ARTICLE

Importance of soil property sampling location in slope stabilityassessmentRui Yang, Jinsong Huang, D.V. Griffiths, Jinhui Li, and Daichao Sheng

Abstract: Site investigations provide characterization of soil properties, but inevitable uncertainty remains at locations thathave not been examined. Only a limited scope of site investigation can be conducted due to budget and time constraints, hencethere are always risks associated with design based on limited investigation information. An efficient geotechnical site investi-gation should involve choosing the optimal number and location of borehole sites to gain adequate information for a given cost.Using a slope as an example, this paper proposes a framework to find the best sampling location that gives the most informationwhile minimizing the probability of making the wrong decisions. The results suggest that the slope crest appears to be theoptimal location to conduct geotechnical site exploration for slope stability assessment.

Key words: geotechnical site investigation, sampling location, slope stability analysis.

Résumé : Les études de site permettent de caractériser les propriétés du sol, mais l’incertitude inévitable demeure aux endroitsqui n’ont pas été examinés. Seule une portée limitée de l’étude du site peut être menée en raison de contraintes de budget etde temps, par conséquent, il y a toujours des risques associés à la conception basée sur des informations limitées provenantd’étude de site. Une étude de site géotechnique efficace devrait impliquer le choix du nombre et de l’emplacement optimaux dessites de forage pour obtenir des informations adéquates à un coût donné. En utilisant une pente comme exemple, ce documentpropose un cadre pour trouver le meilleur emplacement d’échantillonnage qui donne le plus d’informations tout en minimisantla probabilité de prendre de mauvaises décisions. Les résultats suggèrent que la crête de la pente semble être l’emplacementoptimal pour mener l’exploration géotechnique du site pour l’évaluation de la stabilité des pentes. [Traduit par la Rédaction]

Mots-clés : étude géotechnique de site, emplacement d’échantillonnage, analyse de stabilité de pente.

1. IntroductionSlope engineering is a typical geotechnical subject that is dom-

inated by uncertainty. This is due to the fact that soils and rocks intheir natural state often exhibit significant variability from onepoint to another; that is, spatial variability (Vanmarcke 1977). Toanalyse and design a slope, practitioners would ideally like toobtain the soil properties of the whole domain by means of in situ,laboratory or geophysical tests (Kulhawy and Mayne 1990; Lacasseand Nadim 1997; Robertson 2009). Armed with all this informa-tion, practitioners would be very confident of the prediction ofthe slope. However, achieving this goal can be unrealistic andexpensive. Most of the time, the scope of the site investigation isoften governed by how much the client and project manager arewilling to spend, rather than by what is needed to characterize thesubsurface conditions. Expenditure on geotechnical investiga-tions varies considerably, sometimes as low as between 0.025%and 0.3% of the total project cost (National Research Council 1984).Therefore, it is important to select effective site exploration scopeand test methods so that adequate information can be obtainedfor given budgets.

Research on the effectiveness of geotechnical site investiga-tions is rare in the literature. Jaksa et al. (2003) proposed aframework for improving the effectiveness of geotechnical site

investigation. Based on this framework, Jaska et al. (2005) dis-cussed the efficiency of different site investigation scope and testmethods by comparing the pad footing design results based onlimited site investigation with the design results based on com-plete knowledge of a site. The complete knowledge of a site wasobtained by random field simulations. The results showed thatthe design would be more reliable as the scope of site investiga-tion increases. Gong et al. (2014) discussed the relationship be-tween the level of site exploration efforts and the accuracy oftunnelling-induced ground settlement prediction. The resultsshowed that a higher level of site exploration leads to a moreaccurate prediction of ground settlement.

However, for given scope of site investigation and test methods,it is also important to find the best location to conduct the tests sothat the most relevant information can be extracted. Li et al. (2016)discussed the optimal locations and the number of boreholes forslope stability problems by conditional random field simulation.The standard deviation of the factor of safety (FS) of a slope wasused to quantify the uncertainty of the site investigation. Jianget al. (2017) investigated the optimal borehole locations in slopereliability assessment based on maximizing the informationgained for soil properties. Both Li et al. (2016) and Jiang et al. (2017)found that the best sampling location is near the slope crest.

Received 29 January 2018. Accepted 22 May 2018.

R. Yang, J. Huang,* and D. Sheng.* Discipline of Civil, Surveying and Environmental Engineering, Priority Research Centre for Geotechnical Scienceand Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia.D.V. Griffiths. Colorado School of Mines, Golden, CO, USA.J. Li. Department of Civil and Environmental Engineering, Harbin Institute of Technology Shen Zhen Graduate School, Shenzhen, China.Corresponding author: Jinsong Huang (email: Jinsong.Huang@newcastle.edu.au).*J. Huang currently serves as an Editorial Board Member, D. Sheng currently serves as an Editor; peer review and editorial decisions regarding thismanuscript were handled by C. Lake.

Copyright remains with the author(s) or their institution(s). Permission for reuse (free in most cases) can be obtained from RightsLink.

