Assessment of Slope Stability in the MonitoringParameter Space

X. Y. Li1; L. M. Zhang, F.ASCE2; S. H. Jiang3; D. Q. Li, M.ASCE4; and C. B. Zhou5

Abstract: Slope monitoring is routinely conducted, and observational information such as surface/underground displacements, groundwaterlevels, and rock bolt forces at multiple locations is collected. How to make use of the monitoring information to reveal failure mechanisms andassess the slope stability is a key issue in slope engineering. This paper presents a method for assessing the slope stability by integratingmonitoring parameters with physical analysis. The observed information first was used to back analyze the strength and loading parameters,and then the updated basic parameters were used to calculate the factor of safety or failure probability of the slope. The dominant basicparameters whose uncertainties influence the observed results the most were identified from the probabilistic back analysis. Alert levels weredefined in the monitoring parameter space on the basis of a factor of safety or failure probability criterion. A rock slope example was workedout to illustrate the application of the proposed method. DOI: 10.1061/(ASCE)GT.1943-5606.0001490. © 2016 American Society of CivilEngineers.

Author keywords: Slope stability; Structural health monitoring; Reliability; Risk management; Uncertainty principles; Bayesiananalysis.

Introduction

Monitoring and evaluating slope safety is essential for engineeredslopes whose failure can cause catastrophic consequences to lifeand property. The stability of a slope is determined by its geometry,material properties (e.g., cohesion and friction angle), and appliedloads (e.g., groundwater level and surcharge). In this paper, theseparameters are referred to as the basic parameters of the slope.Slope stability can be evaluated in the basic parameter space usinga numerical model or a close-form equation (Hoek and Bray 1974;Wu and Abdel-Latif 2000; Jiang et al. 2014a; Li et al. 2016). Thefailure mechanisms of the slope can be studied in this way. Asan additional safety measure, field monitoring is routinely con-ducted to evaluate the slope stability, provide basis for safety con-trol measures, and mitigate risks of slope failure (Peck 1969; Marr2013). Observational information such as surface/underground dis-placements, groundwater levels, and rock bolt forces is routinelycollected. In this paper, these parameters are referred to as the mon-itoring parameters. The assessment of the slope safety in the mon-itoring parameter space, rather than in the basic parameter space, ispreferred because real-time monitoring provides instant informa-tion for stability evaluation and decision making. The challenge

is how to rationally relate the slope stability to the monitoringparameters with the intrinsic failure mechanisms being considered.

Numerous papers have been published concerning the slope sta-bility evaluation on the basis of observed information, which can bedivided into the following three categories:1. Statistical analysis of historical observation data and empirical

relationships between the slope stability and the observed para-meters (Anbalagan 1992; Dai and Lee 2002; Ermini et al. 2005);

2. Use of the observed information to calibrate or back analyzeparameters for slope stability analysis models (Gilbert et al.1998; Crosta and Agliardi 2003; Zhang et al. 2010a, b; Wanget al. 2013; Jiang et al. 2014b); and

3. Updating of the slope stability with the observed informationwithin a Bayesian framework (Hsiao et al. 2008; Papaioannouand Straub 2012; Peng et al. 2014).Several important issues have not been discussed sufficiently in

the literature and require further investigations.1. The failure mechanisms of the slope may not be considered fully

in empirical relations, leading to difficulties in determining thecauses of changes in the slope performance.

2. Uncertainties associated with the observed information and theslope parameters should be considered rationally because theyaffect the result of the slope stability evaluation.

3. It is often necessary to identify the dominant basic parameterswhose uncertainties have the greatest influence on the observedresults so that proper measures can be taken to regulate thesedominant parameters when necessary.

4. How to set alert levels for taking emergency actions in the mon-itoring parameter space is another concern.The objective of this paper is to present a method for rationally

evaluating the slope stability in the monitoring parameter space.The observed data were first used to back analyze the basicparameters in a Bayesian updating framework that consideredboth the model parameter uncertainty and the model errors, andto identify the dominant basic parameters. Then, the slope stabilitywas analyzed with the updated basic parameters. A rock slopeexample was worked out to illustrate the application of the pro-posed method.

1Ph.D. Candidate, Dept. of Civil and Environmental Engineering,Hong Kong Univ. of Science and Technology, Clear Water Bay 999077,Hong Kong.

