1980). Perhaps the simplest function of this type Is the mean cross- 

 product of terms at given lags k 



^l^ " 5=k ilk ^i • ^^-^ " E[(Xi)(Xi.k)l 



This function (called simply the autocorrelation function In electrical 

 engineering literature; see Bracewell, 1978) Is both unnormallzed and 

 uncentered (the mean Is not removed). Because It depends on absolute 

 magnitude values (In this case, the total water depth), this simple 

 measure can be Improved for the purposes of roughness modelling by 

 removing the sample mean, which yields the au toco variance function 



Y(k) = E[(Xi - X) (Xi-k - X)] 



It Is obvious that at a lag of k-0, (t(0)) Is simply the sample vari- 

 ance. By normalizing the autocovarlance by the sample variance y(0), 

 one derives the normalized autocorrelation function 



p(k) - ir(k)/ 



y(0) 



Notice the equivalence between the normalized autocorrelation function 

 at lag k>l, (p(l)) and the roughness coefficient described previously. 

 In comparing the roughness of two different samples. It Is undesirable 

 to normalize by the sample variance, therefore, the autocovarlance func- 

 tion provides the best measure of variability with frequency (centered, 

 but unnormallzed) . 



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