Computers and Geotechnics 37 (2010) 1015–1022

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Computers and Geotechnics

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

Efficient Monte Carlo Simulation of parameter sensitivity in probabilisticslope stability analysis

Yu Wang ⇑, Zijun Cao 1, Siu-Kui Au 2

Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 April 2010Received in revised form 12 July 2010Accepted 23 August 2010Available online 18 September 2010

Keywords:Probabilistic failure analysisSlope stabilityMonte Carlo SimulationSubset SimulationHypothesis testsBayesian analysis

0266-352X/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compgeo.2010.08.010

⇑ Corresponding author. Tel.: +852 2788 7605; fax:E-mail addresses: yuwang@cityu.edu.hk (Y. Wan

edu.hk (Z. Cao), siukuiau@cityu.edu.hk (S.-K. Au).1 Tel.: +852 3442 6492; fax: +852 2788 7612.2 Tel.: +852 2194 2769; fax: +852 2788 7612.

Monte Carlo Simulation (MCS) method has been widely used in probabilistic analysis of slope stability,and it provides a robust and simple way to assess failure probability. However, MCS method does notoffer insight into the relative contributions of various uncertainties (e.g., inherent spatial variability ofsoil properties and subsurface stratigraphy) to the failure probability and suffers from a lack of resolutionand efficiency at small probability levels. This paper develop a probabilistic failure analysis approach thatmakes use of the failure samples generated in the MCS and analyzes these failure samples to assess theeffects of various uncertainties on slope failure probability. The approach contains two major compo-nents: hypothesis tests for prioritizing effects of various uncertainties and Bayesian analysis for furtherquantifying their effects. Equations are derived for the hypothesis tests and Bayesian analysis. The prob-abilistic failure analysis requires a large number of failure samples in MCS, and an advanced Monte CarloSimulation called Subset Simulation is employed to improve efficiency of generating failure samples inMCS. As an illustration, the proposed probabilistic failure analysis approach is applied to study a designscenario of James Bay Dyke. The hypothesis tests show that the uncertainty of undrained shear strengthof lacustrine clay has the most significant effect on the slope failure probability, while the uncertainty ofthe clay crust thickness contributes the least. The effect of the former is then further quantified by aBayesian analysis. Both hypothesis test results and Bayesian analysis results are validated against inde-pendent sensitivity studies. It is shown that probabilistic failure analysis provides results that are equiv-alent to those from additional sensitivity studies, but it has the advantage of avoiding additionalcomputational times and efforts for repeated runs of MCS in sensitivity studies.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Various uncertainties exist in slope engineering, such as inher-ent spatial variability of soil properties, subsurface stratigraphy,simplifications and approximations adopted in geotechnical mod-els. Effects of these uncertainties on probability of slope failureare often significant, and insight on these effects is of great valuefor understanding failure mechanisms and designing slope reme-dial measures. Several probabilistic methodologies have beendeveloped to incorporate these uncertainties in slope stabilityanalysis, such as the First Order Second Moment (FOSM) method[1–4], First Order Reliability Method (FORM) [5–9], Monte CarloSimulation (MCS) method [10–20], and other methods based onartificial neural network, support vector machine or random set

ll rights reserved.

+852 2788 7612.g), zijuncao3@student.cityu.

[21–25]. Among these methods, MCS method that integrates withdifferent deterministic analysis methods (e.g., limit equilibriummethods [10,11,13], finite element methods [12,14,15]) are gainingpopularity due to their robustness and conceptual simplicity. How-ever, as pointed out by Baecher and Christian [26], although MCSmethod provides a robust and simple way to assess failure proba-bility, it does not offer insight into the relative contributions of var-ious uncertainties to the failure probability and suffers from a lackof resolution and efficiency at small probability levels.

This paper develops a probabilistic failure analysis approachthat makes uses of the failure samples generated in the MCS andanalyzes these failure samples to assess the effects of variousuncertainties on slope failure probability. An advanced Monte Car-lo Simulation called Subset Simulation (Subsim) [27,28] is em-ployed to improve efficiency of generating failure samples inMCS and resolution at small failure probability levels. The paperstarts with mathematical formulation of the approach, includinghypothesis tests for prioritizing effects of various uncertaintiesand Bayesian analysis for further quantifying their effects. SubsetSimulation and its implementation in a commonly-availablespreadsheet environment are briefly described. As an illustration,

1016 Y. Wang et al. / Computers and Geotechnics 37 (2010) 1015–1022

the proposed approach is applied to study a design scenario ofJames Bay Dyke [2,10,14].

2. Probabilistic failure analysis approach

Probabilistic failure analysis is similar to back-analysis [29–31],which is a common analysis procedure in geotechnical engineeringthat intends to find a set of model parameters that would result inthe observed performance of geo-structures. Similarly, probabilis-tic failure analysis aims to identify a group of uncertain parametersthat would significantly affect the slope performance (i.e., theprobability of slope failure). The back-analysis, however, relies onthe observed performance and it is inapplicable when observedperformance of the geo-structures of interest is unavailable (e.g.,during design analysis of a new geo-structure). On the other hand,probabilistic failure analysis makes uses of the failure samples gen-erated in the MCS, and it is readily applicable in design analysis forevaluating the effects of various uncertainties. The proposed prob-abilistic failure analysis approach contains two major components:hypothesis tests for prioritizing effects of various uncertainties andBayesian analysis for further quantifying their effects, which aredescribed in the following two subsections, respectively.

