To Illustrate the Importance of having control over frequency 

 dependence, consider two examples. If a generally flat surface which 

 contains a high frequency roughness component of wavelength X, were sam- 

 pled at spacing X or any integer multiple thereof, each sample would 

 fall at precisely the same depth and yield a variance of zero. Mathe- 



matically, all values of X would be identical, therefore the mean 

 1 " 

 i-1 



X > — £ Xj^ would be equal to all X' s and therefore 



n - 2 

 Z (Xi - X)'' 



Var (Xi) - ^'\. I 



Although this example presents an extreme case, it is obvious that to 

 assure an accurate measure of the variance at a given frequency, the 

 sample spacing must be less than that corresponding wavelength, and the 

 sample length long enough to sample all portions of the cycle. 



A more important shortcoming of these standard dispersion measures 

 occurs at the longer wavelengths. Consider a generally smooth but 

 broadly sloping surface. Examples from the deep sea might be a conti- 

 nental rise or an abyssal fan. Since these dispersion measures record 

 the average variation of individual samples from the sample mean, it is 

 obvious that a relatively long sample would span a greater range of 

 depths and produce a larger dispersion statistic than a smaller sample 

 located in the same data. In this case, the measure of roughness would 

 depend largely on the sample length. 



Many acoustic models of bottom interaction use the more general 

 dispersion measure of the root mean square (RMS) roughness. In these 

 acoustic models, this value represents the RMS variation of depth about 

 some predicted value, normally the smoothed bathymetry. 



12 



