The Impact of Vertical Scaling on the Estimation of Fractal Dimension of Slope Failures in Opencast Mine

Abstract

This study investigates the effect of vertical scaling on fractal dimension (FD) estimation for failure surfaces in opencast mines to improve surface roughness characterization for slope stability analysis. Fractal analysis has become an essential tool in geotechnical engineering, yet the impact of scaling on FD estimation remains underexplored. Seven fractal analysis methods were evaluated: Box Counting Method (BCM), Power Spectrum (PS), Variogram (V), Structure Function (SF), Triangular Prism Method (TPM), Differential Box Counting (DBC), and Detrended Fluctuation Analysis (DFA). These methods were applied using MATLAB to a failure surface from the Mashamba West mine, examining scaling factors from fine to coarse. The results show that vertical scaling significantly affects FD estimates. BCM and DBC exhibited decreasing FD values as scaling increased, with BCM values ranging from1 1.649 at a scaling factor of 0.1 to 1.0367 at 10. In contrast, PS, Variogram, and SF demonstrated increasing FD values, with PS rising from 1.2744 at 0.1 to 2.8701 at 10. TPM produced stable FD values (~2.025) across all scaling factors, indicating robustness to scaling. DFA showed a gradual increase in FD, from 1.666 at 0.1 to 2.6224 at 10, suggesting moderate sensitivity to scaling. These findings highlight the need for careful method selection based on the scale and complexity of the surface, as different methods exhibit varying performance at various scales. This study contributes to a better understanding of the scaling effects on FD estimation and supports a more accurate characterization of failure surfaces in geotechnical applications.