Integrating Soil Parameter Uncertainty into Slope Stability Analysis: A Case Study of an Open Pit Mine in Hungary

Abstract

This study presents a probabilistic geotechnical analysis of the Visonta Keleti-III lignite mining area, focusing on the statistical evaluation of soil parameters and their integration into slope stability modeling. The objective was to provide a more accurate representation of the spatial variability of geological formations and mechanical soil properties in contrast to traditional deterministic approaches. The analysis was based on over 3300 laboratory samples from 28 boreholes, processed through multi-stage outlier filtering and regression techniques. Strong correlations were identified between physical soil parameters—such as wet and dry bulk density, void ratio, and plasticity index—particularly in cohesive soils. The probabilistic slope stability analysis applied the Bishop simplified method in combination with Latin Hypercube simulation. Results demonstrate that traditional methods tend to underestimate slope failure risk, whereas the probabilistic approach reveals failure probabilities ranging from 0% to 46.7% across different sections. The use of tailored statistical tools—such as Python-based filtering algorithms and distribution fitting via MATLAB—enabled more realistic modeling of geotechnical behavior. The findings emphasize the necessity of statistical methodologies in mine design, particularly in geologically heterogeneous, multilayered environments, where spatial uncertainty plays a critical role in slope stability assessments.

1. Introduction

The statistical evaluation of soil parameters is of fundamental importance in slope stability analysis, as the physical and mechanical properties of soil exhibit significant spatial variability. Traditional slope stability analysis methods, such as limit equilibrium approaches, cannot always take into account these uncertainties. Modern approaches, such as Monte Carlo simulation, provide an opportunity for the quantitative estimation and integration of these uncertainties into the design process [1,2]. Uncertainties in shear strength parameters such as soil cohesion and internal friction angle influence safety factors. Monte Carlo simulation enables the implementation of probabilistic analyses, allowing the modeling of the effects of various soil parameters [3,4].

The natural anisotropy of soils, which develops during particle deposition, fundamentally influences slope stability. Traditional isotropic models may significantly overestimate safety factors, whereas considering anisotropy allows for more accurate results [4,5]. Numerical methods, such as the Finite Elements Method (FEM) and the Discrete Elements Method (DEM), can account for the spatial variability of the mechanical properties of soil and rock. Probabilistic analysis techniques, such as the Strength Reduction Techniques (SRT) method, enable a detailed investigation of slope instability processes [6,7]. Statistically based evaluation techniques assist in determining optimal design parameters, thereby reducing the risk of underestimated instability. These methods can be applied to the design of earth dams, mining slopes, as well as road and railway embankments [3,7]. Integrating these approaches into the engineering design process facilitates the development of safer and more cost-effective solutions. It is necessary to understand the impact of soil parameter variability and explore potential risk reduction strategies. Probabilistic modeling is a method that accounts for the uncertainty associated with input parameters during analysis, yielding the statistical distribution of the Factor of Safety (FS) instead of a single deterministic value. The resulting Probability of Failure (PF) has become a widely accepted design criterion [8,9,10]. The basis of the Probability of Failure (PF) concept is the determination of a statistical distribution (probability density function) for each input parameter in a slope stability analysis. The statistical distribution of the Factor of Safety (FS) can be established using a stochastic simulation process. PF quantifies the percentage of cases where FS < 1, indicating potential failure. Soil mechanical analyses often encounter the inherent uncertainty of soil parameters, which significantly impacts slope stability. To account for uncertain soil parameters, researchers have applied a probabilistic approach that has proven to be an effective tool when combined with Monte Carlo or Latin Hypercube simulation and Bishop’s simplified method. The analysis examined critical slip surfaces and the Factor of Safety, highlighting that probabilistic modeling of soil strength improvements and groundwater levels can significantly reduce the risk of slope failure [11]. Natural soil deposition processes often lead to strength anisotropy, which, if neglected, can result in inaccurate slope stability assessments. In the study of He et al. (2022) [12], a generalized anisotropic model was applied to describe the directional variations in the soil’s friction angle. The results indicate that ignoring anisotropy can overestimate the stability factor by up to 32.9%, particularly in the case of gentler slopes. The study emphasizes the importance of explicitly considering soil strength anisotropy to achieve more accurate stability assessments [12]. The stability of mining slopes is of paramount importance due to safety, economic, and environmental considerations. A Synthetic Rock Model (SRM) study aimed to simulate various geological layers and discontinuities accurately. Discrete Elements Methods (DEM) and Strength Reduction Techniques (SRT) enabled a detailed examination of sliding mechanisms. The results confirmed that these numerical models effectively predict the stability of complex geological environments, particularly in identifying slip surfaces and instability processes [13]. This study presents a comprehensive statistical evaluation of soil parameters and a probabilistic slope stability analysis of the Visonta lignite mining area in Hungary, highlighting how advanced data filtering, distribution fitting, and Latin Hypercube simulations can effectively capture the uncertainty of geotechnical parameters and provide a more realistic assessment of slope safety, ultimately revealing an estimated failure probability for one of the critical mining sections. In recent years, several studies have addressed surface lignite mines using probabilistic slope stability analysis to account for geotechnical uncertainties. Vanneschi et al. [14] applied both deterministic and probabilistic LEM and FEM approaches in a Czech lignite mine, modeling soil parameter uncertainty with normal distributions and assessing variability through sensitivity and probabilistic analysis. Masoudian et al. [15] used FEM to analyze spoil heap stability, incorporating material property variability and documenting significant dispersion, highlighting challenges in characterizing heterogeneous materials. Pathan et al. [16] applied probabilistic analysis with Monte Carlo simulation at the Thar coalfield in Pakistan to identify critical slopes and estimate PF values under varying material properties, using stochastic modeling and diverse distribution functions.

2. Geological Settings and Geotechnical Parameters

2.1. Geographical Location

Visonta is located in North Hungary, at the southern foothills of the Northern Medium Mountains, east of Gyöngyös (Figure 1). The Visonta mining site, based on lignite extraction, is located on the northern edge of the Great Hungarian Plain, at the southern foothills of the Mátra Mountains (Figure 2). Since the 1970s, multiple open-pit mining operations have been carried out in the region, utilizing the Visonta lignite deposit. The mining area is a key part of the Northern Hungarian lignite region. Geological research conducted in the 1960s and 1970s indicated that the Mátra and Bükkalja regions contain hundreds of millions of tons of extractable mineral resources suitable for long-term energy utilization. Geomorphologically, the area is highly diverse and can be classified as a low-hill region. Its topography is characterized by decreasing slopes towards the south and rising ridges towards the northwest. The highest part of the mining site, located in the northeastern section, varies between 160 and 180 m above sea level, while its lowest point lies at an elevation of 115 m above sea level in the valley of Tarnóca Creek.

2.2. Interburden and Overburden Layer Characteristics

The overburden above the uppermost coal seam and the interbeds between the individual coal seams consist of loose sediments. The Late Pannonian formation, containing the lignite seams, is overlain by a Quaternary overburden of varying thickness. This overburden consists primarily of various materials, including dark brown, red, reddish brown, yellowish brown, gray, yellow, and black bentonite deposits, often with limestone concentrations and occasionally with limonite and manganese content. In some areas, a sandy, clayey gravel layer can be observed. Some clayey materials have a mosaic structure, which breaks into small (walnut-sized) shiny-surfaced fragments after the loss of the substrate. The thickness of the Quaternary overburden ranges from 5 to 50 m, showing an increasing trend from north to south. The Cserhát–Mátra–Bükkalja lignite-bearing sequence is overlain by inland sand, silt, and clay layers, occasionally interbedded with lignite seams. The coal layers within the sequence are separated by clay and sand layers. In some places, sandstone layers and compact marly silts represent significant resistance to mining.

