Limitations of fractal dimension estimation algorithms with implications for cloud studies

Four fractal dimension estimation algorithms are applied to three different simulated fractal functions over the entire range of fractal dimensions to demonstrate the effect of resolution, fractal type, and algorithm choice on the estimation of fractal dimension. Box counting, horizontal structuring element, variation, and power spectrum algorithms are reviewed and applied to 1-D Takagi, Weierstrass-Mandelbrot, and fractional Brownian motion curves to show that none of them perform consistently with each other or accurately when the correct fractal dimension is known. A better understanding of their limitations is achieved through a sliding window analysis, a procedure that calculates local slopes for linear curve fits. A synergistic approach involving knowledge of algorithm limitations is suggested to make progress toward more reliable estimates of fractal dimension. Implications for cloud studies are considered in lieu of these results. Scale breaks in clouds are investigated to see how well each estimator detects changes from one scaling regime to another. Results are presented to illustrate the importance of correcting errors in the fractal dimension estimation process when modeling cloud radiance fields.