2.1. Hypothesis tests

The effects of various uncertainties on the probability of slopefailure are prioritized by comparing, statistically, failure sampleswith their respective nominal (unconditional) samples. When theuncertainty of an uncertain system parameter has a significant ef-fect on the probability of slope failure, the mean l of failure sam-ples of the parameter differs significantly from the mean l0 of itsunconditional samples. The statistical difference between l andl0 is evaluated by hypothesis tests. A null hypothesis H0 and alter-native hypothesis HA are defined as [32]

H0 : l ¼ l0

HA : l–l0

ð1Þ

Then, a hypothesis test statistic ZH of the parameter is formulated as

ZH ¼l� l0

r=ffiffiffinp ð2Þ

where r is standard deviation of the uncertain parameter and n isthe number of failure samples. Based on Central Limit Theorem,ZH follows the standard Normal distribution when n is large (e.g.,n P 30) [32]. When the failure sample mean l deviates statisticallyfrom the unconditional mean l0 of the parameter, the absolute va-lue of ZH is relatively large. As the absolute value of ZH increases, thestatistical difference between l and l0 becomes growingly signifi-cant, and the effect of the uncertain parameter on failure probabilityalso becomes growingly significant. The absolute value of ZH istherefore formulated in this paper as an index to measure the ef-fects of the uncertain parameters on failure probability and to pri-oritize their relative effects on failure probability. Using theabsolute value of ZH, the uncertain parameters that have significanteffects on failure probability are selected, and their effects are fur-ther quantified using a Bayesian analysis approach described inthe next subsection.

2.2. Bayesian analysis

The failure samples generated in the MCS are further analyzedby a Bayesian analysis to quantify effects of various uncertainties.Let h denote an uncertain parameter selected based on hypothesistests. In the context of the Bayesian Theorem [33,34]

PðFjhÞ ¼ PðFÞPðhjFÞ=PðhÞ ð3Þ

where P(F|h) is the conditional probability density function (PDF) ofslope failure for a given h value; P(F) is the probability of slope fail-ure; P(h|F) is the conditional PDF of h given that the slope has failed;and P(h) is the unconditional PDF of h. As both P(F) and P(h|F) are esti-mated from failure samples of MCS and P(h) is given before MCS, Eq.(3) can be used to estimate P(F|h) using P(h) and P(h|F) obtained fromanalysis of failure samples. Note that P(F|h) is a variation of failureprobability as a function of h, and it can be considered as results ofa sensitivity study of h on slope failure probability. In other words,the probabilistic failure analysis approach presented in this paper,which makes use of the failure samples generated in a single runof MCS for assessment of failure probability, provides results thatare equivalent to those from a sensitivity study, which frequently in-cludes many repeated runs of MCS with different given values of h ineach run. Additional computational times and efforts for repeatedruns of MCS in the sensitivity study can be avoided using the prob-abilistic failure analysis approach described herein.

In addition, Eq. (3) implies that comparison between the condi-tional probability P(h|F) and its unconditional one P(h) provides anindication of the effect of the uncertain parameter h on failure proba-bility. In general, P(F|h) changes as the values of the uncertain param-eter h changes. However, when P(h|F) is similar to P(h), P(F|h) remainsmore or less constant regardless of the values of h. This implies thatthe effect of h on the slope failure probability is minimal. Such impli-cation can be used to validate the prioritization obtained fromhypothesis tests, as shown in the example of James Bay Dyke later.

The resolution of P(F) and P(h|F) is pivotal to obtain P(F|h), and itdepends on the number of failure samples generated in MCS. Asthe number of failure samples increases, the resolution improves.For a given slope stability problem, the value of P(F) is constant,although unknown before MCS. In this case, increasing the numberof failure samples necessitates an increase in the total number ofsamples in MCS. One possible way to improve the resolution is,therefore, to increase the total number of samples in MCS at the ex-pense of computational time. Alternatively, advanced MCS meth-ods can be employed to improve efficiency and resolution atsmall failure probability levels. An advanced MCS called SubsetSimulation (Subsim) [27,28,35,36] is used in this paper to calculatethe failure probability and generate failure samples efficiently forprobabilistic failure analysis.

