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Background and Objective: Higuchi’s method of determining fractal dimension (HFD) occupies a valuable place in the study of a wide variety of physical signals. In comparison to other methods, it provides more rapid, accurate estimations for the entire range of possible fractal dimensions. However, a major difficulty in using the method is the correct choice of tuning parameter (kmax) to compute the most accurate results. In the past researchers have used various ad hoc methods to determine the appropriate kmax choice for their particular data. We provide a more objective method of determining, a priori, the best value for the tuning parameter, given a particular length data set. Methods: We create numerous simulations of fractional Brownian motion to perform Monte Carlo simulations of the distribution of the calculated HFD. Results: Experimental results show that HFD depends not only on kmax but also on the length of the time series, which enable derivation of an expression to find the appropriate kmax for an input time series of unknown fractal dimension. Conclusion: The Higuchi method should not be used indiscriminately without reference to the type of data whose fractal dimension is examined. Monte Carlo simulations with different fractional Brownian motions increases the confidence of evaluation results.
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Fractal Dimensions of Biomass Burning Aerosols from TEM Images Using the Box-Grid and Nested Squares Methods
The fractal dimension is a key parameter in quantifying the morphology of aerosol aggregates, which is necessary to understand their radiative impact. Here we used Transmission Electron Microscopy (TEM) images to determine 2D fractal dimensions using the nested square and box-grid method and used two different empirical equations to obtain the 3D fractal dimensions. The values ranged from 1.70 ± 0.05 for pine to 1.82 ± 0.07 for Eucalyptus, with both methods giving nearly identical results using one of the empirical equations and the other overestimated the 3D values significantly when compared to other values in the literature. The values we obtained are comparable to the fractal dimensions of fresh aerosols in the literature and were dependent on fuel type and combustion condition. Although these methods accurately calculated the fractal dimension, they have shortcomings if the images are not of the highest quality. While there are many ways of determining the fractal dimension of linear features, we conclude that the application of every method requires careful consideration of a range of methodological concerns.
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- Award ID(s):
- 2100708
- PAR ID:
- 10430407
- Date Published:
- Journal Name:
- Atmosphere
- Volume:
- 14
- Issue:
- 2
- ISSN:
- 2073-4433
- Page Range / eLocation ID:
- 221
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/atmos14020221
