How much do the p* (fm) as a group differ from the spectral density, 

 p(f ) ; used in the simulation? The answer to this question was developed 

 for the 10 frequencies, 0.070, 0.072, 0.074, 0.077,^0.083, 0.088, 0.101, 

 0.132, 0.168, and 0.243 sec."^ The 900 values of p*(fm) were ranked for 

 each frequency and the two values with ranks 45^and 855 were selected as 

 estimates of the 5th and 95th percentiles for p*(fj„). These percentile 

 estimates are plotted versus frequency as dots in Figures 61 through 72. 



X. COMPARISON WITH CHI-SQUARED PROBABILITY INTERVALS FOR f (fn,) 



If the sea surface is Gaussian, the spectral density will follow a 

 probability law closely related to a chi-squared random variable with 16 

 degrees of freedom (for the Hurricane Carla estimates) (Borgman, 1972). 

 Symbolically, 



,,. . = X2 (48) 



PlfmJ 16 



where p(fm) denotes the true or population spectral density. Thus, if 

 X?^ ^ „r and Xtz: n qk 'i^"°'te the 5th and 95th percentiles for a chi- 

 squared random variable with 16 degrees of freedom, then: 



.2 r.rf ^ v2 



16 05 P^^mJ Xi6 95 P^V 1 , , 



^^'"•"^ < p(f^) < ^"'^-^^ =0.90 . (49) 



16 •" 16 J 



The interval (xf^^Q^^ p(fn,) / 16 , Xf^^^g^ p(f^) / 16 ) thus provides 



a 90 percent probability interval for f(fm)- However, p(fni) is not 

 known. As an approximation, p(fni) "^ay be substituted for p(f^). An 

 analogous approximation was made in the simulations. The resulting upper 

 and lower limits are plotted as asterisks in Figures 61 through^72 versus 

 each of the selected frequencies. The spectral density values p(fm) are 

 shown in the figures as pluses. 



As was noted in an earlier paper based on an analysis of part of the 

 data (Borgman,' 1972), the chi-squared probability intervals do not differ 

 excessively from the simulated probability intervals. They are both about 

 the same, although there are substantial variations from record to record. 

 The upper bounds show appreciably more scatter than do the lower bounds. 

 Another comparison of the two kinds of probability intervals is given in 

 Figure 73. The two ratios, 



simulation upper bound / p(fm) 



and, 



simulation lower bound / p(fm) > 



64 