3. Subset Simulation

3.1. Algorithm

Subset Simulation is an adaptive stochastic simulation procedurefor efficiently generating failure samples and computing small tailprobability [27,28]. It expresses a small probability event as a se-quence of intermediate events {F1, F2, . . ., Fm} with larger conditionalprobability and employs specially designed Markov chains to gener-ate conditional samples of these intermediate events until the finaltarget failure region is achieved. For the slope stability problem, thefactor of safety (FS) is the key parameter, and let Y = 1/FS be the crit-ical response [20,35]. The probability of Y = 1/FS larger than a givenvalue y (i.e., P(Y = 1/FS > y)) is of interest and let 0 < y1 < y2 <� � � < ym�1 < ym = y be an increasing sequence of intermediate thresh-old values. The sequence of intermediate events {F1, F2, . . ., Fm} arechosen as Fi = {Y > yi, i = 1, 2, . . ., m} for these intermediate thresholdvalues. By sequentially conditioning on the event {Fi, i = 1, 2, . . ., m},the failure probability can be written as

PðFÞ ¼ PðFmÞ ¼ PðF1ÞYm

i¼2

PðFijFi�1Þ ð4Þ

where P(F1) is equal to P(Y > y1) and P(Fi|Fi�1) is equal to{P(Y > yi|Y > yi�1): i = 2, . . ., m}. In implementations, y1, y2, . . ., ym

Y. Wang et al. / Computers and Geotechnics 37 (2010) 1015–1022 1017

are generated adaptively using information from simulated samplesso that the sample estimate of P(F1) and {P(Fi|Fi�1): i = 2, . . ., m} al-ways corresponds to a common specified value of conditional prob-ability p0 (p0 = 0.1 is found to be a good choice) [35,36].

The efficient generation of conditional samples is pivotal in thesuccess of Subset Simulation, and it is made possible through themachinery of Markov Chain Monte Carlo (MCMC). In MCMC,Metropolis algorithm [37] is used and successive samples are gen-erated from a specially designed Markov chain whose limiting sta-tionary distribution tends to the target PDF as the length of Markovchain increases. Details of the computational process of SubsetSimulation and its application to slope stability problems are re-ferred to Au and Beck [27,28] and Wang et al. [20], respectively.

The Subset Simulation provides much more failure samplesthan Direct MCS under the same total number of samples, espe-cially when the failure probability is relatively small. When com-pared with failure samples generated in direct MCS where eachsample carries equal weight in the calculation of P(F) and P(h|F),the samples generated by Subset Simulation are conditional sam-ples and carry different weights for different intermediate eventsFm. Thus, when using these conditional failure samples collectedfrom Subset Simulation to construct the conditional PDF P(h|F) re-quired in Eq. (3), a weighted summation by Total Probability The-orem [33,34] is necessary, which is described in the followingsubsection.

3.2. Estimation of P(h|F) based on conditional failure samples

Consider a Subset Simulation that performs m + 1 levels of sim-ulations. The first level of Subset Simulation is direct MCS, andsamples of the next level are generated conditional on the samplescollected from the previous level. The intermediate threshold val-ues {yi, i = 1, 2, . . ., m} divide the sample space X of an uncertainparameter h into m individual sets {Xi, i = 0, 1, 2, . . ., m}. Accordingto the Total Probability Theorem [34], the failure probability can bewritten as

PðFÞ ¼Xm

i¼0

PðFjXiÞPðXiÞ ð5Þ

where X0 = {Y 6 y1}; Xi, i = 1, . . ., m � 1 is equal to Fi � Fi+1 (i.e.,Xi = {yi 6 Y 6 yi+1}); Xm is equal to Fm (i.e., Xm = {Y P ym}); P(F|Xi)is the conditional failure probability given sampling in Xi; P(Xi) isthe probability of the event Xi. P(F|Xi) is estimated as the fractionof the failure samples in Xi. The failure samples are collected fromsamples generated by Subset Simulation and are based on the per-formance failure criteria (i.e., Y = 1/FS P 1 for a slope stability prob-lem). P(Xi) is calculated as

PðX0Þ ¼ 1� p0

PðXiÞ ¼ pi0 � piþ1

0 ði ¼ 1; . . . ;m� 1ÞPðXmÞ ¼ pm

0

ð6Þ

Note that P(Xi \Xj) = 0 for i – j andPm

i¼0PðXiÞ ¼ 1. When P(F),P(F|Xi) and P(Xi) are obtained, the conditional probability P(Xi|F)is calculated using the Bayesian Theorem

PðXijFÞ ¼ PðFjXiÞPðXiÞ=PðFÞ ð7Þ

Then the conditional PDF P(h|F) of an uncertain parameter h isgiven by the Total Probability Theorem as

PðhjFÞ ¼Xm

i¼0

PðhjXi \ FÞPðXijFÞ ð8Þ

where P(h|Xi \ F) is the conditional probability of h estimated fromfailure samples that lie in Xi. In this paper, P(h|Xi \ F) is estimated

from the failure sample histogram in Xi. The number of bins k in thefailure sample histogram is estimated as [38]

k ¼ 1þ logni2 ð9Þ

where ni is the number of the failure samples in Xi. Using Eqs. (5)and (8), P(F) and P(h|F) can be calculated from Subset Simulation,and subsequently, P(F|h) is calculated using Eq. (3).

