantly from location to location and also with change in the threshold value.

Typical plots of the location parameter for all the 25 locations based on the Weibull distribution for threshold wave height of 1.0 m, 2.0 m, 3.0 m and 4.0 m are given in Figures 21, 22, 23 and 24 respectively. The value of the location parameters do not change much when the locations are changed and the value of the location parameters are closer to the selected threshold values.

The Weibull's location parameter, scale parameter and location parameters are estimated for all the locations and for all the selected threshold values. Table 2a provides the location parameter for all the 25 locations for threshold value of 0.75 m to 3.0 m and table 2b provides the location parameter values for all the 25 locations for threshold value of 3.25 m to 4.5 m. The empty space in the table indicates that there is not enough data points for the analysis. Similarly, table 3a provides the scale parameter for all the 25 locations for threshold value of 0.75 m to 3.0 m and table 3b provides the scale parameter values for all the 25 locations for threshold value of 3.25 m to 4.5 m.

Table 4a provides the shape parameter for all the 25 locations for threshold value of 0.75 m to 3.0 m and table 4b provides the shape parameter values for all the 25 locations for threshold value of 3.25 m to 4.5 m. Table 5a provides the value of coefficient of regression for all the 25 locations for threshold value of 0.75 m to 3.0 m and table 5b provides the value of coefficient of regression for all the 25 locations for threshold value of 3.25 m to 4.5 m. It can be seen from table 5a and 5b that the coefficient of regression for most of the locations and for different threshold values are above 0.08 and are closer to 1.0, which indicates the confidence of using the Weibull's parameter for the prediction of extreme wave heights.

The values in table 2a, 2b, 3a, 3b, 4a and 4b can be used in equation 6 in order to estimate the extreme significant wave heights for different return periods and for different threshold values. Eqn. 6 needs the value of l. The value of l for different locations and for different threshold values is provided in table 6a and 6b. No one can easily estimate the extreme significant wave height for any location in the UAE territorial waters and for any threshold values using these tables.

y = 0.871x - 0.7489R2 = 0.9874

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4

5

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Hs (m)

[In(1

/Q)]^

^(1/

alph

a)

Figure 12. Weibull distribution plot for Location 21 in UAE territorial

waters for threshold value of 1.0 m

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Figure 13. Shape parameter based on Weibull distribution for 25

locations in UAE territorial waters for threshold value of 1.0 m

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Figure 17. Scale parameter based on Weibull distribution for 25


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Figure 21. Location parameter based on Weibull distribution for 25


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4.3 Extreme significant wave height for threshold value of 1.0 m

A typical plot showing the predicted significant wave heights for the different locations based on Weibull distribution for return periods of 12 years, 25 years, 50

years, 100 years and 200 years when the threshold wave height is 1.0 m, are provided in Fig.25. Further details of descriptions and findings are provided in Neelamani et al.[1].

4.4 Effect of threshold value on the predicted extreme significant wave height for 100 year return period

The effect of changing the threshold value on the predicted 100 year return period significant wave height is studied for all 25 locations. Typical plot for location 1, 5, 10, 15, 20, 25 is provided in Figures 26, 27, 28, 29, 30 and 31 respectively. It can be seen that the trend of variation of the predicted wave height for 100 year return period for each location is different when the threshold value is varied. Hence one cannot make any general conclusion on the effect of changing the threshold value on the 100 year return period wave height.

Table 2a. The location parameter (γ) for different threshold values in UAE territorial waters (Threshold value from 0.75 m to 3.0 m)

