Engineering Geology 211 (2016) 150–161

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Engineering Geology

j ourna l homepage: www.e lsev ie r .com/ locate /enggeo

Site-specific characterization of soil properties using multiplemeasurements from different test procedures at different locations – ABayesian sequential updating approach

Zi-Jun Cao a,b, Yu Wang c,⁎, Dian-Qing Li da State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan 430072, PR Chinab State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, PR Chinac Department of Architecture and Civil Engineering, City University of Hong Kong Tat Chee Avenue, Kowloon, Hong Kongd State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan 430072, PR China

⁎ Corresponding author.E-mail addresses: zijuncao@whu.edu.cn (Z.-J. Cao), yu

dianqing@whu.edu.cn (D.-Q. Li).

http://dx.doi.org/10.1016/j.enggeo.2016.06.0210013-7952/© 2016 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 8 January 2016Received in revised form 24 June 2016Accepted 25 June 2016Available online 29 June 2016

Determination of site-specific values of geotechnical parameters (e.g., undrained shear strength, Su) is a key stepin geotechnical analyses and designs at a particular site. This, however, has been a challenging task in geotech-nical practice because of various uncertainties in geotechnical parameters and the fact that the number of testdata obtained during geotechnical site investigation at a site is often too sparse to accurately estimate statisticsof the geotechnical parameters. These issues can be rationally addressed under a Bayesian framework. Thisstudy proposes a Bayesian sequential updating (BSU) approach for probabilistic characterization of geotechnicalparameters based on multi-source information, including the information available prior to the project, referredto as prior knowledge in Bayesian framework, and results of different types of tests that might be performed atdifferent locations in a soil layer within a specific site. In this paper, the proposed BSU approach is formulated forprobabilistic characterization of Su of clay using over-consolidation ratio (OCR), standard penetration test (SPT)data, and cone penetration test (CPT) data. OCR, SPT, and CPT data are sequentially incorporated into the Bayes-ian framework to update statistics (e.g.,mean, standarddeviation, and probability density function) of the Su pro-file. Equations are derived from the proposed approach and are illustrated using real OCR, SPT and CPT data. Theproposed BSU approach is shown to perform satisfactorily and provide insights into evolution of the statistics ofgeotechnical parameters as more results of different testing procedures are used. Such insights allow geotechni-cal practitioners to inspect the quality of information from different testing procedures (including test resultsand transformation models adopted to interpret them) and to identify the most informative test procedure.

© 2016 Elsevier B.V. All rights reserved.

Keywords:Site investigationBayesian approachMulti-source informationUndrained shear strengthUncertainty

1. Introduction

Geo-materials are naturalmaterials. Their parameters (e.g., undrainedshear strength, Su) are affected by various geological processes that theyhave undergone in their geological histories, and hence vary from onesite to another (e.g., Phoon, 2008; Wang et al., 2016). Determining site-specific values of geotechnical parameters is a key step for engineer-ing analyses and designs in engineering geology and geotechnicalengineering. This is, however, not a trivial task in practice, at leastpartially, because of two reasons: 1) various uncertainties exist ingeotechnical parameters, including inherent variability, transforma-tion uncertainty, measurement errors, and statistical uncertainty(e.g., Phoon and Kulhawy, 1999a; Baecher and Christian, 2003);and 2) the number of test results obtained from geotechnical site

wang@cityu.edu.hk (Y. Wang),

investigation for a project is often too limited to accurately deter-mine statistics of the geotechnical parameters, particularly for pro-jects with relatively small or medium sizes. These two issues can berationally addressed under a Bayesian framework, which character-izes various uncertainties in geotechnical parameters using probabi-listic models (e.g., random variables) and explicitly quantifies effectsof sparse site-specific test results on estimates of geotechnical pa-rameters (Wang et al., 2016).

