Influence of Threshold Value on Peak over Threshold Method on the Predicted Extreme Significant Wave Heights in Kuwaiti Territorial Waters

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1 Journal of Coastal Research SI ICS2009 (Proceedings) Portugal ISSN Influence of Threshold Value on Peak over Threshold Method on the Predicted Extreme Significant Wave Heights in Kuwaiti Territorial Waters S. Neelamani Coastal and Air Pollution Department Environment and Urban Development Division, Kuwait Institute for Scientific Research, P.O. Box : 24885, Safat, KUWAIT. nsubram@kisr.edu.kw ABSTRACT NEELAMANI, S., Influence of threshold value on Peak Over Threshold method on the predicted extreme significant wave heights in Kuwaiti territorial waters. Journal of Coastal Research, SI 56 (Proceedings of the 10th International Coastal Symposium), Lisbon, Portugal, ISSN Prediction of extreme waves for different return periods is essential for safe and optimal design of marine structures. Peak Over Threshold (POT) method is widely used for extreme wave height prediction. But the predicted extreme wave height varies, when the threshold wave height is varied in the analysis. The effect of varying the threshold value on the predicted extreme significant wave height is investigated for 19 different locations in the Kuwaiti territorial waters. Threshold value is varied from 0.75 m to 3.0 m with increment of 0.25 m. Gumbel and Weibull extreme value distributions are used for the 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. It is found that there is no definite trend in the predicted extreme significant wave height, when the threshold value is varied. For some locations, the predicted extreme significant wave height has reduced with increased threshold value and for some other locations, the predicted extreme value oscillates with varying threshold values. It is found that the difference between the maximum and minimum predicted 100 year return period significant wave height value is more than 0.5 m for some locations in Kuwaiti territorial waters. For few other locations, the difference is less than 0.25 m. Hence, the extreme wave height value need to be selected carefully, keeping in mind the project costs and risks. ADITIONAL INDEX WORDS: Extreme waves; Kuwaiti territorial waters; Threshold wave height; Gumbel and Weibull distribution; Significant wave height; Hindcasted wave data. INTRODUCTION Reliable estimation of extreme wave heights are essential for safe and cost effective design of marine structures. A lack of the correct estimate will result either in an unsafe marine structure or an over designed and 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 1.5 m or 2 m significant wave height results in an armor unit of weight in the ratio of 3.4:8.0. Hence it is very essential to predict the design wave heights for different return periods. In Kuwaiti territorial waters, most of the coastal structures appear to be over designed, since there is no systematic extreme wave height prediction done during the past for its territorial waters. Different concepts are used for selecting the input data in the extreme value analysis. In the Peak Over Threshold (POT) method, selection of the threshold wave height is an important input during the data preparation stage. The effect of selecting the threshold value needs to be studied in order to know its influence on the predicted extreme wave heights for different return periods, which is not available for the Kuwaiti territorial waters. This is the main motivations for the present study. Work on establishing the extreme wave heights for Kuwaiti territorial waters with threshold wave height of 1.0 m were carried out by NEELAMANI et al. (2007). A marine structure (say a seawater intake system) designed for the design sea state of marine conditions like Gulf of Mexico or Bay of Bengal (the wave climate are severe and cyclone effects are frequent in these seas) need not be adopted for the Kuwaiti marine condition. In the territorial waters of Kuwait, the wave generation is controlled mainly by the limited fetch and shallow water depths. Kuwait is located in the North-Western part of the Arabian Gulf (Fig.1), which is a marginal sea in a typical arid zone and is an arm of the Indian Ocean. It lies between latitudes of and degrees North of the Equator, and longitudes of and degrees East of Greenwich. In Kuwait, the wind direction is usually northwesterly for 43% of the year, increasing to 63% during summer months and decreases to 31% during spring season. Southeasterly winds occur 19% of the year, with max. occurrence of 27% during spring and minimum of 9% during summer (AL-AJMI and MUSTAFA (1998)). 564

