Structural Safety 49 (2014) 65–74

Contents lists available at ScienceDirect

Structural Safety

journal homepage: www.elsevier .com/locate /s t rusafe

Slope safety evaluation by integrating multi-source monitoringinformation

0167-4730/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.strusafe.2013.08.007

⇑ Corresponding author. Tel.: +852 2358 8720; fax: +852 2358 1534.E-mail address: cezhangl@ust.hk (L.M. Zhang).

M. Peng a,b, X.Y. Li b, D.Q. Li c, S.H. Jiang c, L.M. Zhang b,⇑a Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Department of Geotechnical Engineering, Tongji University, Shanghai, Chinab Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kongc State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, China

a r t i c l e i n f o

Article history:Available online 12 December 2013

Keywords:Slope stabilityField monitoringInformation fusionBayesian networkReliability analysisMarkov chain Monte Carlo simulation

a b s t r a c t

A systematic method is presented for evaluating the slope safety utilizing multi-source monitoring infor-mation. First, a Bayesian network with continuously distributed variables for a slope involving the factorof safety, multiple monitoring indexes and their influencing parameters (e.g. friction angle and cohesion)is constructed. Then the prior probabilities for the Bayesian network are quantified considering modeland parameter uncertainties. After that, multi-source monitoring information is used to update the prob-ability distributions of the soil or rock model parameters and the factor of safety using Markov chainMonte Carlo simulation. An example of a slope with multiple monitoring parameters is presented to illus-trate the proposed methodology. The method is able to integrate multi-source information based onslope stability mechanisms, and update the soil or rock parameters, the slope factor of safety, and the fail-ure probability with the integrated monitoring information. Hence the evaluation becomes more reliablewith the support of multiple sources of site-specific information.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction formance. A holistic assessment of the slope safety state may not

Field monitoring is an important means to evaluate the safetystate of slopes, provide basis for slope safety control measures,warn of impending failures and mitigate risks of slope failures[1,2]. Common instruments and measuring indexes in slope mon-itoring are summarized in Table 1 according to Marr [2]. Withappropriate instruments, changes in slope characteristics such assoil stresses, pore water pressures and crack development can bemeasured and used as a basis for evaluating the slope safety.

Einstein and Sousa [1] emphasized that monitoring results andobservations need to be properly interpreted to evaluate slopesafety. Numerous studies have been conducted on the interpreta-tion of slope safety based on monitoring information. Some studiesaimed to find a relationship between rainfall and slope failure withstatistical analysis [3–6] or mechanical analysis [7,8]. It is also be-lieved that slope displacements, especially displacement incre-mental rates, are important indicators to evaluate slope safety[9,10]. Some advanced methods such as GIS and ground-based ra-dar have been attempted for slope safety evaluation and warning[11–13].

Existing interpretation methods typically use one single index(e.g. surface or underground deformation, pore pressure, or rain-fall) as a predictor and hence reveal only one aspect of slope per-

be achieved. Besides, a large portion of monitoring data is oftennot utilized in these methods. Slope safety evaluation using multi-ple sources of monitoring information may be more reasonable,and is the topic of the present research.

Data fusion is a process of integrating multiple sources of dataand knowledge representing the same structure into a consistent,accurate and useful representation [14]. Chang et al. [15] andWong et al. [16] conducted landslide hazard assessment basedon the fusion of multisource remote sensing images of the samesite. To reduce the errors of monitoring information from humanmistakes or faulty equipment, Guo et al. [17] and Peng et al. [18]applied a data-fusion method to filter and extract monitoringinformation from multiple sensors. Available data fusion methodscan improve the quality of monitoring data of the same type (e.g.displacement) by fusing the information from multiple sensors.However, information from different types of monitoring data(e.g. displacement, stress and pore water pressure) may not beintegrated easily and applied to evaluate the slope safety in thesemethods. Artificial neural networks were sometimes used for land-slide susceptibility assessment with multiple pieces of informationof the influential factors (e.g. slope geometry, soil thickness anddrainage) [19]. However, little physical analysis is involved in arti-ficial neural networks; the uncertainties involved are not includedexplicitly either.

In this study, a systematic method of slope safety evaluation ispresented utilizing multi-source monitoring information withBayesian networks. The method is aimed to (1) integrate multi-

Table 1Instruments for monitoring global stability of slopes (modified from Marr [2]).

Instrument type Instrument Measured index

Geometricmeasures

Ground reference point, surveymonument

Settlement andmovement

Settlement plate, platform or cell SettlementVertical inclinometer InclinationHorizontal inclinometer InclinationTilt meter RotationCrack meter WidthExtensometer MovementTime domain reflectrometry (TDR) DistanceAutomated total station Change in positionDifferential Global PositioningSystem (dGPS)

Movement

LIDAR (Light Detection andRanging)

Vertical movement

InSAR (Interferometric syntheticaperture radar)

Surface map

Digital camera with referencetargets

Movement

Liquid level gages SettlementWater related

measuresObservation well Water levelPiezometer Water pressureRain gage Rainfall

Mechanicalmeasures

Strain gage Strain/stressLoad cell ForceAccelerometer Dynamic forcesAcoustic emission monitoring Particles slidingFiber optic sensor Strain or pressureMicro-seismic Energy

Temperaturemeasures

Thermister, thermocouple and TDR TemperatureFiber optic sensor Temperature

Cohesion

……

Friction angle

Deformation

Stress

Pore water pressure

……

Factor of safety

Failure probability

Monitoring information

Slope parameters Slope safety

Back analysis including model uncertainty and

site performance Inference

Fig. 1. Principle of fusion of monitoring data from multiple sources.

66 M. Peng et al. / Structural Safety 49 (2014) 65–74

source information based on slope stability mechanisms; (2)update the soil or rock model parameters with the integrated mon-itoring information; and (3) predict the factor of safety and failureprobability of the slope with the integrated monitoring informa-tion. An example of a slope with multiple monitoring indexes ispresented to illustrate the proposed methodology.

