Response Surface Methods for Slope Reliability Analysis: Review and Comparison

Engineering Geology 203 (2016) 3–14

Contents lists available at ScienceDirect

Engineering Geology

journal homepage: www.elsevier.com/locate/enggeo

Response surface methods for slope reliability analysis: Review and comparison

Dian-Qing Li a,DongZhenga,Zi-JunCaoa,⁎, Xiao-Song Tang a, Kok-Kwang Phoon b a State Key Laboratory of Water Resources and Hydropower Engineering Science, Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering (Ministry of Education), Wuhan University, 8 Donghu South Road, Wuhan 430072, PR China b Department of Civil and Environmental Engineering, National University of Singapore, Blk E1A, #07-03, 1 Engineering Drive 2, Singapore 117576, Singapore article info abstract

Article history: This paper reviews previous studies on developments and applications of response surface methods (RSMs) in Received 1 June 2015 different slope reliability problems. Based on the review, four types of soil slope reliability analysis problems Received in revised form 9 August 2015 are identified from the literature, including single-layered soil slope reliability problem ignoring spatial variabil- Accepted 15 September 2015 ity, single-layered soil slope reliability problem considering spatial variability, multiple-layered soil slope reliabil- Available online 24 September 2015 ity problem ignoring spatial variability, and multiple-layered soil slope reliability problem considering spatial variability, which are referred to as “Type I–IV problems” in this study. Then, the computational efficiency and Keywords: Slope stability accuracy of four commonly-used RSMs (namely single quadratic polynomial-based response surface method Uncertainty (SQRSM), single stochastic response surface method (SSRSM), multiple quadratic polynomial-based response Reliability analysis surface method (MQRSM), and multiple stochastic response surface method (MSRSM)) are systematically Response surface method compared for cohesive and c–ϕ slopes, and their feasibility and validity in the four types of slope reliability Computational efficiency problems are discussed. Based on the comparison, some suggestions for selecting relatively appropriate RSMs Accuracy in slope reliability analysis are provided: (1) SQRSM is suggested as a suitable method for the single-layered soil slope reliability problem ignoring spatial variability (i.e., Type I problem); (2) MQRSM is applicable to the multiple-layered soil slope reliability problem ignoring spatial variability (i.e., Type III problem); and (3) MSRSM is suggested to solve slope reliability problems (including single-layered and multiple-layered slopes) considering spatial variability (i.e., Type II and IV problems). © 2015 Elsevier B.V. All rights reserved.

1. Introduction In addition to the aforementioned reliability methods, response surface methods (RSMs) have been used for slope reliability problems with Reliability analysis of soil slopes has gained considerable attention in implicit performance functions (e.g. Wong, 1985; Xu and Low, 2006; Ji the geotechnical reliability community over the past few decades and Low, 2012; Zhang et al., 2011b, 2013b; Jiang et al., 2014, 2015; Li (e.g., Baecher and Christian, 2003; Low and Tang, 2004; Cho, 2007, et al., 2015a,b; Li and Chu, 2015). RSMs have been proved to be an effi- 2009, 2010, 2013; Fenton and Griffiths, 2008; Ching et al., 2009; Wang cient method for slope reliability analysis. For instance, Wong (1985) et al., 2010, 2011; Ji and Low, 2012; Ji, 2014; Zhang et al., 2013a,b; applied RSM to evaluate the reliability of a homogeneous slope. Xu Jiang et al., 2014, 2015; Li et al., 2011, 2014, 2015a). Many reliability and Low (2006) used RSM to approximate the performance function methods have been proposed for slope reliability analysis in literature, of slope stability in slope reliability analysis, in which the response sur- such as the first-order second moment method (FOSM) (e.g., Christian face is taken as a bridge between stand-alone numerical packages and et al., 1994; Hassan and Wolff, 1999; Duncan, 2000; Xue and Gavin, spreadsheet-based reliability analysis. Recently, several researchers 2007; Suchomel and Mašin, 2010), first-order reliability method (e.g., Zhao, 2008; Li et al., 2013; Samui et al., 2013) proposed a support (FORM) (e.g., Low and Tang, 1997, 2004; Low, 2007; Cho, 2007; Hong vector machine (SVM)-based RSM to approximate implicit performance and Roh, 2008; Ji, 2014; Zeng and Jimenez, 2014), second-order reliabil- function using a small set of actual values of the performance function. ity method (SORM) (e.g., Cho, 2009; Low, 2014), and Monte Carlo Sim- Taking the radial basis function neural network (RBFN) as an approxi- ulation (MCS) (e.g., El-Ramly et al., 2002, 2005; Griffiths and Fenton, mate response surface function for the actual performance function, 2004; Hsu and Nelson, 2006; Cho, 2007, 2010; Huang et al., 2010, Tan et al. (2011) discussed similarities and differences between RBFN- 2013; Tang et al., 2015; Li et al., 2015c) and its advanced variants based RSMs and SVM-based RSMs, which indicated that there is no sig- (e.g., Ching et al., 2009; Wang et al., 2010, 2011; Li et al., 2015d). nificant difference between them. To reduce the number of evaluations of the actual performance function, Tan et al. (2013) proposed two new sampling methods and a hybrid RSM. Similar to SVM-based RSM, rele- ⁎ Corresponding author at: State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan 430072, PR China. vance vector machine (RVM)-based FOSM is adopted to build a RVM E-mail address: [email protected] (Z.-J. Cao). model to predict the implicit performance function and evaluate the

http://dx.doi.org/10.1016/j.enggeo.2015.09.003 0013-7952/© 2015 Elsevier B.V. All rights reserved. 4 D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14 partial derivatives with sufficient accuracy (Samui et al., 2011). In addi- Ji et al. (2012) made a first attempt to solve slope reliability with spatial tion, the artificial neural network (ANN) technique (e.g., Cho, 2009; variability by the RSM with second-order polynomial approximate Chen et al., 2011), the Gaussian process regression (Kang et al., 2015), function without cross terms. Based on the stochastic response surface Artificial bee colony (ABC) algorithm optimized support vector regres- method (SRSM), Jiang et al. (2014) proposed a non-intrusive stochastic sion (SVR) (Kang and Li, 2015), the high dimensional model representa- finite element method for slope reliability analysis considering spatial tion (HDMR) (Chowdhury and Rao, 2010) and neural networks (NN)- variability in shear strength parameters, by which the system reliability based RSM (Piliounis and Lagaros, 2014) can be also used to establish of soil slopes considering spatial variability is evaluated by the multiple a relationship between the factor of safety and soil parameters. Luo stochastic response surface method (e.g., Jiang et al., 2015; Li et al., et al. (2012a,b) and Zhang et al. (2011a) adopted Kriging-based re- 2015a; Li and Chu, 2015). sponse surface to simulate performance functions and demonstrated Based on the above studies, it can be seen that significant advances its applications in solving several geotechnical reliability problems have been made in applications of RSMs in soil slope reliability analysis. (Zhang et al., 2013a). Yi et al. (2015) indicated that the particle swarm Essentially, the RSM uses a computationally efficient model to approxi- optimization–Kriging model has a good curve fitting performance. mate the original analysis model (e.g., limit equilibrium analysis or Recently, an important advance in slope reliability analysis using finite element analysis). Then, slope reliability analysis is carried out RSM is that multiple response surface methods were proposed to eval- based on the explicit performance function represented by the RSM. uate system reliability of slope stability. For example, Zhang et al. Table 1 summarizes the applications of RSM-based reliability methods (2011b) first proposed the multiple response surfaces method for in soil slope reliability analyses. These references are listed in a slope reliability analysis. Ji and Low (2012) constructed a group of strat- chronological order. The soil slope reliability problems concerned in the ified response surfaces corresponding to the most probable failure previous studies on RSMs can be divided into four categories according modes, and slope system reliability is evaluated based on these strati- to the probabilistic model of soil properties (e.g., random variable or ran- fied response surfaces. Using the quadratic polynomial-based response dom field models) and slope types (e.g., single-layered or multiple- surface method, Zhang et al. (2013b) extended the Hassan and Wolff layered): (1) single-layered soil slope reliability problem ignoring method into a practical tool for system reliability analysis. spatial variability (i.e., Type I problem); (2) single-layered soil slope reli- The aforementioned studies have not considered the spatial variabil- ability problem considering spatial variability (i.e., Type II problem); ity when the RSMs are used to evaluate slope reliability problems. (3) multiple-layered soil slope reliability problem ignoring spatial

Table 1 Summary of applications of RSMs in soil slope reliability analyses.

