1. Introduction
The exposure of embankment infrastructure to natural hazards is unavoidable [
1]. This results in the instability of embankment slopes, which has increased in recent times due to human activities [
2] and climate variability [
3]. On a global scale, this problem poses a serious threat to the management and development of railway infrastructure and similar transportation networks [
4]. Recently, the long-term performance of embankment slopes has become a focus of scrutiny [
5,
6]. The consequences of climate change have made rail infrastructure more vulnerable to water-related risks, such as floods [
7], ponding, and fluctuations in water levels [
8].
China has one of the most diversified hydrological systems in the world [
9], with a complex network of rivers, such as the Yellow and Yangtze Rivers, as well as enormous lakes, such as Qinghai Lake and Dongting Lake. The area surrounding Dongting Lake (the study area), near the Nanhu railway embankment between Beijing and Guangzhou provinces, has been affected by the lake’s fluctuations (and resulting groundwater changes), resulting in multiple failures and constant repair. As a result, these situations not only accelerate embankment deterioration but also drive up rehabilitation and maintenance costs [
10]. To address these challenges, several solutions have been used, including the placement of soil reinforcement techniques ranging from fibers and rock mixtures [
11] to steel piling [
12]. However, the control of traffic loads and the consequences of climate change have made these efforts persistently difficult [
13]. Ensuring the long-term stability of embankment slopes under water level fluctuation requires advanced engineering solutions that could capture seasonal and long-term fluctuations while allowing for increased traffic demands at the same time [
14,
15]. An attempt to address these difficulties is crucial to ensuring the reliability and resilience of critical transportation systems [
16]. Recent studies have stressed the relevance of incorporating soil hydromechanical parameters for evaluating the performance of embankment slope constructions under different geohazard conditions [
17,
18,
19,
20]. Using the bibliometric tool VOSviewer 1.6.19 to analyze previous studies from the Scopus database revealed a noticeable trend in slope stability analysis under changing water levels and limited work on sensitivity and optimization of slope stability factors (see
Figure 1).
In addition to external influences, the physical properties of the material structure are internal elements that play an important role in slope stability, as the changes in saturation states of the soil due to seepage fields impact both local and overall slope stability [
17,
21]. Additionally, groundwater table variations could have considerable impact on the slope stability due to its time domain [
22,
23]. Another way to improve the performance of railway embankment slopes is to examine the quantitative relationships between parameters regulating safety coefficients [
24]. Technically, this requires examining the effect of changes in each hydromechanical parameter on the safety factor and knowledge about their dominating roles in stability analysis [
25]. This is critical for developing appropriate mitigation strategies and control guidelines to minimize instability [
26]. Unfortunately, water level changes in railway slope stability is a subject that has received little attention [
27]. Some studies advise that for long-term performance and a robust railway system, sensitivity assessments for each parameter of the railway slope under bidirectional water levels are critical [
28]. Zhou [
17] investigated the sensitivity of variables such as unit weight, internal friction angle, water level, and cohesiveness on the slope stability factor of safety. Using the tree-augmented naive Bayes (TANB) approach, Ahmad [
29] studied the impacts of parameters such as pore–pressure ratio, internal friction angle, cohesion, slope angle, and unit weight on the stability coefficient. Using an artificial neural network, Abdalla [
30] investigated the effects of variables such as internal friction angle, cohesiveness, slope height, and unit weight on the stability factor. Tao [
31] used a support vector machine (SVM) [
32] to evaluate slope parameters. Hu [
33] assessed the sensitivity of unit weight, cohesiveness, and internal friction angle to the slope stability factor of safety using partial Spearman rank (PSS) correlation coefficients.
Zhou et al. [
17] used reliability coefficients with cohesion and internal friction angle as components, following other studies of sensitivity analysis for embankment slope parameters [
34,
35,
36,
37]. However, fluctuating water levels coupled with the static train loads found a sensitivity analysis of railway embankment slopes complicated. To understand their interactions and the influence of input factors on the response variable, it is critical to build a trustworthy and obvious relationship between output and input variables [
38]. Existing studies have concentrated on individual hydromechanical parameters of the soil or external factors (rainfall, water level fluctuation, etc.), with little focus on interactions and sensitivity to railway embankment slopes. Ignoring factor interactions may not accurately depict the nonlinear relationships between components and the slope’s response factor, which is a key drawback in past research. Hence, this study aimed to evaluate the connection between external and internal factors and their impact on the stability of the railway embankments while considering the coupled impacts of the static load stress under fluctuating water levels. To integrate this concept, a method for performing thorough and reliable coupled static stress and seepage sensitivity analyses on railway slopes was developed. Emerging artificial intelligence approaches, such as fuzzy logic (FL), ANN [
39,
40] and gene expression programming (GENXPRO) are regarded as robust and reliable for developing slope stability models, even without prior knowledge of the seepage and stress fields within the railway embankment slope environment. Several strategies for improving the accuracy of ANN model predictions have been proposed [
29], with the back-propagation algorithm (BP) standing out as particularly effective [
41]. To achieve these objectives and establish the integration of both internal and external embankment stability controlling factors, both artificial neural network (ANN) and Box–Behnken design (BBD-RSM) models were selected due to their good precision in the field of civil engineering [
41,
42].
