Introduction

Various issues of uncertainty in slope stability practise emerges primarily due to geological anomalies, data insufficiency, spatial variation in soil properties, uncertainty in slope failure mechanism, simplifying assumptions considered in geotechnical modelling, and human errors in design and construction of geotechnical structures. In a conventional deterministic slope stability analysis, the stability is generally reflected in terms of factor of safety (FoS). However, due to the uncertainties present in input parameters, it is hardly possible to assess the slope failure with a single FoS value. It is found in several studies that two nearly similar slopes, having nearly similar FoS obtained from a deterministic analysis, can have recognizably different failure probabilities due to the uncertainties involved in the geotechnical properties and failure mechanisms [1,2,3,4,5]. Different methods for deterministic slope analyses are prescribed, such as limit equilibrium method (LEM) [6], finite element method (FEM) [7] and finite difference method (FDM) [8]. Even in recent years, although computationally expensive, the application of smooth particle hydrodynamics (SPH) in assessing the stability analyses of slopes under large deformation scenario is gaining grounds [9]. Among these methods, the LEMs are still the most popular among engineers due to its simplicity in application and analyses methodology. However, deterministic slope stability analysis, using LEM, presumes the potential failure surface and other geotechnical properties of the slope that might influence the overall failure behaviour of slope [2]. Moreover, for simplification of the problem, additional approximations are made with respect to externally applied loads (static, seismic or climatic loads) [2, 3]. Therefore, to overcome the limitations of traditional slope stability practise, probabilistic concepts are incorporated in slope stability analysis for better assessment of slope failure while incorporating the variability and uncertainty prevailing in any natural slope.

As compared to deterministic analysis, to characterise the variability in soil properties, the probabilistic analysis requires more data. Furthermore, a strong statistical background is necessary to develop by a good mathematical representation of the spatial variability. The application of statistics in geotechnical engineering and soil property characterization was introduced in the late 1960 [10, 11]. In slope engineering, the probabilistic concept was first introduced in the 1970s [1, 12,13,14]. The advantages of probabilistic slope stability analyses and the importance of incorporating spatial variability in soil properties has long been realised by the researchers [15,16,17,18]. Over the last few decades, the probabilistic concepts for slope stability analysis have evolved [19,20,21,22,23,24,25,26,27,28,29], and serves to be an emerging field of interest for many researchers. Moreover, in the case of limited representative empirical data, probabilistic study of slope is also a useful tool for risk assessment. However, there is a lack of usage of the current advanced probabilistic approaches by the practitioners owing to the inconsistent guidelines to apply the current advanced techniques [2].

This discussion provides a detailed review of probabilistic concepts in different aspects of slope stability practise that has developed over the years. Starting with the basic probability and statistical concepts, various uncertainties involved in the slope stability modelling and the different approaches to encounter these uncertainties in probabilistic slope stability practise are discussed. A brief review is also presented regarding probabilistic slope stability analysis of reinforced/retained natural slopes and the slopes subjected to dynamic excitations.

Different approaches for probabilistic slope stability analysis

There are mainly two categorical approaches for probabilistic study of slope stability. First among them are the approximate methods that includes the first order reliability method (FORM) [21], the mean value first order second-moment reliability method (MVFOSM) [30] and the second order reliability method (SORM) [31]; while the second categorical approach adopts the Monte Carlo simulation (MCS)-based methods [32, 33]. In their primordial form, both the approximate and simulation-based methods make use of deterministic analysis (LEM or FEM-based approach) to estimate the factor of safety, which is further used to assess the failure probability. The following subsections highlight the characteristics of each of the categorical approaches.

Approximate methods-based estimation of failure probability

Among the approximate methods, FORM has been used extensively in several applications in civil and geotechnical engineering [34]. In the field of structural reliability, the Hasofer–Lind reliability index and the First Order Reliability index were developed for approximation of the probability of failure [34, 35]. In this type of assessment, first, the reliability index (β) is estimated by solving an optimisation problem to locate the critical probabilistic slip surface associated with the minimum reliability index. Then, the failure probability is evaluated using Pf = Φ(− β), where Φ(.) denotes the standard normal cumulative distribution function. Several researchers have reported the significance of optimized reliability indices [20, 21, 36, 37]. This technique represents a typical optimisation problem with an implicit equality constraint represented by the limit state function. A limit state function, usually represented as g(x), is a mathematical quantification of the failure of a system. The limit state function is invariably (though arbitrarily) described such that g(x) > 0 specifies a safe state and g(x) < 0 specifies a state of failure. The basic philosophy of this optimization method is to determine the minimum distance of the limit-state surface G(x) = 0 from the origin of the coordinate system to. There are various methods for evaluation of reliability index by solving an optimisation problem, including the evolutionary methods such as genetic algorithms, typically referred as GAs [35, 38]. GAs have gained substantial acceptability as optimisation method due to their inherent capability of identifying the global optima and the ability to deal with multimodal problems to identify multiple potential failures. Other commonly used evolutionary optimization techniques in geotechnical engineering, to name a few, are the particle swarm optimisation [39], ant colony [40] and other nature-inspired algorithms [41]. Any of these optimization problem is defined by the objective function and the set of constraints, both of which are dependent on the parameters used for their definition. For a slope stability problem, these may include the contribution from soil shear strength parameters, geometrical parameters defining the slope, parameters defining the failure mechanism, externally applied ambiental and climatic conditions, and all possible forms of uncertainties associated with these parameters. There is no single straightforward rule as which of the optimization technique would suit best for a problem, and that multiple optimization techniques may prove worthy. However, in general, multiple techniques are generally used with experience, and the one that is able to identify and assess the best possible global optima with a minimum number of iterations and least computation time is generally adjudged the best possible technique. Mbarka et al. [31] coupled different methods of reliability analysis such as MCS, MVFOSM, FORM, SORM and quadrature method (QM) with various mechanical approaches (Caquot–Taylor formula, a LE method of Bishop and a FE model based on the strength reduction method) for a slope stability assessment, considering the cohesion and friction angle of the soil as correlated random variables without any spatial variation of the properties. The study showed that the approximate methods need lesser numbers of iteration of the performance or the objective function (describing the limit state that is usually expressed in terms of FOS) and, hence, are less time consuming than MCS. The study also revealed that if the mechanical model is accurate and the search procedure to find out the minimum reliability index is well defined, these methods result in good approximations of reliability of the slope system. This ongoing research considers the cohesion and friction angle of the soil as correlated random variables without any spatial variation of the properties..