Paper Authors Year Types of response surfaces Single Multiple Spatial Slope type Deterministic slope ID response response variability stability analysis surface surfaces No Yes Single-layered Multiple-layered

1 Wong 1985 Quadratic polynomial √√√ FEM 2 Xu and Low 2006 Quadratic polynomial without cross terms √√ √LEM (Spencer), FEM 3 Zhao 2008 SVM-based response surface √√√√LEM (Simplified Bishop, Spencer) 4 Cho 2009 ANN-based response surface √√ √FDM 5 Chowdhury 2010 High dimensional model representation √√ √LEM (Simplified Bishop, and Rao Janbu, Morgenstern–Price, Spencer, GLE) 6 Chen et al. 2011 SVM-based response surface √√ √LEM (Morgenstern–Price) 7 Tan et al. 2011 RBFN and SVM-based response surface √√ √FEM 8 Samui et al. 2011 RVM-based response surface √√√ LEM (Simplified Bishop) 9 Samui et al. 2013 LSSVM-based response surface √√√ LEM (Simplified Bishop) 10 Luo et al. 2012a Kriging-based response surface √√√√FDM 11 Luo et al. 2012b Kriging-based response surface √√√√FEM 12 Ji et al. 2012 Quadratic polynomial without cross terms √√√ LEM (Spencer) 13 Ji and Low 2012 Quadratic polynomial without cross terms √√ √ LEM (Ordinary, Spencer) 14 Zhang et al. 2011a Kriging-based response surface √√ √FDM 15 Zhang et al. 2011b Quadratic polynomial without cross terms √√√ √ LEM (Morgenstern–Price) 16 Zhang et al. 2013a Classical RSM without cross terms, quadratic √√ √LEM (Simplified Bishop) polynomial without cross terms, Kriging-based response surface 17 Zhang et al. 2013b Quadratic polynomial without cross terms √√ √ LEM (Simplified Bishop) 18 Li et al. 2013 Updated SVM-based response surface √√ √LEM (Simplified Bishop, Spencer) 19 Tan et al. 2013 Quadratic polynomial √√ √LEM (Morgenstern–Price) 20 Piliounis 2014 NN-based response surface √√√ LEM (Simplified Bishop) and Lagaros 21 Jiang et al. 2014 Hermite polynomial chaos expansion √√√ LEM (Morgenstern–Price) 22 Jiang et al. 2015 Hermite polynomial chaos expansion √√√ LEM (Simplified Bishop) 23 Li et al. 2015a Quadratic polynomial without cross terms √√√√ LEM (Simplified Bishop) 24 Li and Chu 2015 Quadratic polynomial without cross terms √√ √ LEM (Ordinary) 25 Yi et al. 2015 PSO Kriging based response surface, classical √√ √FDM RSM without cross terms 26 Kang et al. 2015 GPR-based response surface √√ √LEM (Simplified Bishop) 27 Kang and Li 2015 ABC-SVR response surface √√ √LEM (Simplified Bishop)

Note: SVM = support vector machine; ANN = artificial neural network; RBFN = radial basis function neural network; RVM = relevance vector machine; LSSVM = least square support vector machine; NN = neural networks; PSO = particle swarm optimization; GPR = Gaussian process regression; LEM = limit equilibrium method; FEM = finite element method; FDM = finite difference method. ABC-SVR = artificial bee colony algorithm optimized support vector regression. D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14 5

Table 2 Four types of soil slope reliability problems and the corresponding adopted RSMs.

Problem Response surfaces Uncertainty Types of response surface Uncertainty type models methods propagation methods

I Quadratic polynomial (Wong, 1985) Random Single response surface, multiple MCS, FOSM, SVM-based response surface (Zhao, 2008) variable response surface FORM RVM-based response surface (Samui et al., 2011) Quadratic polynomial without cross terms (Zhang et al., 2011a) Kriging-based response surface (Luo et al., 2012a, b) LSSVM-based response surface (Samui et al., 2013) NN-based response surface (Piliounis and Lagaros, 2014) II Hermite polynomial chaos expansion (Jiang et al., 2014, 2015) Random Single response surface, multiple MCS, FORM Quadratic polynomial without cross terms (Ji et al., 2012; Li et al., 2015a) ﬁeld response surface III Quadratic polynomial without cross terms (Xu and Low, 2006; Zhang et al., 2011b, 2013a,b; Ji Random Single response surface, multiple MCS, FOSM, and Low, 2012; Tan et al., 2013) variable response surface FORM SVM-based response surface (Zhao, 2008; Chen et al., 2011; Tan et al., 2011) ANN-based response surface (Cho, 2009) High dimensional model representation (Chowdhury and Rao, 2010) RBFN-based response surface (Tan et al., 2011) Kriging-based response surface (Zhang et al., 2011a, 2013a; Luo et al., 2012a,b) Quadratic polynomial with cross terms (Tan et al., 2013) Classical RSM without cross terms (Zhang et al., 2013a) Updated SVM-based response surface (Li et al., 2013) PSO Kriging based response surface, Classical RSM without cross terms (Yi et al., 2015) GPR based response surface (Kang et al., 2015) ABC-SVR response surface (Kang and Li, 2015) IV Quadratic polynomial without cross terms (Li et al., 2015a; Li and Chu, 2015) Random Multiple response surface MCS ﬁeld

Note: MCS = Monte Carlo Simulation; FORM = ﬁrst-order reliability method; FOSM = ﬁrst-order second moment.

variability (i.e., Type III problem); and (4) multiple-layered soil slope reli- 2.1. Type I problem: single-layered soil slope ignoring spatial variability ability problem considering spatial variability (i.e., Type IV problem), as shown in Table 2. Type I problem is a relatively simple case where the reliability anal- This paper aims to review previous studies on applications of RSMs in ysis of a homogeneous soil slope is concerned. For such a reliability the four types of slope reliability problems and to investigate the capacity problem, Wong (1985) adopted the quadratic polynomial-based RSM and validity of four commonly-used RSMs in solving different slope reli- for evaluating the reliability of the single-layered soil slope without con- ability problems. The computational accuracy and efﬁciency of the four sideration of spatial variability. Subsequently, the Type I problem has RSMs are systemically compared using cohesive and c–ϕ slope examples. been extensively investigated (e.g., Zhao, 2008; Samui et al., 2011, To achieve such a goal, the paper is organized as follows. In Section 2,four 2013; Zhang et al., 2011a; Luo et al., 2012a,b; Piliounis and Lagaros, types of soil slope reliability problems are brieﬂy introduced. Then, single 2014) using different RSMs, such as SVM-based RSM, RVM-based response-surface methods and multiple response-surface methods are RSM, quadratic polynomial without cross terms, Kriging-based RSM, reviewed in Section 3.InSection 4, four benchmark reliability problems and NN-based RSM. For a homogenous soil slope, the slope failure (including two cohesive slopes and two c–ϕ slopes) are investigated to is dominated by the deterministic critical slip surface because the examine the capacity and validity of the selected RSMs. Finally, several factors of safety (FSs) of other slip surfaces are highly correlated recommendations for selecting relatively appropriate RSMs for different with the FS of the deterministic critical slip surface. Hence, the sys- slope reliability problems are provided. tem failure probability of slope stability can be well approximated by the failure probability of the critical slip surface (Zhang et al. 2. Major soil slope reliability analysis problems 2011b).