4. Implementation of Subset Simulation in spreadsheetenvironment

The Subset Simulation described above has been implementedin a commonly-available spreadsheet environment by a packageof worksheets and functions/Add-In in EXCEL with the aids of Vi-sual Basic for Application (VBA) [35,36]. The package is dividedinto three parts: deterministic model worksheet, uncertainty mod-el worksheet and Subset Simulation Add-In, which are describedbriefly in the following three subsections, respectively.

4.1. Deterministic model worksheet

For a slope stability problem, deterministic model analysis isthe process of calculating factor of safety for a given nominal setof values of system parameters. The system parameters includethe geometry information of the slope and the slip surface, soilproperties and profile of soil layers. In this paper, limit equilibriummethods (e.g., Swedish Circle method, Simplified Bishop methodand Spencer method) [39] are employed to calculate the factor ofsafety for the critical slip surface. The calculation process of deter-ministic analysis is implemented in a series of worksheets assistedby some VBA functions/Add-In [35,36]. From an input–output per-spective, the deterministic analysis worksheets take a given set ofvalues as input, calculate the factor of safety and return the factorof safety as an output.

4.2. Uncertainty model worksheet

An uncertainty model worksheet is developed to generate ran-dom samples of uncertain system parameters that are treated asrandom variables in the analysis. The uncertain worksheet includesdetailed information of random variables, such as statistics, distri-bution type and correlation information. The generation of randomsamples starts with an EXCEL built-in function ‘‘RAND()” for gener-ating uniform random samples, which are then transformed to ran-dom samples of the target distribution type (e.g., normaldistribution or lognormal distribution). If the random variablesare considered correlated, Cholesky factorization of the correlationmatrix is performed to obtain a lower triangular matrix, which isused in the transformation to generate correlated random samples.Details of the random sample generation process are referred to Auet al. [36]. From the input–output perspective, the uncertaintymodel worksheet takes no input but returns a set of random sam-ples of the uncertain system parameters as its output.

When deterministic model worksheet and uncertainty modelworksheet are developed, they are linked together through theirinput/output cells to execute probabilistic analysis of slope stabil-ity. The connection is carried out by simply setting the cell refer-ences for nominal values of uncertain parameters indeterministic model worksheet to be the cell references for therandom samples in the uncertainty model worksheet in EXCEL.After this task, the values of uncertain system parameters shownin the deterministic model worksheet are equal to that generatedin the uncertainty model worksheet, and so the values of the safetyfactor calculated in the deterministic modeling worksheet arerandom.

40-5

20 60 80 100 120 140 160

5

15

25

45

Ele

vati

on(m

)

Distance (m)0

35Embm ankment

Marine ClayyLacustrine Clayy

((((((((((((xxx, y)

Critical slipp circle

31

TiiTT lll

311

ClClaayy CrCruussttSuuMMSSSSuuLL

12.0m

56.0m

8.0mTL

TTTcrDTill

γFγγ ill; φφFill

Fig. 2. The James Bay Dyke (modified after [10]).

1018 Y. Wang et al. / Computers and Geotechnics 37 (2010) 1015–1022

4.3. Subset Simulation Add-In

When the deterministic analysis and uncertainty model work-sheets are completed and linked together, Subset Simulation pro-cedure is invoked for uncertainty propagation. In this paper,Subset Simulation is implemented as an Add-In in EXCEL [35,36].The userform of the Add-In is shown in Fig. 1. The upper four inputfields of the userform (i.e., number of Subset Simulation runs, num-ber of samples per level N, conditional probability from one level tonext p0, the highest Subset Simulation level m) control the numberof samples generated by Subset Simulation. The total number ofsamples per Subset Simulation run is equal to N + mN(1 � p0).The lower four input fields of the userform record the cell refer-ences of the random variables, their PDF values, and the cell refer-ences of the system response (e.g., Y = 1/FS in this paper) and othervariables V (e.g., the samples generated in this paper) of interest,respectively.

After each simulation run, the Add-In provides the complemen-tary cumulative density function (CDF) of the driving variable ver-sus the threshold level, i.e., estimate for P(Y > y) versus y = 1/FS inthis paper, into a new spreadsheet and produces a plot of it [36].Then, based on the output information obtained, the CDF, histo-grams or conditional counterparts (e.g., P(h|F)) of uncertain param-eters of interest are calculated using the procedures and equationsdescribed in the previous section.

5. The James Bay Dyke

As an illustration, the probabilistic failure analysis approach isapplied to analyze a design scenario of the James Bay Dyke. TheJames Bay Dyke is a 50 km-long earth dyke of the James Bay hydro-electric project in Canada. Soil properties and various design sce-narios of the dyke were studied by Ladd et al. [40], Soulié et al.[41], Christian et al. [2], El-Ramly [10], El-Ramly et al. [11] andXu and Low [14]. As shown in Fig. 2, the embankment is 12 m highwith a 56 m-wide berm at mid-height. The slope angle of theembankment is about 18.4� (3H:1V). The embankment is overlyingon a clay crust with a thickness Tcr. The clay crust is underlain by alayer of 8.0-m thick sensitive marine clay and a layer of lacustrine

Fig. 1. The userform of Subset Simulation Add-In.

clay with a thickness TL. The undrained shear strength (i.e., SuM andSuL) of the marine clay and the lacustrine clay were measured byfield vane tests [2,10,40,41]. The lacustrine clay is overlying on astiff till layer, the depth to the top of which is DTill.