Besides its simplicity of implementing in different applications, it is revealed that the results obtained using FORM have higher accuracy when applied to stability of slopes having smaller failure probability or higher reliability index [42]. However, on the same topic, Griffiths et al. [32] has pointed out that FORM is incapable of considering inherent spatial variability in soil properties, thereby resulting in either over-estimation or under-estimation of the probability of slope failure under different conditions. Hence, in reference to the probabilistic analysis, none of the outcomes are ever exact, as the problem is largely dependent on the uncertainty modeling, which subsequently governs the time of computation. Ideally, the exact probability of failure is unachievable, and a trade-off in accuracy would always exist in any realistic probabilistic study. In this context, depending on the complexity of the problem being handled, the approximation methods may fall short of reaching the desired accuracy to some extent, but as the literature indicates, they have proven their worth in assessing the probabilistic slope stability analysis with sufficient dependability. However, it is always suggested to cross-verify any probabilistic assessment problem with more than one technique to identify any marked differences in the outcome, and reach a reasonable dependence on the assessment.

Monte Carlo simulation-based estimation of failure probability

In the MCS-based probabilistic studies for the assessment of the failure of any geotechnical structure, the geotechnical properties are considered as random variables. The ease of computation and simplicity of direct Monte Carlo simulation (MCS) technique makes it preferable over other methods. Based on user-specified ranges of input parameters treated as random variables, MCS generates a set of trial values of the random variable within the specified range, and for each trial value, the random variable estimates the limit state function. The set of trial values for different geotechnical properties generated can be based on the range as reported by several researchers either from laboratory or field test data [17, 18, 37, 42]. The total number of failures occurring in all the trial values of random variable is counted, and the failure probability is expressed as

$$P_{\text{f}} = \frac{{n_{\text{f}}}}{N},$$
(1)

where, nf is the number of failures that occurred in total N number of trials. This method is commonly adopted into different commercially available geotechnical software such as PLAXIS [43], FLAC [44], SVSlope [5], Geostudio [45] and GEO5 [46]. However, a disadvantage of this technique is that it largely depends on the substantial number of trials (iterations) to be undertaken to arrive at the reliable solution, thereby demanding excessive computation time. The accuracy generally increases with increase in number of trials. However, it is necessary to estimate the minimum number of trials of MCS to generate a reliable and consistent result. Repetitive FE analysis is highly time consuming and Pf estimation generally converges within a certain number of trials [3]. Hence, further increase in the number of MCS trials does not improve the accuracy, rather, adversely affect the computational time. For reference, as per the US Army Corps of Engineers [47], Table 1 shows the reliability index (β) and the probability of failure (Pf) of any geotechnical system and their desired performance levels. The value of β generally ranges from 1 to 5, and the corresponding Pf value vary within 0.16–3 × 10–7. Standard geotechnical designs prefer a β value of at least 2 (i.e., Pf < 0.023) for desired performance level better than ‘poor’ [48]. A comparatively small Pf value is of great interest to geotechnical practitioners. To ensure accuracy in the value of failure probability, this direct MCS technique requires a sample number of at least more than ten times the reciprocal value of the desired probability level [49]. Therefore, to obtain a Pf level of 0.001, corresponding to a desired performance level of ‘Above average’ (Table 1), the number of iterations required in direct MCS is more than 10,000. Hence, the direct MCS demands a large amount of computational time as the deterministic analysis of slope stability specifically searches for a number of potential slip surfaces to obtain the minimum FoS. This shortcoming calls for further improvement in the computational efficiency of direct Monte Carlo-based simulation procedures.

Table 1 Relationships between probability of failure, Pf and reliability index, β [47]
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In recent years, an ‘advanced subset simulation’ method is introduced to improve the efficiency of MCS at comparatively small probability levels [48, 50]. Subset simulation is a stochastic simulation technique for efficiently generating failure samples and computing the probability of failure at relatively smaller level. It is based on the concept that a small probability of failure can be represented by a product of larger conditional failure probabilities for some intermediate failure events [51, 52]. Moreover, direct MCS does not offer insight into the effect of different type of uncertainties in the probabilistic study. Therefore, Wang [53] proposed an improvised approach to conduct a sensitivity study of the uncertain parameters within the MCS framework. The variation of slip surface is also stated to be an important factor to be incorporated while considering the spatial variability in MCS-based probabilistic slope stability analysis [48]. In addition, Li et al. [54, 55] proposed that multiple representative slip surfaces can also be used together with MCS to consider multiple failure mode with significantly reduced computational effort. Moving further ahead, instead of adopting several MCS instances to reach a convergence on the factor of safety (FoS) for the probabilistic slope stability analysis, Dyson and Tolooiyan [56] presented a probabilistic slope stability approach using the random finite element method (RFEM) to predict the FoS of each MCS random field instances prior to FE-based simulation on random field similarity measures. This approach resulted in an exceptional reduction of the computational time. Fang et al. [57] proposed a new probabilistic approach for slope stability analysis based on slip-line field theory within MCS framework. The method significantly increases the computational efficiency as this method does not require to compute the FoS and critical failure surface for estimation of probability of failure. The study reports that when the critical slope contour obtained from the slip-line field theory intersects with the toe of the slope, the slope is considered in limiting equilibrium condition. The failure probability estimated using this method is found to agree well with the results obtained from Direct MCS based on the Bishop’s simplified method and FORM.

Uncertainty due to soil heterogeneity

Soil is originated from different geological, environmental and physico-chemical processes. Soil heterogeneity is categorised into two main classes, namely lithological heterogeneity and inherent spatial variability. The lithological heterogeneity is described as one layer of soil overlying another layer, or the presence of pockets of different lithology within a nearly uniform soil mass. Inherent spatial variability refers to the variation of soil properties within one particular soil layer, which originates due to different loading histories and deposition conditions. Earlier, the traditional way to deal with spatial variability was to rely upon a high safety factor, engineering judgement and local experience. However, Morgenstern [58] presented some case studies for various geotechnical engineering applications, where it was clearly highlighted that a complete dependence on the engineering judgment led to noticeably erroneous assessments in 70% of the cases. Hence, it was realised that it is essential to develop more advanced, reliable, and appropriate techniques to address geotechnical uncertainties in safety analysis. In the beginning, to address soil heterogeneity in geotechnical engineering studies, limit equilibrium analysis coupled with MCS method was introduced [5, 59,60,61]. Subsequently, random finite element method (RFEM) coupled with MCS was introduced to address spatial variability of soil into a numerical analysis [3, 45].