Location

Location parameter (γ) for different threshold wave heights HTh =

0.75 m

HTh =

1.0 m

HTh =

1.25 m

HTh =

1.5 m

HTh =

1.75 m

HTh =

2.0 m

HTh =

2.25 m

HTh =

2.5 m

HTh =

2.75 m

HTh =

3.0 m

1 0.78 1.008 1.254 1.525 1.763 2.009 2.348

2 0.774 0.989 1.234 1.515 1.776 2.01 2.301 2.587

3 0.792 1.004 1.245 1.508 1.786 2.063 2.277 2.588

4 0.774 0.992 1.226 1.488 1.743 2 2.258 2.543 2.802

5 0.766 0.98 1.202 1.495 1.739 2.014 2.273 2.476 2.874 3.084

6 0.754 0.98 1.185 1.477 1.755 2.024 2.288 2.491 2.816 3.23

7 0.776 0.97 1.18 1.465 1.747 2.026 2.288 2.53 2.836 3.113

8 0.755 0.956 1.181 1.452 1.727 2.022 2.294 2.532 2.75 3.076

9 0.721 0.937 1.164 1.424 1.697 2.004 2.285 2.534 2.79 3.082

10 0.686 0.93 1.142 1.375 1.664 1.965 2.238 2.499 2.764 3.069

11 0.661 0.89 1.111 1.358 1.653 1.973 2.241 2.516 2.753 3.033

12 0.668 0.883 1.125 1.37 1.656 1.964 2.261 2.532 2.783 3.03

13 0.633 0.895 1.119 1.37 1.655 1.945 2.23 2.479 2.763 3.073

14 0.686 0.881 1.139 1.406 1.66 1.971 2.258 2.54 2.814 3.029

15 0.705 0.949 1.177 1.432 1.701 1.96 2.319 2.545 2.8 2.953

16 0.777 0.978 1.155 1.461 1.746 2.025 2.265 2.568 2.776 3.136

17 0.75 0.966 1.147 1.435 1.733 1.993 2.282 2.553 2.812 2.997

18 0.72 0.952 1.144 1.408 1.707 1.974 2.26 2.551 2.82 3.062

19 0.692 0.915 1.124 1.391 1.676 1.91 2.216 2.509 2.816 3.062

20 0.682 0.901 1.108 1.391 1.652 1.895 2.184 2.47 2.764 2.989

21 0.618 0.86 1.112 1.337 1.603 1.892 2.209 2.478 2.746 3.002

22 0.618 0.878 1.102 1.331 1.632 1.92 2.217 2.454 2.737 3.068

23 0.669 0.931 1.106 1.385 1.661 1.919 2.232 2.487 2.775 3.078

24 0.714 0.938 1.151 1.413 1.674 1.925 2.238 2.564 2.856 3.012

25 0.721 0.972 1.169 1.465 1.744 1.997 2.276 2.541 2.766 3.225

S. Neelamani


Table 2b. The location parameter (γ) for different threshold values in UAE territorial waters (Threshold value from 3.25 m to 4.5 m)

Location

Location parameter (γ) for different threshold wave heights HTh =

3.25 m HTh = 3.5 m

HTh = 3.75 m

HTh = 4.0 m

HTh = 4.25 m

HTh = 4.5 m

1 2 3 4 5 6 3.263 7 8 3.297 9 3.318 3.433 10 3.346 3.473 3.67 11 3.347 3.491 3.657 3.947 12 3.331 3.452 3.693 3.787 13 3.268 3.378 3.681 3.847 14 3.25 3.424 3.676 3.741 15 3.269 3.4 3.601 3.787 16 3.31 3.388 17 3.286 3.448 3.784 18 3.254 3.472 3.755 3.875 19 3.33 3.476 3.745 4.11 4.063 20 3.246 3.373 3.721 3.978 4.249 21 3.255 3.38 3.739 3.985 4.282 4.42 22 3.269 3.414 3.686 4 4.286 4.339 23 3.319 3.426 3.657 3.912 4.135 24 3.227 3.343 3.72 3.894 25 3.352 3.555 3.673

Table 3a: The Scale parameter (b) for different threshold values in UAE territorial waters (Threshold value from 0.75 m to 3.0 m)