2.3. Geological Structure of Lignite Layers

Figure 3 is an actual footage of the Visonta mine site. The photograph illustrates the heterogeneity of the soil strata at the Visonta mine site, highlighting the complex and variable lithological conditions characteristic of the area.

The lignite deposits are found in Pliocene formations, which are overlain by Quaternary cover layers (Figure 4). The structural setting of the area is basin-like, with a basement composed of Miocene andesite formations of the Mátra Mountains. Overlying this basement is a several hundred-meter-thick Pliocene sedimentary sequence, with its Late Pannonian section containing the lignite seams. The sedimentary sequence is characterized by relatively stable depositional conditions, with a general dip direction of NW-SE at an average inclination of 2–3°. Due to variations in stratigraphic dip, some lignite seams extend to the denuded surface of the Late Pannonian formations, where they have been partially eroded. The individual coal seams show significant differences in thickness, quality, and structural characteristics. These lignite seams belong to the Bükkalja Lignite Formation, deposited during the Late Pannonian epoch of the Tertiary period. Sediments accumulate in a shallow basin, fed by weathering products from surrounding mountains. Over time, delta slope facies evolved into floodplain and fluvial-lacustrine deposits, filling the basin by the epoch’s end. Swamps and peatlands formed along basin margins, giving rise to lignite seams from Taxodium swamp forests [18]. Figure 5 displays two geological cross-sections from the Visonta Keleti-III lignite mining area, showing stratigraphic layering based on borehole data. Multiple sedimentary units are represented, including various clays (e.g., fat clay or high plasticity clay, medium plasticity clay, organic and bentonitic clays), silts, silty clays, and lignite seams. Strata are shown with distinct colors. The lignite seams are interbedded with fine-grained sediments and occur at varying depths across sections. Upper layers are predominantly composed of fat clays and silts, while deeper units include high-plasticity and organic clays. The aquifer units consist of granular soils, predominantly sandy layers.

2.4. Geotechnical Investigations

Several investigatory drillings have been made in the area. The geological logs were obtained, containing the identified stratigraphic layers from individual boreholes, along with detailed sample descriptions, age determinations, depth data, sample types, and corresponding sample numbers. These reports contained information such as borehole name, layer name, soil sample color, material characteristics, other relevant properties, layer depth, sample age, and sample number. For disturbed samples, the following parameters were determined: liquid limit (w_L), plastic limit (w_P), relative consistency index (I_c), plasticity index (I_p), and grain size frequency distribution characteristics—percentage distribution of gravel, sand, silt, and clay, as well as characteristic grain diameter (D_m), uniformity coefficient (Cu), and grain size parameters (D₁₀, D₂₀, D₆₀). For undisturbed samples, the available data included water content (w), void ratio (e), degree of saturation (S_r), dry bulk density (r_d), and wet bulk density (r_n). Certain samples underwent shear testing, and results were provided as average values measured on a horizontal shear plane. From these tests, the cohesion values, as well as internal friction angles, were determined. A total of 28 boreholes, drilled between 1988 and 2008 in the Visonta Keleti-III mining area, were examined during the current research. The boreholes are documented with known drilling dates, EOV (X) and EOV (Y) coordinates, height above sea level, and drilling depths.

Table 1. Number of laboratory tests on samples from the Visonta Mining Area.

Borehole		28
Drilling Depth (m)		3305
Grain Size Distribution		1205
Direct Shear Test	Internal Friction Angle f (deg)	237
	Cohesion c (kPa)	238
	Residual Internal Friction frez (deg)	264
	Residual Cohesion crez (kPa)	265
Triaxial Compression Test	Water Content W (%)	196
	Saturated Bulk Density rs (g/cm3)	155
	Void Ratio e	184
	Degree of Saturation Sr	188
	Wet Bulk Density rn (g/cm3)	205
	Dry Bulk Density rd (g/cm3)	197
	Internal Friction Angle f (deg)	166
	Cohesion c (kPa)	166
Natural Water Content–Wn (%)		1265
Median Grain Size–Dm (mm)		1206
Coefficient of Uniformity–Cu		816
Grain Size Corresponding to 10% Finer–D10 (mm)		1206
Grain Size Corresponding to 20% Finer–D20 (mm)		1008
Grain Size Corresponding to 60% Finer (mm)		1171
Liquid Limit–WL (%)		653
Plastic Limit–Wp (%)		654
Consistency Index–Ic		360
Plasticity Index–Ip (%)		1046
Water Content–W (%)		1225
Void Ratio–e (-)		1258
Porosity–n (-)		121
Degree of Saturation–Sr (-)		1221
Dry Unit Weight–rd (g/cm3)		1286
Wet Unit Weight–rn (g/cm3)		1291
Undrained Shear Strength (kPa)		471
Number of Samples		3307

shows the number of laboratory tests completed on soil samples from the Visonta Mining Area.

3. Methodology

After examining the data of the 28 boreholes and more than 3300 samples from the Visonta Keleti-III mining area, soil classification was performed based on the plasticity index and grain size distribution results. Each borehole is marked using a colored symbol on the map in Figure 1a. Their spatial distribution is predominantly concentrated near the cross-sections defining the mine boundary slopes. Although the dataset of 28 boreholes is considerable, it remains insufficient for fully resolving the stratigraphic complexity of the area. Local anomalies or lens-like variations may still go undetected due to the limited spatial resolution. Cohesive soils were classified into five categories based on plasticity index (I_p) values. Sandy silt (I_p = 5–10%) represents low plasticity and weak cohesion. Silt (I_p = 10–15%) has moderate plasticity with slightly higher clay content. Low plasticity clay (I_p = 15–20%) indicates increasing plasticity and cohesiveness. Medium plasticity clay (I_p = 20–30%) shows significant plasticity and is more sensitive to moisture changes. Finally, high plasticity clay (I_p > 30%) exhibits strong swelling and shrinkage behavior. After the classification, statistical analyses were performed focusing on two main categories: shear strength parameters, unit weight, and physical soil parameters. Ultimately, the collected data were used for probabilistic slope stability analysis. This study focuses on examining the stability of 7 sections in the area of the Keleti III lignite mine in Visonta. The 7 sections show boundary slopes, as shown in Figure 1c. The original geometry of the sections is given. In cases where the results indicated an unacceptable likelihood of failure, geometric adjustments—such as the reduction in slope inclination—were considered. A probability-based slope stability test was performed on the completed sections in order to examine the stability of the sections using probabilistic methods. The primary objective of the investigation was to demonstrate the applicability of probabilistic methods in capturing the inherent uncertainties of geotechnical parameters and their impact on slope stability. Slope stability was performed deterministically and probabilistically, and when necessary, the geometry was optimized to reach the EC7 value. During the mining activity, dewatering wells pump out groundwater, and this is the phase when the steepest slope angles occur. After mining, the waste material is returned to the pit to reduce the slope inclination, and the groundwater extraction is stopped. The work chart for the investigation is in Figure 6.

3.1. Methodology of Correlation Analysis

A total of thirteen soil types were analyzed: clayey Sand (clSa), clayey Silt (clSi), Sand (Sa), sandy Clay (saCl), sandy clayey Silt (saclSi), sandy Silt (saSi), sandy silty Clay (saclSi), sandy Gravel (saGr), Silt (Si), silty Sand (siSa), high plasticity Clay, medium plasticity Clay, and low plasticity Clay. The abbreviations used here are derived from the European Soil Classification System (ESCS).