Inherent spatial variability of shear strength parameters of soil

Spatial variation in soil properties is considered to be a significant source of uncertainty in geotechnical engineering problems such as slope stability analyses. At present, incorporation of spatial variability of soil is primarily concentrated on inherent variation of soil property within one nominally homogeneous layer [24, 33, 59, 62,63,64,65]. Ji et al. [59] stated that the probability of failure is overvalued when soil spatial variation is not taken in consideration, i.e., a slope may be much more stable than it is assessed by ignoring the spatial variability. Wang et al. [48] also reported that ignoring spatial variability results in over-estimation of the variance of FoS, which may result in either over- or under-estimation of failure probability of a slope. However, Allahverdizadeh et al. [66] reported the existence of a critical spatial correlation length value that leads to an underestimation of the probability of failure, thereby resulting in a non-conservative design. The critical correlation length resulting in the maximum probability of failure was reported to be of 0.5H–1.0H, where H represents the height of the slope. Similar observation about the critical spatial correlation length was reported by Griffiths and Fenton [24], which is further discussed in a later section. Several researchers have studied the influence of spatial variation in soil shear strength parameters on the failure probability or reliability of slope structures [3, 4, 67,68,69,70]. Low et al. [71] proposed a practical MS-Excel-based programming procedure for assessing the reliability of a slope failure considering spatial variation in soil shear strength parameters. Srivastava and Babu [72] quantified the spatial variability in soil shear strength parameters with the aid of field-test data and estimated the reliability of a soil slope (having spatial variability) against failure. The study revealed that although the deterministic factor of safety value is acceptable, the estimated reliability index indicated the performance level of the given soil slope to be ‘below average’. Griffiths et al. [73] carried out a probabilistic study on infinite slopes to explore the effect of spatial variation of shear strength parameters on reliability-based slope stability analysis. It was highlighted that the first order reliability methods might result in non-conservative estimation because of ignoring the spatial variability in soil. Considering spatial variation in shear strength of soil, Ji et al. [59] conducted a slope stability analysis combining the first order reliability method (FORM) with a deterministic approach, to search for the probabilistic critical slip plane. In this study, two methods, namely the method of auto-correlated slices and the method of interpolated autocorrelations, were adopted. This study was an extension of the method of interpolated autocorrelations of one-dimensional spatial variability of undrained shear strength [74] of a soft foundation supporting an embankment, which has been further improvised to model a two-dimensional spatial variation in reliability-based slope stability analysis. The study revealed that as compared to the effect of spatial variation in horizontal direction, the effect of spatial variation in vertical direction has more significance on the probability of failure of slope structure. This fact has also been reported by the findings of other studies [4]. Zhu et al. [75] utilized the random field analysis approach to study the influence of material heterogeneity of slope considering the spatial variation of permeability function under steady state rainfall infiltration. The parametric study, considering the correlation length of log-permeability of the heterogeneous slope varying from 0.4 to 50 times of the slope height, shows a variation in the matric suction values ranging from 0.5 to 1.25 times of those of a homogeneous case.

Incorporation of spatial variability through probabilistic approach

A continuous random variable can be characterised by a probability density function (pdf). In geotechnical engineering, the Normal or Gaussian distribution functions are most commonly used [76]. As an illustration, if the undrained shear strength (S) of soil be considered as a continuous random variable, the Normal pdf can be represented as fs(s) through the expression

$$f_{{\text{s}}} (s) = \frac{1}{{\sqrt {2\pi \sigma_{{\text{s}}} } }}{\text{e}}^{{ - (s - \mu_{{\text{s}}} )/2\sigma_{{\text{s}}}^{2} }} ,$$
(2)

where, \(\mu_{{\text{s}}}\) and \(\sigma_{{\text{s}}}\) are the mean and standard deviation of S, respectively. Figure 1a shows a typical probability density function for a normally distributed random variable (undrained shear strength, S). The Normal distribution is preferably used to characterise the properties in geotechnical engineering, since the sum of random variables tends to have a normally distributed pdf according to the central limit theorem [77]. However, the disadvantage of adopting a Normal distribution is that it incorporates negative values, which is infeasible for the soil shear strength and other physical parameters in geotechnical engineering. A prudent approach to avoid choosing the negative values for such cases is to characterise the geotechnical properties using a non-negative distribution such as the ‘Lognormal distribution’ [78, 79]. The distribution of the random variable can also be represented in terms of the cumulative distribution function (cdf), as shown in Fig. 1b. The cumulative distribution function (cdf), usually denoted as Fs(s), represents the probability of the random variable S taking a value less than s, and is expressed as

$$P(S < s) = F_{{\text{s}}} (s)$$
(3)
Fig. 1
figure 1

A typical representation of a a probability density function (PDF) and b cumulative density function (CDF)

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Apart from the pdfs and cdfs used to characterize a random variable, it is also desirable to have some quantitative descriptors for the same purpose. The most common characteristic of a random variable (for example, undrained shear strength) is its ‘mean’ or the ‘Expectation’ value, which is denoted as:

$$\mu_{{\text{s}}} {\text{ or }}E\left[ S \right] = \int\limits_{ - \propto }^{ \propto } {sf_{{\text{s}}} (s){\text{d}}s} .$$
(4)

Variance is another important characteristic of a random variable that is indicative of the deviation of the random variable from its mean value. The average deviation value is termed variance, \(\sigma_{{\text{s}}}^{2}\), and the magnitude of its square root is referred as standard deviation (σs). The stated entities are expressed as

$$\sigma_{{\text{s}}}^{2} = E[(s - \mu_{{\text{s}}} )^{2} ] = \int\limits_{ - \propto }^{ \propto } {(s - \mu_{{\text{s}}} )^{2} f_{{\text{s}}} (s){\text{d}}s} ;\quad \sigma_{{\text{s}}} = \sqrt {E[(s - \mu_{{\text{s}}} )^{2} ]} .$$
(5)

Further, for comparing two random variables having different expectation, it is often convenient to normalize the standard deviation by its mean value, and the same is termed as the coefficient of variation (CoV), which is defined as:

$${\text{CoV}} = \frac{{\sigma_{{\text{s}}} }}{{\mu_{{\text{s}}} }}.$$
(6)

Typical ranges of CoV for cohesion and internal friction angles are 0.05–0.5 and 0.02–0.56, respectively [80]. A summary about the typical range of inherent variability of the effective-stress friction angle, its tangent and undrained shear strength is presented in Table 2; further details are available in Phoon et al. [81].