Location

Scale parameter (β) for different threshold wave heights HTh =

0.75 m HTh = 1.0 m

HTh = 1.25 m

HTh = 1.5 m

HTh = 1.75 m

HTh = 2.0 m

HTh = 2.25 m

HTh = 2.5 m

HTh = 2.75 m

HTh = 3.0 m

1 0.428 0.352 0.227 0.215 0.21 0.321 0.26 2 0.57 0.5 0.358 0.263 0.274 0.271 0.291 0.181 3 0.538 0.468 0.335 0.307 0.262 0.17 0.229 0.213 4 0.622 0.542 0.414 0.332 0.337 0.335 0.367 0.255 0.161 5 0.703 0.62 0.503 0.395 0.343 0.296 0.319 0.443 0.291 0.183 6 0.755 0.672 0.551 0.463 0.377 0.322 0.269 0.346 0.461 0.175 7 0.715 0.639 0.529 0.455 0.406 0.32 0.242 0.278 0.409 0.146 8 0.761 0.684 0.543 0.503 0.48 0.383 0.29 0.293 0.445 0.359 9 0.839 0.777 0.625 0.622 0.619 0.499 0.406 0.355 0.342 0.316 10 0.942 0.84 0.763 0.756 0.732 0.661 0.563 0.493 0.425 0.433 11 1.024 0.96 0.836 0.862 0.801 0.731 0.658 0.584 0.524 0.446 12 1.021 0.921 0.835 0.862 0.795 0.738 0.659 0.582 0.497 0.471 13 1.076 0.966 0.872 0.857 0.834 0.781 0.702 0.641 0.562 0.437 14 0.988 0.931 0.834 0.819 0.806 0.738 0.642 0.562 0.477 0.449 15 0.909 0.814 0.726 0.725 0.676 0.68 0.558 0.474 0.478 0.528 16 0.808 0.72 0.655 0.525 0.445 0.351 0.337 0.266 0.352 0.391 17 0.916 0.83 0.768 0.658 0.529 0.486 0.393 0.318 0.293 0.43 18 0.982 0.902 0.831 0.755 0.62 0.545 0.517 0.424 0.349 0.324 19 1.092 1.011 0.947 0.872 0.769 0.724 0.691 0.614 0.479 0.394 20 1.189 1.104 0.999 0.937 0.884 0.837 0.812 0.73 0.658 0.566 21 1.225 1.148 0.993 0.993 0.984 0.923 0.857 0.778 0.709 0.625 22 1.196 1.072 0.974 0.972 0.959 0.884 0.826 0.763 0.705 0.612 23 1.054 0.938 0.895 0.886 0.866 0.809 0.732 0.667 0.577 0.445 24 0.984 0.876 0.788 0.788 0.781 0.759 0.694 0.588 0.437 0.48 25 0.818 0.72 0.645 0.638 0.596 0.588 0.493 0.389 0.452 0.455



Table 3b: The Scale parameter (b) for different threshold values in UAE territorial waters (Threshold value from 3.25 m to 4.5 m)

Location

Scale parameter (β) for different threshold wave heights HTh =

3.25 m HTh = 3.5 m

HTh = 3.75 m

HTh = 4.0 m

HTh = 4.25 m

HTh = 4.5 m

1 2 3 4 5 6 0.21 7 8 0.265 9 0.226 0.201 10 0.294 0.31 0.364 11 0.393 0.262 0.339 0.265 12 0.415 0.344 0.233 0.492 13 0.411 0.541 0.367 0.39 14 0.589 0.51 0.437 0.805 15 0.62 0.614 0.528 0.521 16 0.307 0.26 17 0.499 0.47 0.21 18 0.479 0.372 0.292 0.28 19 0.384 0.347 0.321 0.293 0.49 20 0.554 0.564 0.485 0.417 0.199 21 0.612 0.588 0.498 0.393 0.199 0.245 22 0.572 0.534 0.466 0.337 0.213 0.459 23 0.438 0.439 0.498 0.46 0.28 24 0.535 0.578 0.478 0.524 25 0.44 0.353 0.213

Table 4a: The Shape parameter (a) for different threshold values in UAE territorial waters (Threshold value from 0.75 m to 3.0 m)

