Maximum entropy spectral analysis for the estimation of fractals in topography

Spectral analysis is an important method by which the variation in a data set can be decomposed into waves of different frequencies. In the form of the power spectral density it is usually estimated directly from the data using the fast Fourier transform which often requires considerable pre-processing for accurate calculation. Many geomorphological data, including topography, display a power law/fractal model of a decrease in power spectral density with an increase in frequency. Inaccurate calculation of the power spectral density may result in an incorrect estimation of both the power law exponent and observed fractal dimension. As an alternative, maximum entropy spectral analysis provides, in theory, a more accurate estimate of the power spectrum and therefore a more accurate estimate of the fractal dimension. Results are presented of a maximum entropy spectral analysis of both simulated and real topographic surfaces, with derived calculations of fractal dimension. Although the technique offers certain advantages, and returns accurate estimates of the fractal dimension under certain conditions, these have to be traded off against various methodological difficulties which remain unresolved.