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

Abstract

Fractional 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}, 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, which provides a method to estimate the Hurst exponent.