This paper discusses the use of fractal theory for quantitative analysis of profiles from two-dimensional surfaces. The theoretical relationship between spectral parameters and fractal dimension is examined, including the conditions under which the derived relationship between spectral parameters and fractal dimension by Voss (1985) is valid, and the limitations in making inferences from spectral parameters. Applications of fractal theory to geophysical data are also discussed. In particular, it is shown that an amplitude spectrum with a decay corresponding to a fractal dimension of 1.5 can result from the concatenation of time series with decays corresponding to different fractal dimensions. It has been shown that the spectral density of a fractal distribution will be characterized by a power-law decay, but this paper illustrates that a power-law decay is not sufficient to identify a fractal distribution. Although this paper discusses applications to the study of Earth topography, the results are applicable to the study of any one-dimensional profile. ¿ American Geophysical Union 1989