Table 2 A summary of inherent variation in soil strength properties [81]
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Although it is well recognized that the soil variability is highly site-specific, it is strenuous to acquire site-specific probability distribution for geotechnical properties. The basic probability principles for characterising geotechnical properties, so discussed in this section, are based on the assessment of soil variability from a huge number of data collected from different sites of a large region, or even from various parts of the world. For geotechnical engineering projects at a specific site, the designers are more interested in the variability of geotechnical properties within the particular site, and avoid the variability assessed from sites at other locations. In recent years, several researchers reported that soil property in a specific site does not necessarily follow Normal or Lognormal distributions. In this regard, in recent years, Bayesian methods have been adopted to obtain the site-specific probability distribution from limited site-specific test data [82, 83]. The site-specific probability distribution of any geotechnical property can be expressed as the weighted summation of a number of Normal or Lognormal distributions having different distribution parameters. Thus, based upon the available data, estimating the site-specific probability distribution of geotechnical properties is indicative of estimating an appropriate group of Normal or Lognormal distributions and their respective weights.

Scale of fluctuation

The spatial variation pattern in soil is expressed by the correlation length or the scale of fluctuation [84]. The scale of fluctuation (SoF) indicates the spatial extent within which the soil property is significantly correlated. An approximate way of estimating the vertical SoF is presented by Vanmarcke [84], expressed as:

$$\delta_{{\text{v}}} = 0.8\overline{d} ,$$
(7)

where, \(\delta_{{\text{v}}}\) is the scale of fluctuation in the vertical direction and \(\overline{d}\) is the average of the distance between the intersections of the fluctuating entity with its mean trend, as shown in Fig. 2. The detailed description of the techniques for evaluation of the correlation length are available in literature [17, 22]. A larger correlation distance or higher SoF signifies a smooth variation of properties, whereas a smaller correlation distance signifies an erratic variation of properties within a soil domain, as shown in Fig. 3. A summary of typical SoF for different geotechnical properties is given in Table 3, for which further details are available in Phoon et al. [81].

Fig. 2
figure 2

A typical representation of the estimation of vertical scale of fluctuation

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Fig. 3
figure 3

Sample realizations of a one-dimensional random field X(t) for two different SoF values a θ = 0.04 b θ = 2.0

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Table 3 Summary of SoF for various geotechnical properties [81]
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Coefficient of variation, auto-correlation function and cross-correlation between shear strength parameters

In most of the cases, problems dealt in geotechnical engineering practise considers functions having more than one variable. For instance, the drained shear strength of a soil is a function of both cohesion (c) and the angle of internal friction (φ). In such cases, the joint probability distribution is considered for a particular range of values of the governing parameters [76]. In most of the geotechnical engineering problems involving two variables, there is often a statistical relationship between the variables, which is quantified by their covariance Cov[X, Y], and is represented as

$${\text{Cov}}[X,Y] = E[(X - \mu_{X} )(Y - \mu_{Y} )].$$
(8)

The normalized covariance is called the correlation coefficient (ρ), and is expressed as

$$\rho_{X,Y} = \frac{{{\text{Cov}}[X,Y]}}{{\sigma_{X} \sigma_{Y} }},$$
(9)

where, X and Y are the two random variables, μX and μY are the expectations of the two variables respectively, and σX and σY are their standard deviations, respectively.

In random field theory, the spatial correlation between various soil properties, considered as random variables, is expressed by a spatial autocorrelation (SPAC) function [76, 85]. Table 4 presents different typical autocorrelation functions available in literature for various geotechnical engineering properties, in their one-dimensional (1D) and two-dimensional (2D) representations [76]. The selection of a proper correlation function for a particular site is a challenging task, especially when the obtained site-specific test results are limited as obtained from geotechnical site characterization. Li et al. [64] compared the differences between various 2D theoretical autocorrelation functions and stated that Squared Exponential and Second-Order Markov type autocorrelation functions might be the most appropriate ones to simulate the spatial correlation between different soil properties. Cao and Wang [78] suggested that for probabilistic study of a specific slope, the autocorrelation model should be selected based on the data collected from the given site. Chakraborty and Dey [86] reported the development of 1D stochastic model to characterize inherent spatial variation in SPT data obtained from field test and subsequently estimating the correlation length.

Table 4 Different autocorrelation functions for geotechnical engineering properties [76]
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In geotechnical engineering, the influence of heterogeneity in cohesion (c) and angle of internal friction (\(\phi\)), as well as their correlation, are very significant. Several authors have studied the cross-correlation between various soil properties [3, 5, 11, 16, 37, 67, 87]. While some researchers ignore all possible cross-correlations between the soil shear strength parameters for mathematical convenience [11, 14, 88,89,90,91], some studies considered the cross-correlation between different geotechnical parameters in numerical studies as an important influencing factor [92,93,94,95,96,97,98,99]. The correlation coefficient between cohesion (c) and angle of internal friction (ϕ) has been proposed to be negative by several authors [11, 100,101,102,103,104], which means that at any specific location, a higher magnitude of c is accompanied by a smaller magnitude of ϕ, or vice-versa. It was observed by several authors that the negative correlation between c and \(\phi\) was found to improve the structural reliability as compared to assuming zero correlation coefficient [5, 73, 97]. Javankhoshdel and Bathurst [105] reported that the failure probability of a slope decreases as the values of c and ϕ become more negatively correlated. At the same time, it is understood that a slope with zero or positive correlation between the shear strength parameters is likely to be associated with larger risk of failure, and should be rigorously studied as well. Very few studies have explored the influence of positive correlation between c and ϕ on slope stability [63, 73, 97]. Griffiths et al. [97] conducted analysis on a drained slope having positive correlation between soil shear strength parameters (c′ and ϕ′). It was reported that a positive correlation coefficient resulted in the decrement of the critical coefficient of variation (COV) value, ignoring which would lead to a non-conservative estimate of the failure probability.