Six uncertain system parameters have been considered in liter-ature (e.g., El-Ramly [10], El-Ramly et al. [11] and Xu and Low[14]), including the friction angle /Fill and unit weight cFill ofembankment material, the thickness Tcr of clay crust, the un-drained shear strength SuM of the marine clay, the undrained shearstrength SuL of the lacustrine clay, and the depth of the till layerDTill. During the probabilistic failure analysis of the dyke, the sixuncertain parameters are represented by six independent Gaussianrandom variables [10], respectively. Statistics [i.e., mean, standarddeviation and coefficient of variation (COV)] of these six randomvariables are summarized in Table 1. These statistics are used togenerate random samples for each random variable in uncertainmodel worksheet. Note that thickness of the lacustrine clay TL isan uncertain variable that depends on Tcr and DTill and has a meanof about 6.5 m (see Fig. 2). In addition to these uncertain parame-ters, other system parameters are taken as deterministic, includingan undrained shear strength of 41 kPa for the clay crust and unitweights of 19 kN/m3, 19 kN/m3 and 20.5 kN/m3 for the clay crust,marine clay and lacustrine clay, respectively [10].

Using the soil properties described above, El-Ramly [10] and Xuand Low [14] employed direct MCS methods integrating with limitequilibrium methods and response surface method, respectively, toevaluate failure probability of the dyke. To enable a consistentcomparison with the analyses by El-Ramly [10], the critical slipsurfaces recommended by El-Ramly [10] are adopted in this paper,which is circular and always tangential to top of the till layer andpass through the point (x = 4.9 m, y = 36.0 m). The x coordinate ofthe center is fixed at 85.9 m. For each set of random samples, thecritical slip surface is specified uniquely by the value of DTill. In thispaper, the safety factor of the critical slip surface is calculated bySimplified Bishop method, and two Subset Simulation runs are per-formed. One has the highest simulation level m = 3 and samplenumber N = 1000 per each level, as opposed to m = 4 andN = 10,000 per each level in the other run.

Table 1Soil properties of the James Bay Dyke (modified after [10]).

Soil layers Uncertain systemparameters*

Mean Standarddeviation

Coefficient ofvariation (%)

Embankment /Fill (�) 30.0 1.79 6.0cFill (kN/m3) 20.0 1.10 5.5

Clay crust Tcr (m) 4.0 0.48 12.0Marine clay SuM (kN/m2) 34.5 3.95 11.5Lacustrine clay SuL (kN/m2) 31.2 6.31 20.2Till DTill (m) 18.5 1.00 5.4

* All parameters are modeled as Gaussian random variables. Thickness of theLacustrine clay layer TL is an uncertain variable that depends on Tcr and DTill and hasa mean of about 6.5 m.

Y. Wang et al. / Computers and Geotechnics 37 (2010) 1015–1022 1019

5.1. Simulation results

Fig. 3 shows a typical complementary CDF for Y = 1/FS (i.e.,P(Y > y) versus y), from two Subset Simulation runs with a totalsample number NT = 1000 + 3 � 1000 � (1–0.1) = 3700 (i.e., SubsimRun 1) and NT = 10,000 + 4 � 10,000 � (1–0.1) = 46,000 (i.e., SubsimRun 2), respectively. For comparison, the result from a directMCS with 20,000 samples is also plotted. Three consistent failureprobabilities P(Y P 1) = 0.22%, 0.23%, and 0.25% are estimated,respectively, from direct MCS and two runs of Subset Simulations.In addition, the two runs of Subset Simulations provide results thatare consistent even at low probability levels [e.g., P(Y P y) = 0.01%]where the CDF curve from direct MCS becomes erratic.

Table 2 compares the simulation results with those reported byEl-Ramly [10] and Xu and Low [14]. El-Ramly [10] performed directMCS with the Simplified Bishop method (i.e., the same limit equi-librium method used in this paper) and obtained a failure probabil-ity Pf = 0.24%. This Pf value is almost identical to the average of thethree Pf values obtained in this paper (i.e., 0.22%, 0.23%, and 0.25%in Columns 4–6 of Table 2). In addition, Xu and Low [14] combinedMCS with response surface method to estimate the Pf of the JamesBay Dyke and obtained a Pf value of 0.33%. Although differentdeterministic slope stability analysis methods were used, the Pf

values obtained compare favourably with each other. This impliesthat the probabilistic analysis models for the James Bay Dyke pre-sented in this paper work properly. However, note that the Pf val-ues summarized in Table 2 are obtained under the assumption ofcircular slip surfaces. In contrast, non-circular failure mechanismhas been identified in recent finite element analyses incorporatingspatially varying soil properties [14,19]. When the searching ofnon-circular slip surfaces is included in the MCS, the correspondingPf values will increase.