Bayesian approaches provide a formal and rational vehicle to com-bine multi-source information obtained from site investigation, includ-ing site information available prior to the project (namely priorknowledge) and site-specific test results from multiple testing proce-dures (including test borings, in-situ testing, and/or laboratory testing)(Medina-Cetina and Esmailzadeh, 2014; Zhang et al., 2004, 2016;Wanget al., 2016). Recently, several Bayesian approaches have beendeveloped for geotechnical site investigation, such as those for probabi-listic characterization of sand friction angle (e.g., Wang et al., 2010;Ching et al., 2012; Cao and Wang, 2013), undrained shear strength

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151Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

(e.g., Ching et al., 2010; Cao and Wang, 2014; Müller et al., 2014), un-drained Young's modulus (Wang and Cao, 2013), soil–water character-istic curve (e.g., Chiu et al., 2012), soil types (e.g., Depina et al., 2015),elastic modulus (Feng and Jimenez, 2014) and uniaxial compressivestrength (Wang and Aladejare, 2015) of rocks, etc. Most of these previ-ous studies made use of a single type of test results to probabilisticallyestimate site-specific geotechnical parameters, except a few recentstudies by Ching et al. (2010, 2012) and Müller et al. (2014).

Ching et al. (2010) developed multivariate correlations between Suand test results from multiple testing procedures (including over-consolidation ratio (OCR), standard penetration test (SPT), and conepenetration test (CPT) data) using pairwise correlations. Ching et al.(2012) proposed multivariate correlations to update uncertainties insand friction angle using relative density, mean confining pressure atfailure, SPT data, and CPT data. Müller et al. (2014) extended the multi-variate correlations to incorporate geotechnical spatial variability. Thesemultivariate correlations allowmultiple types of test results to be incor-porated in the site-specific evaluation of geotechnical parameters. Nev-ertheless, they do not contain the information from prior knowledge(e.g., engineering judgments and/or experience) on geotechnical pa-rameters, but only reflect the information from pairwise correlationsused in their developments and the test results involved in the correla-tions. More importantly, using a multivariate correlation requires thatthe multiple types of test results (e.g., OCR, SPT and CPT data) involvedin the multivariate correlation should be obtained at the same location,or at least, in close proximity. This requirement might not be easily sat-isfied in practice because different test procedures are often performedat different locations. How to systematically integrate prior knowledgewith test results from multiple testing procedures performed at differ-ent locations in a soil layer to probabilistically characterize site-specific geotechnical parameters remains an open question.

In addition, the information provided by multiple types of test resultsmay not be consistent due to various uncertainties (e.g., measurement er-rors and uncertainties in transformation models adopted to interpret thetest results) in different testing procedures. Hence, there exist needs ofinspecting the quality of information from different testing procedures(including both test results and their corresponding transformationmodels), and identifying the testing procedures that provide the mostvaluable information on the geotechnical parameter concerned, whencombining the test results from different testing procedures together.This information is particularly useful during interpretation of site inves-tigation data and planning of site investigation.

This paper develops a Bayesian sequential updating (BSU) frameworkthat integrates prior knowledge with multiple types of test results toprobabilistically characterize geotechnical parameters at a particularsite. For illustration, the BSU framework is formulated for probabilisticcharacterization of Su of clay using site-specific OCR, SPT, and CPT data,which can be obtained at different locations and have different quantities.The OCR, SPT, and CPT data are sequentially incorporated into the BSUframework to update statistics (e.g., mean, standard deviation, and prob-ability density function (PDF)) of the Su profile. The sequential updatingprovides insights into evolution of the statistics as more types of test re-sults are used, and it allowspractitioners to inspect the quality of informa-tion from testing procedures and to identify testing procedures thatprovide consistent and/or informative information. The paper startswithdevelopment of theBSU framework and its key components (includ-ing prior distribution and likelihood function), followed by determinationof statistics of the Su profile using Markov Chain Monte Carlo simulation(MCMCS). Finally, the proposed BSU approach is illustrated using real-life data, and effects of the updating sequence are also explored.

2. Bayesian sequential updating (BSU) framework for geotechnicalsite investigation

Let XD denote the design soil property concerned in geotechnical de-sign. To explicitly model the inherent variability of XD in a soil layer, XD

can bemodeled by a randomvariable withmodel parameters (or distri-bution parameters) θ. The information on θ is needed to completely de-fine the probability distribution of XD, and it relies on both priorknowledge and site-specific test data. Let DATAi, i = I, II, …, Nt denoteNt types of site-specific test data obtained from geotechnical site inves-tigation,where the subscript “i” indicates the i-th type of test results andDATAi={DATAi,j, j=1, 2,…, ki} may contain a number ki of data points.It is worth emphasizing that, in engineering practice, the data(e.g., DATAi, i = I, II, …, Nt) from different testing procedures are oftenobtained from different locations in a soil layer, and the number ki ofmeasurements from different types of tests might also be different.

