The full details of the derivation for the m-dependent central limit 

 theorem and the asymptotic chi-squared properties of the spectral esti- 

 mates are given in Appendic C. One useful secondary result, summarized 

 in item (C-8) of the appendix, is that the normed FFT coefficients. 



/Np„ ■ht/2 /Np„ At/ 2 



'\,mQ + l_<m£mQ + r , (71) 



asymptotically follow a multivariate probability law with zero mean and 

 covariance matrix equal to an identity matrix. 



As an illustration of the asymptotic normality, 100 water level ele- 

 vations were selected from the Hurricane Carla data number 6883. The 

 elevations were taken 6 seconds apart from the beginning of the data 

 until 100 elevations were obtained. Also, the 50 pairs of Fourier coef- 

 ficients centered around the frequency associated with the largest spec- 

 tral density value (Fig. 8) were tabulated from the computer listings. 

 The water level elevations and the FFT coefficients were plotted on 

 normal probability paper (after norming them) as shown in Figure 74. An 

 examination of the figure shows that the FFT coefficients follow a rea- 

 sonably straight line while the water level elevations exhibit a concave 

 upward curve. The skewness of the water level elevations was computed to 

 be 0.55 while the FFT coefficients only showed a skewness of 0.03. Thus, 

 the FFT coefficients were reasonably close to normality even though the 

 water level elevations were decidedly nonnormal. 



It would be interesting to explore the normalizing influence of the 

 Fourier transform relative to the other wave records in Hurricane Carla. 

 However, time did not permit that to be included in this investigation. 



XIV. IMPLICATIONS FOR DIRECTIONAL SPECTRUM RELIABILITY 



The asymptotic normality of the finite Fourier transform coefficients 

 are extremely important relative to the estimation of the reliability of 

 the directional spectrum and of other quantities such as energy diffrac- 

 tion computed from the FFT coefficients. The FFT coefficients, at least 

 for large data sets, can be taken as normally distributed and independent. 

 These properties can be carried through the various formulas used in 

 computing the particular quantity to obtain reliability measures for the 

 quantity. 



XV. SUMMARY AND CONCLUSIONS 



1. The statistical properties of the spectral lines for 12 pieces of data 

 measured during Hurricane Carla (8 to 10 September 1961) were examined 

 in detail. The spectral lines were found to show negligible serial 

 correlation. The spectral lines could be reduced to what appears to 

 be random noise by subtracting from the lines the spectral density 

 obtained by smoothing the lines with a moving average and then 



82 