Local averaging and variance reduction

Generally, all engineering properties refer to some sort of locally averaged property. Therefore, instead of investigating the geotechnical properties at discrete locations, it is often significant to investigate the behaviour of the averages of random field [106]. For example, there is higher possibility of slope failure when the applied shear stress exceeds the average shear strength along the critical slip surface, rather than because of the presence of some local weak zones present within the soil mass. Therefore, uncertainty in the average shear strength along the critical failure surface is a more appropriate measure than estimating the uncertainty in shear strength at discreet location within the slope. The moving local average is defined as:

$$X_{T} (t) = \frac{1}{T}\int\limits_{t - T/2}^{t + T/2} {X(\xi ){\text{d}}\xi } ,$$
(10)

where, T is the moving window length and \(X_{T} (t)\) represents local average of \(X(t)\) over a width of T and centred at t, as illustrated in Fig. 4. The first plot, Fig. 4a, shows a random process which is then averaged within a moving window of width T to get the second plot, Fig. 4b. It can be noticed that averaging smoothens the process by reducing its variance. The local averaging process reduced the point variance, and the variance reduction is regulated by a variance function, γ(T), which is expressed as,

$$\gamma (T) = \frac{1}{{T^{2} }}\int\limits_{0}^{T} {\int\limits_{0}^{T} {\rho_{x} } } (\xi - \eta ){\text{d}}\xi {\text{d}}\eta .$$
(11)
Fig. 4
figure 4

A typical effect of local averaging on the variance

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The variance function (Eq. 11) is an average of the correlation coefficient between two subsequent points on the interval [0, T]. If the correlation function diminishes with a steep gradient, i.e., correlation between two points decreases substantially with the separation distance, then γ(T) will be small. In case all the points are perfectly correlated on the interval [0, T], having ρ(τ) = 1 for all τ, then γ(T) will be 1.0, thereby not exhibiting any reduction in variance due to local averaging. Local averaging results in the decrease in the variance of the random process and also damps out the contribution from the high frequency components. Hence, the variance of any geotechnical property, spatially averaged over a particular soil domain, is lower than the variance at discrete locations. The variance decreases with an increase in the extent of the soil domain over which the property is averaged. In case an entity is represented by a normal probability distribution function, local averaging reduces the variance while maintaining the mean unchanged. On the other hand, in case of a log-normal probability distribution function, local averaging results in the reduction in both the mean and the standard deviation, since the mean and variance of a lognormal distribution is influenced by both the mean and variance of the underlying normal distribution [3, 76, 107].

Random finite element method (RFEM)

In geotechnical engineering, to incorporate soil spatial variability for probabilistic analysis, a more appropriate and advanced tool was developed in 1990s, commonly known as ‘random finite element method’ (RFEM) [23, 24]. A soil property is an uncertain quantity at any location within a soil domain and, therefore, is considered as a random variable in RFEM. As an illustration of one-dimensional random field, Fig. 5 shows a typical variation of the cone tip resistance measured during a cone penetration test (CPT).

Fig. 5
figure 5

A typical variation of cone tip resistance with depth

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In the random finite element method, each random variable is characterised using a pdf and is correlated with other random variables at adjoining locations. The set of random variables is characterised by the joint probability distribution function and is represented as a random field. This method combines nonlinear finite element method (FEM) with random field theory. In RFEM, a random field is overlaid upon a finite element mesh and hence, each mesh element behaves as a random variable. This approach succeeded to gain much popularity among researchers and often used in probabilistic slope stability study as this method completely accounts for spatial correlation and local averaging [80]. Further, the method does not make any presumptions regarding the location and shape of the critical slip surface, which is automatically determined using FEM that can search for actual weakest path through the soil domain.

If the mean and covariance of a random field vary with the location, the characterising joint pdf is highly inconvenient to use in mathematical practice as well as to estimate from real field data. Therefore, simplifications of stationarity or statistical homogeneity of random field is essential. A statistically homogeneous or stationary random field indicates that the joint pdf characterising the random field is spatially invariant, i.e., the mean, variance and all the higher order moments are constant at any location within the random field. The correlation between any two random variables entirely depends on their separation distance, and not on their absolute locations. To characterize a stationary random field, the mean, variance and the pattern of soil spatial variability should be known. The pattern of spatial variability in soil can be characterized using autocorrelation function, spectral density function or variance function.

The most commonly used algorithms available in literature to generate multi-dimensional random fields in geotechnical engineering are the moving average (MA), discreet Fourier transformation (DFT), covariance matrix decomposition [44], turning bands (TBM), fast Fourier transformation (FFT) and local average subdivision (LAS) methods [70]. The accuracy of probabilistic study highly depends on the aptness of the algorithm under consideration for generation of random field realisations. Fenton and Griffiths [108] compared different random field generator with respect to their accuracy, efficiency, ease of use and implementation. The FFT, TBM, and LAS methods were found to be much more efficient than the first three methods. It has been highlighted that every method has some advantages and disadvantages, and the selection of random field generator algorithm entirely depends on the particular problem under consideration; further details are furnished in Fenton and Griffiths [108].

Over the years, several researchers have considered the theory of random fields to incorporate the spatial variability of soil properties in geotechnical engineering practise [3, 24, 84, 109]. However, in most of these studies, the spatial variation in soil shear strength was characterised as stationary random field, i.e., the mean of the shear strength parameters is invariant with depth of soil slope. However, it is well understood that the soil properties are non-stationary, i.e., soil property progressively changes with depth from the surface [110]. Several in situ test data showed that soil properties of a homogeneous soil layer exhibit variable trends with depth [18, 67, 111]. Srivastava and Babu [72] reported that for a data exhibiting no trend with depth, the reliability index (β) values underestimate the reliability of slope failure as compared to those obtained with linear trend in data. Further, Li et al. [112] stated that ignoring this increasing trend of shear strength parameters with depth results in overestimation of failure probability. Moreover, the chances of critical slip surface developing at the bottom of the slope decreases significantly when the increasing trend of mean shear strength parameters is considered. In this regard, to incorporate the progressive increment of the various geotechnical parameters with depth, Griffiths et al. [113] introduced the incorporation of non-stationary random fields in probabilistic slope stability studies. Further, Huang et al. [114] investigated the effect of rotated transverse anisotropy, occurring due to various geological processes, combined with non-stationary random field on the reliability of an undrained soil slope stability. The study considered two different cases of non-stationary random field: (a) the trend of soil strength increasing with depth and (b) the trend increasing along the direction perpendicular to bedding. The study revealed that the reliability of the slope stability highly depends on the directions of the trend.

