laal sinusoidal components. Appendix C presents a geometric proof of 

 this relationship. 



In the case where two or more linear trends are present In a sur- 

 face, the form of a(6) will show a simple sinusoid with phase and ampli- 

 tude which can not be uniquely decomposed Into component sinusoids. If 

 an estimate of the azimuth and amplitude of one of the trends could be 

 produced Independently, this component could be removed and the remain- 

 ing component sinusoids analyzed. It should be emphasized that the 

 inability to decompose component trends in no way invalidates the model, 

 it simply complicates the interpretation of the model statistics in 

 terms of formation processes. 



Multiple linear trends can only be decomposed in cases where the 

 trends are sufficiently band-limited to appear as distinct peaks in the 

 frequency spectrum. This approach allows decomposed trends to be 

 uniquely identified by spectra generated in two orthogonal directions, 

 as was shown by Hayes and Conolly (1972). Their work clearly shows such 

 trends in the large scale topography of the Antarctic-Australian 

 Discordance. In examining a great many spectra of small-scale (X <8 km) 

 topography during the present study, no significant spectral peaks have 

 been observed, even in topography which is highly llneated at the larger 

 scales. This result may be due in part to the presentation of these 

 spectra in log-log form. 



Comparison of Theoretical Fbnctional Fbrms with Multibeam Sonar Data 



The previous section developed a simple theoretical model of the 

 effect of linear trends on frequency spectra from profiles sampled at 



87 