Soils are natural materials and their properties are affected by several 2.2. Type II problem: single-layered soil slope considering spatial factors during formation process, such as parent materials, weathering variability and erosion processes, transportation agents, and sedimentation condi- tions. Thus, soils exhibit a large degree of uncertainty (e.g. Cao and Inherentspatialvariability(ISV)isoneoftheprimarysourcesof Wang, 2013, 2014a; Ching and Phoon, 2014; Le, 2014; Li et al., 2014; geotechnical uncertainties, which significantly affects the slope sta- Lloret-Cabot et al., 2014; Jamshidi Chenari and Alaie, 2015). To effectively bility (e.g., Phoon and Kulhawy, 1999; Griffiths and Fenton, 2004; characterize the uncertainty in soil properties, probabilistic models are Cho, 2007, 2010; Srivastava and Babu, 2009; Srivastava et al., 2010; often applied to represent soil properties in slope reliability analyses Huang et al., 2010; Hicks and Spencer, 2010; Wang et al., 2011; Ji (e.g., Christian et al., 1994; Griffiths and Fenton, 2004; Srivastava and et al., 2012, Ji and Low, 2012; Jiang et al., 2014, 2015; Li et al., Babu, 2009; Srivastava et al., 2010; Suchomel and Mašin, 2010; Cao and 2015a; Li and Chu, 2015). When the spatial variability of soil proper- Wang, 2014b). Probabilistic models mainly include random variables ties is considered, there exist multiple failure modes, which will sig- model and random fields (e.g., Chowdhury and Xu, 1995; Phoon and nificantly influence the failure probability of slope stability (Wang Kulhawy, 1999; El-Ramly et al., 2002; Zhang et al., 2011a,b, 2013a,b; et al., 2011). Estimating the reliability of slope stability considering Jiang et al., 2014, 2015; Li et al., 2011, 2014, 2015a; Li and Chu, 2015). spatial variability is, hence, more difficult than that for a single- As shown in Table 2, the problems involved in soil slope reliability analy- layered soil slope ignoring spatial variability. Jiang et al. (2014, ses using RSMs are divided into four categories, which are briefly de- 2015) made use of the Hermite polynomial chaos expansion-based scribed in the following four subsections, respectively. RSM to solve the reliability of a single-layered soil slope considering 6 D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14 spatial variability (i.e., Type II problem). Similarly, Li et al. (2015a) polynomial function (e.g., Bucher and Bourgund, 1990; Zhang et al., used the quadratic polynomial without cross terms-based RSM to 2011b, 2013b): analyze the Type II problem. Xn Xn FSQ ðÞ¼X a þ b x þ c x2 ð1Þ 2.3. Type III problem: multiple-layered soil slope ignoring spatial 0 i i i i i¼1 i¼1 variability where FSQ(X)istheFS for the given slip surface estimated from the Unlike the aforementioned two types of problems, the Type III prob- quadratic polynomial response surface; X =(x1,⋯, xi,⋯, xn)isthe lem focuses on the reliability of a multiple-layered soil slope ignoring vector of input random variables, in which n is the number of the spatial variability in each soil layer. Such a reliability problem has input random variables or the number of random field elements; been extensively investigated in literature, as shown in Table 2.Unlike T a =(a0, b1,⋯, bn, c1,⋯, cn) is the vector of unknown coefficients. the single-layered soil slope, multiple failure modes often exist in the To determine the unknown coefficients, the FS of a slip surface is multiple-layered soil slope (e.g., Zhang et al., 2011b, 2013b; Low et al., evaluated at (2n + 1) points: {μx1, μx2,⋯, μxn}, {μx1 ± kσx1, μx2,⋯, μxn}, 2011; Ji and Low, 2012; Cho, 2013; Zeng and Jimenez, 2014; Kang {μx1, μx2 ± kσx2,⋯, μxn},⋯, and {μx1, μx2,⋯, μxn ± kσxn}, where k is set et al., 2015; Kang and Li, 2015). Only using the deterministic critical as 2 in this study. A regression-based approach is used to compute the slip surface to estimate slope failure probability leads to underestima- unknown coefficients a (e.g., Li et al., 2015a). tion of the slope failure rate because the system effects of all the possible Under the Hermite polynomial chaos expansion, the FS for a given failure modes are not taken into account (e.g., Oka and Wu, 1990; slip surface is calculated by (e.g. Huang et al., 2009; Li et al., 2011; Chowdhury and Xu, 1994, 1995; Zhang et al., 2011b). Therefore, all Al-Bittar and Soubra, 2013; Jiang et al., 2014, 2015): the representative failure modes of a multi-layered soil slope shall be taken into account to achieve a more realistic evaluation of system reli- Xn Xn Xi1 ability of a multi-layered soil slope. FSHðÞ¼ξ a Γ þ a Γ ξ þ a Γ ξ ; ξ 0 0 i1 1 i1 i1i2 2 i1 i2 i1¼1 i1¼1 i2¼1 Xn Xi1 Xi2 2.4. Type IV problem: multiple-layered soil slope considering spatial þ Γ ξ ; ξ ; ξ þ ⋯ ai i 3 i i i variability 1 2 i3 1 2 3 i1¼1 i2¼1 i3¼1 Xn Xi1 Xi2 Xin−1 Type IV problem is the most complicated case among the four þ ⋯ a ;⋯; Γn ξ ; ξ ; ⋯; ξ ð2Þ i1 i2 in i1 i2 in ¼ ¼ ¼ ¼ major problems of soil slope reliability analyses. Such a problem con- i1 1 i2 1 i3 1 in 1 cerns with the reliability of a multiple-layered soil slope considering a ⋯ ⋯ spatial variability. Li et al. (2015a) and Li and Chu (2015) applied where n is the total number of random variables, =(a0,ai1, ,ai1i2, ,in) fi ξ ξ ξ ⋯ ξ quadratic polynomial without cross terms to deal with this problem. is the unknown coef cients to be evaluated; =( i1, i2, , in)isthe Li and Chu (2015) found that when the spatial variability is ignored, vector of independent standard normal variables representing the un- the number of representative slip surfaces is equal to the number of certainties in the input parameters; and Γn(⋅) is the multi-dimensional soil layers. The number of multiple response surfaces also highly de- Hermite polynomials of order p, in which p is taken as 2 in this study. pends on the spatial variability of soil properties. Li et al. (2015a) Eq. (2) is often referred to as the stochastic response surface that proposed a multiple response surface method in which the extended approximates the FS. Cholesky-decomposition technique is used for discretizing the cross- For a single quadratic response surface method (SQRSM), the surro- correlated non-Gaussian random fields. For a multiple-layered soil gate performance function given by Eq. (1) is constructed between the slope considering spatial variability, the number of potential failure minimum factor of safety (FSmin) among all potential slip surfaces and modes and the locations of representative slip surfaces depend on the original random variables. The single stochastic response surface the spatial variability and the stratification of the slope. Multiple po- method (SSRSM) shown in Eq. (2) is also a surrogate model for slope tential failure modes resulted from spatial variability and stratifica- stability analysis that provides an explicit and approximate perfor- tion in the Type IV problem render the slope reliability analysis a mance function to predict the FS of slope stability (e.g., Huang et al., more challenging task. 2009; Li et al., 2011; Jiang et al., 2014, 2015). In the SSRSM, the surrogate

performance function is constructed between the FSmin among all 3. Response surface methods (RSMs) potential slip surfaces and the independent standard normal random variables ξ. The aforementioned four types of slope reliability problems and the corresponding RSMs used in literature are summarized in Table 2.Itis 3.2. Multiple response surface methods (MQRSM & MSRSM) shown that only the quadratic polynomial without cross terms-based RSM (e.g., Xu and Low, 2006; Zhang et al., 2011a,b, 2013a,b; Ji and Unlike the single response surface method, the multiple response Low, 2012; Tan et al., 2013; Li et al., 2015a; Li and Chu, 2015) is used surface methods are often used to solve the slope reliability problem to solve all the four types of slope reliability analysis problems in litera- with multiple failure modes. The multiple response surface methods ture. In addition, the Hermite polynomial chaos expansion-based RSM is consist of the multiple quadratic response surface method (MQRSM) used to analyze the Type II problem by Jiang et al. (2014, 2015) though it and multiple stochastic response surface method (MSRSM), which are can also be applied to the other three types of problems in theory. The briefly introduced as below. remaining part of this paper hence focuses on comparing the feasibility MQRSM makes use of multiple quadratic response surfaces to ap- and validity of the two polynomial-based response surfaces (i.e., the proximate the relationship between FSs of all the potential slip surfaces quadratic polynomial without cross terms and the Hermite polyno- and random variables in their original space, each of which corresponds mial chaos expansion, as shown by bold font in Table 2) in solving to the original performance function of one potential slip surface. After the Type I–IV problems. obtaining multiple quadratic response surfaces, NMC realizations of random fields XNG,F are generated using the extended-Cholesky 3.1. Single response surface method (SQRSM & SSRSM) decomposition technique (e.g., Li et al. 2015a). If spatial variability of NG,V soil properties is not considered, NMC random variable samples X For a potential slip surface of a soil slope, the relationship between are obtained using the Nataf transformation (e.g., Lebrun and Dutfoy, the FS and the input parameters can be approximated by a quadratic 2009; Li et al., 2011; Tang et al., 2013). Substituting NMC realizations of D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14 7 random fields or random variables into multiple quadratic response and it is calculated as (e.g., Schuëller and Pradlwarter, 2007; Au et al., Q ðXNGÞ 2007): surfaces yields NMC estimates (i.e., min FSj )ofFSmin,where j¼1;2;⋯;N s pffiffiffiffiffiffi Ns is the number of potential slip surfaces. Then, the number of FSmin Δ ¼ COV P f Ne ð3Þ values less than 1.0 is determined and is denoted as NF.Thesystem failure probability of slope stability is calculated by P = N ∕ N . f F MC where N is the equivalent number of evaluations of the original perfor- Unlike the MQRSM, the MSRSM is constructed between FS values of e mance function (e.g., limit equilibrium analysis) in each run, which is all potential slip surfaces and the input parameters ξ in independent calculated by. normal space. To construct Ns surrogate performance functions between FS values of all the potential slip surfaces and the input parameters ξ Ne ¼ Npe þ t0∕t ð4Þ when spatial variability of soil properties is considered, Np realizations of independent or cross-correlated random fields are generated using in which N is the number of evaluations of the original perfor- the Karhunen–Loève expansion technique (e.g., Ghiocel and Ghanem, pe mance functions needed to construct single or multiple response sur- 2002; Phoon et al., 2002; Ghanem and Spanos, 2003; Vořechovský, faces; t is the computational time needed to perform one evaluation of 2008; Cho, 2010). Note that the correlation function is needed in gener- the original performance function; and t′ is the computational time of ating random field. For example, the squared exponential (SQX) auto- subsequent MCS based on response surfaces that have been construct- correlation function (ACF) (e.g., Cao and Wang, 2014a; Li et al., 2015a) ed. A relatively small value of Unit COV indicates a relatively high is adopted to model spatial variability in this study. Then the limit equi- computational efficiency. librium methods (e.g., simplified Bishop method) of slope stability with For the soil slope reliability analysis, the limit state function (LSF) is the N realizations of random fields are applied to calculate the FS values p usually defined as g(X)=FS(X) − 1 = 0. The FS(X)isthefactorofsafety of each potential slip surface. The unknown coefficients a in Eq. (2) are calculated by a deterministic slope stability analysis method, e.g., limit determined based on the N random samples ξ and the corresponding p equilibrium method (LEM), which is used to calculate FS values for con- FS values. The above procedure is repeated for N potential slip surfaces. s struction of response surfaces in the four examples. In other words, the Then, the surrogate performance function is constructed between the FS response surfaces in the four examples are calibrated using the limit values and the input parameters ξ for each potential slip surface. If the equilibrium analysis of slope stability. spatial variability of soil properties is not considered, the input soil parameters used for limit equilibrium analysis are obtained via the Nataf 4.1. Applications of RSMs to reliability analysis of two cohesive slopes transformation. After obtaining MSRSM, MCS is carried out to generate N realizations of independent standard normal samples ξ. Substitut- MC 4.1.1. Type I problem: a single-layered cohesive slope (Example #1, no ISV) ing each realization of input samples ξ , k =1,2,⋯, N , into multiple k MC The profile of the undrained slope (Example #1) in a uniform layer is ¼ H stochastic response surfaces leads to NMC values of FSmin min FSj j¼1;2;⋯;Ns shown in Fig. 1, which has been studied by Cho (2010) and Jiang et al. ðξ Þ (2015). The cohesive slope has a height of 5 m, a slope angle of 26.6° k . Similar to MQRSM, the number (i.e., NF)ofFSmin values less than 1.0 and a total unit weight of 20 kN/m3. The undrained shear strength c is determined, and Pf is then calculated as NF ∕ NMC. u is assumed to be lognormally distributed with a mean cu of 23 kPa and a COV of 0.3. Based on the mean value of the undrained shear strength