Relationships were examined between the following parameters: wet bulk density and void ratio, dry bulk density and void ratio, wet bulk density and dry bulk density, and plasticity index and liquid limit. The study was conducted in three phases. The first phase served as a baseline, where all available laboratory results for each parameter related to each soil type were included.

Subsequently, outlier values were filtered for each soil type and for each studied soil parameter using manual, basic filtering based on the Interquartile Range (IQR) method. The IQR filtering method is used to remove data points that significantly differ from the rest, known as outliers. The central part of the data is identified, and the range between the lower and upper quartiles is measured. Based on this range, boundaries are determined, and any values falling outside these limits are excluded, as they are considered unusual or potentially incorrect. This procedure can be performed in Python, and from Microsoft Excel 2016 onwards, the quartile function is also supported for this purpose.

The same procedure was later carried out using a more professional and efficient approach—dynamical filtering—implemented through Python code. The Python program performs iterative outlier detection and visualization on soil-related experimental data. It applies linear regression within each soil type group and iteratively removes statistical outliers based on interquartile range (IQR) analysis of residuals. Each iteration’s regression line is recorded: all intermediate fits are shown as dashed gray lines, while the final fit is highlighted in bold. The tool facilitates reproducible and transparent data cleaning for soil parameter analyses. Dashed gray lines represent earlier regression fits, while the final regression model is shown as a solid blue line (Figure 7). This method allows for very accurate data filtering but carries the risk of excessive data loss. The simple filter called IQR filtering with MS Excel was used during the research since too much data loss can have negative consequences because it reduces the representativeness of the data.

The essential part of the Python script code has been highlighted below.

• def iterative_outlier_filtering(group):
•    group = group.copy()
•    all_outliers = []
•    regression_lines = []
•    num_iterations = 0
•    while True:
•       if len(group) < 2:
•          break
•       slope, intercept, _, _, _ = linregress(group[‘Value1’], group[‘Value2’])
•       regression_lines.append((slope, intercept))
•       num_iterations += 1
•       group[‘Predicted_Value2’] = group[‘Value1’] * slope + intercept
•       group[‘Residual’] = group[‘ Value2’] - group[‘Predicted_ Value2’]
•       q1 = group[‘Residual’].quantile(0.25)
•       q3 = group[‘Residual’].quantile(0.75)
•       iqr = q3 - q1
•       lower_bound = q1 - 1.5 * iqr
•       upper_bound = q3 + 1.5 * iqr
•       group[‘Outlier’] = ~group[‘Residual’].between(lower_bound, upper_bound)
•       outliers = group[group[‘Outlier’]]
•       if outliers.empty:
•          break
•       all_outliers.append(outliers)
•       group = group[~group[‘Outlier’]]
•    group[‘Outlier’] = False
•     if all_outliers:
•       all_outliers_df = pd.concat(all_outliers)
•       all_outliers_df[‘Outlier’] = True
•       final_group = pd.concat([group, all_outliers_df])
•    else:
•       final_group = group
•    final_group = final_group.sort_index()
•    final_slope, final_intercept = regression_lines[−1] if regression_lines else (0, 0)
•    return final_group, final_slope, final_intercept, regression_lines, num_iterations

3.2. Methodology of the Statistical Analysis

In probabilistic slope stability analysis, accurately defining the statistical properties of input parameters is essential. This includes determining the appropriate probability distribution, mean, standard deviation, and range. These parameters ensure that variability is realistically captured in the model. The analysis in this study uses Rocscience software, which limits inputs to seven distribution types: Normal, Lognormal, Triangular, Gamma, Beta, Uniform, and Exponential. Hence, selecting the most representative distribution for each parameter is critical. A custom MATLAB script was developed specifically for this study. It focuses on the seven distributions supported by Rocscience. By estimating parameters, the script fits multiple distributions: normal, lognormal, exponential, gamma, beta, uniform, and triangular. Each model is evaluated using log-likelihood and the Akaike Information Criterion (AIC), which balances model fit with complexity. The best-fitting model is identified based on the lowest AIC value. Visualization includes a histogram with overlaid probability density functions (PDFs) for each distribution, each plotted in a distinct color.

The best-fitting model is emphasized with a thicker line and annotated with the distribution name, AIC, mean, and standard deviation. For improved usability and validation, a parallel Python implementation was created. It mirrors the MATLAB version but includes more robust error handling and input validation.

Figure 8a shows a MATLAB result, while Figure 8b shows the same result calculated using Python code. The two plots show the results of a distribution fitting of a dataset, comparing several probability distributions based on the quality of the fit. In both plots, several distributions (normal, lognormal, gamma, uniform, triangular, and exponential) are fitted to the histogram of the data. The gamma distribution is the best fit, with the lowest AIC value. The gamma distribution closely follows the shape of the histogram, indicating a good fit. In both plots, AIC (Akaike Information Criterion) values were used to determine the fit: lower values indicate a better fit.

The MATLAB and Python code snippets both aim to identify the best-fitting statistical distribution based on the Akaike Information Criterion (AIC), yet exhibit notable differences. The MATLAB script employs a straightforward loop structure, normalizes data specifically for the Beta distribution without parameter constraints, computes log-likelihood directly from fitted distributions, and stores only the best-fitting model parameters and their AIC. In contrast, the Python implementation utilizes explicit normalization parameters (floc, fscale) for Beta distributions, separately manages the triangular distribution as a special case, computes additional statistics (mean, standard deviation) for each fit, systematically captures fitting errors via structured warnings, and returns comprehensive results for all distributions tested, including detailed statistical summaries. Overall, the Python approach provides greater analytical depth, explicit error handling, and enhanced result organization compared to the concise, more simplistic MATLAB methodology.

The essential parts of the codes have been highlighted below.

• MATLAB code for selecting best-fit distribution:

• % Fit each distribution and compute AIC
• for i = 1:size(distributions, 1)
•    distName = distributions{i, 1};
•    distFunc = distributions{i, 2};
•    try
•       if strcmp(distName, ‘Beta’)
•          normalized_data = (data - min(data))/(max(data) - min(data));
•          pd = distFunc(normalized_data);
•       else
•          pd = distFunc(data);
•       end
•       logL = sum(log(pdf(pd, data)));
•       k = length(pd.ParameterValues);
•       AIC = 2 * k - 2 * logL;
•       AIC_values(i) = AIC;
•       if AIC < bestAIC
•          bestAIC = AIC;
•          bestDist = distName;
•          bestParam = pd.ParameterValues;
•          bestPD = pd;
•       end
•    catch ME
•       fprintf(‘Error fitting %s distribution: %s\n’, distName, ME.message);
•    end
• end

• Python code or selecting best-fit distribution:

• def fit_and_compare_distributions(data):
•    results = []
•    for name, dist in distributions.items():
•       try:
•          if name == “beta”:
•             data_min, data_max = np.min(data), np.max(data)
•             normalized_data = (data - data_min)/(data_max - data_min)
•             params = dist.fit(normalized_data, floc=0, fscale=1)
•             ll = np.sum(dist.logpdf(normalized_data, *params))
•             mean, var = dist.stats(*params, moments=“mv”)
•             mean = mean * (data_max - data_min) + data_min
•             std = np.sqrt(var) * (data_max - data_min)
•          elif name == “triangular”:
•             a, c = np.min(data), np.max(data)
•             b = np.mean(data)
•             loc = a
•             scale = c - a
•             params = ((b - a)/scale, loc, scale)
•             ll = np.sum(dist.logpdf(data, *params))
•             mean, var = dist.stats(*params, moments=“mv”)
•             std = np.sqrt(var)
•          else:
•             params = dist.fit(data)
•             ll = np.sum(dist.logpdf(data, *params))
•             mean, var = dist.stats(*params, moments=“mv”)
•             std = np.sqrt(var)
•          k = len(params)
•          aic = 2 * k - 2 * ll
•          results.append({
•             “distribution”: name,
•             “params”: params,
•             “log_likelihood”: ll,
•             “aic”: aic,
•             “mean”: mean,
•             “std”: std,
•          })
•       except Exception as e:
•          warnings.warn(f”Error fitting {name} distribution: {e}”)
•    if results:
•       return results, min(results, key=lambda x: x[“aic”])
•    else:
•       warnings.warn(“No distributions successfully fitted to the data.”)
•       return [], None