Further, it is worth mentioning that although the random field theory is well established in geotechnical slope stability literature, for small or medium sized projects, it is understandable that the data collected from the site is usually sparse. Therefore, random field parameters estimated from such sparse data may contain significant uncertainty resulting in inaccurate estimation of random field samples for slope stability analysis. To overcome such limitations, Wang et al. [115] recently proposed a random field generator based on Bayesian Compressive Sampling (BCS) and Karhunen–Loeve (KL) expansion, which can be suitable for generating random field samples based on sparse measurements obtained from a given site.

As mentioned earlier, lognormal pdf is often chosen in RFEM analysis to avoid negative values for soil properties considered as random variables. However, while using lognormal pdf for a random soil property (say Y), the correlation length relates to X = ln(Y) instead of Y itself. Few researchers [116, 117] stated that the contrast in the correlation length of the underlying Gaussian random field X and the transformed lognormal random field Y should not crucially influence RFEM-based estimations of failure probability. Pula and Griffiths [118] investigated the theoretical relationship between correlation lengths in transformed hyperbolic tangent and lognormal random fields with the underlying Gaussian random fields. The study shows that for COV less than 0.3, the transformed and untransformed correlation lengths are mostly same. An application to a bearing capacity problem shows that the transformations lead to more conservative results in comparison to the untransformed results, however, the differences were very marginal.

A comparative of RFEM with other approaches

In the evolution of different probabilistic techniques, RFEM has gained more popularity in the recent times. This section presents the contrast between the advanced RFEM approach to estimate probability of failure (or reliability) of slopes in comparison to the traditional FORM- or LEM-based probabilistic methods. All the three stated methods predict the probability of failure as opposed to more conventional FoS. However, the predictions of failure probability are found significantly different depending on the magnitude of correlation length [32]. Griffiths et al. [32] showed that based on RFEM analyses, the failure probability is essentially zero when the correlation lengths are small, beyond which it increases rapidly for intermediate values of correlation lengths. For the larger correlation lengths, the failure probability becomes constant and equal in magnitude to that obtained by FORM analyses. A typical illustration of the same is shown in Fig. 6 that depicts the similarity and differences of the outcomes from FORM and RFEM analyses for various correlation lengths. Therefore, it can be stated that FORM results are convergent to RFEM-based outcomes when the correlation lengths are sufficiently large. For smaller correlation lengths, failure probabilities estimated by FORM are found to be conservative. Huang et al. [91] stated that the RFEM can accurately estimate the failure probability of slopes considering a two layered medium. However, FORM was observed to provide less accurate prediction regarding the system reliability of slopes as it primarily targeted the minimum reliability index related to a particular slip surface.

Fig. 6
figure 6

Comparison of FORM and RFEM in terms of the probability of failure for various correlation length

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Javankhoshdel et al. [61] conducted a comparative study on random limit equilibrium method (RLEM) and RFEM to investigate the effect of spatial variability of soil shear strength parameters on the probability of failure of slopes. It is found that the results from 2D RLEM and 2D RFEM analyses, considering isotropic random fields, showed satisfactory agreement in case of slopes with smaller inclinations. However, noticeable differences are seen in case of analyses for large probabilities of failure and steeper slopes. Chakraborty and Dey [119] compared the results obtained from RFEM with those from traditional LEM-based probabilistic approach for slope stability analysis (Fig. 7). The study reveals that the traditional LEM-based probabilistic approach, characterising soil spatial variation using a one-dimensional random field, underestimates the failure probabilities compared to RFEM. The deviation is attributed to the fact that two-dimensional random field can better simulate the uncertainty in field as compared to one-dimensional spatial variation, thereby increases the failure probability. Besides, LEM method assumes the failure mechanism a-priori by deterministic assessment, whereas the RFEM permits the failure plane to evolve in the soil layers through the weakest path. Hence, conventional LEM-based probabilistic approach results in unconservative assessment of slope failure and may lead to inaccurate design solutions. However, given all the shortcomings, LEM-based approach is still widely adopted in probabilistic analysis, simply because of its numerical simplicity and computational economy over RFEM-based approach [5, 105].

Fig. 7
figure 7

Comparison of effect of 1D and 2D spatial variation of soil on probability of failure

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Recently, Li et al. [120] developed a new probabilistic simulation method using subset simulation method with FEM to efficiently estimate the failure probability (Pf) of a slope. This method significantly improves the computational efficiency in the range of smaller failure probability (e.g., Pf < 10−3), thus making FEM-based probabilistic simulation techniques (e.g., RFEM) more feasible in probabilistic slope stability studies for smaller failure probability levels. However, it is noticed that subset simulation in RFEM requires exhaustive computation of FoS repeatedly. Huang et al. [27] proposed a new approach where the subset simulation coupled with RFEM eliminates the requirement for the estimation of FoS. In this method, to measure the safety margin, the value of yield function from an elastoplastic finite element analysis was used instead of the FoS. Very recently, Chwala [121] presented the kinematic failure mechanisms to analyse the stability of slope (comprising spatially variable strength parameters) based on upper bound approach. This approach is observed to be numerically efficient due to the use of rigid block failure mechanisms accompanied by an optimization technique based on the subset simulation.

Geological or lithological uncertainty

As mentioned earlier, the other type of soil heterogeneity originates from geological or lithological heterogeneity. This refers to the case of stratified soil layers or the presence of pockets of various type of soil within a more uniform soil mass. The literature suggests that very less attention has been given to address the effect of geological uncertainty on the probabilistic studies of heterogeneous soil. To deal with such heterogeneity, practising geotechnical engineers mostly depend on local experience and engineering judgment, which can possibly lead to erroneous results in slope stability assessments. Site investigation plays crucial role to interpret such uncertainties in geological and geotechnical practices. Amidst different site investigation methods, borehole exploration is often carried out to assess the subsurface geological model. However, a limited number of borehole exploration is possible for small to medium projects due to project cost and schedule. This leads to collection of geological and geotechnical information at sparse borehole locations, and information are interpreted based on the limited borehole explorations at other locations.