2. Proposed method of slope safety evaluation by integratingmulti-source monitoring information

2.1. The Bayesian network

A Bayesian network is a probabilistic graphical model that rep-resents a set of random variables and their conditional dependen-cies via a directed acyclic graph [20]. A Bayesian network combinesthe knowledge of graph theory and statistics theory. It consists ofnodes (parameters) and arcs/links (inter-relationships) with their(conditional) probabilities, which can be applied to solve uncer-tainty problems by logic reasoning. Bayesian networks have beenproven to be a robust method for reliability analysis and risk anal-ysis, including the integration of multiple sources of information[21–26].

2.2. The framework

Both the factor of safety, FS, and monitoring information areindicators of the performance of a slope of certain parameters(e.g. cohesion, friction angle etc.), as shown in Fig. 1. A change inthe slope parameters will lead to changes in the monitoring infor-mation and slope safety. Based on this, the slope safety evaluationwith multi-source monitoring information is conducted by updat-ing the slope parameters using the monitoring information (backanalysis), and calculating the FS and the failure probability (PF) ofthe slope with the updated slope parameters (inference) as shownin Fig. 1. The main steps of the proposed method are shown inFig. 2:

(1) A causal network is first constructed considering relation-ships among FS, monitoring parameters (M) and slopeparameters (S).

(2) The FS and M as response functions of S, namely, FS(S) andM(S) in Fig. 2, are obtained using finite element analysisand a response surface method.

(3) The prior probabilities of the Bayesian network, namely, theprior probabilities of S, P(S), and the prior conditional prob-abilities of M given S, P(M|S), and FS given S, P(FS|S), areobtained by applying Monte Carlo simulation to theresponse functions, FS(S) and M(S), as shown in Fig. 2. Themodel uncertainties are included in this step.

(4) The posterior probabilities of S, P(S|m), are obtained byupdating the prior, P(S), with available monitoring informa-tion, M = m, as shown in Fig. 2. Markov Chain Monte Carlo(MCMC) simulation is applied in this step since the Bayesiannetwork in this study involves non-normal continuous prob-ability distributions.

(5) Finally, the posterior probability of FS, P(FS|m), is calculatedwith the updated slope parameters as well as the probabilityof slope failure (Fig. 2).

Actually, the updating of the slope parameters and theupdating of the factor of safety, namely, steps (4) and (5), areexecuted at the same time in the computation program. Theyare described in two steps here in order to better understandthe updating mechanisms. The framework of the proposedmethod will be introduced step by step later in the paper. Anexample of a slope with multiple monitoring indexes will bepresented to illustrate the proposed framework toward theend of this paper.

2.3. Constructing a Bayesian network by considering FS andmonitoring parameters

As shown in Fig. 3, a simple casual network is built for a soilslope by considering the soil parameters (i.e. cohesion c and fric-tion angle /), two monitoring parameters (i.e. the vertical displace-ment at point A, DA and the vertical stress at point B, SB), FS and theslope safety state (safe/failed or S/F). In this case, FS, DA and SB aregoverned by c and / if other parameters are well defined (c and /are the parents of FS, DA and SB); S/F depends on the value of FS (FS isthe parent of S/F), which means S/F = safe if FS > 1 and S/F = failed ifFS < 1. c and / are uncertain and difficult to obtain, while DA and SB

can be measured using monitoring instruments. Therefore in theBayesian network, the monitoring information of DA and SB canbe used to update c, /, FS and S/F.

S

M

FS

S

M(S)

FS(S)

P(S)

P(M)

P(FS)

P(S|m)

M=m

P(FS|m)?

P(S|m)

M=m

P(FS|m)

Bayesiannetwork

Response functions

The prior probabilities

Updated slope

parameters

Updated factor of safety

Numerical simulation and response surface

Monte Carlo simulation Markov Chain Monte CarloMethods:

Network and parameters :

Results:

Note: M = monitoring parameters ; S = slope parameters ; FS = factor of safety; m = monitoring values ; M(S) = M as a response function of S; P(S) = the prior probability of S ; P(M) = the prior conditional probability of M given S; P(S|m) = the conditional probability of S given m ; P(FS|m) = the posterior conditional probability of F given m.

Fig. 2. Procedure of slope safety evaluation using multi-source monitoringinformation.

c φ

SBFS

Slip surface

A

BDA

S/F

Cohesion cFriction angle φ

Fig. 3. A Bayesian network related to the soil parameters, FS, and monitoringparameters.

M. Peng et al. / Structural Safety 49 (2014) 65–74 67

2.4. Determination of response functions using finite element analysis

After building the casual network in Fig. 3, the next step is toquantify the network by considering the inter-relationships amongthe nodes or parameters. Theoretically, given c and /, the values ofFS, DA and SB can be calculated as response functions, g(c, /), usingthe finite element analysis method. Normally, g(c, /) is implicit;therefore, reliability analysis directly based on g(c, /) is difficultto conduct because it requires iterative and intensive computa-tions. In this study, g(c, /) is approximated with a third-order poly-nomial function of the following form:

gðc;/Þ ¼Xr

i¼1

a3ix3i þ

Xr

i¼1

a2ix2i þ

Xr

i¼1

a1ixi þ a0 ð1Þ

where r = 2; x1 = c; x2 = /; a3i, a2i, a1i, a0 = regression coefficients.The regression coefficients can be obtained by equating g(c, /) cal-culated using Eq. (1) to those calculated using the finite elementanalysis. The above methodology is often called ‘‘response surfacemethod’’ and has been used frequently in geotechnical reliabilityanalysis [27–29]. After the third order polynomial function is cali-brated, it will be used in MCMC simulation instead of the finite ele-ment method for evaluating the likelihood function. The accuracy ofthe response surface method will be examined and discussed in theexample to be presented.

2.5. Quantification of prior probabilities considering modeluncertainties

The prior probabilities of the Bayesian network are quantifiedwith the aforementioned response functions considering model er-rors. Model error, which is produced when a model is used to sim-ulate a real case, is the difference between the estimated value andthe real value. The model error can be considered by adding a ran-dom item, e, in the model, where e is often assumed as a normalvariate. In this study, two types of model errors are considered.

The first is the numerical model error, which is produced in numer-ical simulation:

gðc;/Þ ¼ Nðc;/Þ þ e1 ð2Þ

where N(c, /) is the numerical result and e1 is the numerical modelerror. The numerical model errors for different monitoring parame-ters may be correlated through the slope parameters.