4. Results

4.1. Correlation Analysis of Soil Physical Parameters

The three plots of Figure 9 show the coefficient of determination (R²) values obtained for three different regression models for different soil types. Each plot shows the R² values for three data treatment approaches: basic, dynamical filtering, and simple filtering. Higher R² values indicate a stronger model fit. Filtering methods, especially dynamical filtering, consistently produce higher R² values for most soil types, suggesting that model performance improves when outliers are removed. High-plasticity clays and medium-plasticity clays generally yield the highest R² values, especially in the dynamical filtered model, indicating that these soil types allow for more predictable relationships between the variables under study. In contrast, sandy silt (saSiS) and Silt (Si) show lower R² values in the basic model, especially for the wet-dry bulk density relationship, suggesting greater data variability or nonlinear behavior in these soils. Predictive relationships between variables are significantly improved when filtering techniques are used. The choice of soil type plays a critical role in the strength of the regression model, with cohesive soils (e.g., clays) providing more reliable results. For practical applications of geotechnical modeling, data filtering is recommended to improve the robustness of regression-based predictions, especially for granular soils with higher natural variability.

Different data filtering methods produce different results, primarily because they remove outliers from the dataset to different degrees. Basic filtering removes less data, resulting in more outliers entering the model and lower correlation (R²) values. Simple filtering removes outliers to a moderate extent, improving the model but retaining most of the data. On the other hand, dynamical filtering treats outliers iteratively and much more strictly, significantly increasing correlation values but causing a large amount of data loss. The result depends largely on the tested soil type and its natural variability. Cohesive soils, such as clays, have a lower natural variance, so more stringent filtering effectively improves the quality of the model while reducing the amount of data. In contrast, granular soils (e.g., sandy silts, sands) have greater natural variability, so overly strict filtering may result in the loss of a lot of data and lead to an overly simplified, idealized model. Results for each filtering method are shown in Figure 9.

Table 2. Data loss during data filtering on Visonta samples.

Soil Type	Wet Bulk Density—Void Ratio			Dry Bulk Density—Void Ratio			Wet Bulk Density—Dry Bulk Density
	No. Samples (Basic)	No. Samples (Simple Filtered)	No. Samples (Dynamical Filtered)	No. Samples (Basic)	No. Samples (Simple Filtered)	No. Samples (Dynamical Filtered)	No. Samples (Basic)	No. Samples (Simple Filtered)	No. Samples (Dynamical Filtered)
Sand (Sa)	60	58	56	59	57	28	59	57	56
sandy Silt (saSi)	206	197	164	206	197	159	206	196	195
Silt (Si)	67	64	63	67	65	56	67	65	64
silty Sand (siSa)	113	107	99	113	108	91	113	105	100
high plasticity Clay	429	413	396	428	414	374	428	421	396
medium plasticity Clay	189	186	175	188	185	179	188	186	176
low plasticity Clay	68	64	65	68	64	59	68	62	61

presents the number of samples used in three different regression analyses—wet bulk density vs. void ratio, dry bulk density vs. void ratio, and wet bulk density vs. dry bulk density—across various soil types and data filtering approaches (basic, simply filtered, and dynamically filtered).The simple filtering method was applied to avoid the excessive data exclusion that may result from a multi-phase filtering procedure. As shown in Table 2, approximately 4% of the dataset was excluded due to the presence of statistical outliers. This level of data reduction did not introduce significant distortion into the calculated Probability of Failure (PF) values. The excluded data primarily represented outliers that, if retained, could have skewed the statistical distributions and, consequently, the outcomes of the simulations. Therefore, the cleaned dataset facilitated a more robust and reliable probabilistic analysis. Rather than introducing bias, the filtering process enhanced the validity and reliability of the PF results. For all soil types and relationships, a reduction in sample size is observed as data filtering is applied, particularly in the dynamically filtered approach. This indicates that outliers or influential data points have been systematically removed to improve model quality. Minimal sample loss is observed in fine-grained soils (e.g., clays), suggesting that these datasets contain fewer extreme outliers, whereas coarser soils (e.g., silty sand) show a more noticeable drop in sample size post-filtering. Data filtering significantly affects the available sample size, but it is necessary to enhance model accuracy, as supported by the higher R² values observed in the filtered datasets.

Clays, especially those with high plasticity, are more stable in terms of data quality, making them preferable for regression-based modeling. Granular soils, due to their higher natural variability, may require more aggressive filtering and careful interpretation to ensure reliable predictions. Thus, the filtering process is justified despite the reduction in sample size, as it contributes to more statistically sound models. The dynamical filtering overestimates the true correlation. Where many values have been filtered out, the difference between the R² values in the diagram is more significant.

4.1.1. Wet Density—Void Ratio

Figure 10 illustrates the relationship between wet bulk density (horizontal axis) and void ratio (vertical axis) for various soil types, each represented by a distinct color. A negative correlation is clearly observed: as wet bulk density increases, the void ratio decreases. Denser soils typically contain less pore space. The distribution of data points suggests that the strength of this relationship varies among soil types. Clayey soils (e.g., high plasticity clay, clayey sand) are associated with higher void ratios at lower wet bulk densities, showing greater variability and a wider spread. The low wet bulk density values for the high plasticity clays are related to high organic content since the organic content reduces the density of the clays. Some clay layers contain different amounts of organic materials. Sands and silts (e.g., sand, sandy silt, silty sand) are clustered at higher densities and lower void ratios, indicating more compacted and less compressible structures. A few outliers can be identified, particularly among clayey sands, suggesting that some variability remains unexplained and may be due to heterogeneity in sample properties or measurement inconsistencies.

Table 3. Equations and correlation coefficient of wet bulk density and void ratio for various soil types.

Name of the Layer	Equation of the Line	Correlation Coefficient
Sand (Sa)	y = −0.8137x + 2.3326	R2 = 0.5826
Silt (Si)	y = −1.2425x + 3.1965	R2 = 0.6201
sandy clayey Silt (saclSi)	y = 0.2077x + 0.1810	R2 = 0.0994
silty Sand (siSa)	y = −1.1796x + 3.0429	R2 = 0.5293
sandy Silt (saSi)	y = −0.8417x + 2.4086	R2 = 0.3972
high plasticity Clay	y = −2.1004x + 4.9338	R2 = 0.8752
medium plasticity Clay	y = −1.5081x + 3.7403	R2 = 0.7106
low plasticity Clay	y = −1.5859x + 3.9098	R2 = 0.5811

presents the linear regression equations for each soil sample, along with the corresponding correlation coefficients.

4.1.2. Dry Density—Void Ratio

Figure 11 displays the relationship between dry bulk density (horizontal axis) and void ratio (vertical axis) for a variety of soil types, each indicated by a different color. A clear inverse relationship is observed: as dry bulk density increases, the void ratio decreases. This trend is consistent with fundamental soil mechanics, where denser soils contain less void space. The data points corresponding to high plasticity clays show a wider spread and are concentrated toward lower dry bulk densities and higher void ratios, indicating a more porous and compressible structure. In contrast, granular soils (e.g., sand, silty sand) are distributed toward the higher density–lower void ratio region, reflecting more compacted conditions with less variability. The tight clustering of points for medium plasticity clays and sands suggests more consistent behavior and potentially better predictability in modeling efforts. A few outliers are present, primarily among clay-rich soils, suggesting heterogeneity in sample characteristics or testing conditions. Low dry bulk density values indicate organic matter content. The higher the organic matter content, the lower the dry bulk density, as mentioned above.

