regression techniques must be selected la order to assure a proper 

 regression fit to the power law form. Of prime concern Is whether the 

 residuals are minimized In log-log space or linear-linear space. The 

 methods used In this study with accompanying theory and computer soft- 

 ware are presented as Appendix A. 



The power law form seems to represent the best model for describing 

 sea-floor topography In the spatial frequency domain. Its simplicity 

 permits the frequency structure of a sample profile to be described 

 using only two parameters, a considerable reduction of the original, 

 deterministic data. Extensive work by Benolt Mandelbrot (1982) has 

 produced a theoretical basis for this consistent relationship, formu- 

 lated In terms of fractal dimension, a parameter functionally related to 

 coefficient b above (see Berry and Lewis, 1980). Bell (1975b) discov- 

 ered the same power law relationship in data from the Pacific Ocean that 

 Including lower spatial frequencies than those of interest in this study 

 (see Fig. 4-4). Notice that Figure 4-4 plots power spectral density, 

 rather than amplitude as in Figure 4-3 and other example spectra. 

 Because the vertical axis in both plots Is logarithmic, this exponentia- 

 tion appears graphically as a linear transformation. Mathematically, 

 the squaring of amplitude represents a simple doubling of the slope, 

 i.e., multiplication of the b term by a factor of two. 



Bell (1975b) finds a fairly consistent relationship at many size 

 scales, with the slope of the log transformed power spectrum of b = 

 -2. Although this value probably represents a good mean estimate, the 

 examination of many spectra in this study will show an accountable var- 

 iation in this value. Berkson (1975) also discovered a significant 

 variation of regression coefficients for spectra generated from dlffer- 



26 