Location

Shape parameter (a) for different threshold wave heights HTh =

0.75 m HTh = 1.0 m

HTh = 1.25 m

HTh = 1.5 m

HTh = 1.75 m

HTh = 2.0 m

HTh = 2.25 m

HTh = 2.5 m

HTh = 2.75 m

HTh = 3.0 m

1 1.3 1.25 1 1 0.9 1.15 1.05 2 1.3 1.3 1.1 0.95 1 1 1.1 0.8 3 1.3 1.3 1.1 1.1 1.05 0.8 0.85 0.8 4 1.3 1.3 1.15 1.05 1.1 1.15 1.3 1.3 1.3 5 1.3 1.3 1.2 1.1 1.05 1 1.05 1.3 1.3 1.3 6 1.3 1.3 1.2 1.15 1.05 1 0.9 1 1.3 0.8 7 1.3 1.3 1.2 1.15 1.15 1.05 0.9 0.95 1.3 0.8 8 1.3 1.3 1.15 1.15 1.2 1.1 0.95 0.95 1.3 1.3 9 1.3 1.3 1.15 1.2 1.3 1.2 1.1 1.05 1.05 1.05 10 1.3 1.25 1.2 1.25 1.3 1.3 1.25 1.2 1.15 1.3 11 1.3 1.3 1.2 1.3 1.3 1.3 1.3 1.3 1.3 1.25 12 1.3 1.25 1.2 1.3 1.3 1.3 1.3 1.3 1.25 1.3 13 1.3 1.25 1.2 1.25 1.3 1.3 1.3 1.3 1.25 1.1 14 1.25 1.25 1.2 1.25 1.3 1.3 1.25 1.2 1.1 1.05 15 1.25 1.2 1.15 1.2 1.2 1.25 1.15 1.05 1.05 1.1 16 1.3 1.3 1.3 1.2 1.15 1.05 1.05 0.95 1.1 1.3 17 1.3 1.3 1.3 1.25 1.15 1.15 1.05 0.95 0.9 1.1 18 1.3 1.3 1.3 1.3 1.2 1.15 1.2 1.1 1 0.95 19 1.3 1.3 1.3 1.3 1.25 1.25 1.3 1.3 1.15 1.05 20 1.3 1.3 1.25 1.25 1.25 1.25 1.3 1.3 1.3 1.2 21 1.3 1.3 1.2 1.25 1.3 1.3 1.3 1.3 1.3 1.25 22 1.3 1.25 1.2 1.25 1.3 1.3 1.3 1.3 1.3 1.3 23 1.25 1.2 1.2 1.25 1.3 1.3 1.3 1.3 1.25 1.1 24 1.25 1.2 1.15 1.2 1.25 1.3 1.3 1.25 1.05 1.1 25 1.25 1.2 1.15 1.2 1.2 1.25 1.15 1 1.1 1.3

S. Neelamani


Table 4b. The Shape parameter (a) for different threshold values in UAE territorial waters (Threshold value from 3.25 m to 4.5 m)

Location

Shape parameter (a) for different threshold wave heights HTh =

3.25 m HTh = 3.5 m

HTh = 3.75 m

HTh = 4.0 m

HTh = 4.25 m

HTh = 4.5 m

1 2 3 4 5 6 0.8 7 8 1.05 9 0.85 0.8 10 1.05 1.1 1.3 11 1.3 1 1.2 1.05 12 1.3 1.15 0.9 1.3 13 1.05 1.3 1.05 1 14 1.3 1.2 1.05 1.3 15 1.3 1.3 1.25 1.05 16 1.3 1.05 17 1.3 1.3 0.8 18 1.2 1 0.8 0.8 19 1.05 1 0.95 0.9 1 20 1.25 1.3 1.3 1.3 0.8 21 1.3 1.3 1.3 1.2 0.8 0.8 22 1.3 1.3 1.25 1.05 0.8 0.95 23 1.1 1.1 1.25 1.2 0.8 24 1.2 1.3 1.2 1.3 25 1.3 1.3 0.9

Table 5a. The Coefficient of Regression (R2) for different threshold values in UAE territorial waters (Threshold value from 0.75 m to 3.0 m)