As shown in Fig. 1, the information on θ provided by DATAi, i = I, II,…,Nt can be sequentially incorporated into a Bayesian framework to up-date the knowledge on θ in a row. The proposed BSU framework con-sists of Nt updating levels, each of which uses one type of test data. Inthis paper, the BSU framework is formulated to probabilistically charac-terize Su in a clay layer using both prior knowledge and three types ofsite-specific test results (including OCR values, SPT N (i.e., N60) valuescorrected at 60% energy ratio, and corrected cone tip resistance q valuesfrom CPT) in the next two subsections.

2.1. Probabilistic modeling of undrain shear strength

Insights from soil mechanics suggest that Su itself is not a fundamen-tal soil property but depends on its corresponding vertical effectivestress, σv0' (e.g., Wroth, 1984; Terzaghi et al., 1996; Mitchell and Soga,2005). Then, the undrained shear strength ratio r = Su ,D/σv0,D' can bemodeled as a random variable (i.e., setting XD = r) in a soil layer,where Su ,D and σv0,D' represent the undrained shear strength and verti-cal effective stress at a given depth D, respectively. Consider, for exam-ple, using a lognormal random variable to characterize the inherentvariability of r in a clay layer (e.g., Lacasse and Nadim, 1996; Baecherand Christian, 2003). Let μr and σv0,D' denote themean and standard de-viation of r (i.e., θ= [μr,σr]), respectively. Then, r can bewritten as (Angand Tang, 2007):

r ¼ Su;D=σ 0v0;D ¼ exp μN;r þ σN;rz� �

ð1Þ

in which z=a standard Gaussian random variable; μN ,r= lnμr−σN ,r2 /2

and σN;r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln ½1þ ðσ r=μrÞ2�

qare the mean and standard deviation of

the logarithm (i.e., ln(r)) of r, respectively. This study applies a randomvariable model to characterize r in a single soil layer, in which r in thewhole soil layer is modeled as a spatially constant random variablewithout explicit consideration of the variation trend of r at differentdepths. The random variable model adopted in this study is generallyreasonable for soft clays (e.g., normal consolidated or slightly over-consolidated clays with OCR varying from 1 to 2). On the other hand,for stiff clays (e.g., heavily over-consolidated clays with OCR varyingfrom 3 to 50), r in a soil layer might vary with depth when OCR varieswith depth. In such a case, r shall be characterized bymore sophisticatedmodels. More data are, of course, needed to determine soil model pa-rameters as the model becomes more sophisticated.

Using Eq. (1), Su ,D is written as:

Su;D ¼ σ 0v0;D exp μN;r þ σN;rz� �

ð2Þ

Eq. (2) indicates that Su ,D increases as the depthD increases becauseσv0,D' generally increases with the depth. As indicated by Eqs. (1) and(2), probabilistic characterization of r and Su ,D needs the knowledgeon μr and σr. The next subsection uses the proposed BSU framework toobtain the updated knowledge on μr and σr based on both prior knowl-edge and three types of test results (i.e., OCR, SPT N60, and CPT q data).

Aim: Update knowledge about model parameters (i.e., ) of the geotechnical design property, XD, using Nt types of site-specific test data (i.e., DATAI, DATAII, …, and DATANt) obtained from geotechnical site investigation.

BSU Procedure:

Updating level I: Prior knowledge (PRIORI)

DATAI

Prior distribution I

Likelihood function I

Posterior knowledge I Posterior distribution I

Likelihood model MLI

&

Updating level II: PRIORII

DATAII

Prior distribution II

Likelihood function II

Posterior knowledge II Posterior distribution II

Likelihood model MLII

&

Updating level Nt: PRIORNt

DATANt

Prior distribution Nt

Likelihood function Nt

Posterior knowledge Nt Posterior distribution Nt

Likelihood model MLNt

&

Outcome: The posterior distribution (i.e., posterior distribution Nt) of which quantitatively reflects the updated knowledge from both prior knowledge (i.e., prior knowledge I) and Nt types of site-specific test data (i.e., DATAI, DATAII, …, and DATANt)

Fig. 1. Conceptual framework of Bayesian sequential updating for geotechnical site investigation.

