The mean absolute deviation (MAD) measures the average absolute deviation of the surface heights from the mean height. It is defined as:

$\text{MAD}=\frac{1}{K}{\displaystyle \underset{i=1}{\overset{K}{\sum }}\left|S_i-S\right|}$ (3-1)

where $S_i$ represents the surface heights at equidistant points $i$, $S$ is the mean surface height, and $K$ is the total number of points. MAD offers a direct measure of roughness and is particularly effective when surface elements are statistically independent. The mean standard deviation (MSD), given by:

$\text{MSD}=\sqrt{\frac{1}{K}{\displaystyle \underset{i=1}{\overset{K}{\sum }}{\left(S_i-S\right)}^2}}$ (3-2)

Equation (3-2) offers insights into the variability of surface heights and is particularly advantageous for assessing the normal distributions in surface-height data. MSD is particularly advantageous for assessing the variability of surface heights when they follow a normal distribution ( Figure 2 ). These metrics are essential for probabilistic slope stability analyses because they quantify the surface variability that affects shear resistance.

To evaluate the localized geometry of the failure surfaces, parameters such as mean peak height ( $M_P$) and mean valley depth ( $M_V$) were calculated as follows:

$M_P=\frac{1}{K_{\text{}p}}{\displaystyle \underset{i=1}{\overset{K_{\text{}p}}{\sum }}{P}_i}$, $P_i=\max \left(0,S_i-S\right)$ (3-3)

$M_V=\frac{1}{K_v}{\displaystyle \underset{i=1}{\overset{K_v}{\sum }}{V}_i}$, $V_i=\max \left(0,S-S_i\right)$ (3-4)

where $K_{\text{}p}$ indicates the total number of peaks in the surface profile, $K_v$ is the total number of valleys in the surface profile, and $P_i$ is the height of the ith peak above the mean height $S$. where $S_i$ is the height of the surface profile at the ith measurement point, $S$ is the mean height of the surface profile, and $V_i$ is the depth of the ith valley below the mean height as in Figure 3 .

Parameters such as the standard deviation of the slope (PS) and curvature (PC) provide insights into the geometric irregularities of failure surfaces. These are defined as:

$\text{PS}=\sqrt{\frac{1}{K}{\displaystyle \underset{i=1}{\overset{K}{\sum }}{\left(\frac{\text{d}{S}_i}{\text{d}x}\right)}^2}}$ (3-5)

$\text{PC}=\sqrt{\frac{1}{K}{\displaystyle \underset{i=1}{\overset{K}{\sum }}{\left(\frac{{\text{d}}^2{S}_i}{\text{d}{x}^2}\right)}^2}}$ (3-6)

$\text{PS}$ is the standard deviation of the slope of the surface profile, which measures the average variation in the slope of the failure surface, reflecting its roughness; $K$ is the total number of equidistant points $i$ sampled along the failure surface profile; $\frac{\text{d}{S}_i}{\text{d}x}$ is the slope of the surface profile at the ith point, representing the first derivative of the surface height ( $S_i$) with respect to the spatial coordinate $x$; $\text{PC}$ is the standard deviation of the curvature of the surface profile, which quantifies the average variation in the curvature of the failure surface, which is indicative of its waviness; $\frac{{\text{d}}^2{S}_i}{\text{d}{x}^2}$ is the curvature of the surface profile at the ith point, representing the first derivative of the surface height ( $R_j$) with respect to the spatial coordinate $x$. These parameters are crucial for characterizing the roughness of the failure surface ( Figure 4 ).