Location

Coefficient of Regression (R2) for different threshold wave heights HTh =

0.75 m HTh = 1.0 m

HTh = 1.25 m

HTh = 1.5 m

HTh = 1.75 m

HTh = 2.0 m

HTh = 2.25 m

HTh = 2.5 m

HTh = 2.75 m

HTh = 3.0 m

1 0.962 0.959 0.97 0.961 0.957 0.932 1 2 0.977 0.98 0.984 0.98 0.978 0.971 0.96 0.947 3 0.975 0.977 0.981 0.974 0.962 0.956 0.95 0.979 4 0.977 0.982 0.984 0.976 0.97 0.953 0.931 0.948 0.621 5 0.978 0.984 0.985 0.983 0.975 0.966 0.933 0.893 0.877 0.897 6 0.979 0.986 0.985 0.985 0.98 0.974 0.958 0.931 0.878 0.902 7 0.981 0.985 0.982 0.979 0.979 0.972 0.955 0.917 0.874 0.941 8 0.984 0.987 0.984 0.983 0.984 0.981 0.973 0.973 0.954 0.97 9 0.987 0.987 0.983 0.983 0.988 0.988 0.985 0.979 0.98 0.976 10 0.987 0.987 0.984 0.982 0.986 0.989 0.987 0.98 0.98 0.986 11 0.986 0.984 0.98 0.982 0.982 0.984 0.984 0.983 0.977 0.964 12 0.985 0.982 0.98 0.983 0.982 0.982 0.982 0.982 0.981 0.978 13 0.985 0.985 0.98 0.985 0.987 0.987 0.987 0.985 0.978 0.965 14 0.986 0.985 0.985 0.987 0.988 0.989 0.988 0.984 0.976 0.963 15 0.989 0.989 0.988 0.988 0.987 0.986 0.98 0.974 0.958 0.945 16 0.979 0.986 0.986 0.987 0.986 0.982 0.976 0.965 0.942 0.844 17 0.981 0.987 0.987 0.988 0.986 0.986 0.985 0.98 0.97 0.961 18 0.983 0.987 0.989 0.989 0.987 0.985 0.988 0.985 0.98 0.971 19 0.985 0.988 0.988 0.987 0.984 0.981 0.988 0.99 0.988 0.985 20 0.987 0.988 0.986 0.985 0.982 0.98 0.982 0.983 0.979 0.97 21 0.986 0.987 0.984 0.983 0.984 0.984 0.986 0.986 0.983 0.977 22 0.986 0.985 0.982 0.982 0.987 0.986 0.987 0.984 0.985 0.986 23 0.985 0.984 0.982 0.986 0.987 0.986 0.989 0.988 0.986 0.98 24 0.988 0.987 0.985 0.986 0.987 0.987 0.986 0.982 0.976 0.967 25 0.989 0.99 0.988 0.99 0.988 0.987 0.982 0.975 0.962 0.969



Table 5b. The Coefficient of Regression (R2) for different threshold values in UAE territorial waters (Threshold value from 3.25 m to 4.5 m)

Location

Coefficient of Regression (R2) for different threshold wave heights HTh =

3.25 m HTh = 3.5 m

HTh = 3.75 m

HTh = 4.0 m

HTh = 4.25 m

HTh = 4.5 m

1 2 3 4 5 6 0.934 7 8 1 9 0.97 0.95 10 0.983 0.985 0.999 11 0.958 0.958 0.946 1 12 0.962 0.954 0.943 1 13 0.946 0.931 0.894 0.813 14 0.938 0.912 0.852 0.621 15 0.937 0.899 0.851 1 16 0.827 1 17 0.947 0.922 0.934 18 0.954 0.958 0.941 0.934 19 0.983 0.982 0.972 0.997 1 20 0.961 0.958 0.942 0.893 0.945 21 0.972 0.97 0.964 0.959 0.951 0.936 22 0.986 0.986 0.981 0.973 0.935 1 23 0.968 0.96 0.927 0.866 0.934 24 0.95 0.937 0.874 0.759 25 0.95 0.941 1

Table 6a. The number of events/year (l) for different threshold value in UAE territorial waters (Threshold value from 0.75 m to 3.0 m)

