where 



if SNR > 



(40) 



a_ , if SNR < 

 and SNR denotes the SNR regarded as a random variable. 



VIII. THE EMPIRICAL PROBABILITY LAW FOR THE SYNffffiTRICALLY NORMED RESIDUAL 

 The cumulative distribution function for the SNR is defined as: 



where P[*] is the probability of the event specified within the brackets. 

 This distribution function can be estimated from the 300 values of the 

 SNR's for each record of Hurricane Carla. Let (SNR)]^ be SNR's ranked 

 in order of increasing size: 



(SNR)^ < (SNR) 2 1 (SNR) 3 < ••• < (SNR)3qq . (42) 



A statistically reasonable estimate of Fgj^[^(w, ) for 



wj^ = (SNR)i, (43) 



is 



^SNR^^k^ = W C44) 



(Gumbel, 1954). Thus, a graph of Fgj^R(wj^) versus wj^ for k = 1 ,2, 3, • • • ,300 



gives the distribution function estimate. The graphs are shown in Figures 

 37 through 48. 



The corresponding probability densities may be obtained by differen- 

 tiating the distribution function numerically. For the present study, 

 this was done by selecting a band on the SNR axis which is 0.5 unit wide 

 and fitting a least square line to all the ranked points lying within the 

 band. The slope of the line is the probability density estimate assigned 

 to the midpoint SNR value for the band. This was repeated for all 300 

 possible midpoints on the SNR axis. The resulting probability densities 

 are given in Figures 49 through 60. 



IX. EMPIRICAL PROBABILITY INTERVALS FOR f (f^,) BY SIMULATION 



Suppose a new spectral density estimate, p*(f^), is developed by 

 simulation from equation (39) by the following procedure. A random num- 

 ber, uniformly distributed on the interval, (0,1), is generated in the 

 digital computer. One of the pieces of record is selected for study and 



50 