Some of the early research reported for characterization of the geological uncertainty are by Evans [122], Tang and Gilbert [123], Halim [124], Hansen et al. [125] and De Marsily et al. [126]. Tang and Gilbert [123] developed a renewal process to incorporate the probabilistic behaviour of a soil domain consisting of two different types of material. Halim [124] modelled the geological anomalies by the means of a Poisson process for evaluating the reliability of geotechnical systems. However, both the processes could only deal with the modelling of the two simplest forms of geological uncertainty. These two processes could not handle soil layers having more than two types of material overlying each other. On the contrary, in real field, geological uncertainty always involves more complex soil formations, with more than two soil types overlying each other.

Recently, to deal with the lithological uncertainty, probabilistic approaches including clustering method [127], Bayesian method [128], wavelet transform modulus maxima method [129] and machine learning-based methods [130] have been developed. The geological structures at the site can be interpreted from the spatial interpolation of the stratigraphic configuration obtained at borehole locations [131,132,133]. The spatial interpolation methods including Kriging [134, 135], inverse-distance weighting [136] and compressive sampling [137] may be utilized for interpretation of properties of the geological strata. Then the uncertainty in the geological configuration can be defined with the probabilistic methods using coupled Markov chain [138,139,140], stochastic Markov random field [141,142,143,144] or using random variables or random fields [3, 4, 19, 21, 25, 110, 145,146,147,148,149,150,151,152].

Huang et al. [91] estimated the system failure probability of two-layered slopes in the framework of Monte Carlo simulations and showed that the RFEM can accurately predict the system reliability of slopes. Qi et al. [140] proposed a more appropriate method of simulating geological uncertainty by the means of coupled Markov chain (CMC) model to assess probability of slope failure, considering the horizontal transition probability matrix (HTPM). Later on, Li et al. [153] extended the work to predict the failure probability of slopes simulating geological uncertainty by an efficient CMC model considering both horizontal transition probability matrix (HTPM) and vertical transition probability matrix (VTPM). Based on the simulated soil heterogeneity, MCS was conducted to estimate the FoS of slope by Finite Element Strength Reduction Method (FE-SRM). Li et al. [142] stated that the influence of the stratigraphic dips in sampled geological strata realizations is highly affected by the input statistical coefficients. In the mentioned approaches, embedment of the complex Markov theory makes it a bit complicated to directly use in the geotechnical practice. Recently, Crisp et al. [154, 155] presented a customised linear interpolation algorithm to characterize the strata boundaries, including a random noise component to simulate the stratigraphic uncertainty. Gong et al. [156] presented a random field approach to characterise the geological uncertainty, wherein an autocorrelation function is used to define the spatial correlation of the stratum existing between different subsurface elements. In a non-borehole element, the probability of existence of a stratum is assessed from the derived spatial correlations; and, for sampling of various realizations of the geological configuration, MCS is used.

Probabilistic slope stability studies under seismic conditions

Earthquake is one of the most significant factors leading to catastrophic failure of slopes in a seismically active region. Hence, analyses for the stability of slopes under earthquake conditions are necessarily important in earthquake prone zones. In geotechnical engineering, the most commonly used seismic slope stability analyses techniques include the stress deformation analysis, permanent deformation analysis and pseudo-static analysis. The stress deformation analyses are conducted using FEM in combination with different constitutive models to incorporate nonlinear material behaviour under seismic motions [157]. Permanent deformation analyses are based on the evaluation of permanent displacements in slope mass due to earthquake force, using the simple sliding block analogy proposed by Newmark [158]. The most common and popular approach of seismic slope stability analysis is the pseudo-static method, in which earthquake forces are considered as equivalent to horizontal and vertical inertia forces acting on the slope [159]. The pseudo-static analyses are primarily improvisations on the LEM-based static slope stability analysis and, in this case as well, the stability of slope is expressed in terms of FoS. In addition to homogeneous slopes, LEM has also been used to assess the deterministic translational stability of multi-layered cover systems (MLCS) by the incorporation of the interface shear strength properties under pseudo-static conditions [160]. Apart from the inherent disadvantages of LEMs, the selection of the proper value of the seismic coefficients is a crucial factor in such analyses, which controls the inertial forces on the soil masses. This method is well known to give conservative results.

Limited literatures are available regarding the probabilistic seismic analysis of slopes. Tsompanakis et al. [161] performed the probabilistic seismic fragility analysis of an embankment, which is a more realistic and efficient approach to interpret the seismic performance and the vulnerability. Xiao et al. [162] presented a probabilistic seismic slope stability analysis considering ground motion parameter in terms of the peak ground acceleration (PGA) in a specified exposure time at a given site. Burgess et al. [79] conducted a probabilistic seismic slope stability analysis by modelling the slope using RFEM. The investigation was conducted to find out the cost analysis and risk-based design, thereby revealing the probable savings if spatial variability is properly incorporated. Although earthquake records are random in nature, the studies mentioned earlier disregards the random nature of the earthquake by considering a constant pseudo-static earthquake coefficient. However, Malekpoor et al. [163] showed that consideration of random variability in the earthquake coefficient results in conservative assessment of the probability of failure of the heterogeneous slope deposit.

However, these adopted approaches primarily do not deal with the actual dynamic response of the slope structure and their consequent deformations during earthquake. Therefore, these approaches are incapable of assessing the actual response of the slope during a seismic event. Therefore, rigorous dynamic analysis needs to be carried out to assess the stability of the slope. To address the random nature of earthquake record, Youssef et al. [96] and Johari et al. [164] assigned a truncated exponential probability distribution function (pdf) for the earthquake acceleration coefficients. Malekpoor et al. [163] also suggested to explore the random nature of earthquake for appropriate assessment of seismic response of slopes for future research in this regard. In his relation, the work done by Xiao et al. [162] and Burgess et al. [79] can be stated as the first step towards a rigorous seismic slope stability analysis within a probabilistic framework.