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Can. Geotech. J. 56: 335–346 (2019) dx.doi.org/10.1139/cgj-2018-0060 Published at www.nrcresearchpress.com/cgj on 30 May 2018.

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However, all these previous studies have not discussed the influ-ence of sampling location on the probability of making the wrongdecision in slope stability analysis, which can in turn lead to over-or under-designs. Overdesign will lead to unnecessary construc-tion costs. Underdesign may reduce the initial construction cost,but unforeseen conditions can lead to significant cost overruns.The most appropriate geotechnical site investigation should min-imize the overall cost overruns. However, to be able to estimatethe overall cost, the probability of over- or underdesign need to bequantified first.

This paper develops a new approach for optimizing samplinglocations in slope reliability assessment based on statistical hy-pothesis testing. The new method can be used to quantify theprobability of over- or underdesign. As will be shown subse-quently, the probability of making a decision error changes as thesampling location varies, and the task is to find the optimal sam-pling location where the error probabilities are minimum. Todemonstrate the proposed approach, an undrained slope is anal-ysed and the importance of sampling locations are shown by com-paring the error probabilities.

2. MethodologySite characterization is a fundamental step when collecting

geotechnical information for proper design, construction, andlong term performance of all types of civil and geotechnical struc-tures. Typical geotechnical site investigation methods includecone penetration tests (CPT), standard penetration tests (SPT), andborehole tests, among others. These test methods only ever sam-ple and test a very small fraction of the site. However, inevitableuncertainty remains at locations that have not been examined. Ifone then designs a slope based on the obtained soil properties, thedesign would be unreliable because the decision about the stabil-ity of slope is made on the basis of a set of samples. The overallobjective of slope design is to ensure that the slope is safe. Thedecision-making process is essentially a hypothesis test (Fentonet al. 2015; Walpole et al. 1993) where the null hypothesis (H0) isthat the slope fails, so that the burden of proof is on showing thatthe alternative hypothesis (H1) is true, at an appropriate level ofconfidence. Generally speaking, two types of errors can be made.The first type of error is that the slope is actually unsafe, but theslope stability analysis suggests that the slope is safe. This istermed here type I error. The second type of error is that the slopeis actually safe, but the slope stability analysis suggests that theslope is unsafe. This is termed here type II error. Type I error isclearly much more serious than type II error because it leads to anunsafe design. The challenge is how to design the site investiga-tion to ensure that the probability of making either type of erroris as small as is acceptable.

Taking an infinite number of samples from the constructionsite (i.e., investigate all subsoils) will eliminate any chance of mak-ing a decision error, but this is neither physically nor economi-cally feasible. This means that some amount of error will alwaysexist, and so it is necessary to relate the error probabilities to thesite investigation strategy to assess the effectiveness of the siteinvestigation. The most reliable site investigation should have thesmallest values of both type I and type II errors.

The probabilities of type I and II errors can be estimated byundertaking Monte Carlo simulation. Figure 1 shows the flowchart of the proposed approach. In general, the whole procedureconsists of two parts: (i) slope stability analysis based on completeknowledge of a slope, and (ii) slope stability analysis based on siteinvestigation. The error probabilities can be estimated by compar-ing the results of these two slope stability analyses. A similarapproach has been used by Fenton et al. (2015) to assess the num-ber of samples required for a reliable quality control program ofcement-based solidification and stabilization.

For a given set of statistics of soil properties (i.e., the mean, thecoefficient of variation, and the spatial correlation length), a slopeis generated by the random finite element method (RFEM)(Griffiths and Fenton 1993). Different geotechnical properties areassigned to each element, depending on the nature of the spatialvariability of the soil profile. This slope is treated as a “real slope”as its properties are known exactly at every location or element.Using the finite element method (FEM), the stability of the slopecan be estimated. Based on the complete knowledge of the slope,this stability assessment reflects the true state of the slope. Thestrength reduction method is usually adopted to determine theFS if the FEM is used for slope stability analysis (Griffiths andLane 1999). However, failure can also be checked without usingstrength reduction method, but by merely checking to see if thecurrent realisation fails to converge, implying stability failure. Asthe search for FS takes much more computational time than di-rect failure assessment (Huang et al. 2017), in this paper the cal-culation for FS is avoided. Instead, a stability index IF definedbelow is used to describe the state of the real slope

(1) IF � �1 failure0 nonfailure

This stability index IF is a “true value”.In reality only a limited amount of sampling can be conducted.

Suppose the site investigation has been carried out to obtain soilproperties at a limited number of locations, as a result of fieldsampling and subsequent laboratory testing. A slope stabilityanalysis needs to be performed based on the obtained properties.To simulate this process, virtually sampling each realisation atselected locations is performed. This is achieved simply by record-ing the soil properties of the simulated real slope at specific loca-tions. The Kriging interpolation approach (Krige 1951) is used inthis study to estimate the soil properties of the entire slope basedon the virtual measurements. In addition, a slope stability analy-sis is conducted based on the Kriged field for each realisation. Thisstability analysis is based on the limited amount of informationobtained from the virtual site investigation. So the stability indexIFk

defined below is an estimation.