2 Influence of threshold value on Peak Over Threshold method The Southeasterly wind has significant fetch length and hence the dominant waves in the Kuwaiti territorial waters are from south east. Kuwait s territorial water extends up to 25 to 30 m water depth. There are nine islands in Kuwait; viz. Boubyan Island, Warba Island, Failaka Island, Miskan Island, Umm Al-Namil Island, Awhah Island, Kubbar Island, Qaruh Island and Umm Al- Maradim Island. Kuwait s territorial waters can be divided into two areas: The shallow Northern area, which is less than five meters deep in most places with muddy bed, and the relatively deep Southern area, which has a bed of sand and silica deposits. Most of Kuwait's ports are located on the Southern shore to take advantage of the deep waters in this area. Figure 1. Kuwait and locations in Kuwaiti territorial waters for analysis Kuwait territorial water is an active navigation routes for crude carriers. Though the territorial water is only about 7,611 square km, the wave climates vary significantly from one location to other due to the significant change in bathymetric conditions. More details of the Oceanographic Atlas of Kuwait can be obtained from AL- YAMANI et al (2004). Projects such as development of Boubyan and Failaka Island are in pipeline. An attempt is made to report the extreme waves in Kuwaiti territorial waters for different return periods. Since the selection of threshold value affects the predicted extreme waves, a detailed analysis is carried out to quantify the effect of threshold value. CAIRES and STERL (2005) have estimated the 100 year return value for significant wave height from the ERA-40 data for the whole oceans of the earth. The wind data used in this study is obtained from grid of 1.5 o x 1.5 o. Unfortunately this course grid can not provide much information for the countries surrounding the Arabian Gulf, since the width of the Arabian Gulf itself is of the order of 1.5 o only. Hence wind data was procured for finer grid size of 0. 5 o x 0. 5 o and the wind speeds are linearly interpolated for grid size of 0.1 o x 0.1 o for running the WAM model for hind casting the significant wave height and for further extreme wave analysis. LITERATURE REVIEW There are a large number of works done around the world on extreme value prediction of winds and waves. GUMBEL (1958) 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 (1973) has discussed Gumbel distribution in the context of extreme wave prediction. Information related to the collection of data samples for extreme value analysis is found from PETRAUSKAS and AAGAARD (1971) and JAHNS and WHEELER (1973). Details on plotting formula used for the extreme wave predictions are available in PETRAUSKAS and AAGAARD (1971). The procedure on extreme wave height predictions are elaborated in SARPKAYA and ISAACSON (1981) and in KAMPHUIS (2000). Extreme value analysis for waves is discussed in detail in MATHIESEN et al. (1994) and GODA et al. (1993). COLES (2001) 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 (1998) and LEADBETTER (1991). These literatures provide information and knowledge for carrying out a detailed extreme value analysis and are used for the present work. INPUT DATA GENERATION For the present work, the wave data is hindcasted using WAM model for a total period of 12 years, starting from 1 st Jan 1993 to 31 st December 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 Kuwaiti territorial waters with a grid size of 0.1 o x 0.1 o. The model was validated using measured data as provided in AL-SALEM et al. (2005). The extreme wave analysis is carried out for 19 different locations as shown in Fig. 1. Each location has a total of 105,192 data points. The longitude, latitude and the water depth of each location is given in Table 1. The water depth around Failaka Island (Location 13, 16, 17, 18 and 19) is shallow compared to the other locations. Table 1: Longitude, Latitude and local water depth at 19 different locations in Kuwaiti territorial waters Location Longitude ( o E) Latitude ( o N) Water depth (m)

3 Neelamani Figure 2. Maximum and average value of the hindcasted significant wave height for Kuwaiti territorial waters based on the hindcasted waves from to The maximum and average significant wave heights for these 19 locations based on the 12 year hindcasted data is provided in Fig.2. The highest max. significant wave height is hindcasted at location 2 (Hs = 3.51 m) and the lowest max. significant wave height is hindcasted at location 17 and 18 (Hs = 1.43 m). Similarly the max. average wave height for 12 year has occurred at location 2 with Hs = m and the minimum average wave height has occurred at location 18 with Hs = m. METHODOLOGY The Gumbel and Weibull distribution is used. The statistics of long term prediction requires that the individual data points 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 (2001)). MATHIESEN et. al. (1994) recommends that the minimum time interval between local maxima be somewhat longer than the time lag for which the auto-correlation function is 0.3 to 0.5. 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 each location, one can expect from 91 to 182 data points per year. For a threshold wave height of 1.0 m, the No. of storm events/year is obtained for each location and is depicted in Fig.3. It can be seen that the maximum value of no. of storms/year is about 60 and the minimum value is even less than 10. It is seen that only location 1,2 and 3 has more than 50 No. of storm events/year with threshold significant wave height of 1.0 m. Location 13, 16, 17, 18 and 19 has less than 10 No. of storm events/year with threshold significant wave height of 1.0 m. This is an important information for marine operation around these locations. The data points used in the POT analysis are the peaks occurring during each storm with threshold wave height of 1.0 m. The total number of data points used for the extreme wave analysis is hence 12 times the No. of storm events/year with threshold Hs value of 1.0 m as provided in the above figure. 