Morphology differences between fractional Brownian motion and the Weierstrass-Mandelbrot function and corresponding Hurst evaluation

Author:

Dong Jiabin,Wu Ying,Jin YiORCID,Liu Shunxi,Zheng Junling,Dong Wenhao

Abstract

AbstractFractional Brownian motion (FBM) and the Weierstrass-Mandelbrot (W-M) function are two important methods for constructing self-affine objects. Accurately characterizing their features, such as the morphology and fractal geometry, is fundamental for their follow-up applications. However, due to the differences between self-similar and self-affine properties, the fractal dimension evaluation is difficult and sometimes unconvincing. In addition, the sampling length and diversity of calculation methods both lead to the non-uniqueness of the fractal dimension. This study compared the morphology differences between FBM and the W-M function and then analyzed the effects of each parameter on their morphology. Comparisons indicated that FBM has fewer control parameters than the W-M function. Finally, from basic fractal geometric properties, we derived the relationship between the measurement scale r and profile lengths L(r) for self-affine profiles and found that it is a complicated form rather than $$L(r)\sim c r^{H-1}$$ L ( r ) c r H - 1 , where H is the Hurst exponent. Meanwhile, the equivalent vertical height v(r) has an intuitive and clear power law relationship to the measurement scale $$v(r)\sim \sigma r^H$$ v ( r ) σ r H , which provides a method to estimate the Hurst exponent.

Funder

Fundamental Research Funds for the Universities of Henan Province

National Natural Science Foundation of China

Publisher

Springer Science and Business Media LLC

Subject

Economic Geology,General Energy,Geophysics,Geotechnical Engineering and Engineering Geology

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