(2) IFk� �1 failure

0 nonfailure

A null hypothesis and alternative hypotheses are defined as

(3)H0 : IF � 0H1 : IF � 1

For each realisation, the comparison between the real stabilityindex, IF, and the estimated stability index, IFk

, can result in threepossible outcomes

(4)

correct decision : IFk� IF

type I error : IFk� IF

type II error : IFk� IF

When IFk� IF, the stability analysis based on limited informa-

tion results in the correct decision. When IFk� IF, the stability

analysis based on limited investigation data suggests the slope issafe when it is actually unsafe. This is an unacceptable type I error,resulting in the worst outcome. When IFk

� IF, the slope stabilityanalysis based on limited investigation data leads to the conclu-sion that the slope is unsafe when it is actually safe. This is anunfavorable type II error, resulting in a higher cost of the project.

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If a slope is generated many hundreds of times, within a MonteCarlo simulation, it is possible to estimate probabilities of decisionerrors. Given the number of simulations nsim, the probabilities oftype I and II errors can be estimated as

(5) p1 �n1

nsim

(6) p2 �n2

nsim

where p1 is the probability of type I error, n1 is the number ofrealisations where IFk

� IF, nsim is the total number of realisations,p2 is the probability of type II error, and n2 is the number ofrealisations where IFk

� IF.The above procedure can be used to estimate the probabilities

of type I and II errors for different sampling locations. By compar-ing the probabilities of type I and II errors, the effectiveness ofsampling locations can be assessed, which is the main objective ofthis paper. The same procedure also can be used to estimate theinfluence of the number of samples on slope stability assess-ments. This will be discussed in a future study.

3. Slope stability analysis based on completeknowledge of slope

In this study, the RFEM is used to perform slope stability anal-ysis based on complete knowledge. The RFEM combines randomfield theory (Vanmarcke 1977) and FEM within Monte Carlo frame-work. For each realisation, local average subdivision (LAS) is usedto generate random fields of soil properties (Fenton andVanmarcke 1990). The LAS method fully accounts for spatial vari-ability and local averaging over each element.

In this paper, the soil properties are assumed to be character-ized statistically by lognormal distributions. A lognormal distri-bution is particularly suitable for the soil parameters that cannottake on negative values because it ranges between zero and infin-ity (Fenton and Griffiths 2008). The lognormally distributed ran-dom field Y is fully specified by its mean �Y, its standard deviation�Y, and its spatial correlation length �Y. It can be translated intonormal space by

(7) X(x) � lnY(x)

where X(x) is normal random field characterized by its mean, �X,and its standard deviation, �X, and Y(x) is lognormal random field.

Fig. 1. Flow chart for calculating probabilities of type I and II errors.

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The correlation between soil properties at different locations isusually characterized by an exponentially decaying correlationfunction

(8) �() � e(2)/�Y

where �() is correlation coefficient between soil properties at twopoints in the random field separated by an absolute distance .

Examination of many in situ test data revealed that soil proper-ties generally fluctuate about a mean trend that typically in-creases with depth (Phoon and Kulhawy 1999). The resultantrandom field is called nonstationary, where the statistics (i.e.,means and standard deviations) are nonconstant over the domainof the random field. In this paper, it is assumed that the mean ofsoil strength increases linearly with depth. Suppose Y0 and Yz arerandom variables at the crest level and depth z, respectively. Themean strength is a linear function of depth according to the equa-tion:

(9) �Yz� �Y0

� tz

where �Yzis the mean strength at depth z, �Y0

is the mean strengthat crest level (i.e., at z = 0), and t is the gradient of mean strength.In this study the standard deviation of the strength is also as-sumed to be a linear function with depth, with a gradient thatresults in a constant coefficient of variation �Y (Griffiths et al. 2015;Zhu et al. 2017).

(10) vY � vY0� vYz

Initially, a homogeneous, stationary, lognormal random field,based on the parameters at crest level, is generated. The parame-ters of the underlying normal distribution at the crest level anddepth z can be obtained as follows:

(11) �X0� �Xz

� �ln�1 � vY2�

(12) �X � ln�Y 12

�X2

The random variable Yz at depth z can be expressed as

(13)

Yz � exp�lnY0 �X0

�lnY0

�Xz� �Xz�

� exp�lnY0 �X0

�X0

�Xz� �Xz�

� exp(lnY0 �X0� �Xz

)

�Xzcan be obtained by eq. (12), then

(14)

Yz � exp�lnY0 �X0� ln�Yz

12

�Xz

2 �� exp�lnY0 �X0

� ln�Yz

12

�X0

2 �� exp[lnY0 ln�Y0

� ln(�Y0� tz)]

� Y0

�Y0� tz

�Y0

It can be seen from eq. (14) that a nonstationary lognormalrandom field can be generated from a stationary lognormal ran-dom field.

The generated random properties are then mapped to the finiteelement mesh and the slope stability analysis follows. This slopestability analysis is based on complete knowledge of soil proper-ties. The stability analysis results reflect the real state of the slopesand will be compared with the stability analysis based on limitedsite investigation to assess the error probabilities.

