1. Introduction
Volcanic edifices have slope failures that range from small rocks falling to massive collapses. Enormous flank (with volume > 0.1 km
3) slides have significantly changed over 200 stratovolcanoes across the globe [
1] and are among the most sudden, damaging, and potentially hazardous volcanic events. As a result of catastrophic side collapses, many volcanoes and stratovolcanoes have changed in big ways worldwide [
2]. The most significant volcanic collapses in history, such as those observed at Bandai (Japan), Mount St. Helens (USA), and Bezymianny (Russia), were initiated by the movement along extensive curved failure surfaces [
3]. In the United States, Mount St. Helens had a terrible fall in 1980 [
4]. Debris avalanches, which can later mobilize into debris disasters, pose major threats to the structure and areas further downslope or downstream when these enormous collapses occur, often involving more than 0.1 km
3 of material. Moreover, a significant number of approximately 700 stratovolcanoes found on Earth present a potential hazard to individuals residing in developing nations. As a result, methods for rapid and cost-effective hazard evaluations are particularly important. In the past, approximately 20,000 individuals have lost their lives by flank collapses [
5]. There exist numerous processes that can lead to the instability of edifices [
3]. The collapse of such structures can be attributed to volcano-specific factors such as magma intrusion, hydrothermal alteration, and thermal pressurization of pore fluids [
6]. Alternatively, it can be caused by more universally recognized factors associated with slope instability, such as increased pore fluid pressures. Mount St. Helens has experienced multiple flank collapses, making it an ideal site for a comprehensive assessment of its stability.
In the past, slope stability assessments have been performed in two dimensions, with the assumption that plane strain conditions are true. Since the 1930s, there has been a lot of interest in slope stability studies [
7]. However, it is important to mention that a three-dimensional (3D) slope assessment procedure would generate accurate results [
8]. This is especially the case when the geometry of the slopes varies widely. In addition, two-dimensional (2D) plane strain assumptions become invalid near the corners of a finite slope [
9]. For simple hillslope shapes, 2D slope stability assessments typically produce lower factor of safety (FOS) values than 3D approaches; however, the variations between these two approaches are often in the range of 10% to 20% [
10,
11]. In the recent past, researchers have investigated 3D slope stability [
8,
12,
13]. Hungr [
14] proposed a method for estimating the FOS of 3D slopes using the 2D Bishop’s approach [
15]. Based on the general limit equilibrium method and as per the work of Lam and Fredlund [
16], Fredlund and Krahn [
17] also proposed a 3D approach. However, several techniques have been devised for analyzing slope stability. The limit analysis method [
18], the limit equilibrium method (LEM) [
19], and the strength reduction method (SRM) [
18] are the common approaches. Among these methods, the LEM and SRM are deterministic because they use predetermined values for individual soil properties to infer the stability of a slope.
Despite providing a conservative analysis, FOS-based approaches have been stated to be ineffective in several cases [
20]. The deterministic approach has a significant drawback when accounting for uncertainties in soil parameters. Specifically, parameters such as cohesion (
c), angle of internal friction (
ϕ), bulk density (
γ), and external loads are not explicitly considered in the FOS approaches [
20]. Furthermore, the determined value of the FOS is frequently employed for a specific objective, such as assessing the long-term stability of slopes, irrespective of the uncertainty associated with the estimation. However, it is not practical to apply the same FOS to several scenarios with varying uncertainty levels. Notably, soil materials are highly complex because of their non-linear stress-strain correlations, elastoplastic behavior under different loading conditions, and time-dependent stress-strain responses [
21]. Thus, it is imperative to conduct a thorough investigation of geotechnical parameters to consider the inherent uncertainties associated with soils during slope stability assessments, and reliability analysis (RA) is considered to be deemed appropriate in such circumstances [
22].
Using probability theory and statistics, uncertainties in soil parameters are rationally comprised of geotechnical analyses [
23]. Reliability index (β) and probability of failure (POF) are commonly employed measures for assessing the performance of geotechnical designs. The probability of not meeting performance standards is known as the POF [
23]. RA of geotechnical structures can be performed using a variety of methodologies, including direct first-order second-moment method (FOSM) [
23], first-order reliability method (FORM) [
24], and Monte Carlo simulation (MCS) method [
25]. The aforementioned techniques employ probabilistic assessments of soil characteristics and sub-soil stratigraphy as input variables and yield the output β and/or POF for a pre-established structures [
23]. Nevertheless, limited attention has been given to the application of these techniques in the context of mitigating mountain slope instability.
Previous approaches have also addressed the use of implicit performance functions. The response surface method (RSM) is one of them [
26]. The implicit performance functions are approximated by RSM using a polynomial function. A fairly precise approximation of β can be generated if the selected polynomial functions fit the limit state well [
27]. In contrast, several machine learning (ML) techniques have previously been employed for slope stability analysis in soil, including multivariate adaptive regression splines (MARS), relevance vector machine (RVM), artificial neural network (ANN), support vector machine (SVM), extreme learning machine (ELM), and various others [
28]. ML algorithms are capable of efficiently simulating slope reliability difficulties by approximating implicit performance functions [
22,
23].