152 Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

2.2. BSU of model parameters of undrained shear strength ratio

Consider, for example, using OCR, SPT N60, and CPT q data as DATAI,DATAII, andDATAIII in updating levels I–III of the proposed BSU frameworkto update the knowledge on μr and σr in this subsection. Although OCR,SPTN60, and CPT q data are used here in updating levels I-III, respectively,for development of the BSU framework, the multiple types of test resultscan be incorporated into the BSU framework in any other order. Effects ofthe updating sequence are further discussed later using real-life data.

As shown in Fig. 1, the updating starts with combining the priorknowledge (i.e., PRIORI) with DATAI (i.e., OCR data). Under the BSUframework, the information provided by PRIORI and DATAI is quantita-tively reflected by the prior distribution (i.e., prior distribution I,P(μr,σr)) and likelihood function (i.e., likelihood function I, P(DATAI | -μr,σr)) in updating level I, respectively. After the prior distribution Iand likelihood function I are obtained, they are combined together toobtain the posterior distribution (i.e., posterior distribution I, P(μr,σr | -DATAI)) using Bayes' Theorem (e.g., Ang and Tang, 2007; Yuen, 2010a):

P μr ;σ rjDATAIð Þ ¼ KIP DATAI jμr ;σ rð ÞP μr;σ rð Þ ð3Þ

where KI=P(DATAI)−1 is a normalizing constant in updating level I,which does not depend on μr and σr. The P(μr,σr |DATAI) quantifies theupdated knowledge (i.e., posterior knowledge I) from both PRIORI andDATAI.

Then, the posterior knowledge I is taken as the prior knowledge(i.e., PRIORII) in updating level II and is further updated using the infor-mation provided by DATAII (i.e., SPT N60 data). The updated knowledge(i.e., posterior knowledge II) on μr and σr in updating level II is quantita-tively reflected by the posterior distribution II, which is expressed as:

P μr ;σ rjDATAII;DATAIð Þ ¼ KIIP DATAIIjμr ;σ r;DATAIð ÞP μr ;σ rjDATAIð Þ ð4Þ

where KII=P(DATAII |DATAI)−1 is a normalizing constant in updatinglevel II; P(DATAII |μr,σr,DATAI)= likelihood function reflecting the infor-mation on μr and σr provided by SPT N60 data; P(μr,σr |DATAI) = priordistribution of μr and σr in updating level II, and it is given by the poste-rior distribution I (i.e., Eq. (3)) since the posterior knowledge I is used asPRIORII.

Similarly, the posterior distribution II given by Eq. (4) is taken as theprior distribution in updating level III, and it is further updated using

153Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

DATAIII (i.e., CPT q data) to obtained the posterior distribution III, whichis written as:

P μr;σ r jDATAIII;DATAII;DATAIð Þ¼ KIIIP DATAIIIjμr;σ r ;DATAII;DATAIð ÞP μr ;σ r jDATAII;DATAIð Þ ð5Þ

where KIII=P(DATAIII |DATAII,DATAI)−1 is a normalizing constant inupdating level III; P(DATAIII |μr,σr,DATAII,DATAI) = likelihood functionreflecting the information on μr and σr provided by CPT q data; P(μr,σr | -DATAII,DATAI)=prior distribution of μr andσr in updating level III, and itis given by the posterior distribution II (i.e., Eq. (4)). Finally, combiningEqs. (3)-(5) gives

P μr ;σ r jDATAIII;DATAII;DATAIð Þ ¼ KIIIKIIKIP DATAIIIjμr ;σ r ;DATAII;DATAIð Þ� P DATAIIjμr ;σ r ;DATAIð ÞP DATAI jμr ;σ rð ÞP μr ;σ rð Þ ð6Þ