3. Results and Discussion
3.1. Parameter Screening with Plackett–Burman (PBD) and RSM Modeling
The global perturbation analysis method examined parameter influences from external and internal sources. Therefore, a screening strategy was used to choose the most sensitive embankment stability coefficient values, followed by an assessment of their total sensitivity. The initial screening of the 14 factors affecting railway embankment stability used PBD by Minitab 19.0 [
17]. These factors include cohesion (
c), elastic modulus (
E), water level rate of change (
v), internal friction angle (
ϕ), train static load (
L), infiltration coefficient (
ks), soil unit weight (
ρs), elevation of fluctuating water level (
H), Poisson’s ratio (
v), residual moisture content (
θr), and van Genuchten fitting parameters (
n,
m, and
α). Zhang [
17] found that
θs,
n,
m,
α, and
θr have little effect on earth material embankment stability and were assumed to be negligible.
This study only examined
L,
v,
ρs,
c,
ks,
ϕ,
E,
H, and
ν. Each variable was tested at two independent levels, with the lower level (−1) set to drop by 10% from the baseline basic working norm and the higher level (+1) set to grow by 10% [
17].
Table 1 shows a 20% difference between the higher and lower levels. A probability value (
p-value) below 0.05 in the statistical analysis showed a significant influence on the railway embankment stability factor, guiding the PBD test factor selection, as shown in
Table 2. These parameters’ primary effects were assessed using PBD and were further evaluated for main effects using BBD-RSM.
The normalized effects show that when rising lake water levels impact the railway slope, the impact of each influencing parameter is ranked in ascending order:
µ < E < v < Ks < H < L < c< ρs< ϕ. Interestingly,
c,
ρs, and
ϕ greatly impact the global static stability and safety factor of the railway embankment slope.
E,
v,
Ks,
H, and
L are less important. PBD results show that when water levels drop, the sensitivity of each parameter is classified as follows:
E < µ < v < Ks < L < ρs < H < c < ϕ. This indicates that
ϕ,
c,
H, and
ρs greatly affect the railway slope’s global static stability and safety factor. The parameters
E,
v,
Ks,
L, and
µ have minimal impact on the global static stability of the railway slope and its factor of safety. A preliminary analysis shows that the mechanical properties (
ϕ,
c, and
ρs) of the railway slope are affected through its global stability and safety factor during varying water levels. Train static loading does not affect the global static slope factor of safety, but it is stronger when water levels rise than decline. The soil coefficient of permeability,
Ks, is less sensitive to the railway slope’s global static stability safety coefficient, possibly due to embankment compaction rates [
17].
In addition, the seepage flux (Ks, H, and v) sensitiveness on the railway slope’s static stability coefficient is greater with falling water levels than with rising levels. The lake’s fast water level variations may explain this (assuming instantaneous pore water pressure response). Thus, the railway embankment slope seepage field characteristics (Ks, H, and v) will need to be monitored for long-term sensitivity. Based on the standardized effects chart, three of the most influential components for the increasing water level scenario and four for the dropping water level scenario were selected for BBD-RSM analysis.
The variables H, ρs, c, and ϕ were found to be critical through a Plackett–Burman design test. We examined stability factors by analyzing response variables for rising (ρs, c, and ϕ) and dropping (ρs, c, H, and ϕ) water levels. Two stages were used for the BBD-RSM tests. Lower levels (-) were 10% lower than the benchmark, whereas high levels (+) were 10% higher. Each variable was evaluated under these conditions. The BBD-RSM produced 17 experimental design runs for rising water levels and 30 for dropping water levels. By contrast, a linear model equation best described the response variables under dropping water levels. A, B, C, and D represent the resulting values of ρs, c, H, and ϕ.
3.1.1. Water Level Stability Factor Responses
A higher stability factor means that the railway slope will perform better. The significance of the hydromechanical properties was evaluated for their principal effects on the overall static stability of the railway embankment slope during both rising and decreasing water phases. The BBD-RSM analysis verified the Plackett–Burman sensitivity analysis results and ordered the primary effects of the parameters as follows: for dropping water levels, ϕ > c > H > ρs, and for rising water levels, ϕ > ρs > c.