The second is the response surface model error, which occurs inthe response surface method:

Nðc;/Þ ¼ Rðc;/Þ þ e2 ð3Þ

where R(c, /) is the simulated result from the response surface ande2 is the response surface model error. Therefore, the real value canbe expressed as

gðc;/Þ ¼ Rðc;/Þ þ e1 þ e2 ð4Þ

With Eq. (4), Monte Carlo simulation can be applied to obtainthe conditional probabilities of FS, DA and SB given c and /.

2.6. Updating slope parameters, FS and F/S with Markov chain MonteCarlo simulation

The slope safety evaluation using multi-source monitoringinformation is conducted by updating the slope parameters, FS

and S/F with site-specific monitoring information. In the Bayesiannetwork as shown in Fig. 3, a target vector of variables, y [i.e.(DA, SB)] is expressed as

y ¼ gðhÞ ¼ gðc;/Þ ð5Þ

where h is a vector of random parameters, which is (c, /) in thiscase. Based on Bayes’ theorem [30,31], the posterior probabilitydensity function of h given the monitoring information can be ex-pressed as

f ðhjyÞ ¼ kLðhjyÞf ðhÞ ð6Þ

where k is a constant to make the probability density function valid,f(h) is the prior probability density function of h and L(.) is a likeli-hood function. If multi-source monitoring information is used [i.e.y = (DA, SB)], the pieces of observed information may be assumedindependent, i.e. LðhjyÞ ¼ LðhjDAÞ � LðhjSBÞ. With the updated h, theposterior distribution of FS can be predicted as

f ðFSjyÞ ¼Z

f ðFSjhÞf ðhjyÞdh ð7Þ

where f(FS|h) is the distribution of FS with parameters h. The meanand variance of FS are updated as

lðFSÞ ¼Z

FSf ðFSjyÞdFS ð8Þ

VðFSÞ ¼Z

F2S f ðFSjyÞdFS � l FSð Þ2 ð9Þ

The probability of slope failure is given by

PðS=F ¼ failedÞ ¼ PðFS < 1Þ ¼Z 1

0f ðFSjyÞdFS ð10Þ

A key part of the above analysis is to update parameters h basedon the monitoring information as shown in Eq. (6). Since f(h) andL(h|y) could be any types of continuous distributions, f(h|y) maynot be obtained with conjugate distributions [30]. In this study,Markov Chain Monte Carlo (MCMC) simulation is introduced tosolve this problem. In the MCMC method, one can use the previoussample values to randomly generate the next sample value, form-ing a Markov chain. The sample data in the Markov chain are

12 m

4.35

m

4 m

Load

Anchored bar

Head block

YA

Rock Rock mass

Free bar

Joint

4 m

3 m

68 M. Peng et al. / Structural Safety 49 (2014) 65–74

proved to follow the required distribution as shown in Eq. (6) [32].Then the sampled data are used to obtain the estimator withMonte Carlo simulation according to Eqs. (7)–(10).

The Metropolis algorithm [32], which is one of the most popularways to generate Markov chains, is adopted in this study to con-duct the MCMC simulation. The Metropolis algorithm can be writ-ten as follows [32,33]:

(1) Start with any initial value h0 satisfying f(h0) > 0. h0 can berandomly drawn from a starting distribution, or simply cho-sen deterministically, for example, as the mean value.

(2) For i = 1,2, . . .,n:

35o

60o

XB

SC

Rock mass

3m

Fig. 4. An anchored rock slope with a single slip surface (based on [34]).

Table 2Parameters of rocks and joints.

(a) Sample a candidate h⁄ from a jumping distribution ortransition kernel J(h⁄|hi�1), which is the probability ofreturning a value of h⁄ given a previous value of hi�1. The-oretically, the jumping distribution could be any distri-bution. The only restriction in the Metropolis algorithmis that the jump density must be symmetric, i.e.,J(h⁄|hi�1) = J(hi�1|h⁄).

(b) Calculate the ratio of the densities

r ¼ f ðh�jyÞf ðhi�1jyÞ

ð11Þ

Rock parameter Value Joint parameter Value

Cohesion (kPa) , c 67 Cohesion (kPa), c LogN(30,*

where f(h|y) is calculated with Eq. (6), therefore,

6)

Friction angle (degree),/

40 Friction angle (degree),/

LogN(25,4)*

Young’s modulus(MPa)

5000 Young’s modulus(MPa)

50

Unit weight (kN/m3) 26 Unit weight (kN/m3) 20Poison’s ratio 0.25 Poison’s ratio 0.3Load (kN/m), P N(300,

30)*

– –

* LogN(l1, r1) means a lognormal distribution with a mean of l1 and a standarddeviation of r1. N(l2, r2) means a normal distribution with a mean of l2 and astandard deviation of r2.

r ¼ kLðh�jyÞf ðh�ÞkLðhi�1jyÞf ðhi�1Þ

¼ Lðh�jyÞf ðh�ÞLðhi�1jyÞf ðhi�1Þ

ð12Þ

(c) Set hi = h⁄ with a probability of min(r, 1); otherwise sethi = hi�1, which means the jump is not accepted. Thiscan be obtained as

hi ¼ h�; if U � r

hi ¼ hi�1; if U > rð13Þ

where U is a random number from the uniform distribution of U(0,1).

(d) Stop the iteration if i = n; otherwise i = i + 1 and go toStep (a).

A good jumping distribution should be chosen to make sure thejumps are not rejected too frequently; otherwise the random walkwill waste much time standing still. The method to select a jump-ing distribution will be demonstrated in the illustrated examples.

Load Cohesion Friction angle

XB SCYAFS

S/F

Note: FS = Factor of safety , YA = Vertical displacement at point A, XB = Horizontal displacement at point B, SC = Horizontal stress at Point C , S/F = Safe or failed

Fig. 5. A Bayesian network that relates the factor of safety to monitoringparameters.