4. Soil slope reliability analysis using RSMs cu,theFSmin is obtained as 1.356 using simplified Bishop method, which is consistent with the value reported by Cho (2010) and Jiang This section will examine the capacity of the aforementioned four et al. (2015). In the calculation, a total number of 4851 potential slip RSMs (i.e., SQRSM, SSRSM, MQRSM, and MSRSM) in solving different surfaces at different locations are generated using the “Entry and Exit” slope reliability problems (i.e., Type I–IV problems). The computational method in SLOPE/W (GEO-SLOPE International Ltd., 2012), as shown accuracy and efficiency of the RSMs are compared and discussed using in Fig. 1. four soil slope examples, namely Examples #1, #2, #3 and #4. For Table 3 summarizes the results of Example #1 for the Type I ̅ ̅ each soil slope example, the mean value P f of slope failure probability problem. It can be seen that the average values P f of the slope failure for each RSM is taken as the average value of Pf obtained from 20 probability are 0.173, 0.173, 0.172 and 0.172 for SQRSM, MQRSM, independent runs of the method. The mean value of P f is an important SSRSM and MSRSM, respectively. These values agree well with 0.173 index in evaluating the accuracy of a given RSM and indicates whether obtained from direct MCS in this study and are slightly lower than the the method is biased or not. The coefficient of variation of Pf, COV[Pf], value 0.186 reported by Cho using FORM (2010). Since the values of ̅ is also obtained using the results from 20 independent runs. The Unit P f obtained from the four RSMs are in good agreement with that obtain- COV (i.e., Δ) is used to quantify the efficiency of RSMs in this study, ed using direct MCS and simplified Bishop method, all of the four RSMs

Fig. 1. The geometry of the slope (Example #1). 8 D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14

Fig. 2. A typical realization of random ﬁeld (Example #1).

Fig. 3. The geometry of three-layered slope and 14,896 potential slip surfaces (Example #2).

provide reasonably accurate estimates of P f of the Type I problem in this cohesive soil layer is discretized into 910 elements with a side length example. The SQRSM has the smallest value of Unit COV = 0.003 among of 0.5 m. Fig. 2 shows a typical realization of the random field of cu. the four RSMs. Compared with the SQRSM, the computational efficiency Table 4 presents the results of the Type II problem in Example #1. of the MCS is low, which produces a relatively high value of Unit COV = When the four types of RSMs are used as surrogate models for the 0.26. Though the four RSMs have comparable accuracy, the SQRSM has implicit LSF for slope reliability analysis, the mean values of slope failure the highest calculation efficiency among the four RSMs. Therefore, it is probability P f are 0.276, 0.082, 0.081 and 0.079 for the SQRSM, MQRSM, ̅ recommended to be used for the Type I problem with a single-layered SSRSM and MSRSM, respectively. The values of P f obtained from cohesive slope ignoring spatial variability. MQRSM, SSRSM and MSRSM generally agree well with 0.076 reported by Cho (2010) (see Table 5), 0.083 from reported by Jiang et al. (2015) 4.1.2. Type II problem: a single-layered cohesive slope (Example #1, (see Table 5), and 0.078 obtained from direct MCS and simplified Bishop consider ISV) method. On the other hand, using SQRSM leads to significant overestima- In this section, the slope profile of the undrained slope shown in tion of slope failure probability in the Type II problem of this example. Fig. 1 is used again. Unlike the Type I problem in Example #1, the spatial This indicates that the SQRSM cannot capture the feature of LSF for a variability of undrained shear strength cu is considered in the Type II soil slope when the spatial variability of undrained shear strength problem. In other words, the Type II problem in Example #1 focuses cu is considered, which subsequently results in a misleading on evaluating the reliability of the undrained slope considering the spa- estimate of slope failure probability. Note that both the MQRSM tial variability of undrained shear strength. The statistical properties of and MSRSM produce relatively small values of Unit COV of 0.446 the other soil parameters remain the same as those in Type I problem. and 0.327, which are significantly smaller than 2.860 by direct

The spatial variability of cu is modeled using a 2D lognormal stationary MCS. Compared with the MQRSM and MSRSM, the SSRSM leads to random field with a horizontal autocorrelation distance θln,h of 20 m, a slightly higher Unit COV of 0.694. Based on these results, the fi and a vertical autocorrelation distance θln,v of 2.0 m. The random field MQRSM and MSRSM are more ef cient than SSRSM and direct MCS. can be simulated using the mid-point method (e.g., Srivastava et al., Thus, both the MQRSM and MSRSM are recommended to be used 2010; Suchomel and Mašin, 2010; Wang et al., 2011), in which the for dealing with the Type II problem.

Fig. 4. A typical realization of random ﬁeld (Example #2). D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14 9

Table 7 presents the results of the Type III problem in Example #2. For comparison, Table 8 also summarizes the reliability results reported

in literature. The mean estimates of slope failure probability P f are 0.216, 0.184, 0.219 and 0.240 for the SQRSM, MQRSM, SSRSM, and MSRSM, respectively. In general, they compare favorably with results (around 0.19) obtained by Zhang et al. (2013b) and Kang et al. (2015), and 0.197 using direct MCS in this study though the estimates

of P f from SQRSM, SSRSM, and MSRSM are slightly higher. It seems that all the four RSMs can provide reasonably accurate estimates of slope failure probability for the Type III problem. Further examination on the Unit COV in Table 7 for the four RSMs shows that the SQRSM and MQRSM have higher computational efﬁciency than SSRSM and MSRSM. Therefore, the SQRSM and MQRSM are suggested to be used to solve the Type III problem in cohesive soils. Fig. 5. The geometry of the slope and 5491 potential slip surfaces (Example #3).

4.1.4. Type IV problem: a three-layered cohesive slope (Example #2, consider ISV)

In the Type IV problem, the undrained shear strength Su1, Su2 and Su3 of three clay layers of Example #2 is modeled by three two-dimensional lognormal stationary random ﬁelds, respectively. The horizontal auto-

correlation distance θln,h and vertical autocorrelation distance θln,v are taken as 20 m and 2.0 m, respectively. The three random ﬁelds of Su1, Su2 and Su3 are assumed to be mutually independent, and are discretized into 495, 939 and 1080 elements, respectively, with a side length of 0.5 m as shown in Fig. 4. Table 9 summarizes the reliability results of the Type IV problem in

Example #2. The mean estimates of slope failure probability P f are 0.027, 0.057, 0.088 and 0.031 for the SQRSM, MQRSM, SSRSM, and MSRSM, respectively. Taking the slope failure probability 0.057 obtained from direct MCS and simplified Bishop method as an “exact” solution, Fig. 6. A typical realization of random field (Example #3). the four RSMs provide estimates of slope failure probability that are generally comparable with that obtained from direct MCS. The SQRSM and MSRSM may slightly underestimate the slope failure probability, 4.1.3. Type III problem: a three-layered cohesive slope (Example #2, no ISV) while the SSRSM may slightly overestimate the probability of failure. The first example, namely Example #1, focuses on the single-layered The slight differences in estimates of slope failure probability between cohesive slope. It is well-known that multiple failure modes may be in- the four RSMs and direct MCS may be attributed to two aspects. First, volved in a multiple-layered slope. For this reason, a three-layered cohe- the potential failure modes for the multiple-layered slopes consider- sive slope (Example #2) is used as the second example to investigate ing spatial variability become more complicated. Second, the SQRSM, the capacity of the four RSMs in solving the Type III problem. Example SSRSM and MSRSM may not completely capture the feature of LSFs #2 is adopted from Feng and Fredlund (2011), Zhang et al. (2013b), Li for the potential slip surfaces. It is noted that the MSRSM has the and Chu (2015) and Kang et al. (2015). The geometry of the slope is lowest value of Unit COV (i.e., 0.69) among the four RSMs, which shown in Fig. 3 and the soil parameters are summarized in Table 6. indicates that the MSRSM is more efficient than the other three The undrained shear strength Su1, Su2 and Su3 of three clayed layers RSMs. It is, therefore, recommended that the MSRSM is used for the are modeled as independent random variables. Based on mean values Type IV problem. of the undrained shear strength, the FSmin is obtained as 1.282 using simplified Bishop method, which is the same as that reported by Kang et al. (2015). As shown in Fig. 3, a total number of 14,896 potential 4.1.5. Suggested RSMs for reliability analysis of cohesive slope slip surfaces are randomly generated to cover the whole potential Taking two cohesive soil slopes as examples (i.e., Examples #1 and failure area using the “Entry and Exit” method in SLOPE/W. #2), four types of slope reliability problems (i.e., Type I–IV problems)