Location

Number of event/year (l) for different threshold wave heights HTh =

0.75 m HTh = 1.0 m

HTh = 1.25 m

HTh = 1.5 m

HTh = 1.75 m

HTh = 2.0 m

HTh = 2.25 m

HTh = 2.5 m

HTh = 2.75 m

HTh = 3.0 m

1 63.25 43.42 28.67 9.417 3.083 0.75 0.167 0.08 0 0 2 74.58 58.33 45.67 27.25 11.5 4.917 1.667 0.67 0.17 0.083 3 74.17 58.5 44.25 23.33 10.17 4 1.333 0.42 0.17 0.083 4 69.42 57.08 45.75 29.25 13.92 6.167 2.167 1.17 0.25 0.083 5 66.75 57.75 49 33.92 21 11.33 4.333 1.5 0.58 0.333 6 65.67 56 50.25 34.92 22.25 13.5 6.75 2.75 0.83 0.417 7 59.92 53.08 45.17 29.5 17.92 11.08 5.833 1.92 0.5 0.333 8 59.67 53.08 46.33 30.17 18.92 12 7.083 3 1.17 0.5 9 58.25 49.25 43.92 29.25 19 13.67 8.833 5.33 2.42 1.083 10 61.42 53.08 44.67 33.25 22.67 15.92 12.08 8.08 5 2.25 11 63.75 55.75 49.67 36.58 26.83 18.75 14.17 10 7.33 4.25 12 60.58 54.83 45.25 33.33 25.5 17.5 12.42 8.92 6.25 3.667 13 64.33 55.67 48.5 38.5 27.92 19.92 14.92 11.1 7.42 4.667 14 61.25 54.83 46 34.67 25.17 17.83 13.17 9.33 6.08 3.667 15 58.17 49.33 42.17 30.5 22.42 14.75 9.75 7.08 3.92 2.417 16 73.17 67.25 59.33 44.5 30 20.17 10.75 5.42 1.92 0.583 17 69.67 63.17 55.75 44.08 34 22.67 15.08 9.08 4.58 1.833 18 68.58 60.33 54.25 43.67 34.83 25.08 15.83 10.4 6.25 3.25 19 65.83 59.58 51.92 43.17 34.58 26.5 17.58 12.8 8.83 6.083 20 65.83 60.83 56.17 45.67 36.5 28.58 20.42 16 11.4 8.667 21 69.67 62.08 56.83 46.92 35.83 27.83 20.33 16.2 12.4 9.333 22 67.17 61 54.75 44.58 32.92 26.08 18.83 15.3 10.8 7.417 23 68.75 60.92 54.67 41.42 30.5 23.67 17.08 13.1 9.08 6.25 24 65.25 59.75 52.92 38.83 27.92 20.5 14.42 9.92 7.08 4.167 25 56.58 47.83 41.25 26.33 17.83 11.42 7.667 5 2.42 0.833

S. Neelamani


Table 6b. The number of events/year (l) for different threshold value in UAE territorial waters (Threshold value from 3.25 m to 4.5 m)

Loca

tion

Number of event/year (l) for different threshold wave heights

H Th =

3.2

5 m

H Th =

3.5

m

H Th =

3.7

5 m

H Th =

4.0

m

H Th =

4.2

5 m

H Th =

4.5

m

1 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 4 0 0 0 0 0 0 5 0.08 0 0 0 0 0 6 0.25 0.08 0 0 0 0 7 0.08 0 0 0 0 0 8 0.25 0.17 0 0 0 0 9 0.58 0.42 0.08 0 0 0 10 1.42 0.83 0.33 0.083 0 0 11 2.08 1.83 0.75 0.25 0.083 0 12 1.83 1.5 0.83 0.167 0.083 0 13 2.92 1.67 1.17 0.583 0.167 0 14 1.67 1.33 0.75 0.25 0.167 0 15 1 0.75 0.42 0.167 0.083 0 16 0.42 0.17 0.17 0 0 0 17 0.75 0.42 0.25 0.083 0 0 18 1.25 0.75 0.33 0.25 0.083 0 19 2.83 2.08 1 0.333 0.25 0.08 20 5.42 3.83 2.08 1.25 0.75 0.25 21 6 4.75 2.67 1.75 1.083 0.5 22 5.17 4.17 2.5 1.417 0.583 0.25 23 3.33 2.5 1.25 0.667 0.25 0.08 24 2.25 1.58 0.75 0.417 0.167 0 25 0.58 0.42 0.17 0.083 0 0