3. Illustrative examples

An anchored rock slope with a joint, which was considered byShukla and Hossain [34], is shown in Fig. 4. The slope has a heightof 12 m and a slope angle of 60�. The joint starts from the toe andextends at a slope of 35�; it turns up vertically at an elevation of7.65 m. The slope is reinforced with 4 rows of anchors 28 mm indiameter. Each anchor is pre-stressed at a force of 20 kN. The gro-uted part has a length of 4 m and a diameter of 120 mm. Theparameters of the rocks and joint are shown in Table 2. The cohe-sion (c) and friction angle (/) of the joint are set as two randomvariables. c and / are assumed to follow lognormal distributionswith means and standard deviations of 30 and 6 for c, and 25and 4 for /. A uniform load (P) is applied on the crest of the slopeand is assumed as a normal variable with a mean of 300 kN/m anda standard deviation of 30 kN/m. Three monitoring points are set tocapture the safety state of the slope: the vertical displacement atpoint A (YA), the horizontal displacement at point B (XB), and thehorizontal stress at point C (SC), as shown in Fig. 4.

3.1. Constructing a Bayesian network

A Bayesian network is constructed to integrate the monitoringinformation and evaluate the slope safety, as shown in Fig. 5. Toillustrate the proposed method, only three uncertain slope param-eters are involved; namely, the load (P), the cohesion (c) and thefriction angle (/) of the joint. In this network, the change of anyof the three parameters would result in changes of the factor of

(a)

6 18 20

(b)

Fig. 7. Results of finite element analysis: (a) strain contours; (b) slope stabilityprofile.

M. Peng et al. / Structural Safety 49 (2014) 65–74 69

safety (FS) and the three monitoring items, namely YA, XB and SC, asshown in Fig. 4. The slope safety (S/F) is related to FS: the slope issafe if FS is larger than 1. A larger YA, XB, or SC would indicate a moredangerous situation or a smaller FS. This can be qualitatively ex-plained within the Bayesian network: a larger YA may be causedby a larger P or a smaller c or /; the larger load or the smaller shearstrength parameter would lead to a smaller FS. This phenomenonwill be quantitatively demonstrated in the next sections.

3.2. Quantification of the Bayesian network

The quantification of the Bayesian network is to find the priorprobabilities for the nodes, which include the probability distribu-tions of the basic nodes (without parents) and the conditionalprobability distributions of the remaining nodes given their par-ents. The probability distributions of the basic nodes, namely theload (P), cohesion (c) and friction angle (/), are shown in Table 2.The conditional probability of S/F given FS is easy to obtain, sinceit fully depends on FS. Thus, a key task in this section is to quantifythe conditional probability distributions for FS, YA, XB and SC giventhe three basic nodes. This is realized using finite element analysisand a response surface method including the model errors.

GeoStudio [35] is a finite element software package, which con-sists of 8 components such as SIGMA/W for stress and displace-ment analysis and SLOPE/W for slope stability analysis. In thisstudy, SIGMA/W is first applied to analyze the stress–strain re-sponse of the slope under the load. Then the stress results are im-ported into SLOPE/W to analyze the slope stability. Within SIGMA/W, the slope is divided into 4322 elements with 4227 nodes, asshown in Fig. 6. For illustrative purposes, a conventional elasticand perfectly plastic model based on the Mohr–Coulomb failurecriterion is adopted to represent the stress–strain behavior of therock. Altogether 125 combinations of P, c and / i.e. the combina-tions of l, l ± r, and l ± 2r, where, l and r are the mean and stan-dard deviation of P or c or /) are considered to obtain the responsefunctions of FS, YA, XB and SC. For example, Fig. 7 shows the simula-tion results of the horizontal strains in the rock mass and the jointand the factor safety at a load (P) of 270 kN/m, a friction angle (/)of 17� and a cohesion (c) of 36 kPa. In this case, the calculated re-sults are YA = 15.2 mm, XB = 14.9 mm, SC = 112.4 kPa and FS = 0.940.

According to Eq. (4), the response function is given by

y ¼ RðP; c;/Þ þ e1 þ e2

¼X3

i¼1

a3ix3i þ

X3

i¼1

a2ix2i þ

X3

i¼1

a1ixi þ a0 þ e1 þ e2 ð14Þ

0 2 4 6 8 10 12 14 16 18 20

Fig. 6. Finite element model for the anchored rock slope.

where y = a vector of FS, YA, XB and SC; x1 = P, x2 = c and x3 = /; R = theresponse functions of y; coefficients a3i, a2i, a1i and a0 are vectorsobtained with the least squares method based on the numericalsimulations as shown in Table 3. To check the accuracy of the re-sponse surface functions, 15 more points are randomly drawn fromthe space of h, and the FS, YA, XB and SC values corresponding to theseparameters are evaluated with both the finite element method andthe third order polynomial function, as shown in Fig. 8. The resultsfrom the two approaches are close but not exactly the same. Hence,the response surface model error, e2, is introduced to characterizethe difference between the response surface method and the finiteelement method. The mean value of e2 is zero. The standard devia-tions of e2 can be estimated with statistical methods, as shown in

Table 3Coefficients for response surfaces.

Coefficient YA XB SC FS

a31 �0.0119 �0.0136 �0.0827 �0.0001a32 0.0146 0.0070 0.2291 0.0001a33 0.0102 0.0194 0.2090 0.0008a21 0.0790 0.1017 0.6887 0.0007a22 �0.1674 �0.1019 �0.7100 �0.0049a23 0.1507 0.1303 1.4936 0.0005a11 3.4278 3.7936 6.7262 �0.0036a12 �1.4040 �1.8430 �10.9293 0.0355a13 �2.3221 �2.5556 �24.1761 0.0776a0 15.5112 15.6990 76.3897 1.0434re2 1.0400 1.2803 8.5492 0.0225R2 0.9712 0.9656 0.9482 0.9679

Fig. 8. Comparison of results calculated using the response function (FS1, SC1, XB1,YA1) and the finite element method (FS2, SC2, XB2, YA2).