Spectral analysis of the failure surfaces is an essential tool for characterizing periodicities and roughness properties. Failure surfaces often exhibit spatial variability that can be decomposed into frequency components, thereby revealing the underlying patterns and effects of geological structures as shown in Figure 5 . For a failure surface, the surface height variation can be expressed as a Fourier series:

$R_m=\frac{a_0}{2}+{\displaystyle \underset{\lambda =1}{\overset{n}{\sum }}\left[a_{\lambda }\cos \left(2\pi \lambda m\right)+b_{\lambda }\sin \left(2\pi \lambda m\right)\right]}$ (3-7)

where $m$ is the spatial distance along the failure surface, $a_{\lambda }$, and $b_{\lambda }$ are the Fourier coefficients representing the contributions of each frequency component. The frequencies of $\lambda =1,2,3,\cdots$ correspond to periodic features of the surface, such as joints, fractures, or bedding planes. The power spectral density (PSD) derived from the Fourier transform quantifies the contribution of each frequency to the total surface variability. It is computed as:

$P_{\omega }=\frac{{\left|R_F\left(\omega \right)\right|}^2}{L^2}$ (3-8)

where $R_F\left(\omega \right)$ is the Fourier transform of $R\left(d\right)$, $\omega$ is the spatial frequency, and $L$ is the length of the analyzed surface profile. The PSD function enables the identification of dominant frequencies associated with structural features that can influence the slope stability. The spectral analysis process for failure surfaces typically involves several preprocessing steps to ensure accurate results. First, the raw surface height data must be detrended to remove non stationary components, such as large-scale inclinations or curvatures unrelated to roughness. The detrended data are then multiplied by a window function, such as a Hamming or parabolic window, to reduce the spectral leakage. Spectral leakage occurs owing to the finite length of the surface profile, which distorts the frequency content in PSD. The dependence of $\log \left(P_{\omega }\right)$ on $\log \left(\omega \right)$ provides valuable insights into surface roughness characteristics. For fractal-like failure surfaces, this relationship is often linear, which indicates self-affine properties. The slope $S$ of the linear segment is related to the fractal dimension $D$ of the surface:

$S=-\left(2D-1\right)$ (3-9)

To simulate the spectral estimators applied to the failure surface, a periodic structure with added random noise $N\left(0,1\right)$ can be used to model the spatial variability of the surface. The simulated failure surface height variation $R\left(d\right)$ along the slope profile can be represented as follows:

$R_m=3\sin \left(2\pi \times 10\cdot m\right)+4\sin \left(2\pi \times 4\cdot m\right)+N\left(0,0.1\right)$ (3-10)

$R_m$ is the surface height variation at spatial distance $m$, $3\sin \left(2\pi \times 10\cdot m\right)$, and $4\sin \left(2\pi \times 4\cdot m\right)$, the periodic components corresponding to structural characteristics such as joint spacing or bedding planes; $N\left(0,0.1\right)$ is a Gaussian random noise component with a mean of 0 and a standard deviation of 0.1, simulating irregularities and stochastic variations typical of natural failure surfaces. This simulated profile captures both periodic geological features and random surface roughness, thereby providing a basis for evaluating the effectiveness of spectral estimators in characterizing the geometric and fractal properties of the failure surface. The choice of $N\left(0,1\right)$ or $N\left(0,0.1\right)$ in a simulation depends on the desired magnitude of noise relative to the signal and the physical characteristics of the modeled failure surface.

Statistical analysis of failure surfaces on slopes provides critical insights into spatial correlations and variability. The primary tools used in this analysis include the autocorrelation function (ACF) and variogram, both of which are used to assess the relationships between surface heights at various spatial separations or lag distances ( Figure 6 ). The autocorrelation function (ACF) quantifies the degree of similarity between surface height measurements separated by a lag distance $t$, which is calculated as:

$\text{ACF}\left(t\right)=\frac{{\displaystyle {\sum }_{i=1}^{N-t}\left(S\left(d_i\right)-S\right)\left(S\left(d_{i+t}\right)-S\right)}}{{\displaystyle {\sum }_{i=1}^N\left(S\left(d_i\right)-S\right)}}$ (3-11)

where $S\left(d_i\right)$ and $S\left(d_{i+t}\right)$ are the surface heights at positions $d_i$ and $d_{i+h}$, $S$ is the mean height, and $N$ is the number of data points. For failure surfaces, ACF reveals how roughness features are spatially correlated, with longer correlation lengths indicating more uniform roughness. The variogram measures the variance in the surface height differences over a range of lag distances. It is defined as:

${\gamma }_t=\frac{1}{2N_t}{\displaystyle \underset{i=1}{\overset{N_t}{\sum }}{\left[S\left(d_i\right)-S\left(d_{i+t}\right)\right]}^2}$ (3-12)

where ${\gamma }_t$ is the variogram value for lag $t$, and $N_t$ is the number of point pairs separated by $t$. For failure surfaces, the variogram is useful for identifying spatial scaling properties and structural anisotropy. Statistical measures, such as variance and covariance, further describe the variability and spatial dependencies of the surface roughness. The covariance function is defined as follows:

$U_t=\frac{1}{N_t}{\displaystyle \underset{i=1}{\overset{N_t}{\sum }}\left[\left(S\left(d_i\right)-S\right)\left(S\left(d_{i+t}\right)-S\right)\right]}$ (3-13)

where $U_t$ represents the covariance between the surface heights at a lag distance $t$. The relationship between the covariance function and variogram is given by:

${\gamma }_t=U_0-U_t$ (3-14)

where $U_0$ is the variance in the surface height data. For failure surfaces, these statistical tools are applied to characterize roughness, detect anisotropic features, and evaluate the degree of spatial correlation, all of which can influence mechanical behavior. For instance, high variability or short correlation lengths in roughness can lead to localized weaknesses, whereas a uniform roughness over longer distances can provide greater shear resistance. The analysis of ACF and the variogram was further extended by fitting fractal or power-law models to quantify the scaling behavior. This is particularly useful for surfaces exhibiting self-affinity, where the relationship between the variogram and the lag distance is as follows:

${\gamma }_t\alpha t^{2H}$ (3-15)

$H$ is the Hurst value, which is related to the fractal dimension $D_f$ of the failure surface. The spatial characteristics of the failure surfaces were effectively captured by applying these statistical methods.

The box-counting method is widely used due to its simplicity in mathematical calculation and empirical estimation (Falconer, 1990). To determine the box-counting dimension $D_B$ of a bounded subset $\Gamma$ in ${ℝ}^n$, a mesh of squares or boxes with side length $\epsilon$ is overlaid on the set, and the number $N\left(\epsilon \right)$ of these elements intersecting $\Gamma$ is counted (Falconer, 1990). By varying the scale $\epsilon$, different values of $N\left(\epsilon \right)$ are obtained. If the surface exhibits fractal behavior, the relationship between $N\left(\epsilon \right)$ and $\epsilon$ is given by:

$N\left(\epsilon \right)~{\epsilon }^{-D_B}$ (4-1)

where $D_B$ denotes the fractal dimension of the failure surface. A logarithmic transformation of this relationship yields a linear plot, where the slope corresponds to the $-D_B$. The box-counting dimension is expressed as follows:

$D_B=-\underset{\epsilon \to 0}{\lim }\frac{\ln N\left(\epsilon \right)}{\ln \epsilon }$ (4-2)

The value $N\left(\epsilon \right)$, representing the number of mesh elements intersecting the set at a given scale $\epsilon$, serves as an indicator of the spatial irregularity or complexity of the failure surface [46] .

In this approach, the log-log relationship between the number of occupied boxes $N\left(\epsilon \right)$ and the inverse of the box size $1/\epsilon$ is plotted. The slope of the linear fit was used to determine the fractal dimension ( $D_B$) as $D_B=-\text{slope}$, resulting in a fractal dimension of approximately 2.0735, as indicated in Figure 7 . This value highlights the complexity and irregularity of the failure surface, suggesting a significant roughness across multiple scales. The strong linearity observed in the log-log plot confirms the fractal nature of the surface, and the calculated dimensions provide valuable insight into its geometric characteristics.