4.5 Predicted minimum and maximum significant wave height for 100 year return period

The predicted minimum and maximum extreme significant wave height at each location, when the threshold value is varied from 0.75 m to 4.5 m is selected. Figure 32 shows the predicted minimum and maximum wave height for 100 year return period for all the 25 locations in the UAE territorial waters.

Figure 33 shows the difference between the Predicted minimum and maximum significant wave height for 100 Year return period in UAE territorial waters.

It can be seen that for few locations (Location 20 and 21), the difference between the maximum and minimum predicted significant wave height is about 1.6 m and for few locations (Location 1,2 and 3) the difference is only of the order of about 0.2 to 0.3 m. Now it is to be remembered that, if the minimum value is selected for design of a marine structure, then the structure may encounter with the maximum predicted wave height value and hence more risk is involved.

1

2

3

4

5

6

7

8

0 5 10 15 20 25Location in UAE territorial waters

Hs

(m)

Tr=12 YearsTr=25 YearsTr=50 YearsTr=100 YearsTr=200 Years

Figure 25. Predicted extreme significant wave heights in UAE

territorial waters for different return periods based on Weibull distribution for threshold value of 1.0 m

2.92.95

33.05

3.13.15

3.23.25

0 0.5 1 1.5 2 2.5


Extr

eme

wav

e he

ight

(m)

Figure 26. Effect of threshold wave height on the predicted extreme

wave height for TR =100 Years at Location 1 in UAE territorial water

33.23.43.63.8

44.24.44.6

0 0.5 1 1.5 2 2.5 3 3.5


Extr

eme

wav

e he

ight

(m)



4.44.54.64.74.84.9

55.15.2

0 1 2 3 4


Extr

eme

wav

e he

ight

(m)





55.15.25.35.45.55.65.75.85.9

0 1 2 3 4 5


Extr

eme

wav

e he

ight

(m)



5.2

5.7

6.2

6.7

7.2

0 1 2 3 4 5


Extr

eme

wav

e he

ight

(m)



4.3

4.5

4.7

4.9

5.1

5.3

0 1 2 3 4


Extr

eme

wav

e he

ight

(m)



1.5

2.5

3.5

4.5

5.5

6.5

7.5

0 5 10 15 20 25Location

Pred

icte

d 10

0 Ye

ar H

s (m

)

Minimum Value

Maximum Value

Figure 32. Predicted minimum and maximum Significant wave

height for 100 Year return period in UAE territorial waters

00.20.40.60.8

11.21.4

1.61.8

0 5 10 15 20 25

Location

The

diffe

renc

e be

twee

n th

e M

ax.

and

Min

imum

pre

dict

ed v

alue

for

100

Year

retu

rn p

erio

d (m

)

Figure 33. The difference between the Predicted minimum and

maximum Significant wave height for 100 Year return period in UAE territorial waters

On the other hand, if the maximum predicted wave height value is used, then the risk can be minimized but the marine structure may be expensive. Hence it is recommended to the user to select a design value in between the minimum and maximum predicted 100 year return period waves by keeping the risk and cost in mind. For example, let us assume that a jacket type crude oil offshore exploitation structure needs to be designed at location 21. Selection of the maximum significant wave height value of 7.17 m instead of the minimum significant wave height value of 5.53 m results in 68% increase in the drag force on the vertical slender members (Since the drag force is proportional to the square of wave height). Similar exercise can be carried out for other type of marine projects at different locations in UAE territorial waters in order to assess the impact of selecting the predicted minimum significant as well as maximum significant wave height for 100 year return period or for other return periods.