Although three types (i.e., Nt =3) of test data are considered in thisstudy, Nt can be a positive integer bigger than 3 in the proposed BSUframework. For a given Nt, Bayesian updating is repeatedly performedNt times to progressively incorporate all the Nt types of site-specifictest data (i.e., DATAi, i = I, II, …, Nt) into the BSU framework. Finally,the information provided by prior knowledge (i.e., PRIORI) that is avail-able prior to the project and the Nt types of site-specific test data arecombined into the posterior knowledge (i.e., posterior knowledge Nt)obtained in the Nt-th updating level and is quantitatively reflected bythe final posterior distribution, e.g., Eq. (6) in this study. As indicatedby Eq. (6), the prior distribution (e.g., P(μr,σr)) and likelihood function(e.g., P(DATAI |μr,σr), P(DATAII |μr,σr,DATAI), and P(DATAIII | -μr,σr,DATAII,DATAI) for updating levels I-III, respectively) of eachupdating level are needed to obtain the final posterior distribution.These components are formulated in the following two sections.

3. Prior distribution

The prior distribution P(μr,σr) can be simply taken as a joint uniformdistribution of μr and σr, and it is written as (e.g., Ang and Tang, 2007):

P μr ;σ rð Þ ¼1

μr;max−μr;min� 1σ r;max−σ r;min

for μ∈ μr;min; μr;maxh i

and σ∈ σ r;min;σ r;max� �

0 others

8<: ð7Þ

where μr,min and μr,max areminimum andmaximumvalues of μr, respec-tively; and σr,min and σr,max are minimum and maximum values of σr,respectively. Note that only the possible ranges (i.e., [μr,min, μr,max] and[σr,min, σr,max]) of model parameters are needed to completely definethe uniform prior distribution herein. The typical ranges of soil propertystatistics (e.g., μr and σr in this study) are usually available in geotechni-cal literature (e.g., Lumb, 1974; Lee et al., 1983; Lacasse and Nadim,1996; Phoon and Kulhawy, 1999a, 1999b; Cao et al., 2016). Althoughthe uniform prior distribution that requires relatively limited priorknowledge on model parameters (e.g., μr and σr) is applied to develop-ing the BSU framework in this paper, more sophisticated prior distribu-tions can also be used in the proposed approach, which, of course,require relatively informative prior knowledge as justifications (Caoet al., 2016).

4. Likelihood functions

As shown in Fig. 1, formulation of the likelihood function for the i-th(i = I, II, or III) updating level requires a likelihood model MLi, whichprobabilistically relates the model parameters (i.e., μr and σr) to thesite-specific test data (i.e., OCR, SPT N60, and CPT q data in updatinglevels I-III, respectively). The likelihoodmodel relies on the probabilisticmodel of the undrained shear strength (e.g., Eq. (2)) and the transfor-mation model that allows derivation of the undrained shear strengthfrom the test data used in the i-th updating level.

During geotechnical site investigation, the undrained shear strengthof clay measured by isotropically consolidated undrained triaxial

compression (CIUC) tests can be related to OCR, SPT N60, and CPT qdata by empirical regressions/transformationmodels deemed appropri-ate at a specific site, which can be obtained by some pre-screeningmodel selection procedure (e.g., Wang and Aladejare, 2015). Consider,for example, the following three empirical regressions in this study(Ching et al., 2010):

ln OCRð Þ ¼ a1 ln Su;D=σ 0v0;D� �

þ b1 þ ε1 ð8Þ

ln N60ð Þ ¼ a2 ln Su;D� �þ c2 ln σ 0v0;D

� �þ b2 þ ε2 ð9Þ

ln qð Þ ¼ a3 ln Su;D� �þ b3 þ ε3 ð10Þ

where a1= 1.563; b1= 1.366; a2=1.633; b2=−3.845; c2=−0.403;a3 = 1.0; b3 = 2.540; ε1, ε2, and ε3 = normal random variables with amean of zero and respective standard deviations σε1 = 0.37, σε2 =0.456, and σε3 = 0.34, and they represent themodeling errors or trans-formation uncertainties associatedwith Eqs. (8)-(10), respectively; q=qT–σv0 in which qT =CPT reading corrected by the pore water pressurebehind the cone and σv0 = in-situ total vertical stress. In this study, thetransformation uncertainty is represented by normal random variables(i.e., ε1, ε2, and ε3 for Eqs. (8)-(10)). For a given transformation model,the transformation uncertainty remains the same for different test loca-tions in a soil layer. From this point of view, the transformation uncer-tainties of the given transformation model at different locations aretaken to be perfectly-correlated in this study. However, it is worthwhileto note that ε1, ε2, and ε3 are from different sources and taken to bemu-tually independent in this study.