4. Slope stability analysis based on siteinvestigation

Within the Monte Carlo simulation, for each realisation, themeasurements are extracted at sampling locations from the gen-erated random field. Then, the Kriging method is used to interpo-late the measurements to estimate the soil properties of thewhole slope. The well-established Kriging method, based onKrige’s empirical work for evaluating mineral resources, and laterformalized by Matheron (1962) into a geostatistical approach, arecommonly used for interpolation between known data (Journeland Huijbregts 1978). In contrast with other common interpola-tion techniques, Kriging can produce site- and variable-specificinterpolation schemes by directly incorporating a model of thespatial variability of the data. As the measured data in this paperare assumed to be lognormally distributed with known statistics,the simple lognormal Kriging (SLK) needs to be used to avoidbiased estimation (Chiles and Delfiner 2009).

4.1. Simple lognormal KrigingTo perform SLK, the mean trend is first removed from the mea-

surements and then translated into normal space by eq. (7). Thetranslated measurements are then interpolated by simple Kriging(SK). The lognormal estimators Y�x� are obtained by taking theexponential of the SK estimators plus the Kriging variance(Rivoirard 1990).

(15) Y(x) � exp�X(x) � ��Err2 /2�

where X�x� is the SK estimator and �Err is the standard deviationof the SK estimator error. To satisfy the unbiased condition, whenthe SK estimator is transformed back to the lognormal estimator,the variance of the estimator error must not be omitted. TheKriged fields are obtained by adding the trend to lognormal esti-mators using eq. (14).

The SK methodology has been described in detail by Voltz andWebster (1990) and Webster and Oliver (1999). So only a briefdescription about SK will be presented here.

4.2. Simple KrigingSimple Kriging (SK) is basically best linear unbiased estimation

with the assumption that the mean, �X, and standard deviation,�X, are constant and known across the entire region of interest.Kriging estimates X(x) at any unknown location using a weightedlinear combination of the values of X at each observation point.Suppose that X1, X2, …, Xn are observations of the normal randomfield, X(x), at known spatial locations x1, x2, …, xn, that is, Xk = X(xk).Then the Kriged estimate of X(x) at x can be expressed as

(16) X(x) � k�1

n

kXk

For the estimator eq. (16) to be unbiased, the mean differencebetween the estimate and the true (but random) value should bezero

(17) E[X(x) X(x)] � E[X(x)] E[X(x)] � k�1

n

k�X �X � 0

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As this must be true for any mean value, �X, the unbiased con-dition reduces to the sum of the weights k and needs to be equalto unity.

(18) k�1

n

k � 1

The estimator error is defined as the difference between theestimate X�x� and its true (but unknown and random value) X(x)

(19) Err � [X(x) X(x)]

The unknown Kriging weights, k, are obtained by minimizingthe variance of the error, which reduces the solution to the matrixequation (Wackernagel 2013)

(20) K� � M

where K and M depend on the covariance structure

(21) �C11 C12 Ê C1n 1É É Ì É ÉCn1 Cn2 Ê Cnn 11 1 Ê 1 0

�� 1

É n

�� � �

C1x

ÉCnx

1�

in which Cij is the covariance between Xi and Xj, and � is a Lagran-gian parameter used to solve the variance minimization problemsubjected to the unbiased condition. The covariance, Cix, appear-ing in the vector on the right-hand side, M, is the covariancebetween the ith observation point and the point at x where thebest estimate is to be calculated. Note that the Kriging matrix, K,only depends on the location of observations and their covari-ances. Thus, the Kriging matrix can be used to obtain the Krigingweights by inverting K only once and then eqs. (21) and (16) can beused repeatedly for different spatial locations.

The variance of the estimator error is given by

(22) �Err2 � E{[X(x) X(x)]2} � �X � �n

T(Kn×n�n 2Mn)

where �n and Mn are the first n elements of � and M and Kn×n is then×n upper left submatrix of K containing the covariances.

5. ExampleTo illustrate the proposed method, an undrained slope is con-

sidered with the profile shown in Fig. 2. The slope is inclined tothe horizontal at angle � = 18.4° (3:1 slope), with a height H = 10 m,and a depth ratio to an underlying firm layer D = 3, the slope hasa soil unit weight �sat (or �) = 20.0 kN/m3, which are all heldconstant. The mean undrained strength increases linearly withdepth. The mean shear strength at the crest level, �cu0

= 18 kPa; thegradient of mean strength, t = 2.4 kN/m3 and is constant; and thecoefficient of variation, vcu

= 0.5. The spatial correlation length is

assumed to be isotropic with �cufixed at 10 m. A finite element

mesh size of 0.5 m × 0.5 m is selected. Based on the parametersgiven above, 20 000 RFEM simulations are performed and theprobability of failure, pf, is found to be 0.1495.

As mentioned above, the aim of this paper is to identify theoptimal location where the error probabilities are a minimum.The minimum corresponding error probability observed in theparametric studies was approximately 0.03. If the desirablemaximum error of the error probability is 0.002, at a confi-dence level 90%, the required number of realisations is 19 686(e.g., Fenton and Griffiths 2008). It can therefore be said that20 000 simulations is adequate to achieve this target errorbound. The number of realisations selected was nsim = 20 000 for allparametric studies.