Table 4. Equations and correlation coefficient of dry bulk density and void ratio for various soil types.

Name of the Layer	Equation of the Line	Correlation Coefficient
Sand (Sa)	y = −0.9326x + 2.1856	R2 = 0.9006
Silt (Si)	y = −0.1054x + 2.4012	R2 = 0.8891
sandy clayey Silt (saclSi)	y = 0.3155x + 0.068	R2 = 0.0067
silty Sand (siSa)	y = −1.0814x + 2.4185	R2 = 0.9179
sandy Silt (saSi)	y = −1.1642x + 2.5428	R2 = 0.9121
high plasticity Clay	y = −1.4969x + 3.1246	R2 = 0.9390
medium plasticity Clay	y = −1.1280x + 2.5379	R2 = 0.9261
low plasticity Clay	y = −1.2018x + 2.6427	R2 = 0.92111

contains the equations of lines of each soil sample, as along with the correlation coefficients.

4.1.3. Wet Bulk Density—Dry Bulk Density

Figure 12 displays the relationship between dry bulk density (x-axis) and wet bulk density (y-axis) for various soil types, each marked with a different color. A strong positive linear relationship is observed: as dry bulk density increases, wet bulk density also increases. This aligns with expectations, as wet bulk density includes both the solid particles and the water content, while dry bulk density considers only the solids. Most soil types follow this linear trend closely, particularly sands, silts, and clays with medium or low plasticity, indicating a consistent relationship across a wide range of soils. High plasticity clays are more widely spread and appear at lower dry bulk densities with correspondingly lower wet bulk densities. This suggests greater variability in water content and structure, which is typical of such fine-grained, compressible soils. The reason for that is the high organic content of some of the high plasticity clay samples, as mentioned before. The tight clustering of sands and silts near the trend line indicates more predictable and uniform behavior. The relationship between dry and wet bulk densities can be reliably modeled with linear regression, particularly for non-cohesive and low-plasticity soils.

Table 5. Equations and correlation coefficient of dry and wet bulk density for various soil types.

Name of the Layer	Equation of the Line	Correlation Coefficient
Sand (Sa)	y = 0.7808x + 0.7282	R2 = 0.7191
Silt (Si)	y = 0.6276x + 0.9829	R2 = 0.8097
sandy clayey Silt (saclSi)	y = 2.7087x + 2.5308	R2 = 0.3727
silty Sand (siSa)	y = 0.5431x + 1.0908	R2 = 0.6044
sandy Silt (saSi)	y = 0.6626x + 0.9133	R2 = 0.5391
high plasticity Clay	y = 0.6780x + 0.9117	R2 = 0.9659
medium plasticity Clay	y = 0.6064x + 1.0197	R2 = 0.8640
low plasticity Clay	y = 0.5348x + 1.1413	R2 = 0.9105

presents the regression equations and corresponding correlation coefficients for each soil sample.

4.1.4. Plasticity Index—Liquid Limit

Figure 13 shows the relationship between the liquid limit and the plasticity index for different soil samples. A reference line—Line A, defined by the equation I_p = 0.73(w_L-20) —is also included to assist in classification, following the widely used Casagrande plasticity chart method. A positive correlation is clearly observed: as the liquid limit increases, the plasticity index also increases, indicating that soils with higher water content at the liquid limit tend to have greater plasticity ranges. The data points are color-coded by soil type, with dark red points representing high-plasticity clays, while cyclamen, gray, and green points representing lower-plasticity silts and clays. Most of the points lie around the A-line, except for the high plasticity clays, mainly with very high liquid limit. Based on the analysis of the Casagrande diagram, it can be determined that some of the high plasticity clays are located below the A-line, which typically indicates an increased organic matter content of the samples.

4.2. Statistical Analysis of Shear Strength Parameters

Figure 14 and Figure 15 show the results of the statistical analysis of shear strength parameters broken down into separate soil types, displayed in box plots or whisker plots, which are commonly used in statistics to visualize the distribution, central tendency, and variability of a dataset. Each box plot summarizes the key characteristics of the data: the central box represents the interquartile range (IQR), which contains the middle 50% of the data, with the lower and upper edges of the box marking the first (Q1) and third quartiles (Q3), respectively. The horizontal line within the box indicates the median (Q2), while the whiskers extend to the smallest and largest values within 1.5 times the IQR from the quartiles. Data points lying outside this range are plotted individually and considered outliers.

The small circles denote these outliers, which fall significantly outside the typical range of values. The “x” markers represent the arithmetic mean of the dataset, providing an indication of the average value for each soil type. These visualizations facilitate a clear comparison of the internal friction angle (ϕ) in Figure 14 and cohesion (c) in Figure 15 across different soil classifications, such as sands, silts, and clays, with varying plasticity.

The boxplot in Figure 14 presents the distribution of friction angle (φ) values for the investigated soil types, providing insight into their shear strength characteristics. Sands (Sa), sandy silts (saSi), and silty sands (siSa) show relatively higher median φ values, generally ranging between 25° and 30°, with some values extending above 35°.

This is expected, as granular soils typically exhibit higher friction angles. Silts (Si) and clays (high, medium, and low plasticity) demonstrate lower φ values, with medians between 10° and 25°, indicating more compressible, less frictional behavior. High plasticity clays have the lowest median and minimum φ values. Variation is higher among fine-grained soils, particularly clays, as seen from the larger interquartile ranges and whiskers, implying greater heterogeneity. These findings highlight the importance of soil classification and testing for proper geotechnical evaluation and design safety.

The boxplot in Figure 15 illustrates the cohesion values (in kPa) for the investigated soil types. It can be observed that sands (Sa), sandy silts (saSi), silts (Si), and silty sands (siSa) exhibit relatively low cohesion values. In contrast, clays, particularly those with high and medium plasticity, demonstrate significantly higher cohesion, with values extending beyond 200 kPa in some cases. Low plasticity clay also shows moderately high cohesion, though with a wider variability. These results indicate that cohesion is strongly influenced by the soil’s plasticity and composition.

4.3. Application of the Statistical Analyses of the Investigated Soils for Probabilistic Slope Stability Calculations

4.3.1. Input Data

Characteristic Soil Properties

The investigatory drillings determined the stratum of the investigated area. Sections for the calculations were determined according to the determined layers. Example sections can be seen in Figure 4. Based on the laboratory results, the characteristic properties (unit weight, cohesion, and angle of friction) were determined for each layer, which is in

Table 6. Characteristic soil properties.

	Unit Weight (kN/m3)	Cohesion (kPa)	Phi (°)
Quaternary fat clay (high plasticity clay)	19.1	103	13
Quaternary clayey Silt (clSi)	20.0	56	22
Quaternary Pannonian sandstone formation	20.0	30	33
Silt (silt) (Si)	20.0	45	22
Silty clay (clayey Silt) (clSi)	20.0	67	16
Lignite seam	13.0	100	26
Silt (silt) (Si)	20.0	42	23
Sandy clay (low plasticity clay)	19.5	56	22
Cover fat clay (high plasticity clay)	19.5	103	13
Intermediate organic fat clay (high plasticity clay)	20.0	87	7
Clayey silt (clSi)	20.0	56	22
Organic silty fat clay (high plasticity clay)	20.3	87	7
Medium clay (medium plasticity clay)	20.0	56	23
Bentonite fat clay (high plasticity clay)	20.0	103	13
Sandy silt (saSi)	20.3	42	23
Aquifer (sand) (Sa)	20.3	20	20
Waste material	17.3	11	28

. The characteristic values were used for deterministic stability calculations and for some layers in the probabilistic stability calculations, where not enough data was available for the probabilistic approach.