Substituting Eq. (2) into Eqs. (8)-(10) gives:

ln OCRð Þ ¼ a1μN;r þ b1 þ a1σN;rzþ ε1 ð11Þ

ln N60ð Þ ¼ a2μN;r þ a2 þ c2ð Þ ln σ0v0;D

� �þ b2 þ a2σN;rzþ ε2 ð12Þ

ln qð Þ ¼ a3μN;r þ a3 ln σ0v0;D

� �þ b3 þ a3σN;rzþ ε3 ð13Þ

Eqs. (11)-(13) relate model parameters (i.e., μr and σr) to ln(OCR),ln(N60), and ln(q) in a probabilistic manner, respectively. For a givenset of μr and σr, ln(OCR), ln(N60), and ln(q) are independent becauseε1, ε2, and ε3 are independent in this study. These equations are thentaken as likelihood models to formulate respective likelihood functionsin updating levels I–III in this study. Although Eqs. (8)-(10) are adoptedto develop the likelihood models Eqs. (11)-(13), other theoretical and/or empirical correlations between the design soil property of interestand site observation data reported in the literature (e.g., Kulhawy andMayne, 1990; Phoon and Kulhawy, 1999b; Ching et al., 2010, 2012)can also be applied in the proposed BSU framework, which, of course,lead to different formulations of likelihood models and likelihood func-tions. It is always prudent to use the correlations that are deemed ap-propriate at a specific site (e.g., Cao and Wang, 2014).

In updating level I, Eq. (11) is taken asMLI to formulate the likelihoodfunction (i.e., P(DATAI |μr,σr)).When the inherent variability is assumedto be independent of the transformation uncertainty (i.e., z is indepen-dent of ε1 in Eq. (11)), ln(OCR) is a Gaussian random variable with a

mean of a1μN ,r+b1 and standard deviation offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21σ

2N;r þ σ2ε1

q. Therefore,

P(DATAI |μr,σr) can be written as:

P DATAIjμr;σ rð Þ ¼ ∏kI

j¼1

1ffiffiffiffiffiffi2π

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21σ

2N;r þ σ2ε1

q exp −12

ln OCRð Þ j− a1μN;r þ b1� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21σ

2N;r þ σ2ε1

q264

37528><

>:

9>=>;

ð14Þ

where DATAI={ln(OCR)1, ln(OCR)2, ... , ln(OCR)kI} is a set of ln(OCR)values measured at kI different locations in a clay layer. As indicatedby Eq. (14), theDATAI={ln(OCR)1, ln(OCR)2, ... , ln(OCR)kI} are simplified

154 Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

as kI independent realizations of the Gaussian random variable ln(OCR)in this study. Such a simplification can be justified when the test loca-tions are not close, and hence, correlation among different test data isminimal.

In updating level II, Eq. (12) is taken as MLII to formulate the likeli-hood function (i.e., P(DATAII|μr,σr,DATAI)). Similarly, it can be reasonedthat ln(N60) at a given depth is a Gaussian random variable with ameanof a2μN ,r+(a2+c2) ln(σv0,D' )+b2 and standard deviation offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22σ

2N;r þ σ2ε2

qin MLII. Since μN ,r and σN ,r are functions of μr and σr

(see Eq. (1)), the Gaussian PDF of ln(N60) at a given depth, whereln(σv0,D' ) is considered as constant, is uniquely determined for a givenset of μr and σr according to MLII. Then, P(DATAII|μr,σr,DATAI) is simpli-fied as P(DATAII|μr,σr), which is written as:

P DATAIIjμr ;σ r ;DATAIð Þ ¼ P DATAIIjμr;σ rð Þ

¼ ∏kII

j¼1

1ffiffiffiffiffiffi2π

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22σ

2N;r þ σ2ε2

q exp −12

ln N60ð Þ j−a2μN;r− a2 þ c2ð Þ ln σ0v0;D

� �−b2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a22σ2N;r þ σ2ε2

q264

37528><

>:

9>=>;

ð15Þ

whereDATAII={ln(N60)1, ln(N60)2, ... , ln(N60)kII} is a set of ln(N60) valuesmeasured at kII different locations in a clay layer. The DATAII={ln(N60)1, ln(N60)2, ... , ln(N60)kII} in Eq. (15) are, again, considered as kIIindependent realizations of ln(N60) obtained at test locations that arenot close.

