Thus, U^ is the cosine transform while Vj^ is the sine transform of the 

 water level elevations. The spectral lines are then computed by the 

 formula. 



P(fml = (Urn + v2)/(NAt) , (5) 



where 



^m = NAT ^^^ 



for m=0,l ,2,3, . . . , N-1. These N spectral lines can be computed with 

 great rapidity on a digital computer with the fast Fourier transform (FFT) 

 procedure (Cooley and Tukey, 1965; Robinson, 1967). 



The frequency 



f^y = l/2At (7) 



is called the Nyquist frequency. A symmetry relation 



PCfm) = P(fN-m) W 



holds because of intrinsic mathematical properties of equations (4) , (5) , 

 and (6). Hence, it is only necessary to specify p(fni) for 0<^m<^N/2 (i.e., 

 for frequencies between zero and the Nyquist frequency) . The spectral 

 density defined in equation (5) is defined by analogy to the conventions 

 used by Blackman and Tukey (1958) to be a two-sided spectral density in 

 which only the right-hand side is reported. That is, the total variance 



ZN-l /N 

 p(fjjj)Af. 



The second procedure, involving the fast Fourier transform, was 

 selected for the spectral computations for Hurrican Carla because it 

 incurred less loss of information than the covariance procedure. The FFT 

 method permits the inspection of all 2,048 spectral lines for frequencies 

 up to the Nyquist frequency before the lines are averaged to yield a 



smooth estimate, p(f), of the spectral density. The covariance method 

 gives only the smoothed spectral density. 



However, conclusions concerning statistical confidence for spectral 

 estimates based on data computed with the FFT method are valid also for 

 spectral estimates based on the covariance method. The two procedures 

 for computing the spectral density give essentially the same result. 



III. ESSENTIAL EQUIVALENCE OF THE FFT AND COVARIANCE METHODS 



The covariance procedure gives spectral estimates which approximate a 

 smoothing of the population or "true" spectral density. The effective 

 width of the smoothing is approximately l/(2k„At), where k„At is the 