Table 2 also compares the number of failure samples in directMCS (e.g., Columns 2 or 4) with that in Subset Simulations (e.g.,

0.001

0.01

0.1

1

10

100

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

y=1/FS

P(Y

>=y)

(%

)

Subsim Run 1 3700 Samples

Subsim Run 2 46000 Samples

Direct MCS 20000 Samples

Fig. 3. Complementary CDF plot from simulations.

Table 2Comparison of simulation results.

Simulation results El-Ramly [10] Xu and Low [14] Dir

Failure probability Pf (%) 0.24a 0.33b 0.2Number of failure samples, NF 48 N/A 44Number of Total Samples, NT 20,000 N/A 20,Percentage of failure samples, NF/NT (%) 0.24 N/A 0.2

a Pf is calculated by MCS methods integrating with the simplified Bishop method.b Pf is calculated by MCS methods integrating with the response surface method.

Columns 5 or 6). For a total sample number NT = 20,000, directMCS leads to only 48 or 44 failure samples. In contrast, Subset Sim-ulations with m = 3 and NT = 3700 (i.e., Run 1 in Column 5) or m = 4and NT = 46,000 (i.e., Run 2 in Column 6) result in a failure samplenumber of 1128 and 20,482, respectively. This comparison clearlyshows that Subset Simulations significantly improve the efficiencyof generating failure samples, which enables generation of a largenumber of failure samples with relative ease and makes probabilis-tic failure analysis feasible. Table 2 also shows that, as the value ofm increases (e.g., from 3 in Column 5 to 4 in Column 6), the effi-ciency increases as well (e.g., the percentage of failure sample in-creases from 30.5% to 44.5%).

6. Probabilistic failure analysis results

With the large number of failure samples generated from Sub-set Simulations, probabilistic failure analysis are performed forthe James Bay Dyke, including hypothesis tests for identifyingkey uncertainties that have significant effects on slope failureprobability and Bayesian analysis for further quantifying theireffects.

6.1. Hypothesis test results

Based on the failure samples generated from Subsim Run 1, thehypothesis test statistics ZH defined by Eq. (2) are calculated andshown in Fig. 4 for all uncertain parameters. The absolute valuesof ZH varies from less than 2 for the thickness of clay crust Tcr toabout 95 for undrained shear strength of the lacustrine clay SuL.The decreasing order of the ZH absolute values is: SuL, DTill, cFill,SuM, /Fill, and Tcr. This implies that the uncertainty of SuL has themost significant effects on the slope failure probability, while theuncertainty of Tcr contributes the least to the failure probability.This result is consistent with that reported by El-Ramly [10] whoemployed an Excel spreadsheet – based MCS program @RISK [42]and compared the spearman rank correlation coefficients for vari-ous uncertain parameters to show that SuL and Tcr are the most andleast influential parameter, respectively. The results can be vali-dated by an independent sensitivity studies on SuL and Tcr, whichare described in next subsection.

6.2. Validation of hypothesis test results

Sensitivity studies are performed to explore the effect of SuL andTcr uncertainties on slope failure probability. The coefficient of var-iation (COV) of SuL and Tcr is varied in the sensitivity studies from0.15 to 0.50 and from 0.05 to 0.25, respectively. The SuL COV rangeadopted in this study follows the typical range of COV of undrainedshear strength of clay measured by vane shear tests [43], and theCOV of Tcr varies from half to about twice of the value reportedby El-Ramly [10]. Other parameters, including the means of SuL

and Tcr, remain unchanged in the sensitivity studies. About 40additional Subset Simulation runs are performed to validate thehypothesis test results, and their results are shown in Fig. 5a and

ect MCS with EXCEL Subsim with EXCEL Run 1 Subsim with EXCEL Run 2

2a 0.23a 0.25a

1128 20,482000 3700 46,0002 30.5 44.5

0 10 20 30 40 50 60 70 80 90 100

Absolute Value of ZH

Tcr

SuL

γFill

φ Fill

SuM

DTill

Fig. 4. Summary of absolute values of ZH.

1020 Y. Wang et al. / Computers and Geotechnics 37 (2010) 1015–1022

b for Tcr and SuL, respectively. In addition, sensitivity studies onCOVs of Tcr and SuL are also carried out using the Excel spreadsheet– based MCS program @RISK [42] and a FORM calculation spread-sheet developed by Low and his coworkers [5–8]. Fig. 5 includesthe results from @RISK and FORM for comparison.