2Professor, Dept. of Civil and Environmental Engineering, Hong KongUniv. of Science and Technology, Clear Water Bay 999077, Hong Kong(corresponding author). E-mail: cezhangl@ust.hk

3Lecturer, Nanchang Univ., Jiangxi 430000, P.R. China.4Professor, State Key Laboratory of Water Resources and Hydropower

Engineering Science, Wuhan Univ., Wuhan 330000, P.R. China.5Professor, Nanchang Univ., Jiangxi 430000, P.R. China.Note. This manuscript was submitted on May 17, 2015; approved on

December 31, 2015; published online on March 29, 2016. Discussion per-iod open until August 29, 2016; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Geotechnical andGeoenvironmental Engineering, © ASCE, ISSN 1090-0241.

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Slope Stability Evaluation in the MonitoringParameter Space

Slope Stability in the Basic Parameter Space

Geotechnical parameters are inherently variable because of thecomplex natural geological processes. Therefore, the basic param-eters of the slope, X ¼ ðX1;X2; : : : ;XnÞ, are taken as random var-iables. The joint prior distribution ofX is denoted as fXðXÞ, whichcan be obtained through laboratory tests, published values, or ex-pert judgments. The factor of safety of the slope (FS) is a functionof the basic parameters

Fs ¼ gðXÞ þ ε ð1Þwhere gðXÞ = calculation model, which can be a numerical modelor a close-form equation; and ε = another random variable for char-acterizing the uncertainty associated with the calculation model.Consequently, FS is no longer a deterministic value, but a randomvariable. Its mean value can be calculated as

μFS¼

Z þ∞−∞

Z þ∞−∞

½gðXÞ þ ε�fXðXÞfεðεÞdxdε ð2Þ

where fεðεÞ = probability density function of ε.The mean factor of safety does not fully reflect the safety status

of the slope when uncertain parameters are considered (Ang andTang 2007; Duncan 2000). Failure probability is a better indicatorin such a case. It is defined as the probability that the slope will failwith all the uncertainties being considered, which is calculated as

Pf ¼ PrðFS ≤ 1Þ

¼Z þ∞−∞

Z þ∞−∞

I½gðXÞ þ ε ≤ 1�fXðXÞfεðεÞdxdε ð3Þ

where I½gðXÞ þ ε ≤ 1� = indicator function defined as

I½gðXÞ þ ε ≤ 1� ¼�1 gðXÞ þ ε ≤ 1

0 gðXÞ þ ε > 1ð4Þ

In this paper, failure is the factor of safety that is not larger thanunity. Sometimes a serviceability limit state criterion is used to de-fine failure. In such a case, failure can be defined as Yi ≥ Yi;cri, inwhich Yi is a key monitoring parameter and Yi;cri is the criticalvalue of Yi. Correspondingly, the indicator function in Eqs. (3)and (4) will be equal to 1 if Yi ≥ Yi;cri, and 0 for other cases.One important issue is how to choose a proper problem-specificcritical value Yi;cri, which will be discussed in section Alert Levelsin the Monitoring Parameter Space. To avoid ambiguity, failure issaid to have occurred if the limit-state criterion is violated, unlessstated otherwise in the following sections.

Probabilistic Back Analysis with MonitoringParameters

The monitoring parameters, Y ¼ ðY1;Y2; : : : ; YmÞ, which are theresponse quantities of the slope, also are determined by the basicparameters

Yi ¼ hiðXÞ þ εi þΔi ¼ hiðXÞ þ ζi ð5Þwhere hiðXÞ = calculation model; εi andΔi = error terms caused bythe model and the measurement process, respectively; and ζi ¼εi þΔi = total error. To evaluate the slope stability with the mon-itoring parameters, a space transformation from Y to FS has to beconducted. Fig. 1 shows a sketch of such a transformation. It is

composed of two steps: (1) from space Y to space X, which isa back analysis process; and (2) from space X to space FS, whichis a forward analysis process following Eq. (1).