In this study, the models were chosen based on the highest-order polynomial, in which extra terms are significant and are not aliased. For this investigation, the dropping water level was studied using the same procedure (assuming an instantaneous pore water pressure change). High R-squared values, approaching unity, indicate that the quadratic and two-factor interaction (2FI) equations for rising and falling water level models, respectively, have been appropriately calibrated to the actual data. The rising water level stability factor response demonstrated a significantly high F-value of 6173.65 and a p-value < 0.0001. The model’s p-value was <0.05, indicating that an F-value of this magnitude has a 0.01% probability of occurring owing to noise.
The results show that the model is competent for exploring the experimental space to determine the ideal conditions for railway slope design for rising water levels. The study shows that the SFRES was influenced in the following ascending order: internal friction angle (B) with an F-value of 3.52 × 10
5, unit weight (C) with a 1.46 × 10
5 F-value, cohesion (A) with a 9.6 × 10
3 F-value, AC with an 84 × 10
2 F-value, C2 with a 5.1 × 10
2 F-value, and BC with an 8.1 × 10
1 F-value, while AB, A2, and B2 are not significant to the stability factor of the railway embankment slope (Equation (8)).
The LF
p-value was zero, indicating that the model LF is not significant, and the difference between the adjusted and predicted coefficient of determination (
R2) < 0.2 indicate reasonable agreement. This implies that the model is suitable for exploring the experimental parameters to find the best conditions for designing the railway slope for a falling water level. It is evident that the stability factor of the railway embankment slope was influenced in the following ascending order: internal friction angle (C) with an F-value of 2.1 × 10
4, cohesion (B) with a 1.1 × 10
3 F-value, water level (A) with a 4.3 × 10
2 F-value, unit weight (D) with a 2.0 × 10
2 F-value, BD with a 4.3 F-value, AB with a 3.5 F-value, and BC with 1.7 F-value, while AC, BC, AD, and CD are not significant to the stability factor of the railway embankment slope. The Equation (9) represents the regression model developed to predict the stability factor of the railway embankment slope during falling water levels.
3.1.2. Validation and Selection of BBD-RSM Model
Table 3 shows the Box–Behnken design response surface methodology (BBD-RSM) ANOVA results at a 95% confidence interval. The adjusted
R2 values for rising water level (RWL) and falling water level (FWL) responses are 0.99(99) and 0.99(25). The predicted
R2 values are 0.99(98) and 0.97(80), respectively. This analysis indicates that the two values closely agreed, suggesting the model’s validity. The small difference between the adjusted and predicted values, both <0.2, confirms the model’s excellent predictability.
3.1.3. Actual vs. Predicted Investigative Plots for the Responses
The model plots the FWL safety coefficient’s observed vs. expected responses in
Figure 3c. The graphic shows the accurate prediction and use of the FWL safety coefficient in the railway embankment slope response model. The datapoints align well with the line of best fit, indicating good agreement between actual and predicted outcomes [
41].
Figure 3d depicts a randomly and sparsely scattered residual plot for the FWL safety coefficient response model, showing that errors are not correlated and have equal discrepancies. The normal residual plot is appropriate because most responses cluster around the line of fit, validating the accuracy and reliability of the generated model (
Figure 3). Strength lines represent lines of best fit, while blue, green, and red dots denote the level of agreement of predicted values corresponding to low, medium, and high values of the factor of safety, respectively.
3.1.4. FWL and RWL Stability Factor Response
Three-dimensional and perturbation response graphs from the BBD-RSM show how input parameters affect output responses. The charts show how input factors affect output. Each model shows input parameters on the Y and X axes and the output (safety factor) on the Z.
Figure 4 shows the interactions among FWL parameters, (a)
c vs.
H, (b)
ϕ vs.
H, (c)
ρs vs.
H, (d)
ϕ vs.
c, (e)
ρs vs.
c, and (f)
ρs vs.
ϕ. The a–f signified how the combined variables influenced the factor of safety. It can be seen from the plots that factor of safety (FS) increases with increase in
ϕ and
c, and decreases with increase in
ρs and
H. The model’s 3D contours for AC, BC, AD, and CD are relatively flat, suggesting little interaction between these factors. This shows that these two elements’ magnitudes affect the response behavior less. The vivid blue area of the plot denotes a region with minimal influence on the factor of safety, while the reddish-orange area indicates a region of optimal influence. Also the contour plots show response values in blue, green, and red classes. The colors indicate low, moderate, and very ideal interaction [
41].
Figure 5 depicts the interactions among RWL parameters in 3D for (a)
ϕ vs.
c, (b)
ρs vs.
c, and (c)
ρs vs.
ϕ, while (d)
ϕ vs.
c, (e)
ρs vs.
c, and (f)
ρs vs.