5. CONCLUSIONS AND RECOMMENDATIONS

The effect of varying the threshold value on the predicted extreme significant wave height is investigated. UAE territorial water is used for this case study. 25 different locations are selected for the analysis. Threshold value is varied from 0.75 m to 4.5 m with increment of 0.25 m. Gumbel and Weibull extreme value distributions are used for the extreme wave analysis and Weibull distribution is found to be better for all the locations. The value of shape parameter is varied from 0.8 to 1.3 with increment of 0.05 and the best shape parameter is selected (Based on the highest coefficient of regression value) for obtaining the location parameter and scale parameter of the Weibull distribution. The following are the conclusions obtained out of this investigation.

1. There is no definite trend in the predicted extreme significant waves for all the selected locations, when the threshold value is varied from 0.75m to 4.5m. For most of the locations, the predicted extreme significant wave height reduces with

S. Neelamani


increased threshold value and for some location, the predicted extreme significant wave height value oscillates when the threshold value is changed from 0.75 to 4.5m.

2. The difference between the maximum and minimum predicted 100 year return period significant wave height value is about 1.6m for some locations in the UAE territorial waters and for few locations, it is only about 0.2 to 0.3m.

3. For a typical location in the territorial waters of UAE, it is recommended that the user/owner of the project to select a design value in between the minimum and maximum predicted significant wave height for different return periods by keeping the risk and project cost in mind.

4. A large number of marine projects are in progress and many new projects are being planned for the near future in the UAE territorial waters. The results of the present study will be highly useful for the risk based analysis and design of marine structures in these projects.

Acknowledgement The authors wish to express their gratitude to Kuwait Foundation for Advancement of Science (KFAS), Kuwait for their financial sponsorship. We express our thanks for Kuwait Institute for Scientific Research, Kuwait for providing all the infrastructure facilities to carry out this work. The author also expresses his sincere thanks to his colleagues Dr. Karim Rakha for providing the raw data and Eng. Khaled Al-Salem for providing his user friendly interactive software tool for the extreme wave analysis.

References 1. Neelamani, S., Al-Salem, K. and Rakha, K. (2006).

Extreme Water Waves in the UAE Territorial Waters. Emirates Journal for Engineering Research, Vol.11 (2), 37-46.

2. Neelamani, S., Al-Salem, K. and Rakha. K. (2007). Extreme Waves for Kuwaiti Territorial Waters, Ocean Engineering, Pergaman Press, UK, Vol. 34, Issue 10, July 2007, 1496-1504.

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Annexure A Gumbel distribution

The Gumbel distribution is given as

P = exp [-exp{-(H-g)/b}] (1)

where

P = Probability of non exceedence (P=1-Q) g = Location parameter and b = Scale parameter.

Eqn. 1 is linearised by taking logs on both sides twice and it becomes

In[In(1/P)] = (1/b) H –(g/b) (2)

In[In(1/P)] is called as the reduced variate of Gumbel distribution.

For the data set described, a least square best line fit is made and the value of b and g is obtained from the slope and intercept of the best line fit.

Weibull distribution

The Weibull distribution is a three parameter distribution and is given as

P = 1 – exp[-{(H-g)/b}α] (3)

which may also be expressed as

Q = exp[-{(H-g)/b}α] (4)

where α is the shape parameter, which can be varied from 0.8 to 1.3. For the purpose of linear transformation, take logarithm on both sides and after reorganization, we get

[In(1/Q)] 1/α = (1/b) H –(g/b) (5)

[In(1/Q)] 1/α is called as the reduced variate of Weibull distribution.

Again, for the data set described, a least square best line fit is made and the value of b and g is obtained from the slope and intercept of the best line fit. As discussed, the shape parameter α is varied from 0.8 to 1.3 in step of 0.05 and the best line fit is selected based on the highest value of correlation coefficient.

EFFECT OF THRESHOLD VALUES ON THE PREDICTED …EFFECT OF THRESHOLD VALUES ON THE PREDICTED EXTREME WAVES-A CASE STUDY IN UAE TERRITORIAL WATERS ... Burj Al Arab, Mina Siyahi, Jebel

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