Fig. 7. The geometry of the slope and 25,947 potential slip surfaces (Example #4). 10 D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14

Fig. 8. A typical realization of random ﬁeld (Example #4).

are investigated using the SQRSM, MQRSM, SSRSM and MSRSM. The 4.2. Applications of RSMs to reliability analysis of two c–ϕ slopes recommendations for selecting the relatively appropriate RSMs for each type of problem are summarized in Table 10. Note that the RSMs 4.2.1. Type I problem: a single-layered c–ϕ slope (Example #3, no ISV) marked with “√” indicate that they can provide reasonably accurate es- The above two examples (e.g., Examples #1, #2) are under timates of slope failure probability in the corresponding slope reliability undrained condition, where ϕ is taken as 0. Hence, simplified Bishop problems. As mentioned previously, the Unit COV is an important indi- method is identical to the ordinary slice method for Examples #1 and cator for evaluating computational efficiency. The computational effi- #2. In such a case, the performance function G for a potential slip surface ciency associated with the four RSMs is ranked from 1 to 4. As the is linear (Chowdhury and Xu, 1994). For further illustration, a single- rank number increases from 1 to 4, the corresponding RSM becomes layered c–ϕ slope (Example #3) shown in Fig. 5 with a height of 10 m less efficient. Based on the trade-off between computational accuracy and a slope angle of 45° is explored, which has been analyzed by Cho and efficiency, the appropriate RSMs are recommended for different (2010) and Jiang et al. (2015). Statistics of soil parameters are shown slope reliability problems, as shown in Table 10. For the single-layered in Table 11. Based on the mean values of soil parameters, the FSmin is cohesive soil slope reliability problem ignoring spatial variability, calculated using simplified Bishop method, and it is equal to 1.206, the SQRSM is recommended. For the single-layered cohesive soil slope which agrees well with those (i.e., 1.206 and 1.204, respectively) reliability problem considering spatial variability, both the MQRSM reported by Jiang et al. (2015) and Cho (2010). and MSRSM can be applied. In addition, both the SQRSM and MQRSM Table 12 summarizes the results of the Type I problem in Example are applicable to the Type III problem in cohesive soils, i.e., multiple- #3 illustrated by a single-layered c–ϕ slope ignoring spatial variability layered cohesive soil slope reliability problem ignoring spatial variability. For multiple-layered cohesive slope reliability analysis considering Table 5 spatial variability, the MSRSM can be adopted. Results for Type II problem obtained from the references (Example #1, ISV).

Method Pf Source MCS (100,000 samples) 0.076 Cho (2010) Table 3 MSRSM 0.079 Jiang et al. (2015) Results for Type I problem: a single-layered cohesive slope (Example #1, No ISV). MCS (1000 samples) 0.083 Jiang et al. (2015)

Methods Equivalent number of Mean COV[Pf]Unit Source performance function value of P f COV evaluations of each run Table 6 Statistical properties of soil parameters in Example #2. SQRSM 3.02 0.173 0.002 0.003 This study γ 3 MQRSM 4.47 0.173 0.003 0.006 Slope layers Variable Unit weight, (kN/m ) Undrained strength, Su (kPa) SSRSM 3.01 0.172 0.022 0.038 Mean COV Distribution MSRSM 3.57 0.172 0.022 0.042 MCS 100.00 0.173 0.03 0.26 Clay 1 Su1 18 18 0.3 Lognormal Clay 2 Su2 18 20 0.2 Lognormal Note: The method marked by bold font is the recommended RSM. This usage is also valid Clay 3 Su3 18 25 0.3 Lognormal for the results shown in Tables 3, 4, 7, 9, 12, 13, 16 and 17.

Table 4 Table 7 Results for Type II problem: a single-layered cohesive slope (Example #1, ISV). Results for Type III problem: a three-layered cohesive slope (Example #2, No ISV).

Methods Equivalent number of Mean COV[Pf] Unit Source Methods Equivalent number of Mean COV[Pf] Unit Source performance function value of P f COV performance function value of P f COV evaluations of each run evaluations of each run

SQRSM 1821.30 0.276 0.008 0.337 This study SQRSM 7.05 0.216 0.002 0.005 This study MQRSM 1823.75 0.082 0.010 0.446 MQRSM 15.61 0.184 0.002 0.009 SSRSM 1000.04 0.081 0.022 0.694 SSRSM 10.04 0.219 0.534 1.694 MSRSM 1001.47 0.079 0.010 0.327 MSRSM 15.37 0.240 0.522 2.047 MCS 1000.00 0.078 0.090 2.860 MCS 100.00 0.197 0.095 0.952 D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14 11

Table 8 Table 11 Results for Type III problem obtained from the references (Example #2, No ISV). Statistical properties of soil parameters in Example #3.

Methods Pf Source Parameter Mean COV Distribution Correlation coefﬁcient MCS with 10,000 samplings 0.187 Zhang et al. (2013b) Cohesion, c (kPa) 10 0.3 Lognormal r = −0.7 Extended Hassan and Wolff method with 10,000 0.184 Zhang et al. (2013b) Friction angle, ϕ (°) 30 0.2 Lognormal samples Unit weight, γ (kN/m3)20 ―― ― MCS using GPR-based RSM with 10,000 samples 0.186 Kang et al. (2015) Note: The symbol “―” denotes the result is not applicable. MCS using GPR-based RSM with 20,000 samples 0.185 Kang et al. (2015) MCS using GPR-based RSM with 100,000 samples 0.186 Kang et al. (2015)

Table 12 Results for Type I problem: a single-layered c–ϕ slope (Example #3, No ISV).

Table 9 Equivalent number of Mean Results for Type IV problem: a three-layered cohesive slope (Example #2, ISV). Methods performance function COV[P ] Unit COV Source value of P f evaluations of each run f Methods Equivalent number of Mean COV[Pf] Unit Source performance function value of P f COV SQRSM 5.04 0.043 0.005 0.011 evaluations of each run MQRSM 8.25 0.042 0.005 0.014 SSRSM 6.01 0.040 0.752 1.843 This study SQRSM 5029.43 0.027 0.03 2.34 This study MSRSM 7.91 0.079 1.824 5.129 MQRSM 5033.10 0.057 0.02 1.43 MCS 250.00 0.039 0.131 2.067 SSRSM 800.08 0.088 0.08 2.19 MSRSM 801.50 0.031 0.02 0.69 MCS 1000.00 0.057 0.08 2.54 takes precedence over computational efficiency in choosing suitable RSMs among the four ones. Although the values of Unit COV of SSRSM ϕ of c and . The average values of failure probability P f obtained from and MSRSM are 2.56 and 3.57, respectively, which are higher than SQRSM, MQRSM and SSRSM are 0.043, 0.042 and 0.040, respectively, those of SQRSM and MSRSM, they still have distinct superiority over which agree well with 0.039 obtained using direct MCS and simplified direct MCS in the view of computational efficiency. Based on the Bishop method. However, the probability of failure (i.e., 0.079) obtained computational accuracy and efficiency, both of SSRSM and MSRSM are from MSRSM is obviously higher than that of direct MCS (e.g., 0.039). In suggested methods for dealing with the Type II problem in c–ϕ soils. addition, it is shown that SQRSM has the smallest Unit COV among the four RSMs (see column 5 of Table 12). It is recommended to be used 4.2.3. Type III problem: a multiple-layered c–ϕ slope (Example #4, no ISV) for the Type I problem in c–ϕ soil (i.e., single-layered c–ϕ slope ignoring Fig. 7 shows the geometry of the last slope example (Example #4), soil variability) though MQRSM and SSRSM also provide reasonably i.e., the Congress Street Cut, which has been analyzed by Oka and accurate slope failure probabilities for this type of problem. Wu (1990), Chowdhury and Xu (1994, 1995), Ching et al. (2009), Chowdhury and Rao (2010), and Ji and Low (2012). The cross section 4.2.2. Type II problem: a single-layered c–ϕ slope (Example #3, consider of the slope consists of an upper layer of sand at the top and three clay ISV) layers, each of which has its corresponding c and ϕ. Table 15 presents In the Type I problem of Example #3, the ISV of c and ϕ is not taken the statistics of shear strength parameters (i.e., c and ϕ)andunitweight into account. In the Type II problem, their ISV is characterized by two (γ), which are adopted from Oka and Wu (1990) and Chowdhury and Xu (1994). Based on the mean values of soil properties, the FS is cross-correlated 2D lognormal stationary random fields c and ϕ. θln,h min calculated using simplified Bishop method, and it is 1.392. and θln,v of both c and ϕ are taken as 20 m and 2.0 m, respectively. As shown in Fig. 6, the random fields are discretized into 1210 elements Table 16 shows the results of the Type III problem in Example #4. with a side length of 0.5 m. Compared with the result (i.e., 0.010) obtained from direct MCS and Table 13 summarizes the results of the Type II problem in Example simplified Bishop method, MQRSM provides an un-biased estimate of ̅ #3. Table 14 also summarizes the reliability results obtained from the Pf in the sense that the P f from MQRSM is equal to 0.010. As shown in other references for comparison. Compared with results (i.e., 0.0041) Table 16, MQRSM provides the most accurate estimate of slope failure obtained from direct MCS in this study and those (around 0.004) report- probability among the four RSMs. From the perspective of computation- ed by Cho (2010) and Jiang et al. (2015), SSRSM and MSRSM provide al accuracy, the estimates of slope failure probability from SQRSM, reasonably accurate estimates (i.e., 0.0052 and 0.0051, respectively) of SSRSM and MSRSM are somewhat biased. Note that the value of Unit slope failure probability. By contrast, both SQRSM and MQRSM overes- COV of MQRSM (i.e., 0.04) is much smaller than those of SRSM, timate the failure probability for the Type II problem in Example #3. MSRSM, and direct MCS (i.e., 4.20, 5.57, 9.98, respectively). This indi- This is mainly attributed to the fact that SQRSM and MQRSM cannot rea- cates that MQRSM can efficiently evaluate the slope failure probability sonably approximate the LSF of slope stability for a soil slope system for the Type III problem in c–ϕ soils compared with SRSM, MSRSM, when the spatial variability of c and ϕ is considered. In the view of com- and direct MCS. Based on both computational accuracy and efficiency, putational accuracy, SSRSM and MSRSM are appropriate methods for MQRSM is recommended to be used to solve the Type III problem in the Type II problem in c–ϕ soils. Note that computational accuracy c–ϕ soils.