5.1. Influence of sampling location on error probabilitiesThe proposed method is adopted to explore the optimal sam-

pling location for the stability analysis of slopes. Suppose only onesampling, such as cone penetration tests (CPT), standard penetra-tion tests (SPT) or borehole, is performed to obtain the soil prop-erties along a vertical line. The spatial correlation length ofundrained shear strength is taken as �cu

= 10 m. Figures 3 and 4show the influence of the sampling location on the probability oftype I and II errors. Each point on the plot is obtained using 20 000realisations. Figure 3 shows that as the sampling location movesfrom the left to the right of the slope, the probability of type Ierror first decreases and then increases. It reaches the minimumvalue near the slope crest. Similarly, the probability of type IIerror reaches the minimum value when sampling is performedaround the slope crest. Both the probabilities of type I and type IIerrors are a minimum when sampling is performed at the slopecrest. This indicates that the slope crest appears to be the optimallocation to conduct site investigation. It is also interesting to notefrom Figs. 3 and 4 that both the probabilities of type I and II errorsare relatively small when the sampling location is conducted be-tween the slope crest and the toe.

Typical type I and II errors are selected from the 20 000 simula-tions. The strength reduction method is performed for theseparticular simulations to clearly show the failure mechanism.Figure 5 shows a typical case where type I error occurs. If theproperties of the whole slope are known, as shown in Fig. 5a, thestability analysis suggests that the FS is 0.8281, which meansthe slope is unsafe. Figures 5 and 6 depict the variation of un-drained shear strength and have been scaled in such a way thatthe dark and light regions depict “strong” and “weak” soils, re-spectively. Black represents the strongest element and white isthe weakest in the particular realisation. Only one sampling (nearthe slope crest) is performed at the location indicated by the redarrow in Fig. 5a. A column of undrained shear strengths is obtainedat that location. The Kriging method is then used to estimate theundrained shear strengths of the whole slope. Figure 5b shows a greyscale of the estimated undrained shear strengths constrained by themeasurements. The slope stability analysis based on limited infor-

Fig. 2. Finite element mesh.

10m

5m

27.5m 30m 20m

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mation suggests that the FS is 1.2188. This is an unacceptable type Ierror, which may lead to unsafe design.

Figure 6 shows a typical type II error. The slope is safe, but thestability analysis based on limited information suggests that theslope is unsafe. It would lead to overdesign, which increasesthe project cost.

5.2. Influence of spatial correlation length on errorprobabilities

The results presented in section 5.1 are based on a certain levelof spatial variability (i.e., �cu

= 10 m). On the one hand, if the spatialvariability is not significant, one sampling test would be enoughto obtain reliable measurements of the properties of the whole

Fig. 3. Influence of sampling location on probability of type I error. [Colour online.]

Fig. 4. Influence of sampling location on probability of type II error. [Colour online.]

Fig. 5. Typical type I error (�cu= 10 m): (a) real undrained slope is unsafe (FS = 0.8281); (b) slope stability analysis based on limited information

suggests slope is safe (FS = 1.2188). [Colour online.]

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slope. On the other hand, if the spatial variability is significantone site investigation test may not be sufficient to provide enoughinformation for reliable stability assessment. In the followingstudies, the influence of spatial correlation length on the proba-bilities of type I and II errors is investigated. According to Phoon

and Kulhawy (1999), the spatial correlation length is reported tobe in the range of 2–60 m. The ranges of spatial correlation lengthadopted in parametric studies are usually wider (e.g., Griffithset al. 2009). In this study, the spatial correlation length is assumedto be isotropic and varied in the range �cu

= (1, 5, 10, …, 100 m). All

Fig. 6. Typical type II error (�cu� 10 m): (a) real undrained slope is safe (FS = 1.125); (b) slope stability analysis based on limited information

suggests slope is unsafe (FS = 0.9219). [Colour online.]

Fig. 7. Probability of slope failure for different spatial correlation lengths (�cu).

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other statistics remain the same as described in the previous sub-section. It is assumed that sampling is conducted at the best loca-tion (i.e., near the slope crest).

Before investigating the influence of the spatial correlationlength on the probabilities of type I and II errors, its effect on theprobability of failure is examined. The computed probability offailure for a range of spatial correlation lengths is given in Fig. 7.The results show that the probability of failure increases when thespatial correlation length increases. Similar results have been re-ported by Griffiths and Fenton (2004) and Griffiths et al. (2009).The results in Fig. 7 will be used later to explain the influence ofspatial correlation length in the Kriging estimation on the errorprobabilities.