ϕ represents the 2D. The a–f signified how the combined variables influenced the factor of safety for both 3D and 2D surfaces. From the 2D plots, it is shown that the FS is maximized when
ϕ = 37.4 °C,
c = 15.4 Kpa, and
ρs = 1710 kg/m
3. Both 2D and 3D response surface diagnostic plots show sloping flat and color margin contour profiles, indicating a strong joint interaction between independent variables [
53]. The charts show that each of these factors significantly affects safety. The safety factor increases as unit weight decreases by 10–20%. An increase in the input variables cohesion and internal friction angle, is positively correlated with the stability factor. The BBD-RSM perturbation analysis shows an inverse relationship between unit weight and the two remaining input variables (cohesion and internal friction angle). The flatness of the 3D surface suggests that changes in one parameter and its effects on the factor of safety are relatively constant across the range of values of the second parameter, and vice versa, while the overall effects of these parameters are significant.
3.2. The Modeling and Optimization of Artificial Neural Networks
Building an artificial neural network required identifying model ambiguity and choosing a network design to reduce input integration mistakes and improve efficiency [
53]. A neural network (ANN) was used to assess hydromechanical parameter efficacy [
54,
55]. The RWL stability coefficient input parameters are cohesion, internal friction angle, and unit weight (
Table 4). The FWL stability coefficient input factors are cohesion, internal friction angle, water level, and unit weight. Due to differential initial conditions, the output factors, RWL stability coefficient, and FWL stability coefficient were examined separately. A “feed-forward back-propagation” network design was used to handle the modeling system’s complexity and parameter selection [
56]. The FWL network had a 4-8-1-layer pattern with eight hidden layer neurons to connect the input and output layers.
Table 4 shows the statistical model values used for all neuron architectural standards (
Figure 6).
The statistical analysis showed a strong positive correlation between the actual and predicted responses of the artificial neural network (ANN) model at a 95% confidence interval, indicating high agreement. The RWL had
R2 values of 0.99(99), 0.97(70), and 0.99(91), and FWL had
R2 values of 0.99(97), 0.97(71), and 0.99(56) as training, testing, validation, respectively (
Table 5). The output of each hidden layer neuron was used to determine the optimal artificial neural network (ANN) model’s predicted results. In Equations (10) and (11), the optimized ANN-based model-predicted responses were determined with the use of output from individual neuron in the hidden layer, by adding the individual neuron and its weight.
where,
represents the hidden layer output,
represents the corresponding input variable value, and
bh and
bi are the biases hidden layers and the input, respectively [
57]. Sensitivity analysis was carried out based on the relative importance of individual operating variable (input parameter) on the ANN model output (
Y).
Sensitivity was evaluated on the proposed approach [
44], Equation (12), which uses the neural network weight indices (i.e., layer and input weights). The fundamental principle is based on the use of the connection weights predicted by the model and then computed using Equation (12) as follows:
where
denotes the j
th variable relative importance (%) on the ANN-based model output,
Ni,
Nh are the input layer and hidden neurons, respectively, and W represents the ANN model connection weight. The matrices
o,
i, and
h stand for the output, input, and hidden layers, respectively, while
n,
k, and
m, are the neuron numbers at the output, input, and hidden layers, respectively (
Figure 7 and
Figure 8).
In the RWL scenario, cohesion, internal friction angle, and unit weight were identified as significant factors in determining the stability factor. In the FWL scenario, cohesion, internal friction angle, water level, and unit weight were key. Notably, the internal friction angle had the most substantial impact, accounting for 62.8% for RWL and 53.1% for FWL. Following this, unit weight contributed 22.4% for RWL and 5.2% for FWL, while cohesion accounted for 17.8% for RWL and 32.6% for FWL and water level accounted for 10.1% for FWL. This analysis underscores the crucial role of the internal friction angle in driving stability factors.
3.3. Performance Comparison between BBD-RSM and ANN Models
Numerous researchers have shown that design within response surface methodology (BBD-RSM) and artificial neural network (ANN) modeling for sensitivity analysis is effective [
41,
44,
58]. This work shows forecast stability coefficients with varying water levels using BBD-RSM and ANN. To assess their predictions’ accuracy, the study compared predicted and actual data and calculated absolute relative error (ARE).
Figure 9 shows the comparison between the data from the ANN and BBD-RSM models and the actual data to determine model accuracy. Scatter plots comparing each value to the number of runs showed that model-predicted values matched actual values. This comparison study found that the BBD-RSM model aligned data better than the ANN model. Thus, the BBD-RSM model slightly outperformed the ANN model in data representation.
The study also found that BBD-RSM models predicted responses marginally better. However, both the BBD-RSM and ANN models accurately predicted numerical findings and validated them across diverse data combinations utilized to investigate water level changes, whether rising or falling. The appropriateness of the BBD-RSM and ANN-based models was further examined using
R2,
MRE, and
RMSE (
Table 6). The significant relationship between these indicators shows the models can imitate real-world outcomes. However, the time lag between surface water and groundwater should be further investigated to validate the results of this study.