Fig. 5a shows that, when the COV of Tcr varies from 0.05 to 0.25,the slope failure probability fluctuates between 0.19% and 0.43%.For comparison, the baseline failure probability (i.e., about 0.24%corresponding to the values summarized in Table 1) is also in-cluded in the figure. The failure probabilities from sensitivity studyusing Subset Simulations fall around the horizontal line of 0.24%,

0.01

0.1

1

10

100

0.05 0.10 0.15 0.20 0.25

COV of Tcr

Fai

lure

Pro

bab

ility

, P(F

) (%

) SubsimFORMisk

0.24%

COV=12.0% used by El-Ramly [10]

@ Risk

(a) Thickness of clay crust, Tcr

0.01

0.1

1

10

100

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

COV of SuL

Fai

lure

Pro

bab

ility

, P(F

) (%

) Subsim

FORM

Risk

0.24%

COV=20.2% used by El-Ramly [10]

@ Risk

(b) Undrained shear strength of the lacustrine clay, SuL

Fig. 5. Effects of coefficient of variation (COV) for different system parameters.

and the failure probability is insensitive to the uncertainty on Tcr.The results from Subset Simulations are in good agreement withthose from @RISK and FORM. This validates the results fromhypothesis tests that the Tcr uncertainty has the least effect onthe failure probability. It is interesting to note that the probabilityof slope failure should, theoretically, increase as the COV of Tcr in-creases from 0.05 to 0.25, as shown by the FORM results [i.e., arather slight increase of the Pf values shown by the open circlesin Fig. 5a when the COV of Tcr increases from 0.05 to 0.25]. The ef-fect is, however, so minimal that it is dominated by the MCS‘‘noise” (i.e., the random fluctuations of the Pf values obtained fromSubset Simulations and @RISK).

On the other hand, Fig. 5b shows that the slope failure probabil-ity increases as the COV of SuL increases. The results from SubsetSimulations are in good agreement with those from @RISK andFORM. When the COV of SuL is 0.15, the probability is less than0.1%. When the COV of SuL increases to 0.50, the slope failure prob-ability increases by two orders of magnitude (i.e., increases toabout 10%). The failure probability varies significantly with thechange of the SuL COV. This agrees well with the results fromhypothesis tests that the uncertainty of SuL has significant effecton the slope failure probability. Such agreement further validatesthat the hypothesis test procedure proposed in this paper properlyprioritizes effects of various uncertainties on failure probability.

6.3. Bayesian analysis results

Based on the failure samples generated from Subsim Run 2, aBayesian analysis is performed using Eqs. (3), (7), and (8) accord-

0.001

0.01

0.1

1

10

100

17 18 19 20 21 22 23 24

DTill(m)

Fai

lure

Pro

babi

lity

(%)

Parametric

Series4P(F|DTill) estimated from Bayesian analysisP(F|DTill) estimated from sensitivity study

(b) θ = depth of the till layer, DTill

(a) θ = undrained shear strength of the lacustrine clay, SuL

0.001

0.01

0.1

1

10

100

12 14 16 18 20 22 24

SuL(kN/m2)

Fai

lure

Pro

babi

lity

(%)

Parametric

Series4P(F|SuL) estimated from Bayesian analysisP(F|SuL) estimated from sensitivity study

Fig. 6. Bayesian analysis results for different system parameters h.

Y. Wang et al. / Computers and Geotechnics 37 (2010) 1015–1022 1021

ingly. Fig. 6 shows the Bayesian analysis results by open circles forSuL and DTill, which have been identified from the Hypothesis tests(see Section 6.1) as the two most influential uncertain parameters.Note that the conditional probability [i.e., P(F|SuL) in Fig. 6a andP(F|DTill) in Fig. 6b] obtained from Bayesian analysis is a variationof failure probability as a function of SuL or DTill. Fig. 6a shows that,as SuL increases from 12 kPa to 24 kPa, the slope failure probabilitydecreases from more than 10% to less than 0.1%. Similarly, Fig. 6bshows that, as DTill increases from about 18 m to 23 m, the slopefailure probability increases from about 0.1% to about 10%. It isobvious that the values of SuL and DTill have significant effects onslope failure probability, and such effects can be quantified explic-itly from the Bayesian analysis of failure samples.

0.01

0.1

1

10

100

12 14 16 18 20 22 24

SuL(kN/m2)

Pro

bab

ility

(%

)

Series4

Series1P(SuL)

P(SuL|F)

(b) θ = undrained shear strength of the lacustrine clay, SuL

(a) θ = thickness of clay crust, Tcr

0.01

0.1

1

10

100

2 3 4 5 6

Tcr(m)

Pro

bab

ility

(%

)

Series4

Series1P(Tcr)

P(Tcr|F)

0.001

0.01

0.1

1

10

100

17 18 19 20 21 22 23 24

DTill(m)

Pro

bab

ility

(%

)

Series6

Series4P(DTill)

P(DTill|F)

(c) θ = depth of the till layer, DTill

Fig. 7. Conditional probability density function (PDF) (P(h|F)) for different systemparameters h.