The influence of the spatial correlation length on probability oftype I error is shown in Fig. 8. As the spatial correlation lengthincreases, the probability of type I error first increases and thendecreases. For example, the probability of type I error increases

from 0, at spatial correlation length of 1 m, to a maximum value of0.1087 at a spatial correlation length of 15 m, and then decreasesto 0.05875 when the spatial correlation length reaches 100 m.Figure 9 indicates the influence of the spatial correlation lengthon the probability of type II error. Similar to Fig. 8, the probabilityof type II error starts at 0.00035, increases to 0.0501, and thendrops down to 0.03785 for �cu

= 1, 20, and 100 m, respectively.The aforementioned results can be explained as follows. In the

limiting case, when �cu¡ 0, points in the field are independent,

local averaging removes all variance, and the mean tends to themedian. Thus the slope is homogeneous in the horizontal direc-tion. At the other extreme, when �cu

¡ ∞, points in the field areperfectly correlated with each other and the slope is homoge-neous in the horizontal direction too. This means in both casesone set of vertical samples is sufficient to represent the undrainedstrength of the entire slope. Then probabilities of type I and IIerrors are both zero. At intermediate spatial correlation lengths,

Fig. 8. Influence of correlation length (�cu) on probability of type I error. [Colour online.]

Fig. 9. Influence of correlation length (�cu) on probability of type II error. [Colour online.]

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there must be a worst case scenario where type I and II errors aremaximum.

The highest probability of type I error occurs at a spatial corre-lation length of 15 m and the highest probability of type II erroroccurs at a spatial correlation length of 20 m. These results sug-gest that when the spatial correlation length is 1.5–2 times that ofthe slope height, caution should be taken to design the site inves-tigation scope so that the error probabilities are within an accept-able range.

5.3. Influence of spatial correlation length for Krigingestimation on error probabilities

In the previous two subsections (i.e., 5.1 and 5.2), it was assumedthat the statistics of the random field are known when undertak-ing Kriging. In reality, the spatial correlation length requiresmore tests to estimate than the mean and standard deviationrequire. In this section, the spatial correlation length in Kriging isassumed to be unknown. The effect of spatial correlation length

in Kriging on the error probabilities is investigated. The spatialcorrelation length of random field is fixed at 10 m (�cu

= 10 m) andthe spatial correlation length in Kriging �k is varied in the range�k = {1, 5, 10, …, 100 m}.

Figures 10 and 11 show the influence of �k on the probability oftype I and II errors, respectively. It can be seen from Fig. 10 that theprobability of type I error decreases as �k increases. However, itcan be seen from Fig. 11 that the probability of type II error in-creases as �k increases. These results can be explained as follows.Because all the statistical properties are fixed and �cu

= 10 m, theprobability of failure is 0.1495 according to Fig. 7. However, whenthe spatial correlation length in Kriging is increased, the proba-bility of failure estimated by Kriging will also increase, as sug-gested by Fig. 7. This means that the probability of type I errordecreases and the probability of type II error increases when �kincreases.

Because the actual correlation length is rarely, if ever, known atany site, it is more important to choose the spatial correlation

Fig. 10. Influence of Kriging correlation length (�k) on probability of type I error. [Colour online.]

Fig. 11. Influence of Kriging correlation length (�k) on probability of type II error. [Colour online.]

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length in Kriging to minimize the probabilities of both type I andII errors. However, the results show that there is a trade-off be-tween the type I and II errors. On the one hand, if a large spatialcorrelation length in Kriging is chosen, the design would be overlyconservative. On the other hand, if a small spatial correlationlength in Kriging is chosen, the design may be unsafe. Cautionmust be taken to choose an appropriate spatial correlation lengthwhen Kriging.

5.4. Influence of factor of safety based on mean on errorprobabilities

The influence of FS based on the mean strength on error prob-abilities is investigated by varying the mean strength, but main-taining all other parameters constant. It is assumed that samplingis performed at the best location. Figures 12 and 13 show theinfluence of the mean strength at the crest level on the probabil-

ities of type I and type II errors, respectively, for �cu= 10 m and �k =

10 m. When the mean strength at the crest level increases from�cu0

= 0.1 kPa to �cu0= 18 kPa, the FS based on the mean increases

from 0.86 to 1.52. The corresponding probability of type I errorstarts at 0.14, increase to 0.38, and then drops to 0.10. It reaches amaximum of 0.38 when FS = 1.09. Similarly, the highest probabil-ities of type II error are observed when FS = 1.31. As shown inFig. 14, the highest probability of making a wrong decision (eithertype I or type II error) occurs when FS = 1.13.

The results can be explained as follows. Each Monte Carlo sim-ulation yields a linearly increasing strength profile, but they areall different from each other. When the mean strength of the soilproperties is very small, both the stability analyses based on com-plete knowledge and site investigation data will suggest the slopeis unsafe. In this case, the probabilities of type I and type II errors

Fig. 12. Influence of factor of safety on probability of type I error. [Colour online.]

Fig. 13. Influence of factor of safety on probability of type II error. [Colour online.]

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will be small. Similarly, when the mean strength of the soil prop-erties is very large both the stability analyses based on completeknowledge and site investigation data will suggest the slope issafe. In this case the probabilities of type I and II errors will be alsosmall. There must be an intermediate factor of safety for whichboth error probabilities reach the maximum.