The semi-variogram method by Oliver and Webster (1986) offers an alternative approach to estimating fractal dimensions [47] . The semi-variogram ${\gamma }_t$ is computed as in Equation (3-12), and for fractal profiles, fractional Brownian motion (FBM) corresponds to a specific form of the semi-variogram given by Equation (3-15) following the work of Mandelbrot et al. [47] [48] . To minimize variability, the analysis is typically restricted to a fraction (e.g., 10% or 25%) of the data length and employs logarithmic lag spacing to improve the robustness of least-squares regression [47] . This method is particularly effective in identifying the fractal properties of surface profiles, allowing for accurate characterization of the roughness associated with failure surfaces, as shown in Figure 8 .

The slope of this linear fit was used to calculate the Hurst exponent ( $H$), which is related to the fractal dimension ( $D$) using the equation $D=2-H$, yielding a fractal dimension of approximately 1.5634, as shown in Figure 8 . This value indicates that the failure surface was self-affine and exhibited moderate roughness and heterogeneity. The deviations from linearity observed at larger lag distances are likely to be due to boundary effects or data limitations.

This study used high-resolution profiles to investigate the surface roughness and fractal properties of the failure surface of the Mashamba West mine. A comprehensive approach was developed to analyze the different aspects of the failure surface. Thus, it provides valuable information regarding the characteristics of the failure surface and its significance in slope stability analysis. Surface roughness analysis ( Figure 2 ) shows that the surface has moderate variability and localized heterogeneity at surface elevations. The mean absolute deviation (MAD) of 0.83, and the mean standard deviation (MSD) of 1.04, indicating significant microroughness on the failure surface. These metrics can be used to quantify the degree of surface irregularity and compare the roughness of different mining regions. Analysis of the topographical extremes ( Figure 3 ) revealed a mean peak height (MP) of 2.4828 and mean valley depth (MV) of −4.7747, which, in general, describes the shape of the topography with respect to the contact area variability. These extreme features are where stress concentration and weakness in the slope structure are likely to occur, and therefore, may significantly affect the overall stability. Geometric irregularities were assessed based on the profile slope and curvature variability ( Figure 4 ). A standard deviation of the profile slope (PS) of 1.04 and curvature variability (PC) of 2.963 indicated substantial surface geometry irregularities relevant to deformation behaviors and sliding mechanisms. These parameters contain information about the complexity of the failure surface topography and slope movement dynamics rate. From the power spectral density (PSD) analysis ( Figure 5 and Figure 6 ), a dominant frequency of 0.014 was found, corresponding to periodic geological structures. This reveals recurring patterns on the failure surface, which might be related to the geological formations or structural features underlying them. The log-log PSD plot had a linear slope of 0.076, consistent with the established fractal models, indicating that the object or pattern is self-similar at different scales. Fractal dimension calculations were performed using two methods to ensure robust results. The semi-variogram method yielded a fractal dimension of 1.5634 ( Figure 7 ).

By contrast, the box-counting method yielded a value of 2.0735 ( Figure 8 ). Both results confirm that the failure surfaces are self-affine, implying that the surface roughness has similar patterns at different scales. This fractal behavior has important implications for the scale-dependent properties of the slope and its failure mechanisms. The quantified parameters can be used to develop improved numerical slope stability models, enhance risk assessment procedures, and help develop appropriate mitigation measures. Therefore, the research findings can be directly applied to real-world engineering problems. The fractal nature of the failure surface suggests that similar patterns may also appear at different scales, which is important for both local- and extensive-scale slope stability studies. This investigation provides a detailed characterization of the fractal properties of the failure surface of the Mashamba West mine.