Numerous ML methods, such as ANN, MARS, RVM, and ELM, have previously been employed to tackle various engineering issues [
28], including RA of different soil slopes [
22]. Radial basis function networks and ANNs were utilized by Deng [
29] and Deng et al. [
30], respectively, for structural RA. Cho [
31] used RSM based on ANN to carry out probabilistic slope analyses. Erzin and Cetin [
32] used ANN to calculate the FOS of a soil slope. Several researchers have also examined slope reliability analysis using RVM, SVM, MARS, and other ML approaches [
23]. Kang et al. [
33] employed replacement models with updated SVMs and two swarm intelligence methods, viz., particle swarm optimization (PSO) and artificial bee colony (ABC) algorithms. Zhao [
20] used SVM for slope RA. RA of soil slopes using RVM-based RSM, multi-kernel RVM, and enhanced FOSM were also performed [
34].
Nonetheless, it should be noted that earthquakes can trigger a considerable number of earth and rockslides, leading to an extensive destruction of prominent structures such as hills, hill-highways, railway tracks, dam reservoirs, etc. Note that, the stability of a slope is significantly influenced by the geometry of its shape and the physical properties of the existing soils. Slope failures can cause ground deformation, which in turn can damage structures. Instability is a common problem in hilly areas because of their geodynamic and structural makeup and the effects of factors like extreme rainfall, urbanization, and other influences [
35]. High-quality roads and highways are desperately needed in these mountainous areas to facilitate easier travel, more visitors, and risk-free development efforts. The significance of performing slope stability analyses that consider the spatial variability of geotechnical characteristics inside a prominent volcanic formation like Mount St. Helen is of utmost relevance.
Furthermore, existing literature exhibits no prior utilization of high-performance ML approaches for probabilistic evaluations of Mount St. Helens in seismic and non-seismic conditions. Thus, considering the above points, probabilistic assessments of Mount St. Helens were carried out in this study using a hybrid intelligence approach of ANN and a meta-heuristic approach. For this purpose, the details of slope geometries provided by Reid et al. [
36] were used for modeling spherical failure surfaces using the 3D Bishop approach. For estimating FOS, Scoops3D, an open-source platform, was used, followed by probabilistic assessments in seismic and non-seismic conditions.
2. Research Significance
Numerous studies have emphasized the use of ANNs in the domains of engineering and science [
28]. The ability of ANNs to represent non-linear problems without considering a functional relationship between input and output is a significant advantage over other ML algorithms. Furthermore, the output generation is unaffected by one or more corrupted cells. Despite these advantages, ANN possesses notable limitations, including challenges related to the entrapment in local minima and the occurrence of overfitting. Moreover, the challenge of accurately determining the precise global minimum can lead to unfavorable outcomes [
28]. To address these issues, researchers employed various optimization algorithms (OAs), including PSO, ABC, and genetic algorithm (GA) [
28]. Due to their robust global search capabilities, OAs can iteratively optimize the learning parameters of ANNs, resulting in enhanced prediction performance. Over the past decade, there has been a significant increase in the utilization of hybrid ANNs, such as ANN-ABC, ANN-GA, ANN-PSO, etc., for addressing various problems [
28,
37]. Tun et al. [
38] used GA to assess 3D slopes with several failure regions. Nevertheless, it should be noted that there is a lack of research on the reliability of slope analysis for the Mount St. Helens utilizing hybrid ANNs. Moreover, no study has been conducted on the evaluation of hybrid ANNs that are built using different groups of OAs for slope stability analysis in cone-shaped terrains.
Thus, this study aims to address the gap in the literature by using hybrid ANNs built with different groups of OAs to perform a probabilistic analysis of Mount St. Helens. Specifically, eight distinct OAs viz., ALO, BBO, CPA, DE, EO, FF, GA, and PSO (see
Section 5.2 for details) were used to optimize weights and biases ANNs, resulting in eight hybrid ANNs, viz., ANN-ALO, ANN-BBO, ANN-CPA, ANN-DE, ANN-EO, ANN-FF, ANN-GA, and ANN-PSO. The computational findings were used to select the most effective hybrid ANN model for performing RA of Mount St. Helens under seismic and non-seismic conditions. The outcomes of the current study were compared to the findings of Tun et al. [
38] and evaluated in the subsequent sections.