6.4. Validation of Bayesian analysis results

Note that variations of failure probability as a function of SuL orDTill shown in Fig. 6 can also be obtained from sensitivity studies onSuL or DTill, which frequently include many repeated runs of MCSwith different given values of SuL or DTill in each run. Therefore,to validate the Bayesian analysis results, about 40 additional Sub-set Simulation runs are performed with different given values ofSuL or DTill in each run. Fig. 6 also includes results from these addi-tional sensitivity runs by open triangles. The open triangles followtrends similar to the open circles (i.e., the Bayesian analysis results)and plot closely to the open circles as well. This validates that aBayesian analysis of failure samples generated in MCS or SubsetSimulations provides results that are equivalent to those fromadditional sensitivity studies. In addition, the Bayesian analysishas the advantage of avoiding additional computational timesand efforts for repeated runs of MCS or Subset Simulations in thesensitivity studies.

As mentioned before, Eq. (3) implies that comparison betweenthe conditional probability P(h|F) and its unconditional one P(h)provides an indication of the effect of the uncertain parameter hon failure probability. This offers a means to check the Bayesiananalysis results with those from hypothesis tests. Fig. 7 comparesthe conditional probabilities [i.e., P(Tcr|F), P(SuL|F) and P(DTill|F)]for Tcr, SuL and DTill and their unconditional ones [i.e., P(Tcr), P(SuL)and P(DTill)]. Fig. 7a shows that P(Tcr|F) and P(Tcr) are almost iden-tical for different Tcr values, which is consistent with the hypothe-sis test results that Tcr has the least effect on failure probability. Incontrast, the hypothesis tests show that SuL and DTill are the twomost influential uncertain parameters. Fig. 7b and c show thatP(SuL|F) and P(DTill|F) differ significantly from their respectiveunconditional one [i.e., P(SuL) and P(DTill)]. The Bayesian analysis re-sults agree well with the hypothesis test results.

7. Conclusions

This paper developed a probabilistic failure analysis approachthat makes use of the failure samples generated in the MCS andanalyzes these failure samples to assess the effects of variousuncertainties on slope failure probability. The approach containstwo major components: hypothesis tests for prioritizing effects ofvarious uncertainties and Bayesian analysis for further quantifyingtheir effects.

A hypothesis test statistic ZH was formulated to evaluate thestatistical difference of failure samples with their respective nom-inal (unconditional) samples. As the absolute value of ZH increases,the statistical difference between failure samples and their respec-tive nominal samples becomes growingly significant, and the effectof the uncertain parameter on failure probability also becomesgrowingly significant. Therefore, the absolute value of ZH is usedas an index to measure the effects of the uncertain parameterson failure probability and to prioritize their relative effects on fail-ure probability.

A Bayesian analysis approach was developed to further quantifyeffects of the uncertain parameters that have been identified fromthe hypothesis tests as influential parameters. Equations were de-rived for estimating conditional PDF [i.e., P(F|h)] of slope failure fora given value of uncertain parameter h. As P(F|h) is a variation offailure probability as a function of h, it can be considered as resultsof a sensitivity study of h on slope failure probability. In otherwords, a Bayesian analysis of the failure samples provides resultsthat are equivalent to those from additional sensitivity studies. Inaddition, it has the advantage of avoiding additional computationaltimes and efforts for repeated runs of MCS or Subset Simulations inthe sensitivity studies. Furthermore, it was shown that comparison

1022 Y. Wang et al. / Computers and Geotechnics 37 (2010) 1015–1022

between the conditional probability P(h|F) and its unconditionalone P(h) provides an indication of the effect of the uncertainparameter h on failure probability. This offers a means to checkthe Bayesian analysis results with those from hypothesis tests.

The resolution of P(F) and P(h|F) is pivotal to obtain P(F|h), and itdepends on the number of failure samples generated in MCS. Anadvanced Monte Carlo Simulation called Subset Simulation wasemployed to improve efficiency of generating failure samples inMCS and resolution at small failure probability levels. Subset Sim-ulation algorithm and its implementation in a commonly-availablespreadsheet environment were briefly described.

As an illustration, the proposed probabilistic failure analysis ap-proach was applied to study a design scenario of James Bay Dyke.The hypothesis tests show that the uncertainty of SuL has the mostsignificant effect on the slope failure probability, while the uncer-tainty of Tcr contributes the least to the failure probability. Thehypothesis test results are very consistent with results from inde-pendent sensitivity studies. Such agreement validates that thehypothesis test procedure proposed in this paper properly priori-tizes effects of various uncertainties on failure probability.

A Bayesian analysis was performed to quantify explicitly the ef-fects of SuL and DTill, which have been identified from the Hypoth-esis tests as the two most influential uncertain parameters. It isshown that the slope failure probability change significantly asthe values of SuL and DTill change. The Bayesian analysis resultshave also been validated against those from independent sensitiv-ity studies. In addition, a cross-check between the hypothesis testresults and the Bayesian analysis results shows that they agreewell with each other.

It is worthwhile to note that, although the proposed approachwas developed together with a slope stability analysis problem,the approach is general and applicable to other types of geotechni-cal analyses and engineering problems.

Acknowledgements

The work described in this paper was supported by a grant fromthe Research Grants Council of the Hong Kong Special Administra-tive Region, China [Project No. 9041484 (CityU 110109)] and astrategic research grant from City University of Hong Kong (ProjectNo. 7002455). The financial support is gratefully acknowledged.

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