Table 10 Recommendations of selecting appropriate RSMs for four types of reliability problems of a cohesive slope.

Problem type Indicator SQRSM MQRSM SSRSM MSRSM Recommended method

Accuracy √√ √√ Single-layered slope ignoring spatial variability (I) SQRSM Efficiency 1 2 3 4 Accuracy √√√ Single-layered slope considering spatial variability (II) MQRSM, MSRSM Efficiency 2 3 4 1 Accuracy √√ √√ Multiple-layered slope ignoring spatial variability (III) SQRSM, MQRSM Efficiency 1 2 3 4 Accuracy √√ √√ Multiple-layered slope considering spatial variability (IV) MSRSM Efficiency 4 2 3 1 12 D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14

Table 13 Table 17 Results for Type II problem: a single-layered c–ϕ slope (Example #3, ISV). Results for Type IV problem: a three-layered c–ϕ slope (Example #4, ISV).

Equivalent number of Equivalent number of Mean Unit Mean Unit Methods performance function COV[P ] Source Methods performance function COV[P ] Source value of P f COV value of P f COV evaluations of each run f evaluations of each run f

SQRSM 4541.61 0.0828 0.01 0.96 SQRSM 6918.84 0.0543 0.02 1.93 MQRSM 4547.58 0.0112 0.06 3.71 MQRSM 7064.21 0.0021 0.11 9.31 SSRSM 1000.23 0.0052 0.08 2.56 This study SSRSM 1000.07 0.0072 2.16 68.42 This study MSRSM 1020.06 0.0051 0.11 3.57 MSRSM 1013.77 0.0015 0.07 2.15 MCS 2500.00 0.0041 0.22 10.99 MCSa 70,000 0.0027 0.07 19.40

Note: a A single MCS with 70,000 samples. Table 14 Results for Type II problem obtained from the references (Example #3, ISV). –ϕ Methods Pf Source 4.2.5. Suggested RSMs for reliability analysis of c slope Similar to the recommendations shown in Section 4.1.5,therecom- MCS (50,000 samples) 0.0039 Cho (2010) MSRSM 0.0049 Jiang et al. (2015) mendations for selecting the relatively appropriate RSMs for slope reli- MCS (10,000 samples) 0.0044 Jiang et al. (2015) ability problems in c–ϕ soils are summarized in Table 18.TheSQRSMis MCS (50,000 samples) 0.0039 Jiang et al. (2015) recommended for the single-layered c–ϕ soil slope reliability problem ignoring spatial variability. For the single-layered c–ϕ soil slope reliability problem considering spatial variability, both SSRSM and MSRSM can 4.2.4. Type IV problem: a multiple-layered c–ϕ slope (Example #4, consider be applied. The MQRSM is applicable to the multiple-layered c–ϕ soil ISV) slope reliability problem ignoring spatial variability. For the multiple- This subsection focuses on the multiple-layered c–ϕ soil slope con- layered c–ϕ soil slope reliability problem considering spatial variability, sidering spatial variability (Type IV problem in Example #4). The the MSRSM is recommended. example of Congress Street Cut is, again, used while the spatial variability of c and ϕ in each soil layer is explicitly modeled using a 2D lognor- 5. Summary and conclusions mal stationary random field with a horizontal autocorrelation distance θ of 20 m and a vertical autocorrelation distance θ of 6.0 m. As ln,h ln,v This paper reviewed previous studies on applications of RSMs in the shown in Fig. 8, the random fields in three soil layers are discretized different slope reliability problems and systematically investigated the into 434, 773 and 522 elements with a side length of 0.5 m, respectively. capacity and validity of four response surface methods (i.e., SQRSM, Table 17 shows results of the Type IV problem in Example #4. As SSRSM, MQRSM, and MSRSM) to solve the reliability of slope stability. shown in Table 17, the values (i.e., 0.0543 and 0.0072, respectively) of Based on the review, four types of soil slope reliability analysis problems slope failure probability obtained using SQRSM and SSRSM are higher were identified from the literature, including single-layered soil slope than that (i.e., 0.0027) from direct MCS and simplified Bishop method reliability problem ignoring spatial variability, single-layered soil slope in this study. On the other hand, MQRSM and MSRSM provide estimates reliability problem considering spatial variability, multiple-layered soil (i.e., 0.0021 and 0.0015, respectively) of slope failure probability that slope reliability problem ignoring spatial variability, and multiple- are generally comparable with that from direct MCS. In addition, the layered soil slope reliability problem considering spatial variability. Unit COV (i.e., 2.15) of MSRSM is much smaller than that (i.e., 9.31) of These problems are referred to as “Type I–IV problems” in this study, re- MQRSM. Hence, it has a higher computational efficiency. Based on the spectively. For each type of the problem, the computational accuracy trade-off between the computational accuracy and efficiency, MSRSM and efficiency of SQRSM, SSRSM, MQRSM, and MSRSM are systematical- is suggested to solve the reliability of slope stability for the Type IV prob- ly explored and compared for cohesive and c–ϕ soil slopes, respectively. lem in c–ϕ soils. Based on the comparison, some recommendations for selecting relatively appropriate RSMs for different slope reliability analysis problems in slope engineering practice are provided, as shown in Table 19. Table 15 In summary, for a single-layered homogenous soil slope ignoring Statistical properties of soil parameters in Example #4. spatial variability (i.e., Type I problem), the critical deterministic slip Angle of internal friction, γ Cohesion, c (kPa) surface is the dominating slope failure mode. Thus, the SQRSM provides Slope Unit weight, ϕ (°) layers (kN/m3) reasonably accurate estimate of slope failure probability and has a high Mean COV Distribution Mean COV Distribution computational efficiency. When the spatial variability of soil properties Sand 21 0 ―― 30 ―― is taken into account in reliability analysis of a single-layered soil slope Clay 1 19.5 55 0.37 Lognormal 5 0.2 Lognormal (i.e., Type II problem), the number of potential failure modes increases Clay 2 19.5 43 0.19 Lognormal 7 0.21 Lognormal compared with those of the homogenous soil slopes. In such a case, Clay 3 20 56 0.2 Lognormal 15 0.24 Lognormal the MSRSM can be applied to solve the slope failure probability. In addition, stratification of soils also affects the potential failure modes of a soil Table 16 slope and usually leads to multiple failure modes in the Type III prob- Results for Type III problem: a three-layered c–ϕ slope (Example #4, No ISV). lem, for which the MQRSM is recommended. When both stratification

Equivalent no. of and spatial variability are involved in soil slope reliability analysis Mean Unit Methods performance function COV[P ] Source (i.e., Type IV problem), the MSRSM is the suggested method that can value of P f COV evaluations of each run f capture the performance functions well and has a good computational SQRSM 13.04 0.057 0.005 0.02 efﬁciency. MQRSM 23.19 0.010 0.009 0.04 SSRSM 28.03 0.086 0.793 4.20 This study MSRSM 37.82 0.106 0.743 4.57 Acknowledgments MCSa 20,000 0.010 0.071 9.98

Note: This work was supported by the National Science Fund for Distin- a A single MCS with 20, 000 samples. guished Young Scholars (Project No. 51225903), the National Basic D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14 13

Table 18 Recommendations of selecting appropriate RSMs for four types of reliability problems of a c–ϕ slope.