Probabilistic Soil Properties

In

Table 7. Best-fitting probability distributions for unit weight across soil types using multiple statistical methods.

	Clayey Sand (clSa)	Sand (Sa)	Sandy Clayey Silt (saclSi)	Sandy Silt (saSi)	Silt (Si)	Silty Sand (siSa)	High Plasticity Clay	Medium Plasticity Clay	Low Plasticity Clay
CumFreq [Mean Absolute Error (MAE)]	Lognormal	Lognormal	Normal	Normal	Normal	Lognormal	Lognormal	Lognormal	Lognormal
EasyFit [Kolmogorov–Smirnov]	Uniform	Triangular	Uniform	Gamma	Beta	Gamma	Beta	Beta	Beta
EasyFit [Anderson Darling]	Gamma	Gamma	Uniform	Normal	Beta	Lognormal	Beta	Gamma	Lognormal
EasyFit [Chi Squared]	N/A	Beta	N/A	Lognormal	Lognormal	Normal	Beta	Gamma	Beta
MATLAB [Akaike information criterion (AIC)]	Gamma	Lognormal	Gamma	Normal	Normal	Lognormal	Normal	Lognormal	Lognormal
Python [Akaike information criterion (AIC)]	Lognormal	Normal	Lognormal	Normal	Normal	Normal	Normal	Normal	Lognormal

, the best-fitting statistical distributions for the unit weight of various soil types can be seen, determined using multiple methods: CumFreq (MAE), EasyFit (Kolmogorov–Smirnov, Anderson–Darling, Chi-Squared), and Akaike Information Criterion (AIC) via MATLAB and Python. Lognormal, Normal, Gamma, and Beta distributions are predominantly identified. Lognormal and Normal distributions are frequently favored, especially by MAE and AIC-based evaluations. For example, low plasticity clay and silty sand are consistently best described by the Lognormal distribution across most methods. Beta distributions are commonly associated with high-plasticity clays. Table 7 shows the results of the probabilistic analyses.

Table 8. Probabilistic values of unit weight (kN/m³).

	Distribution	Mean	Std.dev.	Abs.max.	Rel.max.	Abs.min.	Rel.min.
Quaternary fat clay (high plasticity clay)	Normal	18.57	1.48	21.90	3.33	14.90	3.67
Quaternary clayey silt (clSi)		20.00
Quaternary Pannonian sandstone formation		20.00
Silt (silt) (Si)	Normal	19.56	0.83	21.64	2.08	17.54	2.02
Silty clay (clayey silt) (clSi)		20.00
Lignite seam		13.00
Silt (silt) (Si)	Normal	19.56	0.83	21.64	2.08	17.54	2.02
Sandy clay (low plasticity clay)	Lognormal	19.78	0.62	21.40	1.62	18.60	1.18
Cover fat clay (high plasticity clay)	Normal	18.57	1.48	21.90	3.33	14.90	3.67
Intermediate organic fat clay (high plasticity clay)	Normal	18.57	1.48	21.90	3.33	14.90	3.67
Clayey silt (clSi)		20.00
Organic silty fat clay (high plasticity clay)	Normal	18.57	1.48	21.90	3.33	14.90	3.67
Medium clay (medium plasticity clay)	Lognormal	19.70	0.88	21.99	2.29	17.36	2.34
Bentonite fat clay (high plasticity clay)	Normal	18.57	1.48	21.90	3.33	14.90	3.67
Sandy silt (saSi)	Normal	19.04	0.70	20.70	1.66	17.34	1.70
Aquifer (sand) (Sa)	Lognormal	18.99	0.68	20.33	1.34	17.56	1.43
Waste material		17.30

summarizes the input parameters used in the probabilistic analysis. The distributions were selected based on the results of the MATLAB calculations or, where applicable, the most frequently occurring distribution type.

Table 9. Best-fitting probability distributions for shear strength parameters (f, c) by soil type and statistical method.

	High Plasticity Clay		Medium Plasticity Clay		Low Plasticity Clay		Sandy Silt (saSi)		Silt (Si)		Silty Sand (siSa)		Sand (Sa)
	phi [deg]	c [kPa]	phi [deg]	c [kPa]	phi [deg]	c [kPa]	phi [deg]	c [kPa]	phi [deg]	c [kPa]	phi [deg]	c [kPa]	phi [deg]	c [kPa]
CumFreq [Mean Absolute Error (MAE)]	Normal	Normal	Normal	Gamma	Normal	Normal	Normal	Normal	Lognormal	Normal	Normal	Normal	Normal	Lognormal
EasyFit [Kolmogorov–Smirnov]	Beta	Gamma	Normal	Gamma	Beta	Uniform	Triangular	Normal	Lognormal	Uniform	Normal	Uniform	Beta	N/A
EasyFit [Anderson Darling]	Beta	Gamma	Gamma	Gamma	Triangular	Normal	Triangular	Normal	Lognormal	Uniform	Normal	Normal	Normal	N/A
EasyFit [Chi Squared]	Triangular	Triangular	Beta	Gamma	Beta	Triangular	Triangular	Normal	Exponential	Lognormal	Lognormal	Beta	N/A	N/A
MATLAB [Akaike information criterion (AIC)]	Gamma	Normal	Normal	Gamma	Normal	Normal	Normal	Normal	Gamma	Gamma	Normal	Gamma	Normal	Lognormal
Python [Akaike information criterion (AIC)]	Gamma	Gamma	Uniform	Gamma	Uniform	Uniform	Uniform	Uniform	Lognormal	Gamma	Uniform	Uniform	Lognormal	Lognormal

summarizes the best-fit probability distributions for shear strength parameters of the investigated soil types using several goodness-of-fit methods: CumFreq (MAE), EasyFit (Kolmogorov–Smirnov, Anderson–Darling, Chi-Squared), and Akaike Information Criterion (AIC) applied via MATLAB and Python. Gamma and Normal distributions are most frequently selected, particularly for clay soils. Uniform and Triangular distributions are more often associated with sandy and silty soils, especially in EasyFit methods. Lognormal and Exponential distributions are mainly linked to Silt and Silty Sand for phi, especially in AIC-based evaluations.

Table 10. Probabilistic Cohesion (kPa) values.

	Distribution	Mean	Std.dev.	Abs.max.	Rel.max.	Abs.min.	Rel.min.
Quaternary fat clay (high plasticity clay)	Normal	91.4	43.1	207.4	116	1.7	89.7
Quaternary clayey silt (clSi)		56.0
Quaternary Pannonian sandstone formation		30.0
Silt (silt) (Si)	Gamma	46.1	31	87.1	41	6.6	39.5
Silty clay (clayey silt) (clSi)		67.0
Lignite seam		100.0
Silt (silt) (Si)	Gamma	46.1	31	87.1	41	6.6	39.5
Sandy clay (low plasticity clay)	Normal	58.3	28.7	110.6	52.3	10.9	47.4
Cover fat clay (high plasticity clay)	Normal	91.4	43.1	207.4	116	1.7	89.7
Intermediate organic fat clay (high plasticity clay)	Normal	91.4	43.1	207.4	116	1.7	89.7
Clayey silt (clSi)		56.0
Organic silty fat clay (high plasticity clay)	Normal	91.4	43.1	207.4	116	1.7	89.7
Medium clay (medium plasticity clay)	Gamma	86.3	49.9	207	120.7	1.7	84.6
Bentonite fat clay (high plasticity clay)	Normal	91.4	43.1	207.4	116	1.7	89.7
Sandy silt (saSi)	Normal	30.9	17.2	70	39.1	1.3	29.6
Aquifer (sand) (Sa)	Lognormal	21.3	8.4	32.3	11	13.5	7.8
Waste material		11.0

and

Table 11. Probabilistic values of internal friction angle Phi (deg) values.

