maximum lag used in the covariance estimate (i.e., k^ is the maximum value 



of k used in computing equation (1)) and At is the time increment between 

 water level measurements (Blackman and Tukey, 1958). (See App. A for the 

 definition of effective width.) If "hamming" smoothing, 



p^(f^) = (l/4)p(f^_l) + (l/2)p(f^) + (l/4)p(f^^^) , (9) 



is then made, the effective smoothing width for the estimates, p2(f-j,), 

 is changed to l/(kj^At) (Blackman and Tukey, 1958). 



The FFT method also yields estimates which are smoothed versions of 

 the true underlying population spectral density. This can be seen from 

 the following derivation. 



The true or population spectral density is defined as the integral 

 Fourier transform of the theoretical covariance function. That is, 



c(T) = E[n(t) n(t + T)] , (10) 



where E[*] denotes the expectation operator and 



C(T) e '-^" dx (11) 



.oo 



(Blackman and Tukey, 1958). Since C(t) is symmetric about T = for 

 stationary stochastic processes, it follows that equation (11) reduces to: 



p(f) = / C(T) cos (2TTfT) dT . (12) 



/. 



Equations (1) and (2) are obvious analogues to equations (10) and (12) 

 It is not immediately obvious that the same thing can be said for equa- 

 tions (4) and (5), although it is true there also. To see this, let 



N-1 



^k = N E ^n^n+k ' fl^) 



n=0 



where N = 4,096 in the context of the Hurricane Car la records and the n 

 for n^N and n<0 required for the computation of equation (13) are 

 defined by periodicity as: 



'n-N 



(14) 



Equation (13) will give almost the same estimate of the covariance function 

 as equation (1) for 0<k<N/2. There is, of course, a little distortion 



12 



