Fully describing the variability of a process with its autocovarl- 

 ance function requires values at all n lags. By taking the Fourier 

 transform of either the au toco variance or the unnorraallzed autocorrela- 

 tion function, the process can be expressed more concisely as a function 

 of frequency. This measure is the well known power spectral density 

 function and can be estimated directly from the data with Fourier trans- 

 forms. Besides providing a concise expression for the roughness as a 

 function of frequency, the power spectrum also has many useful proper- 

 ties which are described in Chapter 6 of Bracewell (1978). One theorem 

 of particular interest is the derivative theorem which states that if a 

 function f(x) has the transform F(s), then its derivative f ' (x) has the 

 transform i2irsF(8). In this application, given the power spectrum of 

 depths as a measure of roughness, the distribution of slopes (first 

 derivative of depth) can be directly calculated. 



An additional measure of bottom roughness, which is particularly 

 favored by those interested in acoustic modelling of bottom interaction, 

 is the two-point conditional probability distribution function 

 P(hi,h2|ri»r2). This function defines the probability of measuring two 

 heights (hj^ and h2) given two distance vectors (rj^ and r2). TWo consid- 

 erations make this approach intractable. First, the description of the 

 function requires a large number of parameters to be retained. Sec- 

 ondly, complete two-dimensional data are required to adequately generate 

 the function. This type of survey data is rarely available and only in 

 areas which have been surveyed using multlbeam sonar. 



The power spectral density function appears to be the best choice 

 of statistical measure for bottom roughness, and it will underlie the 

 stochastic model generated in this study. For convenience, the ampll- 



15 