6. Data Description and Modelling
The geology of Mount St. Helens appeared more consistent before it collapsed due to volcanic eruptions than that of surrounding stratovolcanoes. Voight et al. [
4] described the physical properties of debris avalanche materials. Intact edifice rock was found to have an average unit weight of 24 kN/m
3, and average values of
and
c were 40° and 1000 kN/m
2, respectively. This information was used to perform the slope stability analysis. Following the normal distribution sampling technique, a total of 100 samples were generated using mean values of
c = 1000 kN/m
2,
ϕ = 40°; and
γ = 24 kN/m
3. Subsequently, the FOS of the slope was determined using Scoops3D for the generated samples with five distinct values of seismic coefficient (k
e), viz., 0 (for non-seismic case), 0.05, 0.10., 0.15, and 0.20, referred to as Set 1 to Set 5, respectively. Descriptive details of the 500 samples (i.e., 100 samples against each k
e value) are presented in
Table 1. The present study incorporates certain simplifications, including the assumption of homogenous material properties, intact rock mass, the exclusion of groundwater effects, and the inclusion of seismic loading. However, according to
Table 1, the parameter
c varies from 809.77 kN/m
2 and 1195.17 kN/m
2. The ranges for the remaining parameters,
ϕ,
γ, and k
e are 35.18° to 49.40°, 22.02 kN/m
3 to 25.99 kN/m
3 and 0 to 0.20, respectively. The other descriptive details can be seen in
Table 1.
Stage I: After FOS estimations, the primary database of 500 records was finalized. This database was normalized randomly between 0 and 1 and then divided into training and testing subsets. Using 5-fold cross-validation, 80% of the entire dataset was used for training, i.e., 400 samples, whereas the remaining 20%, i.e., 100 samples, were used for testing. Notably, the training subgroup was used to construct hybrid ANNs, while the testing subgroup was used for validation. After model construction, multiple performance matrices, namely mean absolute error (MAE), Nash-Sutcliffe efficiency (NSE), performance index (PI), coefficient of determination (R2), RMSE, and weighted mean absolute percentage error (WMAPE), were determined and assessed. Subsequently, the best-forming hybrid ANN was selected for the probabilistic assessment of Mount St. Helens.
Stage II: Subsequent to the selection of the best-performing model, RA was performed in seismic and non-seismic conditions. This was achieved by producing different sets of input parameters with different coefficient of variation (COV) values. The following stage involves the normalization of the new dataset based on the original input variables (see
Table 1). The best-obtained paradigm was then used to generate FOS, followed by a probabilistic assessment of the slope.
Figure 8 shows the entire process of FOS estimation and probabilistic analyses of Mount St. Helens.
8. Summary and Conclusions
RA was carried out on the real topography of Mount St. Helens, which collapsed in 1980 due to an eruption, and the results are described in this study. Initially, the FOS was determined by performing a series of calculations in Scoops3D with various input parameters, such as rock characteristics that vary with location and seismic coefficient. The next step was to construct eight hybrid ANNs, using FOS as an output and the corresponding rock parameters and ke as inputs. After performance assessment, the best-performing model, i.e., ANN-FF (based on performance in the testing phase with R2 = 0.9996 and RMSE = 0.0042), was chosen for probabilistic analyses the Mount St. Helen in seismic and non-seismic scenarios.
For this purpose, five distinct COV combinations were investigated for five different ke values, varying between 0 and 0.20. According to the experimental results, the POF varies in the range of 1.1 × 10−44 to 1.7 × 10−04 and 2.1 × 10−38 to 0.19 in non-seismic and seismic conditions, respectively. Even at the high COV and seismic levels, the POF was between 9.7 × 10−20 and 0.19. These results indicate that the failure probability of the slope is negligible even at higher COV levels. Thus, it is deduced that if Mount St. Helens does not erupt, COVs will have little effect on the POF. Given the uncertainty of rock properties, the suggested ANN-FF-based RA is determined to be an appropriate solution for calculating the POF of the Mount St. Helen.
Furthermore, the proposed technique demonstrates an accurate estimation of the FOS of the slope irrespective of seismic conditions. The proposed ANN-FF has the following advantages: (i) improved generalization, (ii) faster convergence, and (iii) higher prediction accuracy in both phases. However, the suggested ANN-FF model has higher computational cost than other hybrid ANNs built in this study. In addition, the above investigation was carried out using the upper and lower ranges of rock parameters (as detailed in
Table 1). Hence, it can be deduced that the above analysis may not yield satisfactory results beyond these values, which can be considered as one of the limitations of the present study. However, further assessment is required for this case. Therefore, the following points should be considered as the future scope of the study: (i) implementation of improved mechanism to reduce the computational cost of ANN-FF paradigm; (ii) in-depth assessment of slope failure probability at high COV and larger range of c, Ø, and γ parameters, (iii) adoption of Mononobe-Okabe trick combined with seismic actions for an in-depth assessment, and (iv) a comprehensive assessment of other hybrid models of ANN, RVM, and ELM constructed with different group of OAs. Nevertheless, per the author‘s knowledge and literature review, this study is the primary implementation of the ANN-FF model to perform probabilistic analyses of the Mount St. Helens in seismic and non-seismic conditions.