Uncertainty in fractal dimension estimated from power spectra and variograms

The reliability of using fractal dimension (D) as a quantitative parameter to describe geological variables is dependent mainly on the accuracy of estimated D values from observed data. Two widely used methods for the estimation of fractal dimensions are based on fitting a fractal model to experimental variograms or power-spectra on a log-log plot. The purpose of this paper is to study the uncertainty in the fractal dimension estimated by these two methods. The results indicate that both spectrum and variogram methods result in biased estimates of the D value. Fractal dimension calculated by these two methods for the same data will be different unless the bias is properly corrected. The spectral method results in overestimated D values. The variogram method has a critical fractal dimension, below which overestimation occurs and above which underestimation occurs. On the bases of 36,000 simulated realizations we propose empirical formulae to correct for biases in the spectral and variogram estimated fractal dimension. Pitfalls in estimating fractal dimension from data contaminated by white noise or data having several fractal components have been identified and illustrated by simulated examples.