RMS roughaess 



ill (Xj -Xj)' 

 n - 1 



f/2 



where X, represents a predicted value of depth at point i. By measuring 

 the roughness relative to a predicted depth, rather than a simple mean 

 as in the case of the standard deviation, the effect of long wavelength 

 slopes on sampling is reduced. This improved measure does not, however, 

 provide control over the distribution of roughness with frequency. 

 Since the reflection or scattering of an incident acoustic signal on a 

 surface is dependent upon spatial frequency, this often used parameter 

 appears inadequate. 



Another possible measure of roughness is termed the "roughness 

 coefficient" by Bloorafield (1976), and has the form 



^^ (X, - X, ,)^ 



gj (x^ - x)2 



Because this measure (also referred to as the von Neumann ratio and the 

 Durbin-Watson statistic) is normalized by the total sum of squares of 

 the residuals, the dependence on sample length is minimized. However, 

 because this measure also depends on the squared differences of adjacent 

 points, it measures only the variability of the signal at a wavelength 

 corresponding to the sampling Interval. This roughness coefficient then 

 is essentially a ratio of the high frequency variability of a signal to 

 its long-term trend, an interesting statistic, but not adequate for a 

 complete stochastic model. 



Several statistical functions exist which describe the sample vari- 

 ability as a function of discrete data spacings or lags (see Chatfield, 



13 