Methodologies for back analysis can be classified into twogroups: deterministic methods (Wesley and Leelaratnam 2001;Tiwari et al. 2005) and probabilistic methods (Gilbert et al. 1998;Zhang et al. 2010a, b; Wang et al. 2013). Deterministic methodsaim to find a best set of basic parameters that would result in themonitoring results, whereas probabilistic methods recognize thatnumerous combinations of basic parameters can result in the samemonitoring results, although their relative likelihoods are different.Commonly used probabilistic back analysis methods in geotechni-cal engineering include the Kalman filter approach (Yang et al.2011), the maximum likelihood method (Wang et al. 2014), andthe Bayesian method (Zhang et al. 2010b). In this paper, to transferthe space from Y to X, a probabilistic back analysis approach onthe basis of the Bayesian framework was applied. The approachconsidered the model and parameter uncertainties, and had no limi-tation on the number of observed data.

Suppose the monitoring result is Y ¼ y. The likelihood functionfor one piece of measurement to occur can be expressed as

LðXÞ ¼ Pr½h1ðXÞ þ ζ1 ¼ y1jX� ¼ Pr½ζ1 ¼ y1 − h1ðXÞ�¼ C1fζ1 ½y1 − h1ðXÞ� ð6Þ

where C1 = proportionality coefficient; and fζ1ð•Þ = probabilitydensity function of ζ1. Eq. (6) is valid on the basis of theassumption that X and ζ1 are independent. When multiple piecesof measurements are available, if the additional uncertainties asso-ciated with different measurements are further assumed to be inde-pendent, the likelihood of observing these measurements can beexpressed as

LðXÞ ¼ Prf½ζ1 ¼ y1 − h1ðXÞ� ∩ : : : ∩ ½ζm ¼ ym − hmðXÞ�g

¼Ymi¼1

Pr½ζi ¼ yi − hiðXÞ� ¼ C2

Ymi¼1

fζi ½yi − hiðXÞ� ð7Þ

where C2 = proportionality coefficient.According to Bayes’ theorem, the joint distribution of X can be

updated (referred to as the posterior distribution) as

Fig. 1. Sketch of space transformation from the monitoring parameters,Y, to the factor of safety, FS

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fXjYðXÞ ¼ fXðXÞLðXÞRþ∞−∞ fXðXÞLðXÞdx ð8Þ

Substituting Eq. (7) into Eq. (8) yields

fXjYðXÞ ¼ fXðXÞQmi¼1 fζi ½yi − hiðXÞ�Rþ∞−∞ fXðXÞQmi¼1 fζi ½yi − hiðXÞ�dx ð9Þ

Coefficient C2 vanishes because it is the same in both the nom-inator and the denominator in Eq. (9). The denominator in Eq. (9) isa constant, which makes the probability density function valid.

Evaluation of the statistics of the updated X (e.g., mean valuesand standard deviations) can be carried out on the basis of Eq. (9)using such simulation methods as the Markov Chain Monte Carlo(MCMC) simulation (Gilks et al. 1996). The difference between theprior and posterior distributions indicates the effect of the uncertainparameter on the observed results, which can be used to identifydominant basic parameters. An indicator ZH is used to characterizethe statistical difference between the prior and posterior distribu-tions as follows (Wang et al. 2010):

ZH ¼ μ 0 − μ0

σ 0 ð10Þ

where μ0 and μ 0 = prior and posterior mean values, respectively;and σ 0 = posterior standard deviation. Because σ 0 is always pos-itive, a positive ZH indicates that the mean value of the parameterincreases after updating, whereas a negative ZH indicates that themean value of the parameter decreases after updating. The absolutevalue of ZH reflects the statistical difference between the prior andposterior distributions. As the absolute value of ZH increases, thestatistical difference becomes significant, which implies that theeffect of the uncertain parameter on the observed results alsobecomes more significant.

Slope Stability in the Monitoring Parameter Space

The mean factor of safety can be updated after incorporating theprobability distribution of the basic parameters, updated by themonitoring parameters. It can be obtained by replacing fXðXÞ inEq. (2) by fXjYðXÞ in Eq. (9)

μFSjY¼y

¼Rþ∞−∞

Rþ∞−∞ ½gðXÞ þ ε�fXðXÞQmi¼1 fζi ½yi − hiðXÞ�fεðεÞdxdεRþ∞−∞ fXðXÞQm

i¼1 fζi ½yi − hiðXÞ�dxð11Þ

Similarly, the failure probability also can be updated by replac-ing fXðXÞ in Eq. (3) by fXjYðXÞ in Eq. (9)