70 M. Peng et al. / Structural Safety 49 (2014) 65–74

Table 3. According to Tang et al. [36] and Zhang et al. [37], the mod-el error, e1, of FS from slope stability analysis can be assumed as abiased normal distribution, N(�0.025, 0.054). Zhang and Chu [38]found that estimated displacements of soil and rock in pile engi-neering follow an unbiased normal distribution with a coefficientof variation (COV) of 0.2–0.3. Thus, in this study, each of the modelerrors of YA, XB, and SC is assumed to follow a normal distribution,N(0, 0.25l), where l is the estimated mean value of YA, or XB, orSC. Conducting Monte Carlo simulation (1 million samplings) withEq. (14), the prior distributions of YA, XB, SC and FS are obtained, asshown in Fig. 9. In this case, the mean and standard deviation ofthe prior FS are 1.04 and 0.09. This only shows a general under-standing of FS without any monitoring information. The FS can beupdated with the monitoring information of YA, XB or SC, which willbe illustrated in the next section.

0 2 4 6 8 10

x 104

150

200

250

300

350

400

Sample number

Load

(kN

/m)

200 240 280 320 360 400

40004000

8000

12000

Load (kN/m)

Num

ber

of s

ampl

es

μ=296.52σ=23.57

(a) (b)

3.3. Evaluation of slope safety with monitoring information

MCMC simulation is used to update the slope parameters (i.e. P,c and / in this study), FS and PF with the monitoring information.The jumping function J(h⁄|hi�1) needs to be determined before con-ducting the MCMC simulation as introduced earlier. Here hi�1 is thecurrent sampled vector of (P, c, /) and h⁄ is the candidate vector of(P, c, /) generated from hi�1 with the jumping function. Theoreti-cally, J(h⁄|hi�1) could be an arbitrary distribution. However, accord-ing to the Metropolis algorithm [33], J(h⁄|hi�1) must be chosen as

0 10 20 30

4000

8000

12000

YA (mm)

Num

ber

of s

ampl

es

μ=15.58σ=4.43

0 10 20 30

4000

8000

12000

XB (mm)

Num

ber

of s

ampl

es

μ=15.83σ=5.04

(a) (b)

0 50 100 150

4000

8000

12000

SC

(kPa)

Num

ber

of s

ampl

es

μ=78.05σ=27.51

0.7 1.0 1.3 1.6

8000

12000

16000

FS

Num

ber

of s

ampl

es

μ=1.04σ=0.09

(c) (d)

Fig. 9. Prior distributions of: (a) YA; (b) XB; (c) SC and (d) FS.

symmetrical to keep calculations efficient, i.e. J(h⁄|hi�1) = J(hi�1|h⁄).One way to obtain a symmetrical distribution is to set h(h⁄ � hi�1)as a zero-mean normal distribution N(0, C), in which h() could beany function and C is the covariance matrix. In this case, the distri-butions of both g(h⁄ � hi�1) and g(hi-1 � h⁄) are identical and followN(0, C). Let us set g(h⁄ � hi�1) as h⁄ � hi�1. Thus, h⁄ follows a normaldistribution, N(hi�1, C). Zhang et al. [32] suggested that good calcu-lation efficiency can be attained when C = 0.5Ch, in which Ch is thecovariance matrix of the prior distribution of h. Therefore, in thisstudy, the jumping distribution is set as a multivariate normal dis-tribution with the mean vector being the current variate vector (i.e.hi�1) in the Markov chain, and the covariance matrix being 0.5Ch. Inthis case, the jumping distribution has a constant covariance ma-trix but a varying mean vector, since hi�1 changes along the Markovchain. According to the procedure of MCMC simulation, the distri-butions of P, c and / are updated. Fig. 10 shows the updated P, cand / with the monitoring values of YA = 15 mm, XB = 15 mm andSC = 80 kPa. The samples for MCMC move randomly and achievestationary states efficiently. The updated means and standard devi-ations of P, c and / are 296.52 and 23.57 kN/m, 29.92 and 5.81 kPa,and 24.80� and 3.04�, respectively. Note the prior distributions of P,c and / are N(300, 30), LogN(30, 6) and LogN(25, 4). The standarddeviations of the updated parameters are smaller than those ofthe priors, hence the uncertainties are reduced when the monitor-ing information is included. With the updated P, c and /, the up-dated factor of safety is easily obtained according to Eq. (14). Inthis case, the updated mean and standard deviation of FS are 1.06and 0.08; the corresponding failure probability is PF = 0.227.

3.4. Sensitivity of slope safety to monitoring information

Generally, the slope would be more dangerous when YA, XB andSC become larger. As shown in Fig. 11, FS decreases and PF increaseswith an increase of any of these three monitoring parameters. Thisis because that larger displacements or stresses reflect a larger loador a smaller cohesion or friction angle of the joint, which then leadsto a smaller FS and a larger PF. The curves of YA and XB are quite

0 2 4 6 8 10

x 104

10

20

30

40

50

60

Sample number

Coh

esio

n (k

Pa)

10 20 30 40 50 60

4000

8000

12000

Cohesion kPa

Num

bero

f sam

ples

μ=29.92σ=5.81

(c) (d)

0 2 4 6 8 10

x 104

10

20

30

40

50

Sample number

Fric

tion

angl

e (d

egre

e)

10 20 30 40 50

4000

8000

12000

16000

Friction angle (degree)

Num

ber

of s

ampl

es

μ=24.80σ=3.04

(e) (f)

Fig. 10. Samples for MCMC simulation: (a) P, (c) c and (e) /; and the correspondinghistograms (b) P, (d) c and (f) /.

0 10 20 30 40 500.7

0.8

0.9

1.0

1.1

1.2

FS

YA (mm)

FS

0.0

0.2

0.4

0.6

0.8

1.0

PF

PF

(a)

0 10 20 30 40 500.7

0.8

0.9

1.0

1.1

1.2

FS

XB (mm)

FS

0.0

0.2

0.4

0.6

0.8

1.0

PF

PF

(b)

0 50 100 150 200 2500.7

0.8

0.9

1.0

1.1

1.2

FS

SC (kPa)

FS

0.0

0.2

0.4

0.6

0.8

1.0

PF

PF

(c)

Fig. 11. Variations of FS and PF with one monitoring parameter: (a) YA; (b) XB; and(c) SC.

0 10 20 30 400.85

0.90

0.95

1.00

1.05

1.10

1.15

FS

XB (mm)

Without information of YA

YA=12 mm

YA=18 mm

YA=24 mm

Fig. 12. Sensitivity of FS to YA and XB.