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 and 3.25 m were used and the extreme value analysis is repeated for each threshold values in order to understand the effect of threshold value on the predicted extreme waves of different return periods. Figure 3: No. of storm events/year with threshold significant wave height of 1.0 m in Kuwaiti territorial waters The data points for each location are arranged in the descending order. The probability of exceedence, Q is calculated using the formula Q = (i-c 1 )/(N+c 2 ) + (1) Where, i is the Rank and N is the total number of data points c 1 =0.44 and c 2 = 0.12 for Gumbel distribution and c 1 =0.20+ (0.27/α) and c 2 = (0.23/α) for Weibull distribution, 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 (2000)). No. of Events/Year for Different Threshold Wave Height at Different Locations The number of events/year for different threshold wave height at location 1, 3, 8, 15, 16 and 19 is given in Fig.4. This figure provide vital statistical information, which are essential for different type of marine works around these locations (constructions, operations, maintenance of coastal structures etc) in the Kuwaiti territorial waters. Prediction of the Wave Height for the Selected Return Period The return period, T R and the probability of exceedence are linked by the following expression Q = 1 / (l T R ) (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 of H s for each location in Kuwaiti territorial waters. Since the data is for a total duration of 12 years, the value of l can be calculated immediately. Now according to Gumbel distribution, the wave height expected for a selected return period H TR can be estimated as follows: H TR = g -b In[In(1/P)] (3) i.e. H TR = g -b In[In{(lT R )/(lt R -1)}] (4) According to the Weibull distribution, the wave height expected for a selected return period H TR can be estimated from the following formula: H TR = g +b [In(1/Q)] 1/α (5) i.e. H TR = g +b [In(lT R )] 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. 566

4 Influence of threshold value on Peak Over Threshold method Figure 4: The number of events/year for different threshold wave heights in location 1, 3, 8, 15, 16 and 19 RESULTS AND DISCUSSIONS The following are the steps used for the long-term prediction of waves in Kuwaiti territorial waters:- a. The data set for each location is obtained based on peak over threshold value of significant wave height from 0.75 m to 3.25 m with an interval of 0.25 m and for all 19 locations for the hindcasted data for the period from to Steps 'b' to 'f' is carried out for each data sets pertaining to a threshold wave height and repeated for all the 11 different threshold wave heights selected. b. The wave heights obtained at each location is arranged in descending order. c. 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. d. 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). e. 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 (NEELAMANI et al. 2008). f. 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 (Neelamani et al. (2007)) and hence all the extreme wave prediction is carried out by using Weibull distribution. The predicted wave heights for the different locations based on Weibull distribution for different return periods when the threshold wave height is 1.0 m, are provided in Fig.5. It is found that the locations 1, 2 and 3 (around Quaro Island) is expected to have the high long term significant wave heights in Kuwait (more than 3.5 m). The locations around Failaka Island (Location 13, 16, 17 and 18 in Fig.1) are expected to experience long term significant wave heights of less than 2.0 m. Plans to develop Failaka Island as a tourism hub in Kuwait is on pipeline and hence the results will be useful. Effect of Threshold Wave Height on the Predicted Wave Height for 100 Year Return Period The effect of changing the threshold wave height value on the predicted 100 year return period significant wave height is studied for all 19 locations. Typical plot for location 1, 3, 8, 15, 16, 19 is provided in Fig. 6. 