PfjY¼y

¼Rþ∞−∞

Rþ∞−∞ I½gðXÞþε≤1�fXðXÞQmi¼1fζi ½yi−hiðXÞ�fεðεÞdxdεRþ∞−∞ fXðXÞQm

i¼1fζi ½yi−hiðXÞ�dxð12Þ

Both Eqs. (11) and (12) can be calculated using a reliabilitymethod, such as Monte Carlo simulation (MCS), importance sam-pling, and subset simulation. In the following, PfjY¼y is calculatedby MCS. Eq. (12) can be approximated by

PfjY¼y ¼PNMCS

t¼1 fI½gðXðtÞÞ þ εðtÞ ≤ 1�Qmi¼1 fζi ½yi − hiðXðtÞÞ�gPNMCS

t¼1 fQmi¼1 fζi ½yi − hiðXðtÞÞ�g

ð13Þ

where NMCS = number of simulations used in the MCS; and XðtÞ

and εðtÞ = samples generated from fXðXÞ and fεðεÞ in the tth sim-ulation, respectively. The procedure is summarized as follows:1. Set t ¼ 1, nominator ¼ 0, and denominator ¼ 0.2. Sample XðtÞ and εðtÞ from fXðXÞ and fεðεÞ, respectively. Cal-

culate gðXðtÞÞ andQ

mi¼1 fζi ½yi − hiðXðtÞÞ�. Set denominator ¼

denominator þQmi¼1 fζ i ½yi − hiðXðtÞÞ�.

3. If gðXðtÞÞ þ εðtÞ ≤ 1, set nominator ¼ nominatorþQmi¼1 fζi ½yi − hiðXðtÞÞ�. Otherwise, set nominator ¼ nominator.

4. Go to Step (5) if t ¼ NMCS. Otherwise, set t ¼ tþ 1, and go toStep (2).

5. Calculate the updated failure probability: PfjY¼y ¼nominator=demominator.

Alert Levels in the Monitoring Parameter Space

In risk management, several alert levels are established to representdifferent slope performance states and response plans. For example,Marr (2011, 2013) suggested a green-yellow-red light system, inwhich the performance state is evaluated by the monitoring infor-mation. The threshold level and limit level are defined and used tointerpret the monitoring data. Below the threshold level is the greenlight zone, which indicates that the system is in a normal state with-out any anomalous or undesirable behavior. No action is required.In between the threshold level and the limit level is the yellow lightzone, which means that the system performance is outside the rangeexpected in the design. Prompt actions should be taken, such asdouble checking the measurements, investigating the causes, in-creasing the monitoring frequency, and rehabilitating the system.Above the limit level is the red light zone, which indicates thatthe safety of the system is in question with its performance deterio-rating and not controllable. Immediate actions should be taken torestore the system to a safer condition.

Establishing the threshold level and limit level is usually prob-lem specific, which needs to consider the cost of failure and followsome stability guidelines. In most cases, the threshold level andlimit level are defined on the basis of the factor of safety; e.g., FS ¼1.2 represents the threshold level, whereas FS ¼ 1.0 represents thelimit level. With the relationship between the mean factor of safetyor probability of failure, and the monitoring parameters derived inthe previous section [Eqs. (11) and (12)], the alert levels also can bedefined in the monitoring parameter space. For example, by settingμFS

¼ 1.2, one can solve Eq. (11) and get yi, which is the thresholdvalue of the monitoring parameter. Similarly, by setting μFS

¼ 1.0,one can get the limit value of the monitoring parameter. Thethreshold and limit values of the monitoring parameter obtainedin this way also are uncertain, which are affected by the variationsof the basic parameters and the model errors. If the limit valueof the monitoring parameter is observed, it only indicates thatthe mean factor of safety of the slope is 1.0, but the slope stillmay survive because the actual factor of safety may be larger than1.0 because of variations. Similar procedures can be followed toobtain the threshold and limit levels in the monitoring parameterspace that correspond to specified values of failure probability bysolving Eq. (12).

Illustrative Example

A case study is presented to illustrate the application of the pro-posed method to a rock slope. The stability of the slope was evalu-ated on the basis of monitored surface displacements, rock boltforces, and groundwater levels. The basic parameters were back

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analyzed from the monitoring parameters. Alert levels in themonitoring parameter space also were obtained.