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

5 10 15 20 25 30

5

10

15

20

25

30

PF (%)

XB (

mm

)

YA (mm)

182634425058667482

(a) (b)

Fig. 13. Variations of FS and PF with YA and XB: (a) FS and (b) PF.

M. Peng et al. / Structural Safety 49 (2014) 65–74 71

close, because the displacements at points A and B, either in thehorizontal or in the vertical direction, are similarly influenced bythe slope parameters. This does not suggest that one of thesetwo monitoring items is redundant. As shown in Fig. 12, withoutthe information of YA, FS decreases with XB rapidly. This is becauseFS only relies on XB. With the information of both YA and XB, the up-dated slope parameters integrating these two monitoring valueswill lead to a gentler curve for FS. As YA increases from 12 to24 mm, FS decreases obviously. When both the two monitoring in-dexes are large, we may have more confidence that the slope ismore unsafe.

There are intersections between the curves with and withoutincluding the information of YA in Fig. 12. For example, the FS withYA = 18 mm is smaller than that without the information of YA

when XB is relatively small, (e.g., XB < 20 mm), and vise verse. Thatis because YA = 18 mm is a dangerous sign for the situation onlywith the information of a small XB value, e.g. XB < 20 mm; whileit may be a safe sign for the situation only with the informationof a large XB value, e.g. XB > 20 mm. The fusion of the two monitor-ing indexes can also be illustrated in a straight-forward way asshown in Fig. 13. Larger values of YA and XB imply a more danger-ous state.

An important feature of the present method using Bayesian net-works is that several different types of monitoring information (i.e.,displacement, stresses, etc.) can be integrated into the slope-safetyevaluation. Fig. 14 shows the results of two types of monitoringinformation: displacements (YA and XB) and rock stress (SC). Gener-ally, the rock stress is more sensitive to the load than the resistanceparameters of the rock joint. Therefore, a larger SC reflects a larger Pmore than a smaller c or /. The FS and Pf values in Fig. 14 are lessscattered than those in Fig. 13, which means the uncertainty is re-duced with the information of SC. With an increase of SC, the slopebecomes less safe. In this figure, the safest state is when YA and XB

are both 3 mm and SC is 30 kPa. In this case, the FS is 1.158 and thePF is 0.108. The most dangerous situation is when YA and XB areboth 30 mm and SC is 130 kPa; the corresponding FS is as smallas 0.897 and PF as large as 0.861.

In addition to the monitoring information, other qualitativeinformation like engineering judgment can also be taken into theslope-safety evaluation in the present method. For example, if anengineer believes the slope is more likely to be safe based on his/her experience, then the information of ‘‘safe’’ or ‘‘FS > 1’’ can be ta-ken into the Bayesian network, as shown in Fig. 15(a) and (b). Com-paring to Fig. 14(c) and (d), with this engineering judgment, FS

becomes larger, PF becomes smaller, and the uncertainties of FS

and PF are further reduced. If there is a heavy rain and the engineerbelieves the slope is more likely to fail, then the information of‘‘failure’’ or ‘‘FS < 1’’ can also be inputted into the Bayesian network,as shown in Fig. 15(c) and (d).

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

SC=30kPa

5 10 15 20 25 30

5

10

15

20

25

30

PF (%)

XB (

mm

)

YA (mm)

182634425058667482

SC=30kPa

(a) (b)

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

SC=80kPa

5 10 15 20 25 30

5

10

15

20

25

30

PF (%)

XB (

mm

)

YA (mm)

182634425058667482

SC=80kPa

(c) (d)

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

SC=130kPa

5 10 15 20 25 30

5

10

15

20

25

30

PF (%)

XB (

mm

)

YA (mm)

182634425058667482

SC=130kPa

(e) (f)

Fig. 14. Variations of FS and Pf with SC given YA and XB: (a) and (b) SC = 30 kPa; (c)and (d) SC = 80 kPa; (e) and (f) SC = 130 kPa.

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

SC=80kPa

FS>1

5 10 15 20 25 30

5

10

15

20

25

30

PF (%)

XB (

mm

)

YA (mm)

182634425058667482

SC=80kPa

FS>1

(a) (b)

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

SC=80kPa

FS<1

5 10 15 20 25 30

5

10

15

20

25

30

PF (%)

XB (

mm

)

YA (mm)

182634425058667482

SC=80kPa

FS<1

(c) (d)

Fig. 15. Role of engineering judgment: (a) and (b) FS > 1; (c) and (d) FS < 1.

Load Cohesion Friction angle

XB SCFS

S/F

YA

εA εΒ εC

Other factors Model

Fig. 16. Correlations among the model errors in the Bayesian network.

72 M. Peng et al. / Structural Safety 49 (2014) 65–74

4. Discussions

4.1. Correlation among monitoring parameters

In practical cases, the monitoring parameters may be correlatedwith each other. The correlations among the monitoring parame-ters are considered in the Bayesian network in two types: the cor-relations among the deterministically estimated monitoringparameters and the correlations among the model errors of themonitoring parameters. This can be explained with a modifiedequation from Eq. (14):

y ¼ RðP; c;/Þ þ e ð15Þ

where y = a vector of the three monitoring parameters, YA, XB andSC; e = the model errors of y, namely e1 and e2 in Eq. (14). Thecorrelations among the model errors are limited to those from

numerical analysis (e1). The correlations among the response sur-face model errors (e2) are not considered in this study since thistype of model errors are generated from regression analyses of sep-arate response data.

The monitoring parameters are firstly correlated through theresponse functions with the same model parameters, namely P, cand /. For example, a large YA implies a large P or a small c or /,which may in turn lead to a large XB or SC. Besides, the monitoringparameters may also be correlated through the model errors. Forexample, a change in the model error of YA may lead to changesin the model errors of XB and SC. The correlations among the modelerrors can be modeled with a modified Bayesian network as shownin Fig. 16. The model errors, eA, eB and eC, are influenced by themodel parameters (i.e., P, c and /), the chosen model and other fac-tors (e.g., finite element mesh, boundary conditions, and unknownfactors). Note that YA, XB and SC influence the model errors as eachof the model errors is assumed to follow a normal distribution, N(0,0.25l), where l is the deterministically estimated value of YA, orXB, or SC. Based on the concepts of Bayesian networks [20], if thefactors influencing the model errors remain unknown, eA, eB andeC are correlated; if all the factors are known, however, eA, eB andeC will become independent. Thus, eA, eB and eC are independent gi-ven the load, the cohesion and friction angle of the soil, and thesimulation model. This Bayesian network can be simplified asFig. 5 by integrating eA, eB and eC into YA, XB and SC, respectively.In this study, the node ‘‘other factors’’ in Fig. 16 is not considered.