Problem type Indicator SQRSM MQRSM SSRSM MSRSM Recommended method

Accuracy √√ √ Single-layered slope ignoring spatial variability (I) SQRSM Efficiency 1 2 3 4 Accuracy √√ Single-layered slope considering spatial variability (II) SSRSM, MSRSM Efficiency 1 4 2 3 Accuracy √ Multiple-layered slope ignoring spatial variability (III) MQRSM Efficiency 1 2 3 4 Accuracy √√ Multiple-layered slope considering spatial variability (IV) MSRSM Efficiency 1 3 4 2

Table 19 References Selected RSMs for soil slope reliability problems. Al-Bittar, T., Soubra, A.H., 2013. Bearing capacity of strip footings on spatially random soils Recommended Problem type using sparse polynomial chaos expansion. Int. J. Numer. Anal. Methods 37 (13), method 2039–2060. Single-layered slope ignoring spatial variability (I) SQRSM Au, S.K., Ching, J., Beck, J.L., 2007. Application of subset simulation methods to reliability – Single-layered slope considering spatial variability (II) MSRSM benchmark problems. Struct. Saf. 29 (3), 183 193. Multiple-layered slope ignoring spatial variability (III) MQRSM Baecher, G.B., Christian, J.T., 2003. Reliability and Statistics in Geotechnical Engineering. John Wiley and Sons, New York. Multiple-layered slope considering spatial variability (IV) MSRSM Bucher, C.G., Bourgund, U., 1990. Afastandefficient response surface approach for structural reliability problems. Struct. Saf. 7 (1), 57–66. Cao, Z.J., Wang, Y., 2013. Bayesian approach for probabilistic site characterization using cone penetration tests. J. Geotech. Geoenviron. 139 (2), 267–276. Research Program of China (973 Program) (Project No. 2011CB013506), Cao, Z.J., Wang, Y., 2014a. Bayesian model comparison and selection of spatial correlation the National Natural Science Foundation of China (Project No. functions for soil parameters. Struct. Saf. 49, 10–17. 51329901, 51409196) and the Natural Science Foundation of Hubei Cao, Z.J., Wang, Y., 2014b. Bayesian model comparison and characterization of undrained shear strength. J. Geotech. Geoenviron. 140 (6), 04014018. Province of China (Project No. 2014CFA001). Chen, C.F., Xiao, Z.Y., Zhang, G.B., 2011. Time-variant reliability analysis of three- dimensional slopes based on Support Vector Machine method. J. Cent. South Univ. 18, 2108–2114. Ching, J.Y., Phoon, K.K., 2014. Transformations and correlations among some clay Appendix A. List of symbols parameters—the global database. Can. Geotech. J. 51 (6), 663–685. Ching, J.Y., Phoon, K.K., Hu, Y.G., 2009. Efficient evaluation of reliability for slopes with circu- lar slip surfaces using importance sampling. J. Geotech. Geoenviron. 135 (6), 768–777. Cho, S.E., 2007. Effects of spatial variability of soil properties on slope stability. Eng. Geol. Symbol Description 92 (3), 97–109. Cho, S.E., 2009. Probabilistic stability analyses of slopes using the ANN-based response RSM response surface method surface. Comput. Geotech. 36 (5), 787–797. FOSM first-order second moment method Cho, S.E., 2010. Probabilistic assessment of slope stability that considers the spatial FORM first-order reliability method variability of soil properties. J. Geotech. Geoenviron. 136 (7), 975–984. SORM second-order reliability method Cho, S.E., 2013. First-order reliability analysis of slope considering multiple failure modes. MCS Monte Carlo Simulation Eng. Geol. 154, 98–105. SQRSM single quadratic response surface method Chowdhury, R., Rao, B.N., 2010. Probabilistic stability assessment of slopes using high MQRSM multiple quadratic response surface method dimensional model representation. Comput. Geotech. 37 (7), 876–884. SSRSM single stochastic response surface method Chowdhury, R.N., Xu, D.W., 1994. Slope system reliability with general slip surfaces. Soils MSRSM multiple stochastic response surface method Found. 34 (3), 99–105. SVM support vector machine Chowdhury, R.N., Xu, D.W., 1995. Geotechnical system reliability of slopes. Reliab. Eng. RBFN radial basis function neural network Syst.Saf.47(3),141–151. RVM relevance vector machine Christian, J.T., Ladd, C.C., Baecher, G.B., 1994. Reliability applied to slope stability analysis. – ANN artificial neural network J. Geotech. Eng. 120 (12), 2180 2207. HDMR high dimensional model representation Duncan, J.M., 2000. Factors of safety and reliability in geotechnical engineering. J. Geotech. Geoenviron. 126 (4), 307–316. NN neural networks El-Ramly, H., Morgenstern, N.R., Cruden, D.M., 2002. Probabilistic slope stability analysis FS factor of safety for practice. Can. Geotech. J. 39 (3), 665–683. FSmin minimum FS among Ns potential slip surfaces El-Ramly, H., Morgenstern, N.R., Cruden, D.M., 2005. Probabilistic assessment of stability ISV inherent spatial variability of a cut slope in residual soil. Geotechnique 55 (1), 77–84. XNG,F fi non-Gaussian random eld in original space Feng, T., Fredlund, M., 2011. SVSLOPE: Verification Manual. SoilVision Systems Ltd., NG,V X non-Gaussian random variable in original space Saskatoon. NMC number of Monte Carlo Simulation Fenton, G.A., Griffiths, D.V., 2008. Risk Assessment in Geotechnical Engineering. John Ns number of potential slip surfaces Wiley and Sons, New York.

Np number of realizations of random ﬁelds GEO-SLOPE International Ltd., 2012. Stability modeling With SLOPE/W: An Engineering

Ne equivalent number of evaluations of the original performance function Methodology, May 2012 Edition. GEO-SLOPE International Ltd., Calgary, Alberta, Canada. Npe number of original performance function evaluations in a run Ghanem, R.G., Spanos, P.D., 2003. Stochastic Finite Elements: A Spectral Approach. 2nd ed. NF number of FSmin values less than 1.0 Dover Publications, New York. Pf probability of failure Ghiocel, D.M., Ghanem, R.G., 2002. Stochastic finite-element analysis of seismic soil– P average value of failure probability obtained from 20 independent runs f structure interaction. J. Eng. Mech. 128 (1), 66–77. I{·} indicator function Griffiths, D.V., Fenton, G.A., 2004. Probabilistic slope stability analysis by finite elements. ξ vector of independent standard normal variables J. Geotech. Geoenviron. 130 (5), 507–518. fi COV coef cient of variation Hassan, A.M., Wolff, T.F., 1999. Search algorithm for minimum reliability index of earth Δ Unit COV slopes. J. Geotech. Geoenviron. 125 (4), 301–308. t′ computational time of MCS procedure Hicks, M.A., Spencer, W.A., 2010. Influence of heterogeneity on the reliability and failure t computational time of one original performance function evaluation of a long 3D slope. Comput. Geotech. 37 (7), 948–955. LEM limit equilibrium method Hong, H.P., Roh, G., 2008. Reliability evaluation of earth slopes. J. Geotech. Geoenviron. SRM strength reduction method 134 (12), 1700–1705. LSF limit state function Hsu, S.C., Nelson, P.P., 2006. Material spatial variability and slope stability for weak rock masses. J. Geotech. Geoenviron. 132 (2), 183–193. θln,h horizontal autocorrelation distance Huang, J.S., Griffiths, D.V., Fenton, G.A., 2010. System reliability of slopes by RFEM. Soils θln,v vertical autocorrelation distance Found. 50 (3), 343–353. 14 D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14

