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Weibull and Gamma Distributions for Wave Parameter Predictions

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Weibull and Gamma Distributions for Wave Parameter Predictions J. Ind. Geophys. Union ( January 2005 ) Vol.9, No.1, pp.55-64 Weibull and Gamma distributions for Wave Parameter Predictions S.P.Satheesh, V.K.Praveen, V.Jagadish Kumar, G.Muraleedharan1 and P.G.Kurup2 Department of Physical Oceanography, Cochin University of Science and Technology, Cochin-682016 1Centre for Atmospheric Sciences, IIT Delhi, New Delhi-110016 2Amrita Institutions, Ashram Lane, Azad Road, Cochin-682017 e-mail:[email protected] ABSTRACT Distribution of maximum wave height follows the model derived on the concept that the individual zero up-crossing wave heights follow the Weibull law. Better results are obtained when a depth factor included to accommodate shallow water attenuation effects. Certain wave height parameters such as mean maximum wave height, most frequent maximum wave height, extreme wave height, return period of an extreme wave and probability of realising an extreme wave in a time less than the designated return period are estimated and compared with computed values giving reliable results. Predicted significant wave heights are comparable with the computed value with maximum deviation of 0.21m. Theoretical and empirical supremacy of Weibull is re-established. A 100.0% empirical support is obtained in the simulation of zero up-crossing wave periods by Gamma at 0.05 level of significance. A Weibull-Gamma (with non-zero correlation) joint distribution simulates more effectively the real 3D plot. INTRODUCTION was offered in terms of the intensity function (Muraleedharan, Unnikrishnan Nair & Kurup 1993). Unlike in deep water, the wave climate in shallow A modified Weibull distribution which includes water exhibits spatial variations owing to complex the depth factor to accommodate the shallow water transformation processes. Coastal engineering projects attenuation effects is used to fit the shallow water require reliable wave data for design and maximum wave height distribution. Significant wave implementation. heights computed from zero up-crossing wave heights In order to understand the wave phenomenon are utilised to calibrate the depth factor. correctly, it is necessary to carry out field measurements The Erlang model is suggested for simulating the and then try to express the physical processes in terms distribution of zero up-crossing wave periods.Various of mathematical equations. If the properties of waves wave period statistics are predicted and empirically at a given location are available, then it is possible to validated (Unnikrishnan Nair, Muraleedharan & Kurup select the suitable mathematical models to describe 2003). them. The model thus selected may help in predicting For a complete description of wind generated the statistical wave characteristics at a given location. random waves, it is necessary to consider wave height Such simulated wave characteristics are becoming an and period simultaneously. Serious consideration accepted practice in coastal and ocean engineering. must be given to the combined effect of height and Of the various theoretical models available for period. Wave data measured in the ocean often show modelling ocean waves, only the Rayleigh distribution that the height and period of incident waves are not is derived mathematically on the basis of physical statistically independent. All the existing joint processes governing the wave heights (Longuet-Higgins distribution models of wave heights and periods are 1952). Yet we need a model that can meet the twin derived on the basis that there exists zero correlation objectives of a) accommodating the Rayleigh between them. Many researchers have pointed out that distribution whenever the basic assumptions that there exists a significant correlation between them. justify it are satisfied, b) fitting data situations under The Weibull-Gamma joint distribution including more general conditions. Thus we arrive at Weibull correlation effects is recommended for the simulation model for simulating the wave height distribution. of surface plots of real wave situations of wave height- Another possible explanation for the Weibull curve period distributions. 55 S.P.Satheesh et al. MATERIALS AND METHODS 770E, 40X40, NPOL Atlas) enclosing Valiathura coastal station are used for validating the Erlang model(or Long-term visually estimated wave heights and periods Gamma). This model is suggested by mathematical (deep water wave data) off Goa (NPOL Atlas 1978) and logic for simulating the significant wave period recorded shallow water wave data off Alleppey and distribution. The validation results show its Valiathura (CESS Wave Data 1984)are considered in applicability. this study. Monthly averaged visually estimated wave height distributions off Goa (Grid.9, 130-170N, 730- Modified Weibull model 770E, 40x40, NPOL Atlas)) are fitted to Rayleigh, Weibull, Gumbel and Exponential models. The The modified Weibull model with depth factor for offshore of Goa is enclosed by Grid.9. maximum wave height distribution is given as The visually estimated(appro.