In practice, the model errors may be influenced by many factorsand may not be modeled in a simple way as in Fig. 5. In this case,the correlations among the model errors can be described using acorrelation matrix. Thus, the likelihood function in Eq. (6) is a jointdistribution. For example, when only two monitor parameters, XB

and YA, are considered, the likelihood function is given by

LðhjXB ¼ xB;YA ¼ yAÞ/ f ½gXB

ðhÞ þ eXB ¼ xB; gYAðhÞ þ eYA

¼ yA�/ f ½eXB ¼ xB � gXB

ðhÞ; eYA ¼ yA � gYAðhÞ�

/ feXB;eYA½xB � gXB

ðhÞ; yA � gYAðhÞ�

ð16Þ

where xB and yA = monitored values and h = model parameters.Fig. 17 shows the influence of the correlation coefficient betweenthe model errors of XB and YA, qðeXB ; eYA Þ, on FS. As qðeXB ; eYA Þ changesfrom 0 to 0.7, the uncertainty of FS becomes smaller and smaller.This is because XB and YA with a strong correlation cannot changeas freely as those with a weak correlation, leading to a smalleruncertainty of FS.

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

(a) (b)

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

5 10 15 20 25 30

5

10

15

20

25

30

FS

XB (

mm

)

YA (mm)

0.900.930.960.991.021.051.081.111.14

(c) (d)

Fig. 17. Influence of correlation among the model errors of XB and YA to FS: (a)qðeXB ; eYA

Þ ¼ 0; (b) qðeXB ; eYAÞ ¼ 0:3; (c) qðeXB ; eYA

Þ ¼ 0:5; (d) qðeXB ; eYAÞ ¼ 0:7.

M. Peng et al. / Structural Safety 49 (2014) 65–74 73

4.2. Features of the present method

Compared to the existing studies, the proposed method of slopesafety evaluation using multi-source monitoring information hasthe following features:

(1) Multiple sources of monitoring information can be takeninto account. Bayesian networks are a very powerful tool for theintegration of multi-source monitoring information. Physical anal-ysis of slope stability with either analytic analysis or numericalanalysis can be involved to obtain the prior probabilities. Monitor-ing information, engineering judgment and statistical data can alsobe used to update the networks and improve the safety evaluation.Within this new method, the uncertainties can be largely reducedand the evaluation will be more reliable with the support of multi-ple sources of information.

(2) The model uncertainties can be considered. Previous studiesshow that model errors cannot be ignored. The proposed methoddeals with the correlated model uncertainties within the Bayesiannetwork using MCMC simulation. The Bayesian method has beenproven to be a useful tool to estimate the model errors by compar-ing the monitoring records and the model estimates [30,35,36].Note that the Bayesian method is a special case of a Bayesian net-work by setting the parameters as the parent random variables.

4.3. Limitations of the present method

The present work has several limitations that should be im-proved in the future:

(1) An engineered slope may involve several failure surfaces andseveral failure mechanisms such as excavation stress relief,seepage, earthquakes and anchor corrosion. How to identifylikely failure mechanisms from monitoring data and how todeal with the system reliability issues involved [39] are sub-jects of further research.

(2) The time effect of monitoring information needs to be con-sidered, which may be caused by creep, aging anchors, crackdevelopment and so on.

(3) The spatial variation of soil parameters should be consideredin practical applications. Stochastic finite element methodsmay be used to establish the response functions of the factorof safety and monitoring parameters considering the spatialvariability of soils and rocks.

(4) The correlations among the model errors of different moni-toring parameters need to be further evaluated withrecorded monitoring data in specific case studies. This maybe realized by comparing simulated results with recordeddata using the Bayesian method.

5. Conclusions

This paper presents a systematic method for slope safety evalu-ation utilizing multi-source monitoring information. The followingconclusions can be drawn:

(1) A Bayesian network is constructed by considering the inter-relationships among soil parameters, loads, monitoringinformation and the factor of safety. The prior probabilitiesare quantified with finite element analysis and a responsesurface method including model and parameteruncertainties.

(2) Markov chain Monte Carlo simulation is used to update theBayesian network with multi-source monitoring informa-tion. The probability distributions of the parameters in theBayesian network can be any types of distributions.

(3) Different sources of monitoring information can be takeninto account in the slope safety evaluation in the proposedmethod, including multiple types of monitoring data, engi-neering judgment and statistical data. More reliable evalua-tion is obtained with the support of multiple sources of site-specific information.

(4) The spatial variation of soil parameters and the correlationsamong different monitoring parameters in specific engineer-ing cases should be considered in the future to make theslope safety evaluation method more robust.

Acknowledgements

The research reported in this paper was substantially supportedby the National Basic Research Program (973 Program) (No.2011CB013506) and the Natural Science Foundation of China(Nos. 51129902 and 51225903).

References

[1] Einstein H, Sousa R. Warning system for natural threats. In: Editors: Nadim F,Pottler R, Einstein H, Klapperich H, Kramer S, editors. Proceeding of the 2006ECI conference on Geohazards (Keynote), Lillehammer, Norway; 2006.Available from: <http://dc.engconfintl.org/cgi/viewcontent.cgi?article=1009&context=geohazards>.

[2] Marr WA. Instrumentation and monitoring of slope stability. In: Proceedings ofthe geo-congress, San Diego, California: Reston, ASCE; 2013, p. 2231–52,Geotechnical Special Publication No. 231.

[3] Keefer DK, Wilson RC, Mark RK, Brabb EE, Brown III WM, Ellen SD, et al. Real-time landslide warning during heavy rainfall. Science 1987;238(4829):921–5.

[4] Finlay PJ, Fell R, Maguire PK. The relationship between the probability oflandslide occurrence and rainfall. Can Geotech J 1997;34(6):811–24.