Probabilistic study of reinforced/retained natural slopes

Since the early 1980s, application of foreign reinforcements (natural or geosynthetic-based) have gained popularity as a measure of slope stabilization and enhancing the stability of slopes. A lot of studies are available in literature encompassing different types of analytical and numerical approaches that have been adopted in this regard including limit equilibrium methods, finite difference methods and finite element methods. Most of the studies focussed on deterministic methods of analyses, while a very few studies are available in literature involving probabilistic analysis of reinforced slopes. At the beginning, traditional deterministic limit equilibrium methods for unreinforced slopes had been improvised to incorporate the contribution of reinforcement layers in soil structures [165,166,167]. The stability of reinforced slope or embankments were investigated in terms of the modification and enhancement in the factor of safety. Low and Tang [74] presented a limit equilibrium model for reinforced embankments on soft ground allowing tension crack in the embankment, tensile reinforcement at the base of the embankment and a nonlinear undrained shear strength profile in the soft foundation. A practical reliability evaluation procedure was adopted for the study. Kitch [34] conducted probabilistic analyses of two steep reinforced slope problems using commercially published guidelines based on deterministic LEMs. The study reported the computation and comparison of the reliability of internal failure modes (passing within the reinforced zone) and composite failure modes (passing mostly outside of the reinforced zone). Cherubini [168] designed an anchored sheet pile wall considering the spatial variability in soil. Li and Liang [169] developed a reliability-based computational algorithm to stabilize an unstable slope with piles in the probabilistic framework using MCS-based LE approach. Zhang et al. [170] adopted an existing reliability analysis method for unreinforced slopes to estimate reliability of slopes reinforced with piles. However, it is worth noticing that both the mentioned studies ignored the inherent spatial variability of soil slope considering the soil shear strength parameters only as random variables. Other researchers [171,172,173] investigated the stability of geogrid reinforced slope while accounting for the soil spatial variability. Recently, Chen et al. [28] presented a RLEM-based approach for pile-reinforced slope stability study in spatially variable soil. However, in all the above studies, the adopted traditional limit equilibrium approach has the limitation of presuming the critical failure slip surface as well as the magnitude and distribution of incipient tensile forces in the reinforcement. Hence, there is an ardent need to assess the stability of such reinforced slopes using a more rational finite element-based method including rigorous probabilistic approaches. It is also worth noting that probabilistic analysis of reinforced or retained natural slopes under seismic conditions has not yet been taken into account in the research domain of probabilistic slope analysis.

Critical summary and future prospects

It is explicitly understood that conventional limit-equilibrium or finite element based analyses, incorporating deterministic soil parameters, fails to address the stability and failure of actual slopes due to uncertainties associated to geotechnical properties. Under such circumstances, the incorporation of parametric uncertainty through probabilistic analysis of the slope stability becomes imperative. Over the last decades, probabilistic study in slope engineering is continuously developing with newer understanding and incorporation of realistic intricacies. Given the uncertainties associated with the properties of the soil forming the complex lithology of slope, the probability of failure or reliability of slope structure can be underestimated or overestimated. In this regard, this article critically highlights the salient works related to probabilistic slope stability analyses and their conceptual-cum-technical evolution, along with the future scopes to address important slope stability issues.

The overview of the literature indicates that there exist two main approaches, namely ‘approximate methods’ and ‘simulation-based methods’, for incorporating uncertainty in soil properties in a probabilistic slope stability analysis. Approximate methods, such as FORM, does not incorporate spatial variability, and is found to either overestimate or underestimate the probability of slope failure. On the other hand, simulation-based methods, such as MCS, is capable of conveniently incorporating the soil spatial variability. However, it is largely dependent on the number of iterations to achieve a desired level of accuracy, and hence is computationally expensive. This limitation is overcome by employing the advanced subset simulation-based MCS technique, which proves to be capable of capturing slope failure with sufficient accuracy, even at relatively smaller probability levels. Rigorous researches are further required in this direction for improving the accuracy in detecting of slope failure at lower probability levels and reduced computational time, which is coveted by the computational geotechnical engineers.

In probabilistic slope stability analysis, both LEM and FEM approaches can be adopted to estimate the probability of slope failure. To overrule the inherent limitations of deterministic approach, Random Finite Element Method (RFEM) is mostly adopted in recent times. RFEM considers the shear strength parameters of soil as random fields to incorporate their spatial variability, and seek out the critical slip surface and the associated failure mechanism as an outcome of the analysis. It is noticed that in comparison to RFEM, the LEM-based probabilistic slope stability techniques, considering one-dimensional random fields, underestimates the probabilities of failure. A comparison between the outcomes of FORM and RFEM indicates that the former gives conservative results for smaller correlation lengths, while they provide agreeable results at larger correlation lengths. The literature highlights that a 2D random field can simulate the field uncertainty more realistically than the 1D spatial variation, thereby increasing the reliability of the assessment. While incorporating uncertainties in multiple parameters, it is important to include their cross-correlations if the parameters are interrelated to or dependent on each other (for example, the shear strength parameters of soil). While the angle of internal friction and cohesion of soil are generally negatively correlated, specific site-dependent cases of soil heterogeneity and lithology might lead to positive correlations as well. Although most of the literature deals with negative correlation between shear strength parameters, the positively correlated instances of spatial variability of soil did not attract much attention, even though they can lead to a noticeable different understanding to the probabilistic failure of slopes. Further, most of the studies have considered stationary random fields to simulate the soil uncertainty using the random field theory, ending up in prescribing an overestimated failure probability. Very few recent studies have attempted to incorporate non-stationary random fields, although significant research is required in this direction to address the slope failure probabilities in a more realistic manner. It is worth noting that soil heterogeneity, in terms of spatial variability, has drawn the attention of many researchers. Nonetheless, ground uncertainty due to lithological heterogeneity remains to be widely ventured for efficient slope stability assessment, thereby demanding more rigorous studies in this direction.

The seismic response of slopes is even more complicated given the presence of variability of soil shear strength parameters. Very few literatures are available on probabilistic seismic stability studies of slopes. Until date, most of such attempts are only based on the pseudo-static earthquake condition, which has its inherent shortcoming in representing the actual dynamic response of slopes during and post-occurrence of an earthquake. Further rigorous research should be directed to understand the influence of uncertainty and spatial variability of the soil shear strength parameters, as well as the uncertainties associated with the seismic motion, on the rigorous dynamic analysis and seismic response of slopes.

With extended urbanization in hilly terrains, the demand for hillslope excavations are gradually on the rise. The vulnerability of such toe-excavated slopes increase manifold, and such slopes need to be protected by proper and selective retention measures. Most of the studies regarding the stability analysis of cut slopes adopted deterministic LEM-based approaches. Probabilistic studies in reinforced or retained natural slopes is significantly lacking. As per the authors’ knowledge, no such studies are yet explored where spatial variability is incorporated with the aid of random field theory. Further probabilistic seismic studies for the analysis of cut-slope stability and artificially constructed MLCS is need to be widely and necessarily ventured.