This study conducted a comprehensive analysis of the failure surface of the Mashamba West mine, unveiling the fractal characteristics of the surface and quantifying its surface roughness. The surface roughness analysis revealed high micro-roughness, as indicated by the mean absolute deviation (MAD) of 0.83 and a mean standard deviation (MSD) of 1.04. These values signify the variability of surface elevations, a measure of the surface irregularity of geological surfaces formed by weathering, faulting, and jointing processes. From these metrics, it can be deduced that the failure surface exhibits moderate variability in surface roughness and possesses complex and heterogeneous features that are of significant importance in the study of topographical intricacies of natural failure surfaces. Understanding extreme topographical features, such as a mean peak height (MP) of 2.4828 and a mean valley depth (MV) of −4.7747, is crucial. These features not only confirm the presence of high and low points on the failure surface but also underscore the importance of considering the entire spectrum of surface characteristics in any geotechnical analysis. Surface geometry, which can vary with the contact area and mechanical interactions within the rock mass, is best understood when extreme features are identified. Fractal analysis using the semi-variogram and box-counting methods revealed the self-affine nature of the failure surface. The fractal dimensions of 1.5634 and 2.0735 prove that the surface has scale-invariant properties of surface roughness; the same roughness patterns are found at different scales. This finding is consistent with previous research on geological surfaces, which found that fractal geometry can replicate the irregularity and complexity of rock masses and failure surfaces. The fractal nature of the surface indicates that small-scale irregularities such as asperities may have a structure similar to larger-scale topographical features. Thus, fractal analysis can help characterize the entire range of surface geometries. Power spectral density (PSD) analysis also supported fractal characterization and revealed a dominant frequency of 0.014, corresponding to periodic geological structures on the surface. Identifying such periodic features implies that the failure surface is formed by underlying geological formations that impose periodic patterns on the surface. These findings are similar to those of Wu et al. [49] [50] , who reported periodic features in other geologically complex environments. This insight is important because it shows how geological structures may affect surface roughness and its evolution. The integration of fractal analysis into surface characterization represents a significant advancement over the traditional methods. Although measures such as MAD and MSD provide limited information about the surface, fractal analysis offers a more comprehensive understanding of natural surfaces. By simulating long-range correlations and scale-dependent properties, fractal analysis revealed the complex nature of surface roughness. This is particularly important in geotechnical engineering, where many surfaces exhibit fractal-like characteristics because of the complex geological processes that produce them.

In general, high-precision quantification and characterization of surface roughness are crucial for a better understanding of natural rock masses and their behavior under different loading conditions. Although this study did not directly examine the relationship between surface roughness and slope stability, detailed surface characterization suggests the influence of roughness features on mechanical properties such as shear strength, friction, and deformation. Furthermore, based on the fractal nature of the failure surface, it may be possible to develop methods for predicting how surface irregularities at different scales affect the behavior of rock masses during mining activities, inspiring further research and practical applications. This research highlights the potential of combining fractal analysis with other geotechnical methods to develop more realistic models of rock-mass behavior. For instance, when used in conjunction with numerical modeling techniques, fractal analysis can help identify how surface roughness affects the mechanical response of rock masses under different environmental and loading conditions. Moreover, using fractal dimensions in modeling can enhance the reliability of stability assessments and lead to improved predictions of potential failure areas in open-pit mining. Other research could also focus on how surface roughness changes over time, especially in response to weathering, changes in stress, or excavation. This research can provide valuable information on the long-term behavior of open-pit slopes and the formation of failure surfaces. Furthermore, by integrating fractal analysis with real-time monitoring technologies, such as LiDAR or photogrammetry, it is possible to develop dynamic assessments of surface changes that can improve the predictive capability of slope stability models, opening up exciting avenues for future research. This study successfully demonstrated the use of advanced surface-roughness characterization techniques, particularly fractal analysis, to understand the complexity of failure surfaces. The present study offers a suitable foundation for future research and applications in geotechnical engineering, with the possibility of improving the accuracy and reliability of models describing the behavior of failure surfaces in open-pit mining.