Huang, J.S., Lyamin, A.V., Griffiths, D.V., Krabbenhoft, K., Sloan, S.W., 2013. Quantitative Luo, X.F., Cheng, T., Li, X., Zhou, J., 2012b. Slope safety factor search strategy for multiple risk assessment of landslide by limit analysis and random fields. Comput. Geotech. sample points for reliability analysis. Eng. Geol. 129, 27–37. 53, 60–67. Luo, X.F., Li, X., Zhou, J., Cheng, T., 2012a. A Kriging-based hybrid optimization algorithm Huang, S.P., Liang, B., Phoon, K.K., 2009. Geotechnical probabilistic analysis by collocation- for slope reliability analysis. Struct. Saf. 34 (1), 401–406. based stochastic response surface method: an Excel add-in implementation. Georisk Oka, Y., Wu, T.H., 1990. System reliability of slope stability. J. Geotech. Eng. 116 (8), 3(2),75–86. 1185–1189. Jamshidi Chenari, R., Alaie, R., 2015. Effects of anisotropy in correlation structure on the Phoon, K.K., Kulhawy, F.H., 1999. Characterization of geotechnical variability. Can. stability of an undrained clay slope. Georisk 9 (2), 109–123. Geotech. J. 36 (4), 612–624. Ji, J., 2014. Asimplified approach for modeling spatial variability of undrained shear Phoon, K.K., Huang, S.P., Quek, S.T., 2002. Implementation of Karhunen–Loeve expan- strength in out-plane failure mode of earth embankment. Eng. Geol. 183, 315–323. sion for simulation using a wavelet-Galerkin scheme. Probab. Eng. Mech. 17 (3), Ji, J., Low, B.K., 2012. Stratified response surfaces for system probabilistic evaluation of 293–303. slopes. J. Geotech. Geoenviron. 138 (11), 1398–1406. Piliounis, G., Lagaros, N.D., 2014. Reliability analysis of geostructures based on Ji, J., Liao, H.J., Low, B.K., 2012. Modeling 2-D spatial variation in slope reliability analysis metaheuristic optimization. Appl. Soft Comput. 22, 544–565. using interpolated autocorrelations. Comput. Geotech. 40, 135–146. Samui, P., Lansivaara, T., Bhatt, M.R., 2013. Least square support vector machine applied to Jiang, S.H., Li, D.Q., Cao, Z.J., Zhou, C.B., Phoon, K.K., 2015. Efficient system reliability slope reliability analysis. Geotech. Geol. Eng. 31 (4), 1329–1334. analysis of slope stability in spatially variable soils using Monte Carlo simulation. Samui, P., Lansivaara, T., Kim, D., 2011. Utilization relevance vector machine for slope J. Geotech. Geoenviron. 141 (2), 04014096. reliability analysis. Appl. Soft Comput. 11 (5), 4036–4040. Jiang, S.H., Li, D.Q., Zhang, L.M., Zhou, C.B., 2014. Slope reliability analysis considering Schuëller, G.I., Pradlwarter, H.J., 2007. Benchmark study on reliability estimation in higher spatially variable shear strength parameters using a non-intrusive stochastic finite dimensions of structural systems—an overview. Struct. Saf. 29 (3), 167–182. element method. Eng. Geol. 168, 120–128. Srivastava, A., Babu, G.S., 2009. Effect of soil variability on the bearing capacity of clay and Kang, F., Li, J.J., 2015. Artificial bee colony algorithm optimized support vector regression in slope stability problems. Eng. Geol. 108 (1), 142–152. for system reliability analysis of slopes. J. Comput. Civ. Eng. http://dx.doi.org/10.1061/ Srivastava, A., Babu, G.S., Haldar, S., 2010. Influence of spatial variability of permeability (ASCE)CP.1943–5487.0000514. property on steady state seepage flow and slope stability analysis. Eng. Geol. 110 Kang, F., Han, S.X., Salgado, R., Li, J.J., 2015. System probabilistic stability analysis of soil (3–4), 93–101. slopes using Gaussian process regression with Latin hypercube sampling. Comput. Suchomel, R., Mašin, D., 2010. Comparison of different probabilistic methods for predicting Geotech. 63, 13–25. stability of a slope in spatially variable c–φ soil.Comput.Geotech.37(1–2), 132–140. Le, T.M.H., 2014. Reliability of heterogeneous slopes with cross-correlated shear strength Tan, X.H., Bi, W.H., Hou, X.L., Wang, W., 2011. Reliability analysis using radial basis parameters. Georisk 8 (4), 250–257. function networks and support vector machines. Comput. Geotech. 38 (2), Lebrun, R., Dutfoy, A., 2009. A generalization of the Nataf transformation to distributions 178–186. with elliptical copula. Probab. Eng. Mech. 24 (2), 172–178. Tan, X.H., Shen, M.F., Hou, X.L., Li, D., Hu, N., 2013. Response surface method of reliability Li, D.Q., Jiang, S.H., Cao, Z.J., Zhou, C.B., Li, X.Y., Zhang, L.M., 2015a. Efficient 3-D reliability analysis and its application in slope stability analysis. Geotech. Geol. Eng. 31 (4), analysis of the 530 m high abutment slope at Jinping I Hydropower Station during 1011–1025. construction. Eng. Geol. 195, 269–281. Tang, X.S., Li, D.Q., Zhou, C.B., Phoon, K.K., 2015. Copula-based approaches for evalu- Li, D.Q., Jiang, S.H., Cao, Z.J., Zhou, W., Zhou, C.B., Zhang, L.M., 2015b. A multiple response- ating slope reliability under incomplete probability information. Struct. Saf. 52, surface method for slope reliability analysis considering spatial variability of soil 90–99. properties. Eng. Geol. 187, 60–72. Tang, X.S., Li, D.Q., Zhou, C.B., Phoon, K.K., Zhang, L.M., 2013. Impact of copulas for model- Li, D.Q., Tang, X.S., Phoon, K.K., 2015c. Bootstrap method for characterizing the effect of ing bivariate distributions on system reliability. Struct. Saf. 44, 80–90. uncertainty in shear strength parameters on slope reliability. Reliab. Eng. Syst. Saf. Vořechovský, M., 2008. Simulation of simply cross correlated random fields by series 140, 99–106. expansion methods. Struct. Saf. 30 (4), 337–363. Li, D.Q., Xiao, T., Cao, Z.J., Zhou, C.B., Zhang, L.M., 2015d. Enhancement of random finite Wang, Y., Cao, Z.J., Au, S.K., 2010. Efficient Monte Carlo simulation of parameter sensitivity element method in reliability analysis and risk assessment of soil slopes using Subset in probabilistic slope stability analysis. Comput. Geotech. 37 (7), 1015–1022. Simulation. Landslides http://dx.doi.org/10.1007/s10346-015-0569-2. Wang, Y., Cao, Z.J., Au, S.K., 2011. Practical reliability analysis of slope stability by ad- Li, D.Q., Chen, Y.F., Lu, W.B., Zhou, C.B., 2011. Stochastic response surface method for vanced Monte Carlo simulations in a spreadsheet. Can. Geotech. J. 48 (1), 162–172. reliability analysis of rock slopes involving correlated non-normal variables. Comput. Wong, F.S., 1985. Slope reliability and response surface method. J. Geotech. Eng. 111 (1), Geotech. 38 (1), 58–68. 32–53. Li, D.Q., Qi, X.H., Phoon, K.K., Zhang, L.M., Zhou, C.B., 2014. Effect of spatially variable shear Xu, B., Low, B.K., 2006. Probabilistic stability analyses of embankments based on finite- strength parameters with linearly increasing mean trend on reliability of infinite element method. J. Geotech. Geoenviron. 132 (11), 1444–1454. slopes. Struct. Saf. 49, 45–55. Xue, J.F., Gavin, K., 2007. Simultaneous determination of critical slip surface and reliability Li, L., Chu, X.S., 2015. Multiple response surfaces for slope reliability analysis. Int. J. Numer. index for slopes. J. Geotech. Geoenviron. 133 (7), 878–886. Anal. Methods 39 (2), 175–192. Yi, P., Wei, K.T., Kong, X.J., Zhu, Z., 2015. Cumulative PSO-Kriging model for slope reliabil- Li, S.J., Zhao, H.B., Ru, Z.L., 2013. Slope reliability analysis by updated support vector ity analysis. Probab. Eng. Mech. 39, 39–45. machine and Monte Carlo simulation. Nat. Hazards 65 (1), 707–722. Zeng, P., Jimenez, R., 2014. An approximation to the reliability of series geotechnical Lloret-Cabot, M., Fenton, G.A., Hicks, M.A., 2014. On the estimation of scale of fluctuation systems using a linearization approach. Comput. Geotech. 62, 304–309. in geostatistics. Georisk 8 (2), 129–140. Zhang, J., Huang, H.W., Phoon, K.K., 2013a. Application of the Kriging-based response sur- Low, B.K., 2007. Reliability analysis of rock slopes involving correlated nonnormals. Int. face method to the system reliability of soil slopes. J. Geotech. Geoenviron. 139 (4), J. Rock Mech. Min. 44 (6), 922–935. 651–655. Low, B.K., 2014. FORM, SORM, and spatial modeling in geotechnical engineering. Struct. Zhang, J., Huang, H.W., Juang, C.H., Li, D.Q., 2013b. Extension of Hassan and Wolff method Saf. 49, 56–64. for system reliability analysis of soil slopes. Eng. Geol. 160, 81–88. Low, B.K., Tang, W.H., 1997. Probabilistic slope analysis using Janbu's generalized Zhang, J., Zhang, L.M., Tang, W.H., 2011a. Kriging numerical models for geotechnical procedure of slices. Comput. Geotech. 21 (2), 121–142. reliability analysis. Soils Found. 51 (6), 1169–1177. Low, B.K., Tang, W.H., 2004. Reliability analysis using object-oriented constrained Zhang, J., Zhang, L.M., Tang, W.H., 2011b. New methods for system reliability analysis of optimization. Struct. Saf. 26 (1), 69–89. soil slopes. Can. Geotech. J. 48 (7), 1138–1148. Low, B.K., Zhang, J., Tang, W.H., 2011. Efficient system reliability analysis illustrated for a Zhao, H.B., 2008. Slope reliability analysis using a support vector machine. Comput. retaining wall and a soil slope. Comput. Geotech. 38 (2), 196–204. Geotech. 35 (3), 459–467.