Slope Model

A hypothetical rock slope (Fig. 2), which is revised from Shuklaand Hossain (2011) and Jiang et al. (2014b), was adopted in thispaper to illustrate the proposed method. The slope had a height of12 m and an inclination angle of 60°. A joint was present within therock mass. It started from the toe of the slope and extended at aninclination angle of 35°. It intersected a vertical tension crack at adepth of Z ¼ 4.35 m. The distance between the top of the tensioncrack and the crest of the slope was 4 m. The tension crack wasfilled with water to depth Zw. The groundwater table was assumedto be a polyline BOC in Fig. 2. A ratio rw ¼ Zw=Z was used tocharacterize the groundwater condition (Low 2007), e.g., rw ¼ 0when the tension crack was dry, whereas rw ¼ 1 when the tensioncrack was fully filled. According to Low (2007) and Li et al.(2011), rw can be modeled by a truncated exponential distribution.In this paper, an exponential distribution with a mean of 0.1, trun-cated within an interval (0, 1), was used to model rw. The top ofthe slope was subjected to surcharge p, which follows a normaldistribution with a mean of 200 kPa and a standard deviation of20 kPa. To reinforce the slope, four rows of rock bolts were in-stalled at a spacing of 2.5 m in both the horizontal and vertical di-rections. The effective anchoring length into the rock mass aftercrossing the joint was 4 m. The angle between the rock bolts andthe normal direction of the joint was 35°. The rock bolts were as-sumed to be hot-rolled ribbed steel bars with a diameter of d ¼28 mm, a Young’s modulus of 200 GPa, and a yield stress ofσt ¼ 400 MPa. Therefore, the tensile yield force was Ft ¼ σtπd2=4 ¼ 246 kN. The bond stiffness, bond strength, bond friction an-gle, and bond diameter were 1 GPa, 100 kN=m, 40°, and 120 mm,respectively. The strength and deformation parameters of the rockmass and the joint are listed in Table 1. Among them, the cohesion(c), friction angle (ϕ), and Young’s modulus (E) of the joint aremore variable, so they were set as random variables. Therefore,there were five basic parameters for this slope:X ¼ ðc;ϕ;E; rw;pÞ.

Their prior distributions are summarized in Table 2. It is likely thatc and ϕ are negatively correlated (Low 2007; Zhang et al. 2010a).For simplicity, all the basic parameters were assumed to be inde-pendent from each other so that their joint distribution was theproduct of their marginal distributions. The Young’s modulus wastreated as a basic parameter because it is a key parameter that af-fects slope deformations.

To monitor the performance of the slope, several types of mon-itoring data were collected, including the vertical displacement ofthe joint at the top (YA), the horizontal displacement of the toe (XB),the depth of groundwater table (H), and the axial force of the rockbolt in the top row (F). Therefore, four monitoring parameterswere involved in this case: Y ¼ ðYA;XB;H;FÞ. Because it is not

12 m

Z = 4.35 m

35o

60o

p

Rock bolt

XB

YA

Rock mass

Joint

4 m

Rock mass

Zw

H

F

B

OC

Fig. 2. Geometry of the rock slope

Table 1. Properties of the Rock Mass and the Joint

Parameter Rock mass Joint

Cohesion (kPa) 67 cFriction angle (°) 40 ϕYoung’s modulus (MPa) 5,000 EPoisson ratio 0.25 0.3Dilation angle (°) 0 0Tensile strength (MPa) 5 0Density (kg=m3) 2,600 2,000

Table 2. Prior Distributions of the Basic Parameters

Parameter Distribution Mean SDCoefficient of

variation

c (kPa) Normal 20 6 0.3ϕ (°) Normal 32 6.4 0.2E (MPa) Normal 50 10 0.2rw Truncated exponential 0.1 -p (kPa) Normal 200 20 0.1

Note: rw is truncated within (0, 1).

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a real-world slope problem, hypothetical values were assigned tothe monitoring parameters in the following sections.

