for tides and depth by proper mathematical filters, the autocovariance is 

 then computed. 



The autocovariance or autocorrelation Cn (x) is defined as 



Cn(T) = <[ni(t). ni(t - i)) 



(5) 



Here x is a lag interval having values k»At where k is a positive integer 

 and ni(t) is the time history of water level fluctuations, assumed sta- 

 tistically stationary and oscillating about zero as a mean value. / \ 

 designates an average taken over all values ni(t). The subscript eleven 

 (as Cii) is introduced here to prevent ambiguity with the cross-correlation 

 between two time series introduced in the following sections. 



The spectrum Si(a) is then defined as the cosine transform of the 

 autocovariance 



Si (a) = 



Ci 1 (t ) • cos(ax ) • dx 



(6) 



Jc 



This is usually plotted against frequency f = a/2iT giving 



Si (f) = 4 



Cii(x) . cos(2iTfx) 



dx 



(7) 



Now Cii(x) has units of (length) and dx has units of time. Si(f) is thus 

 a measure of the energy per unit frequency and is sometimes referred to as 

 the estimate of the energy density-. 



Since the autocovariance and the spectrum are Fourier transforms of 

 each other, the spectrum can be expressed in more compact notation using, 

 two-sided forms (-<=° to +<^) with exponential kernels (see Cox, 1962, p. 755) 

 However, because the autocovariance function is symmetrical, the same re- 

 sults are obtained from twice the value of the one-sided cosine transform. 

 One-sided forms are used in this and following developments because they 

 follow more closely the actual manipulations performed by the digital 

 computer. 



b. Error and A I iasing . The spectral analysis yields estimates 

 of the mean energy attributable to sinusoidal wave components in contigu- 

 ous elemental frequency bands of width Af. The width of the band is de- 

 termined by the data sampling interval At and the number of chosen lags m 

 by the relation 



Af = 



2Atm 



The estimates of energy density are usually given in units of the 

 mean-square elevation <^ n^^ per Af. The estimates are subject to errors, 

 whose probability distribution is Gaussian (normal). The confidence 



22 