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. Figure 5: Predicted extreme significant wave heights in Kuwaiti territorial waters for different return periods based on Weibull distribution Figure 6: Effect of changing the threshold wave height value on the predicted extreme wave height for 100 year return period for locations 1, 3, 8, 15, 16 and 19. Predicted Minimum and Maximum Significant Wave Height for 100 Year Return Period For each location, the extreme wave height for different return periods is estimated by varying the threshold value from 0.75 m to 3.0 m. Then for each location, the min. and max. value of the predicted extreme significant wave height is selected for different return periods. A plot showing the predicted min. and max. significant wave height for 100 year return period for all the 19 locations in the Kuwaiti territorial waters is provided in Fig. 7. It can be seen that for few locations, the difference between the maximum and minimum value is more than 0.5 m and for few locations the difference is even less than 0.25 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 wave height value and hence some risk is involved. On the other hand, if the maximum value is used then the risk can be minimized but the marine structure may be more expensive. Hence it is up 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 breakwater needs to be designed at location 2. Selection of the maximum significant wave height value of 4.42 m instead of the minimum significant wave height value of 3.83 m results in 54% increase in the weight of the armor unit. Similar exercise can be carried out for other type of marine projects at different locations in Kuwaiti territorial waters. 567

5 Neelamani Figure 7: Predicted minimum and maximum Significant wave height for 100 Year return period in Kuwaiti territorial waters CONCLUSIONS AND RECOMMENDATIONS The following are the conclusions obtained out of this investigation: There is no definite trend in the predicted extreme significant waves when the threshold value is varied from 0.75 m to 3.0 m. For some locations, the predicted extreme significant wave height reduces with increased threshold wave height value and for some location, the predicted extreme wave height value oscillates when the threshold value is changed from 0.75 to 3.0 m. The difference between the maximum and minimum predicted 100 year return period significant wave height value is more than 0.5 m for some locations in Kuwaiti territorial waters and for few locations the difference is even less than 0.25 m. For a typical location in the territorial waters of Kuwait, the user/owner of the project need to select the 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 coastal projects are in progress and many new projects are planned for the near future in the Kuwaiti territorial waters. The results of the present study will be useful for the risk based analysis and for optimal design of marine structures in the Kuwaiti territorial waters. LITERATURE CITED AL-AJMI, D.N., and MUSTAFA, A., Weather and Climate parameters and Environment factors. (In Arabic). AL-SALEM, K., RAKHA, K.A., SULISZ, W., and AL-NASSAR W Verification of a WAM Model for the Arabian Gulf. Arabian Coast 2005 Conf., Dubai, Nov., AL-YAMANI, F.Y., BISHOP, J., RAMADHAN, E., AL- HUSAINI, M., and AL-GHADBAN, A., Oceanographic Atlas of Kuwait's Waters. Kuwait Institute for Scientific Research, Kuwait. 203 pp. CAIRES,S., and STERL,A., Year return value estimates for Ocean wind speed and significant wave height from the ERA-40 data. Journal of Climate. Vol. 18, pp COLES,S., An introduction to Statistical modeling of extreme values. Springer-Verlag, 208 pp. FERREIRA, J.A., and GUEDES SOARES, C., An application of the peaks over threshold method to predict extremes of significant wave height. J. Offshore Mech. Arct. Eng., Vol.120, pp GODA, Y., HAWKES, P., MANSARD, E., MARTIN, M.J., MATHIESEN, E., PELTIER, E., THOMPSON, E., and VAN VLEDDER, G., Intercomparison of extremal wave analysis methods using numerically simulated data. Proc. 2 nd Int. Symp. On Ocean Wave Measurement and Analysis, ASCE, New Orleans, pp GUMBEL, E.J., Statistics of Extremes. Columbia University Press, New York. JAHNS, H.O. and WHEELER,J.D., Long-term wave probabilities based on hindcasting of severe storms. Journal of Petrol. Tech., Vol.25, pp KAMPHUIS, J.W., Introduction to Coastal Engineering and Management. Advanced series on ocean engineering-vol.16, World Scientific, Singapore Chapter.4. Long term wave analysis. pp LEADBETTER, M.R., On a basis for "peak over threshold" modeling. Stat. Prob. Lett., Vol.12, pp MATHIESEN, M., HAWKES, P., MARTIN, M.J., THOMPSON, E., GODA, Y., MANSARD, E., PELTIER, E., and VAN VLEDDER, G., Recommended practice for Extreme wave analysis. J. Hydraulic Research., IAHR, Vol. 32, pp NEELAMANI, S., AL-SALEM, K., and RAKHA, K., Extreme waves for Kuwaiti territorial waters". Ocean Engineering, Vol. 34, Issue 10, July 2007, pp NEELAMANI, S., AL-SALEM, K., and RAKHA, K., Effect Of Threshold Value On The Predicted Extreme Waves In Kuwaiti Territorial Waters, Technical Report, KISR, October 2008, 35 pages. PETRAUSKAS, C. and AAGAARD, P., Extrapolation of historical storm data for estimating design wave height. Journal of Soc. Petrol. Eng., Vol.11, pp SARPKAYA, T., and ISAACSON, M. DE ST. Q., Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold Company, New York, USA. ST. DENIS, M., Some cautions on the Employment of the spectral technique to describe waves of the sea and the response thereto of oceanic systems. Proc. Offshore Technology Conference, Houston, Paper No. OTC 1819, pp ACKNOWLEDGEMENTS The authors wish to express their gratitude to Kuwait Foundation for Advancement of Science for partially sponsoring the project. 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 software tool for the extreme wave analysis. 568