Fractal Maps Research Articles

Fractal mapping of digitized images: application to the topography of Arizona and comparisons with synthetic images : Huang, J; Turcotte, D L J Geophys ResV96, NB6, June 1989, P7491–7495

Fractal mapping of digitized images: application to the topography of Arizona and comparisons with synthetic images : Huang, J; Turcotte, D L J Geophys ResV96, NB6, June 1989, P7491–7495

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

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.

Fractal Dimension in the Large Scale Distribution of Galaxies

In this work, we start from the definition of fractal dimension and by the number counting of galaxies to investigate that if the large scale distribution of galaxies has really a fractal structure. A fractal dimension in a distribution of objects is defined as where N(R) is the number of objects within scale R. To find out D from number counting we have to pay attention to that the distribution of galaxies could not be a regular fractal map. The N(R) would understand as a statistical average. To avoid the edge effect appearing in the number counting we compare it with that obtained from an average by many times Monte Carlo sampling which is a uniform distribution and would have Dr = 3. From eq. (1) we get