	Distribution	Mean	Std.dev.	Abs.max.	Rel.max.	Abs.min.	Rel.min.
Quaternary fat clay (high plasticity clay)	Gamma	11.00	5.80	23.50	12.50	0.00	11.00
Quaternary clayey silt (clSi)		22.00
Quaternary Pannonian sandstone formation		33.00
Silt (silt) (Si)	Lognormal	15.20	10.40	36.00	20.80	1.60	13.60
Silty clay (clayey silt) (clSi)		16.00
Lignite seam		26.00
Silt (silt) (Si)	Lognormal	15.20	10.40	36.00	20.80	1.60	13.60
Sandy clay (low plasticity clay)	Normal	17	8.4	30	12.96	2.4	14.64
Cover fat clay (high plasticity clay)	Gamma	11.00	5.80	23.50	12.50	0.00	11.00
Intermediate organic fat Clay (high plasticity Clay)	Gamma	11.00	5.80	23.50	12.50	0.00	11.00
Clayey silt (clSi)		22.00
Organic silty fat clay (high plasticity clay)	Gamma	11.00	5.80	23.50	12.50	0.00	11.00
Medium clay (medium plasticity clay)	Normal	15.4	7.2	33.30	17.86	1.8	13.64
Bentonite fat clay (high plasticity clay)	Gamma	11.00	5.80	23.50	12.50	0.00	11.00
Sandy silt (saSi)	Normal	22	7	33.8	11.78	9.02	13
Aquifer (sand) (Sa)	Normal	25.80	3.10	28.80	3.00	21.50	4.30
Waste material		28.00

show the results of the analysis for the cohesion and angle of friction.

4.3.2. Results of the Slope Stability Analysis

The introduced statistical data processing was used in practice to determine the stability of the slopes of the open-pit lignite mine. Both deterministic and probabilistic slope stability calculations were performed using the Slide2 software on the seven sections, which are shown in Figure 1. The results demonstrate the difference between the two methods and the importance of the probabilistic approach.

Deterministic Calculation Results

The results of the deterministic analysis are encouraging; the slope proved to be stable for all sections. Due to the minimum safety factor of 1.35, slope optimization was later performed on Sections 3, 4, and 7, but this is not discussed in this study.

Table 12. Deterministic test results of all sections.

	Section 1	Section 2	Section 3	Section 4	Section 5	Section 6	Section 7
FS (basic) [-]	1.951	1.379	1.305	1.226	1.653	1.583	1.334
FS (optimized) [-]	1.951	1.379	2.108	1.646	1.653	1.583	1.826

presents the Factor of Safety (FS) values obtained from both the basic and optimized deterministic analyses for each Sections. In the case of Section 3, Section 4, and Section 7, geometric modifications were required to achieve an FS above the threshold of 1.35, indicating that the original geometry did not meet the stability criterion.

Figure 16 shows the deterministic calculation result of Section 1. The Factor of Safety (FS = 1.951) is above the desired value; the slope is stable according to the result.

Probabilistic Calculation Results

Table 13. Probabilistic test results of all sections.

		Section 1	Section 2	Section 3	Section 4	Section 5	Section 6	Section 7
Basic Geometry	FS [-]	2.024	1.219	1.172	1.052	1.731	1.582	1.334
	FS (mean)	1.504	1.041	0.949	0.866	1.195	1.522	1.334
	PF [%]	10.8%	42.8%	53.5%	67.5%	25.7%	0.80%	0.0%
	RI (normal)	1.359	0.136	−0.164	−0.428	0.635	4.067	7687.336
	RI (lognormal)	1.559	0.002	−0.325	−0.586	0.578	4.947	8849.537
Optimized Geometry	FS [-]	2.024	1.219	2.081	1.646	1.731	1.582	1.826
	FS (mean)	1.054	1.041	1.538	1.209	1.195	1.522	1.826
	PF [%]	10.8%	42.8%	10%	29%	25.7%	0.80%	0.0%
	RI (normal)	1.359	0.136	1.429	0.609	0.635	4.067	258.712
	RI (lognormal)	1.559	0.002	1.663	0.542	0.578	4.947	344.357

presents the results of the probabilistic slope stability study of the Visonta Keleti-III lignite mine. The FS (Factor of Safety), PF (Probability of Failure), and RI (Reliability Index) values in the table were determined for the different sections based on the Bishop simplified method and Latin Hypercube simulation. Latin Hypercube analysis is an advanced version of Latin Hypercube simulation that samples the entire set of input parameter values in a structured manner, thus increasing the efficiency and accuracy of the simulation. FS is the result of a deterministic calculation run with the Mean values of the parameters. FS (mean) is the expected FS value obtained from the results of 1000 runs, where the Mean value takes on a different value in each run. The RI is the reliability index; it also indicates the stability of the slope. Based on the FS values, Section 3 (FS = 2.081) is the most stable section, while Section 2 (FS = 1.219) is the most critical. In terms of PF, i.e., the Probability of Failure, Section 2 is particularly risky, as it shows a 42.8% failure probability. The results of the research confirm that traditional deterministic models cannot reliably handle the spatial uncertainty of geotechnical parameters in the Visonta mine area. The probabilistic approach using Latin Hypercube simulation provides a more detailed and realistic assessment of slope stability, which is essential for safe mine design.

After the statistical analysis was completed and the statistical parameters were obtained, the practical implementation followed. In addition to the statistical parameters, there was a need for characteristic values. The Visonta Mining Area was very densely stratified, as shown in Figure 4 and Figure 5, so for several layers, there were only characteristic input data. For the rest, we calculated with probabilistic values. The interpretation of the results of the probabilistic slope stability calculation is presented in Section 1 of the Visonta Mining Area. Figure 17 displays a probabilistic analysis of the slope stability of this section using the Bishop simplified method.

Figure 18 shows a histogram of the Factor of Safety (FS) with relative frequency on the y-axis. The distribution is right-skewed, indicating a larger concentration of simulations resulting in FS values between approximately 1.0 and 2.0, with a peak near 1.7. The orange bars on the left (FS < 1.0) represent failure cases where the slope is not considered stable. The blue bars correspond to stable conditions (FS ≥ 1.0). A fitted probability density function (PDF) curve is overlaid in gray, approximating the distribution trend.

The results indicate that while the majority of simulations show a stable slope, a non-negligible Probability of Failure exists. This probabilistic approach highlights the variability and uncertainty in slope stability due to input parameter distributions.

Figure 19 illustrates the Cumulative Distribution Function (CDF) of the Factor of Safety (FS) calculated using the Bishop simplified method. The x-axis represents the Factor of Safety, and the y-axis shows the corresponding cumulative probability. The curve exhibits a typical sigmoidal shape, characteristic of cumulative distributions. At FS = 1.0, the cumulative probability is approximately 0.10, indicating a 10% Probability of Failure.