=H , Grid.17, NPOL (Muraleedharan, Unnikrishnan Nair & Kurup 2001) s Atlas) and recorded off Valiathura (May-October, CESS n b - h + h Data) significant wave height distributions are utilised a d []()n l to validate the simulation capability of the Weibull F h = 1 − .......... .......... .......... ....(1 ) model. Long-term recorded shallow water maximum wave where F(h)-Weibull distribution function, a scale height distributions off Alleppey(80-120N, 750-770E, parameter, b-shape parameter, n-sample size, h-is the water depth = 5.5m) from May-October (1981-84) are maximum wave height, d- is the water depth, h-mean simulated by a modified Weibull model including a maximum wave height. The various design wave depth factor to accommodate the shallow water height parameters derived were (Muraleedharan, attenuation effects. Unnikrishnan Nair & Kurup 1999) As a case study individual zero up-crossing shallow water wave heights obtained from wave recorder charts Most frequent maximum wave height (H ) m f m by the zero up-crossing technique for the month of January 1981 off Valiathura (water depth=5.5m) by The most frequent maximum wave height is using a digitising software package ACECAD are also 1 analysed to compute the significant wave height(H ) 1 b h s H = a 1− − and are then compared with the estimated value mfm nb d .. .(2) obtained from the parametric relation derived from the Weibull distribution giving reliable accuracy. Mean of maximum wave height The zero up crossing wave period distribution of 1 ()− 1 ()− r+1 1 the active south-west monsoon season off Valiathura nà n n 1 à 1 à a b b b H max = − +L+ .......................(3) (May to October, 80 26 N, 760 54E, Water depth = b 1! 1 1 2!×2 b n b 5.5m, CESS Data) are tried to be simulated by available theoretical models such as Gamma, Rayleigh, r = 1,2,3 exponential and Bretschneider. The simulation capability of the various models are tested using root Significant wave height mean square relative error (e ) and relative bias (e ) rms mean (Roelvink 1993). The supremacy of Gamma over other 1 a 1 competing models is established. H = a × (ln 3) b − DF − SC + × 3× I .................. (4) s ln 3 Then various significant wave periods(T ,T b b 1/3 1/ , .T , ..) are computed from the above zero up- ∞ 4 1/10 l−t × p−1 × crossing wave period data and are compared with where I(x)(P) = ∫ t dt , is the incomplete estimated values from the expression derived from Gamma function,x DF-depth factor(ratio of mean wave Erlang(special case of Gamma for its shape parameter height to water depth), SC-shoaling coefficient ( = equal to the nearest integer, considered for its 1.15). (Kurian 1987). simplicity) giving acceptable accuracy. The accuracy of the predicted design wave Then the data of the visually and monthly averaged parameters is evaluated by root mean square relative wave periods(appro.= T ) from Grid.17 (50-90N, 730- error (e ) and relative bias (e ) (Roelvink 1993) i.e s rms mean . 56 Weibull and Gamma distributions for Wave Parameter Predictions 1 ∑ []P − C 2 Hence root mean square relative error ε = N rms 1 á−1 i ()ë j ∑ C t N ∑∑ ∑ []− −1 i==0 j 0 j! ε = P C m(t) = t + ë relative bias (mean error) mean ....... (7) ∑ C á−1 ()ë i ∑ t where , = i! = i 0 N total number of data. The various predicted significant wave periods (T = 1/ P predicted value , T , T .) derived from the expression (7) are 3 1/4 1/10 C = computed value compared with the computed values from the recorded wave data off Valiathura during the southwest Prediction of various significant wave periods monsoon seasons and its accuracy is estimated using e and e . rms mean The prediction of various significant wave periods is given as (Unnikrishnan Nair, Muraleedharan & Kurup Distribution model for significant wave period. 2003) : The probability density function of the Erlang An Erlang distribution model is suggested theoretically model is given to be for the significant wave period distribution from the −ë t mathematical logic (Unnikrishnan Nair, á á− l f ()t =ë ⋅t 1 , t >0, á, ë >0 ........................... (5) Muraleedharan and Kurup 2003) ()á−1 ! t m`(t) as a model for wave period. The distribution function F()t = 1 − F ()t = exp − ∫ dt, .......(8) is obtained as, 0 m(t) − t ) α−1 l−ët where, m`()t is the derivative of m(t) = − ()ë i ≥ ........................ (6) F(t) 1 ∑ t , t 0, F(t) is the distribution function i=0 i! ∞ 1 Validation of the Model m(t) = E()T = t + ( )× F(t)× dt T > t () ∫ F t t Visually estimated wave period data for Grid.17 λ and α are the scale and shape parameters respectively. compiled for ten years(NPOL 1978) are utilised to á−1 validate the simulation capability of the Gamma ∑()ët i × l −ët model for significant wave period distribution. where, F(t) = i=0 Valiathura recording station is located in this grid.
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