A two-dimensional finite difference analysis program, FLAC(Itasca 2000), was used to analyze the slope. Both the rock massand the joint were modeled by the elastic-perfectly plastic Mohr-Coulomb model. The parameter properties are listed in Table 1. Theinitial stress first was generated by turning on the gravity. Thegroundwater condition was modeled by a groundwater table. Then,cable elements were added to simulate the rock bolts. Afterwards,the surcharge on the top of the slope was applied, and the displace-ments and the rock bolt forces were calculated. Finally, the strengthreduction method (Griffiths and Lane 1999) was used to calculatethe factor of safety of the slope. Fig. 3 shows the maximum shearstrain increment and the displacement vector of the slope near fail-ure. It is shown that the most critical slip surface was coincidentwith the joint. The maximum axial force of each rock bolt occurrednear the joint, because the displacements in this zone were rela-tively large.

Two possible external factors that may trigger slope failure areincrease of the surcharge and rising of the groundwater level. Theeffect of the surcharge on the slope performance was investigatedby applying different surcharges to the slope, whereas other param-eters were kept constant. The results are plotted in Fig. 4. As thesurcharge increased, the factor of safety decreased. The displace-ments increased with increasing surcharge slowly at the beginning,and rapidly increased as the slope approached failure (the factor ofsafety is close to unity), which is consistent with the rock massbehavior in a plastic constitutive model. The axial force of therock bolt increased with the surcharge. It reached the tensile yieldforce (246 kN) before the slope failed. The effect of the ground-water level also was investigated in a similar way, as shown inFig. 5. As the groundwater level rose, both the displacements and

Max.

(a)

(b)

. shear strain increment 0.00E+00 1.00E-02 2.00E-02 3.00E-02 4.00E-02 5.00E-02 6.00E-02 7.00E-02 8.00E-02

Fig. 3. (a) Maximum shear strain increment; (b) displacement vector ofthe slope near failure

0 50 100 150 200 250 300 350

0

5

10

15

20

Dis

plac

emen

t (m

m)

Surcharge, p (kPa)

XB

YA

1.0

1.2

1.4

1.6

1.8

FS

Fac

tor

of s

afet

y, F

S

0 50 100 150 200 250 300 350

0

50

100

150

200

250

Roc

k bo

lt fo

rce,

F (

kN)

Surcharge, p (kPa)

1.0

1.2

1.4

1.6

1.8

Fac

tor

of s

afet

y, F

SFFS

Fig. 4. Effects of surcharge on the slope performance

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

Dis

plac

emen

t (m

m)

Groundwater level ratio, rw

XB

YA

1.0

1.2

1.4

1.6

1.8

FS

Fac

tor

of s

afet

y, F

S

0.0 0.2 0.4 0.6 0.8 1.0

0

50

100

150

200

250

Roc

k bo

lt fo

rce,

F (

kN)

Groundwater level ratio, rw

1.0

1.2

1.4

1.6

1.8

Fac

tor

of s

afet

y, F

S

FFS

Fig. 5. Effects of groundwater level on the slope performance

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the rock bolt force increased, whereas the factor of safety de-creased, indicating that the slope became less stable.

Prior Reliability of the Slope

The prior mean factor of safety and prior failure probability of theslope can be estimated by Eqs. (2) and (3), respectively. The cal-culation model, gðXÞ, in these two equations, was substituted bythe numerical model for the slope. When the basic parametersX ¼ðc;ϕ;E; rw;pÞ were specified and input into the model, the factorof safety was calculated. However, challenges remained in the com-putational time and resources required in Monte Carlo simulations,in which a substantial number of model runs are needed. The re-sponse surface method is one of the most popular methods to solvesuch computationally intensive reliability problems (Bucher andBourgund 1990; Zhang et al. 2010b). The idea is to replace thenumerical model by an approximated less-expensive polynomialequation, so that the computational cost can be greatly reduced.In this paper, a second-order polynomial response surface functionof the following form was used:

gðXÞ ¼ α0 þX5i¼1

αi1Xi þX5i¼1

αi2X2i ð14Þ

where Xi ¼ ith component of X; and α0, αi1, and αi2 = regressioncoefficients. A total of 52 design points were adopted on the basisof a central composite design technique (Aslan 2008) and used todetermine the regression coefficients by the least squares method.Thereafter, the response surface function instead of the numericalmodel was used in the analysis. The use of the response surfacemethod introduced extra uncertainty, which is regarded as part ofthe model uncertainty and considered in the error term, ε. Toquantify the model uncertainty is difficult and often requires alarge number of actual measurements (Zhang et al. 2009; Chinget al. 2010), which was beyond the scope of this study. Becausethe illustrative example is not a real-world slope problem, themodel uncertainty was estimated on the basis of the calculationerror of the response surface function and the assumed error ofthe numerical model. In this paper, ε was assumed to follow anormal distribution with a mean of 0 and a standard deviationof 0.05. Finally, Monte Carlo simulations with 500,000 sampleswere conducted according to Eqs. (2) and (3), which resulted in aprior mean factor of safety of 1.11, and a prior failure probabilityof 24.9%.