The graph in Figure 20 illustrates the convergence behavior of the Probability of Failure (PF) calculated via the Bishop simplified method as a function of the number of Latin Hypercube samples. The x-axis represents the number of samples, while the y-axis shows the corresponding PF (%). Initial estimates (<100 samples) show high variability and instability, as expected with low sample sizes. Around 300–400 samples, the PF begins to stabilize, with fluctuations decreasing progressively. From approximately 600–700 samples onwards, the PF converges around 11%, indicating statistical stability of the estimated Probability of Failure. This behavior confirms that at least several hundred samples are required for reliable PF estimation in slope stability analysis using probabilistic methods. This plot is essential for validating the adequacy of the amount of samples in probabilistic geotechnical modeling.

5. Discussion

The Probability of Failure (PF) results from the probabilistic stability study conducted in the Keleti III part of the Visonta Mining Area highlight the risk assessment limitations of traditional deterministic methods and confirm the need for a probabilistic approach. The effectiveness of probabilistic slope stability studies is also supported by other research, especially studies using metaheuristic methods, such as Zeng et al. [19], where hybrid optimization algorithms were used to assess slope safety more reliably. The spatial variability of physical soil parameters fundamentally affects the reliability of slope stability studies. Therefore, probabilistic analyses complementing deterministic approaches are necessary for safe planning.

The acceptability of failure probability is a widely discussed issue. There is no agreed consensus, only recommendations and tables [10,20,21,22,23,24,25,26,27,28,29,30,31,32]. In

Table 14. Recommended slope stability criteria from the literature.

Literature Source	Min.FS [-]	Max.PF [%]	Conditions of Applicability
Read and Stacey (2009) [24]	1.3–1.5	<5	Overall slopes with high failure consequences
Priest and Brown (1983) [20]	>2	<0.3; P (FoS < 1.5) < 5	Medium-sized (50–100 m) and high slopes (<150 m) carrying major haulage roads or underlying permanent mine installations with “very serious” failure consequences
DME (1999) [33]	>2	<0.3	Permanent pit walls near public infrastructure with “high” failure consequences
Hormazabal et al. (2011) [26]	>1.5	<3	Slopes affecting infrastructure or public roads
Wesseloo and Read (2009) [25]	1.2–1.3	<10	Inter-ramp (high consequence of failure)

, the most relevant recommendations have been summarized.

Kirsten (1983) [34] provides a practical interpretation of PoF levels, linking them to slope service life, liability, and monitoring needs. High PoF implies short-term, high-risk conditions requiring strict controls, while low PoF values indicate long-term stable slopes with minimal monitoring. The matrix by B. M. Adams (2015) [27] offers a structured method for selecting acceptable FoS and PoF values based on failure consequences and design confidence, distinguishing between slope types by design life, and prescribing appropriate risk management measures.

Numerous interpretations of PF are apparent after reviewing several articles related to PSSA from the past decades. In their article, Ng and Kok Shein [35] considered a value below 5% safe, while above 25% they considered a high risk. Penalba et al. [36] in their work considered a 6% failure probability acceptable. Bi et al. [37] are quite permissive, stating that slopes below 50% are safe. Moradi et al. [38] maximized the allowable failure probability at 5%. In their case study in Taiwan, Wang et al. [39] considered the failure probability of 17.3% to be too high. Hamedifar et al. [40] considered 30% to be dangerous. According to Kulatilake et al. [41], 5% is the permissible limit, and Chaulagai [42] had the same opinion. Mandal et al. [43] said that the slope was not considered safe for a PF = 15%. Neman et al. [44] 2018 set a PF value of 10% as the upper limit. However, Obregon [45] determined the maximum at 20%. Sitharam [46] quotes Sjöberg [47], meaning that the PF obtained of 7.5% is acceptable but close to the 10% limit. Rafiei [48] claimed that PF should be below 10–15%. Mathe and Ferentinou [49] and Sachpazis [50] capped the maximum allowable value of PF at 5%. Sdvyzhkova et al. [51] assessed a PF value of 12.55% as a significant slip hazard. Do et al. [52] recommend a 10–20% allowable PF. However, Li et al. [53] wrote that a PF greater than 5–10% is no longer acceptable. Nguyen et al.’s [54] are even more stringent; according to their results, PF = 2.2% cannot be considered safe. Abhijith et al. [55] interpret a PF greater than 17% as a potential danger zone, and this was based on the recommendation of the US Army Corps of Engineers (1997) [56].

These studies show no final consistent agreement on the maximum value of PF. It is important to keep in mind Ferreira’s [32] statement, after Wesseloo and Read (2009) [25], that the minimum FS and maximum PF requirements must be evaluated simultaneously. The calculated FS and PF of the slope of the Visonta lignite mine also show that it is not easy to evaluate the long-term stability according to the results. When the FS was around 1.50, which was considered stable, the PF was between 0 and 11%, which can be stable but bigger than the 5% limit, which was considered the upper limit of the safe zone in some of the above-mentioned studies. When the SF was 1.33 at Section 7, then the PF was 0%, so it can be considered stable. There have been landslides in Visonta. Around 2010, significant landslides occurred in the open-pit lignite mine in Visonta, especially in the area around the western waste heap. As a result of the landslides, temporary restrictions on mining operations were ordered, and geotechnical monitoring systems were reinforced.

For a more exact evaluation of the PF results, it is beneficial to define PF threshold values according to engineering risk categories. Based on a review of the above-mentioned international literature and Adams [27], Gibson [10], Wesseloo, and Read [25], it is suggested to use the following indicative thresholds: for critical slopes—where failure may endanger human life or vital infrastructure—a PF of ≤5% is recommended. For general slopes or temporary structures, a PF of ≤15% may be acceptable, provided the design includes adequate monitoring and maintenance strategies.

In this case study, according to Adams [27], the accepted risk can be moderate, so the upper limit of the PS should be 10%.

6. Conclusions

The paper presents a comprehensive probabilistic geotechnical study of the Visonta Keleti-III lignite mining area in Hungary, emphasizing the statistical evaluation of soil parameters and their integration into probabilistic slope stability analysis. Relationships between key variables—such as bulk density, void ratio, and plasticity index—were investigated, revealing strong correlations, especially within cohesive soil types. Following the identification of parameter interdependencies, the study examined the statistical distribution characteristics of key input parameters, including unit weight, cohesion (c), and friction angle (φ). Over 3300 laboratory samples collected from 28 boreholes were analyzed, and outlier filtering was conducted using interquartile range (IQR) analysis. Distribution fitting was performed using tools such as CumFreq 1.3.1.0., EasyFit 3.0, MATLAB R2023a, and Python 3.11. Among the tested distributions, Gamma, Normal, and Lognormal were most frequently selected, though their appropriateness varied depending on soil type and parameter. Cohesive soils—particularly clays—exhibited more predictable statistical patterns and yielded improved model fits following filtering. The paper introduces a probabilistic slope stability analysis in a stratified and geologically heterogeneous environment. The probabilistic modeling was carried out using the Bishop simplified method in combination with Latin Hypercube Sampling (LHS), offering improved sampling efficiency and convergence behavior. All of the seven sections were examined, but Section 1 was presented extensively. Convergence analysis of the Probability of Failure (PF) indicated stabilization after 400–500 simulations, validating the sufficiency of the sample size and reliability of the probabilistic results. Finally, the study underscores the engineering significance of adopting a probabilistic approach in slope stability evaluation. While deterministic models may overlook parameter uncertainty and spatial variability, the integrated probabilistic framework—augmented with advanced data processing and customized scripting—enables a more realistic and defensible assessment of slope behavior. In the case of Visonta’s Section 1, the estimated Probability of Failure of 11% suggests marginal stability and highlights the need for potential design mitigation. The developed computational tools and workflow, compatible with industry-standard software such as Rocscience, provide a replicable and scalable methodology for future geotechnical risk assessments in similarly complex geological settings.