Fractal mapping of digitized images: Application to the topography of Arizona and comparisons with synthetic images

Abstract

The Earth's topography generally obeys fractal statistics; after either one‐ or two‐dimensional Fourier transforms the amplitudes have a power law dependence on wave number. The slope gives the fractal dimension, and the unit wave number amplitude is a measure of the roughness. In this study, digitized topography for the state of Arizona (7 points/km) has been used to obtain maps of fractal dimension and roughness amplitude. The roughness amplitude correlates well with variations in relief and is a promising parameter for the quantitative classification of geomorphology. Significant variations in fractal dimension are also found. For Arizona the mean fractal dimension for two‐dimensional Fourier spectral analyses is D = 2.09; for one‐dimensional Fourier spectral analyses the mean fractal dimension is D = 1.52, close to the Brown noise value D = 1.5. Synthetic two‐dimensional images have also been generated for a range of D values. For D = 2.1, the synthetic image has a mean one‐dimensional spectral fractal dimension D = 1.56, consistent with our results for Arizona. These results are also consistent with those of previous authors and show that it is not appropriate to subtract one from the two‐dimensional fractal dimension of topography in order to obtain the one‐dimensional fractal dimension.