[5] Au SWC. Rain-induced slope instability in Hong Kong. Eng Geol1998;51(1):1–36.

[6] Brunetti MT, Peruccacci S, Rossi M, Luciani S, Valigi D, Guzzetti F. Rainfallthresholds for the possible occurrence of landslides in Italy. Nat Hazards EarthSyst Sci 2010;10(3):447–58.

[7] Chen H, Lee CF. A dynamic model for rainfall-induced landslides on naturalslopes. Geomorphology 2003;51(4):269–88.

[8] Zhang LL, Zhang J, Zhang LM, Tang WH. Stability analysis of rainfall inducedslope failure: a review. Proc Inst Civil Eng Geotech Eng 2011;164(5):299–316.

[9] Dixon N, Spriggs M. Quantification of slope displacement rates using acousticemission monitoring. Can Geotech J 2007;44(8):966–76.

[10] McCombie PF. Displacement based multiple wedge slope stability analysis.Comput Geotech 2009;36(1–2):332–41.

[11] Dai FC, Lee CF. Landslide characteristics and slope instability modeling usingGIS, Lantau Island. Hong Kong. Geomorphol 2002;42(3–4):213–28.

[12] Bozzano F, Mazzanti P, Prestininzi A, Mugnozza GS. Research and developmentof advanced technologies for landslide hazard analysis in Italy. Landslides2010;7(3):381–5.

74 M. Peng et al. / Structural Safety 49 (2014) 65–74

[13] Casagli N, Catani F, Del Ventisette C, Luzi G. Monitoring, prediction, and earlywarning using ground-based radar interferometry. Landslides2010;7(3):291–301.

[14] Liggins ME, Hall DL, Llinas J. Handbook of multisensor data fusion: theory andpractice. second ed. London: Taylor & Francis; 2009.

[15] Chang YL, Liang LS, Han CC, Fang JP, Liang WY, Chen KS. Multisource datafusion for landslide classification using generalized positive Boolean functions.IEEE Trans Geosci Remote Sens 2007;45(6):1697–708.

[16] Wong MS, Nichol J, Shaker A, Hui CF. Data fusion using aerial photographs andsatellite images for detailed landslide assessment. Int J Image Data Fusion2011;2(2):181–90.

[17] Guo K, Peng JB, Xu Q, Yuan Y. Application of multi-sensor target tracking tomulti-station monitoring data fusion in landslide. Chin J Rock Soil Mech2006;27(3):479–81 [in Chinese].

[18] Peng P, Shan ZG, Dong YF, Jia HB, Hou JX. Application of multi-sensor valuationfusion theory to monitoring dynamic deformation of landslides. Chin J EngGeol 2011;19(6):928–34 [in Chinese].

[19] Ermini L, Catani F, Casagli N. Artificial neural networks applied to landslidesusceptibility assessment. Geomorphology 2005;66(1–4):327–43.

[20] Jensen FV. Bayesian networks and decision graphs. New York: Springer; 2001.[21] Mahadevan S, Zhang R, Smith N. Bayesian networks for system reliability

reassessment. Struct Saf 2001;23(3):231–51.[22] Grêt-Regamey A, Straub D. Spatially explicit avalanche risk assessment linking

Bayesian networks to a GIS. Nat Hazards Earth Syst Sci 2006;6(6):911–26.[23] Straub D, Der Kiureghian A. Combining Bayesian networks with structural

reliability methods: methodology. J Eng Mech 2010;136(10):1248–58.[24] Straub D, Der Kiureghian A. Combining Bayesian networks with structural

reliability methods: application. J Eng Mech 2010;136(10):1259–70.[25] Peng M, Zhang LM. Analysis of human risks due to dam break floods – part 1: a

new model based on Bayesian networks. Nat Hazards 2012;64(2):1899–923.[26] Peng M, Zhang LM. Analysis of human risk due to dam break floods – part 2:

application to Tangjiashan landslide dam failure. Nat Hazards2012;64(1):903–33.

[27] Wong FS. Slope reliability and response surface method. J Geotech GeoenvironEng 1985;111(1):32–53.

[28] Bucher CG, Bourgund U. A fast and efficient response surface approach forstructural reliability problems. Struct Saf 1990;7(1):57–66.

[29] Rajashekhar MR, Ellingwood BR. A new look at the response surface approachfor reliability analysis. Struct Saf 1993;12(3):205–20.

[30] Ang AHS, Tang WH. Probability concepts in engineering, emphasis onapplications in civil & environmental engineering. second ed. NewYork: Wiley; 2007.

[31] Tang WH, Stark TD, Angulo M. Reliability in back analysis of slope failures.Soils Foundations 2009;39(5):73–80.

[32] Zhang LL, Zhang J, Zhang LM, Tang WH. Back analysis of slope failure withMarkov chain Monte Carlo simulation. Comput Geotech 2010;37(7–8):905–12.

[33] Gelman BA, Carlin BP, Stem HS, Rubin DB. Bayesian dataanalysis. London: Chapman & Hall; 2004.

[34] Shukla SK, Hossain M. Stability analysis of multi-directional anchored rockslope subjected to surcharge and seismic loads. Soil Dyn Earthquake Eng2011;31(5–6):841–4.

[35] Geo-SLOPE International Ltd., GeoStudio 2007 Help. Calgary, Alberta, Canada;2007.

[36] Tang WH, Zhang J, Zhang LM. Characterizing geotechnical model uncertainty.In: Furuta, Frangopol, Shinozuka, editors. Proceedings of the 10th internationalconference on structural safety and reliability, Osaka, Japan. London: Taylor &Francis Group; 2009. p. 114–23.

[37] Zhang J, Zhang LM, Tang WH. Bayesian framework for characterizinggeotechnical model uncertainty. J Geotech Geoenviron Eng2009;135(7):932–40.

[38] Zhang LM, Chu LF. Calibration of methods for designing large-diameter boredpiles: serviceability limit state. Soils Foundations 2009;49(6):897–908.

[39] Ang AHS, Tang WH. Probability concepts in engineering planning and design.Decision risk and reliability, Vol. II. Berlin: Springer; 1984.