where the subscript R identifies the pdf as Rayleigh. Equation 9 can be 

 written in dimensionless form by multiplying both sides by H^^^ to form 





(10) 



17. The cumulative probability function for the Rayleigh distribution 

 is found in dimensionless form by integrating Equation 10 with respect to 

 H/Hj^3 over the limits from zero to H/H . The result is 



'4h1J ' 



(H/Hrms) ,-^-^. 



The Rayleigh exceedence probability function is found by substituting 

 Equation 11 in Equation 4 to yield 



[B 



1 - a-W»-.'' (12) 



Modified Rayleigh Model 



18. The Modified Rayleigh pdf is the deepwater asymptotic form of the 

 Beta-Rayleigh model introduced by Hughes and Borgman (1987) . They defined the 

 Modified Rayleigh pdf as 



where the subscript MR stands for Modified Rayleigh, P is the gamma 

 function (see Abramowitz and Stegun 1970) , and a is a parameter defined as 



fMrmal' . 1 



(14) 



In Equation 14, a depends on H^^^ , which stands for root -mean- quad wave 

 height and is computed from a set of N observed wave heights H^^ through 

 the expression 



[ii-r 



(15) 



10 



