46. If more than one set of wave heights is available, a better 

 estimate of the pdf of the underlying process is found from the average number 

 J^ of contributions to each range bin. Since H^^^ is unlikely to vary much 

 from run to run, the effect is simply to increase the sample population of 

 wave heights. Such an increase gives a better pdf estimate because there are 

 more degrees of freedom for the population of each range bin. 



47. The same set of normalized wave heights E^/U^^^ can be used to 

 estimate the cumulative or exceedence probability functions. If the H^/H^^^ 

 are ordered from smallest to largest, the probability that any of the wave 

 heights is less than the height of wave n is the fraction n/(N + 1) . This 

 computation provides an estimate of the cumulative distribution function. The 

 data estimate of exceedence probability QQ(HyH^„3) is one minus the cumula- 

 tive distribution at height n , or 



K.'-^rmsJ 



1 - rrr <^^) 



48. A third set of statistics that are of use is the average of the 

 highest fraction r of an observed set of waves, denoted H^"^^ . One of the 

 most commonly cited members of this group is the average of the highest one- 

 third of the waves in a given sample. This is given the symbol H'-"-'^^ 

 following the notation of Longuet-Higgins (1952). Note that the SPM (1984) 

 and some other references use the symbol H^^j , instead. From the set of 

 normalized, ordered wave heights H^/H^.^^ , the computation of the j"*^ dis- 

 crete, monotonically increasing fraction rj is given by 



r. = i (65) 



J N 



and the average of that fraction of the highest waves is 



Hllii = 1 y ^Izlll (66) 



The result of Equation 66 can be interpolated if a particular desired fraction 

 does not fall on one of the discrete fractions r^ . 



49. For the Rayleigh and Modified Rayleigh models being tested here, 

 the corresponding average heights are found following the derivation given by 



26 