6. Summary and conclusionsIn this paper, the importance of sampling location on slope

stability assessment is examined by Monte Carlo simulation. Anundrained slope with a mean strength that increases with depth isconsidered. Based on the results obtained in this paper, the fol-lowing conclusions can be drawn:

1. For the given slope, both the probabilities of type I and type IIerrors vary perceivably with the sampling location and reachthe minimum value in the vicinity of the slope crest. It issuggested that the slope crest appears to be the optimal loca-tion to conduct a geotechnical site exploration for slope sta-bility assessment.

2. The influence of spatial correlation length on probabilities oftype I and II errors are investigated. It is shown that there is aworst case spatial correlation length where both type I and IIerrors are a maximum. The worst case spatial correlationlength is 1.5–2.0 times the slope height. Additional site inves-tigations are needed to reduce the error probabilities in stabil-ity assessment when the level of spatial variability is withinthis range.

3. The influence of spatial correlation length when Kriging isused on the probabilities of type I and II errors are investi-gated. The results show that the probability of type I errordecreases as �k increases, but the probability of type II errorincreases as �k increases. An appropriate spatial correlationlength needs to be selected to strike a balance between thedesign safety level and cost.

4. The influence of factor of safety based on the mean strengthon error probabilities are investigated. The results show thatthe highest decision error probability occurs when the factorof safety is close to 1. In other words, when the safety level is

marginal more sampling should be conducted to ensure thatthe design is reliable.

5. In this paper, it is assumed that only one penetration soundingis conducted. The influence of additional soundings on theerror probabilities will be investigated in a future study.

AcknowledgementsThe authors wish to acknowledge support from the Australian

Research Council Centre of Excellence for Geotechnical Scienceand Engineering and National Natural Science Foundation ofChina (Project No. 51679117). The first author wishes to acknowl-edge support from the China Scholarship Council and The Univer-sity of Newcastle.

ReferencesChiles, J.-P., and Delfiner, P. 2009. Geostatistics: modeling spatial uncertainty.

John Wiley & Sons.Fenton, G.A., and Griffiths, D.V. 2008. Risk assessment in geotechnical engineer-

ing. Wiley.Fenton, G.A., and Vanmarcke, E.H. 1990. Simulation of random fields via local

average subdivision. Journal of Engineering Mechanics, 116: 1733–1749. doi:10.1061/(ASCE)0733-9399(1990)116:8(1733).

Fenton, G.A., Liza, R., Lake, C.B., Menzies, W.T., and Griffiths, D.V. 2015. Statis-tical sample size for quality control programs of cement-based “solidifica-tion/stabilization”. Canadian Geotechnical Journal, 52(10): 1620–1628. doi:10.1139/cgj-2013-0478.

Gong, W., Luo, Z., Juang, C.H., Huang, H., Zhang, J., and Wang, L. 2014. Optimi-zation of site exploration program for improved prediction of tunneling-induced ground settlement in clays. Computers and Geotechnics, 56: 69–79.doi:10.1016/j.compgeo.2013.10.008.

Griffiths, D., and Fenton, G.A. 1993. Seepage beneath water retaining structuresfounded on spatially random soil. Géotechnique, 43: 577–587. doi:10.1680/geot.1993.43.4.577.

Griffiths, D., and Fenton, G.A. 2004. Probabilistic slope stability analysis by finiteelements. Journal of Geotechnical and Geoenvironmental Engineering, 130:507–518. doi:10.1061/(ASCE)1090-0241(2004)130:5(507).

Griffiths, D., and Lane, P. 1999. Slope stability analysis by finite elements. Géo-technique, 49: 387–403. doi:10.1680/geot.1999.49.3.387.

Griffiths, D., Huang, J., and Fenton, G.A. 2009. Influence of spatial variability onslope reliability using 2-D random fields. Journal of Geotechnical and Geoen-vironmental Engineering, 135: 1367–1378. doi:10.1061/(ASCE)GT.1943-5606.0000099.

Griffiths, D., Huang, J., and Fenton, G. 2015. Probabilistic slope stability analysisusing RFEM with non-stationary random fields. In Geotechnical safety. Edited byV. Risk, T. Schweckendiek, A.F. van Tol, D. Pereboom, et al. pp. 704–709.

Huang, J., Fenton, G., Griffiths, D.V., Li, D., and Zhou, C. 2017. On the efficient

Fig. 14. Influence of factor of safety on error probabilities.

Yang et al. 345

Published by NRC Research Press

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y C

OL

OR

AD

O S

CH

OO

L O

F M

INE

S on

03/

19/1

9Fo

r pe

rson

al u

se o

nly.

estimation of small failure probability in slopes. Landslides, 14(2): 491–498.doi:10.1007/s10346-016-0726-2.

Jaksa, M., Kaggwa, W., Fenton, G., and Poulos, H. 2003. A framework for quan-tifying the reliability of geotechnical investigations. In Proceedings of the 9thInternational Conference on the Application of Statistics and Probability inCivil Engineering.