Slope Stability Evaluated in the Monitoring ParameterSpace

The relationship between the monitoring parameters and the basicparameters, e.g., hðXÞ in Eq. (5), was established through thenumerical model. Similarly, the response surface functions wereobtained to avoid the expensive computational cost. Afterwards,Monte Carlo simulations were conducted according to Eqs. (11)and (12) to obtain the mean factor of safety and the failure prob-ability. The model uncertainties of YA, XB, H, and F were assumedto follow normal distributions with mean values equal to 0, andstandard deviations equal to 1 mm, 1 mm, 0.2 m, and 20 kN, re-spectively. Such assumptions were made on the basis of the calcu-lation errors of the response surface functions, the assumed errorsinduced by the numerical model, and the measurement processattributable to the lack of information for satisfactory evaluationof the model uncertainties.

The cases in which there was only one monitoring parameterwere studied first. For instance, assume YA is the only monitoring

parameter. By assigning different hypothetical values to YA[i.e., yYA

in Eqs. (11) and (12)], one can obtain the correspondingmean factor of safety and failure probability, so that the relationsbetween YA and μFS

, and between YA and Pf, can be obtained. Theresults are shown in Fig. 6(a). As YA increased, μFS

decreased andPf increased, indicating that the slope became increasingly unsta-ble. The relationship between YA and μFS

is quite linear, whereasthat between YA and Pf is quite nonlinear. An obvious turning pointin which Pf starts to speed up was found.

After considering the observed results, the updated distributionsof the basic parameters also were analyzed. MCMC, which is apowerful method to generate samples from any probability distri-butions, first was applied to generate samples of the basic param-eters from the posterior distribution on the basis of Eq. (9). Thebasic idea of MCMC is to draw samples from an arbitrary distri-bution, and then correct the samples on the basis of the probabilitydensity function of the target distribution. If the number of samplesis large enough, the corrected samples will finally converge to thetarget distribution (Gilks et al. 1996). The detailed application ofMCMC in geotechnical problems can be referred to Zhang et al.(2010b) and Cao and Wang (2014). In this paper, 500,000 MCMCsimulations with the Metropolis algorithm (Metropolis et al. 1953),which is a popular way to construct a Markov Chain, were applied.Afterwards, statistical analysis was conducted to calculate the up-dated mean values and standard deviations, which were then com-pared with the prior ones. Fig. 6(b) plots the statistical indicator ZH[Eq. (10)] of different basic parameters. It is shown that as YA in-creased, the mean values of c, ϕ, and E decreased, and those of rwand p increased, indicating that a larger displacement can be causedby a smaller c, ϕ, or E value, and a larger rw or p value. The

(a)

SF

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Vertical displacement at Point A, YA (mm)

c

Er

w

p

Fig. 6. Slope performance evaluated on the basis of YA (a) Pf and μFS;

(b) ZH

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absolute value of ZH was the largest for ϕ, hence it was the dom-inant parameter for the observed displacement. Fig. 7 shows theslope performance evaluated on the basis of XB, H, and F, respec-tively. The relationships between μFS

and each of the monitoringparameters were quite linear, whereas those between Pf and each ofthe monitoring parameters were strongly nonlinear. Because H andrw are closely related [rw ¼ Zw=Z ¼ ðZ −HÞ=Z], the measured Honly affected the distribution of rw, and had little influence on otherbasic parameters [Fig. 7(b)]. The ϕ seemed to be the dominantparameter for the observed displacements and rock bolt forcesbecause its absolute value of ZH was the largest.