Jaska, M., Goldsworthy, J., Fenton, G., Kaggwa, W., Griffiths, D., Kuo, Y., andPoulos, H. 2005. Towards reliable and effective site investigations. Géotech-nique, 55: 109–121. doi:10.1680/geot.2005.55.2.109.

Jiang, S.-H., Papaioannou, I., and Straub, D. 2017. Optimizing borehole locationsfor slope reliability assessment. In Proceedings of Geo-Risk 2017.

Journel, A.G., and Huijbregts, C.J. 1978. Mining geostatistics. Academic Press.Krige, D.G. 1951. A statistical approach to some basic mine valuation problems

on the Witwatersrand. Journal of the Southern African Institute of Miningand Metallurgy, 52: 119–139.

Kulhawy, F.H., and Mayne, P.W. 1990. Manual on estimating soil properties forfoundation design. Electric Power Research Inst., Palo Alto, Calif. (USA); Cor-nell University, Ithaca, New York (USA). Geotechnical Engineering Group.

Lacasse, S., and Nadim, F. 1997. Uncertainties in characterising soil properties.Publikasjon-Norges Geotekniske Institutt, 201: 49–75.

Li, Y., Hicks, M., and Vardon, P. 2016. Uncertainty reduction and sampling effi-ciency in slope designs using 3D conditional random fields. Computers andGeotechnics, 79: 159–172. doi:10.1016/j.compgeo.2016.05.027.

Matheron, G. 1962. Traité de géostatistique appliquée. Paris, Éditions Technip.National Research Council. 1984. Geotechnical site investigations for under-

ground projects. National Academy Press.Phoon, K.-K., and Kulhawy, F.H. 1999. Characterization of geotechnical variabil-

ity. Canadian Geotechnical Journal, 36(4): 612–624. doi:10.1139/t99-038.Rivoirard, J. 1990. A review of lognormal estimators for in situ reserves. Mathe-

matical Geology, 22: 213–221. doi:10.1007/BF00891825.Robertson, P.K. 2009. Interpretation of cone penetration tests — a unified

approach. Canadian Geotechnical Journal, 46(11): 1337–1355. doi:10.1139/T09-065.

Vanmarcke, E.H. 1977. Reliability of earth slopes. Journal of the GeotechnicalEngineering Division, ASCE, 103: 1247–1265.

Voltz, M., and Webster, R. 1990. A comparison of kriging, cubic splines and classi-fication for predicting soil properties from sample information. EuropeanJournal of Soil Science, 41: 473–490. doi:10.1111/j.1365-2389.1990.tb00080.x.

Wackernagel, H. 2013. Multivariate geostatistics: an introduction with applica-tions. Springer Science & Business Media.

Walpole, R.E., Myers, R.H., Myers, S.L., and Ye, K. 1993. Probability and statisticsfor engineers and scientists. Macmillan, New York.

Webster, R., and Oliver, M.A. 1999. Geostatistics for environmental scientists,Wiley Online Library.

Zhu, D., Griffiths, D., Huang, J., and Fenton, G. 2017. Probabilistic stability anal-yses of undrained slopes with linearly increasing mean strength. Géotech-nique, 67(8): 1–14. doi:10.1680/jgeot.15.P.014.

List of symbols

Cij covariance between observationsCix covariance between observation point and estimate pointCu undrained shear strengthD foundation depth ratioE expectation

Err simple Kriging estimator errorFS factor of safetyH slope height

H0 null hypothesisH1 alternative hypothesisIF real stability index

IFkestimated stability index

i simple counterK Kriging matrix

Kn×n n×n upper left submatrix of KM covariance between observation point and intermediate

pointMn first n elements of M

n number of observationsn1 number of realisations where IFk

� IFn2 number of realisations where IFk

� IFnsim number of simulations

p1 probability of type I errorp2 probability of type II errorpf probability of failuret strength gradient

vcucoefficient of variation of undrained shear strength

vY lognormal coefficient of variationvY0

lognormal coefficient of variation at crest levelvYz

lognormal coefficient of variation at depth zXi observations

X(x) normal random fieldX�x� simple Kriging estimator

x spatial position of observationxi spatial position of observationY0 lognormal random variable at crest levelYz lognormal random variable at depth z

Y(x) lognormal random fieldY�x� lognormal Kriging estimator

z depth below crest� unknown simple Kriging weight matrix

�n first n elements of � k unknown simple Kriging weight� soil unit weight

�sat saturated soil unit weight� Lagrangian parameter

�cuspatial correlation length of undrained shear strength

�k spatial correlation length for Kriging�Y lognormal spatial correlation length

�cu0mean undrained shear strength at crest level

�X normal mean strength�X0

normal mean strength at crest level�Xz

normal mean strength at depth z�Y lognormal mean strength�Y0

lognormal mean strength at crest level�Yz

lognormal mean strength at depth z�() correlation coefficient between properties assigned to

two points�Err standard deviation of simple Kriging estimator error�lnY0

equivalent standard deviation at crest level�X normal standard deviation�X0

normal standard deviation at crest level�Xz

normal standard deviation at depth z�Y lognormal standard deviation

absolute distance between two points in random field� slope angle

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