Cases with multiple monitoring parameters also were investi-gated. The mean factor of safety and failure probability evaluatedon the basis of two monitoring parameters, YA and XB, are plottedin Fig. 8; whereas those evaluated by three monitoring parameters,XB, H, and F, are plotted in Fig. 9. It is shown that larger values ofYA, XB, F, or a smaller H will lead to a smaller mean factor ofsafety and a larger failure probability. The μFS

changed nearly lin-early with the monitoring parameters, whereas the change of Pfwas nonlinear. The proposed method was not limited to three mon-itoring parameters.

Alert Levels Determined by the Monitoring Parameters

The alert levels can be established on the basis of a factor of safetycriterion or a failure probability criterion. They can be representedby one or multiple monitoring parameters. First, consider the alertlevels represented by a single monitoring parameter, YA, on the ba-sis of the factor of safety criterion. Assume that for the slope con-sidered, a factor of safety of 1.2 denotes the threshold level,whereas a factor of safety of 1.0 denotes the limit level. As shownin Fig. 6(a), the corresponding threshold value and limit value of YAcan be determined, which are 4.1 mm and 7.2 mm, respectively. IfYA is smaller than 4.1 mm, the slope is in the green light zone; if YAis between 4.1 and 7.2 mm, the slope is in the yellow light zone; andif YA is larger than 7.2 mm, the slope is in the red light zone. Similarprocedures can be followed to determine the threshold valuesand limit values of other single monitoring parameters on the basisof either the factor of safety criterion or the failure probabilitycriterion.

The alert levels also can be represented by multiple monitoringparameters. For example, Fig. 10 shows the alert levels determinedby XB and YA on the basis of the factor of safety criterion[Fig. 10(a)], and the failure probability criterion [Fig. 10(b)]. It

-3

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9 12

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60 120 18

F (kN)

60 120 18F (kN)

80 2400.8

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80 240

Fig. 7. Slope performance evaluated on the basis of (a) XB; (b) H; (c) F

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10 0

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SF

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5

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1.0

Fig. 8. Slope performance evaluated on the basis of YA and XB: (a) μFS;

(b) Pf

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is assumed that for the slope considered, a failure probability of0.01 denoted the threshold level, whereas a failure probabilityof 0.4 denoted the limit level. The three performance state zonesof the slope can be identified from Fig. 10. The data pair (XB,YA) was used to determine the current performance state of theslope, which makes better use of the monitoring information.Because nearly real-time monitoring data are available, it is veryefficient to assess the slope safety status with such alert levelsin the monitoring parameter space.

Conclusions

This paper proposes a method for assessing the slope stability in themonitoring parameter space. Multiple pieces of observed informa-tion can be used efficiently to evaluate the stability of the slopeconsidering failure mechanisms. A rock slope example was pre-sented to illustrate the application of the proposed method. Thefollowing conclusions can be drawn:1. The slope stability is linked to the monitoring parameters

through the basic parameters. To evaluate the slope stability withthe monitoring parameters, one needs to first back analyze thebasic parameters from the observed information, and then usethe updated basic parameters to calculate the factor of safety orfailure probability of the slope.

2. How a basic parameter influences the observed results can beinvestigated by comparing the posterior and prior distributions.

The difference between the prior and posterior distributionsindicates the effect of the uncertain parameter on the observedresults, which can be used to identify the dominant basic para-meters. For the slope in this paper, increases in the observeddisplacements and rock bolt forces were caused by an increasedsurcharge, rising of the groundwater level, and decreases of thestrength and deformation parameters of the joint. The frictionangle of the joint was the dominant basic parameter whose un-certainty affects the observed displacements and rock bolt forcesthe most.

3. The alert levels for slope safety management can be establishedon the basis of the factor of safety criterion or the failure prob-ability criterion. They can be expressed in the monitoring para-meter space, which makes it efficient to assess the slope safetystatus directly with the monitoring information.

Acknowledgments

The research reported in this paper was substantially supportedby the National Basic Research Program of China (Project No.2011CB013506), the Research Grants Council (RGC) of the HongKong SAR (Grant Nos. HKUST6/CRF/12R and 16212514), theNational Science Fund for Distinguished Young Scholars (ProjectNo. 51225903), and the Natural Science Foundation of HubeiProvince of China (Project No. 2014CFA001).

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Fig. 9. Slope performance evaluated on the basis of XB, H and F:(a) μFS

; (b) Pf Fig. 10.Alert levels determined by YA and XB on the basis of (a) factorof safety criterion; (b) failure probability criterion

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