Finally, Eq. (13) is taken asMLIII to formulate the likelihood function(i.e., P(DATAIII |μr,σr,DATAII,DATAI)) in updating level III, and it is rea-soned that ln(q) in Eq. (13) is a Gaussian random variable with amean of a3μN ,r+a3 ln(σv0,D' )+b3 and standard deviation offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23σ

2N;r þ σ2ε3

q. Similar to updating level II, the Gaussian PDF of ln(q)

at a given depth is uniquely determined for a given set of μr and σr ac-cording to MLIII, and a set of CPT q values (i.e., DATAIII={ln(q)1, ln(q)2, ... , ln(q)kIII}) measured at kIII different locations in a claylayer are viewed as kIII independent realizations of ln(q). Then,P(DATAIII |μr,σr,DATAII,DATAI) is simplified as P(DATAIII |μr,σr) and iswritten as:

P DATAIIIjμr;σ r ;DATAII;DATAIð Þ ¼ P DATAIIIjμr ;σ rð Þ

¼ ∏kIII

j¼1

1ffiffiffiffiffiffi2π

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23σ

2N;r þ σ2ε3

q exp −12

ln qð Þ j−a3μN;r−a3 ln σ0v0;D

� �−b3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a23σ2N;r þ σ2ε3

q264

37528><

>:

9>=>;

ð16Þ

where DATAIII={ln(q)1, ln(q)2, ... , ln(q)kIII} is a set of CPT q values mea-sured at kIII different locations in a clay layer.

Using Eqs. (7) and (14)-(16), the posterior distributions(e.g., P(μr,σr |DATAI), P(μr,σr |DATAII,DATAI), and P(μr,σr | -DATAIII,DATAII,DATAI)) of μr and σr in updating levels I–III are obtained,and they reflect the updated information on μr and σr from multiplesources (including PRIORI, OCR, SPT N60, and CPT q data). The next sec-tion uses the updated information to determine the PDF of r and gener-ates equivalent samples of r and Su for their probabilisticcharacterizations.

5. Probabilistic characterization of undrained shear strength usingequivalent samples

Using the Theorem of Total Probability (e.g., Ang and Tang, 2007),the PDF of r for a given set of prior knowledge (i.e., PRIORI) and site-specific test data (i.e., DATA) is expressed as:

P rjDATA; PRIORIð Þ ¼ ∬P rjμr;σ rð ÞP μr ;σ rjDATA; PRIORIð Þdμrdσ r ð17Þ

in which DATA are, respectively, taken as {DATAI}, {DATAI, DATAII}, and{DATAI, DATAII, DATAIII} in updating levels I-III of the proposed BSU frame-work; P(r |μr,σr) is a lognormal PDF with a mean μr and standard devia-tion σr; P(μr,σr |DATA,PRIORI) is the updated PDF of μr and σr based onPRIORI and DATA. When the symbol “PRIORI” is dropped according to the

conventional Bayesian notation, P(μr,σr |DATA,PRIORI) is simplified asP(μr,σr |DATA), and it is given by Eqs. (3)-(5) in updating levels I-III ofthe proposed BSU framework. Substituting Eqs. (3)-(5) into Eq. (17)gives the updated PDFs of r in the three updating levels. Asmore differenttypes of test results are incorporated into the BSU framework, the PDF of ris progressively updated from P(r |DATAI,PRIORI) in updating level I toP(r|DATAIII,DATAII,DATAI,PRIORI) in updating level III.

For each updating level, the PDF of the random variable r in Eq. (17)can be depicted numerically using a large number of r samples generat-ed byMarkov ChainMonte Carlo Simulation (MCMCS). MCMCS is a nu-merical process that simulates a sequence of samples of a randomvariable (e.g., r) as a Markov Chain with the PDF (e.g., Eq. (17) for r) ofthe random variable as the Markov Chain's limiting stationary distribu-tion (e.g., Beck and Au, 2002; Robert and Casella, 2004; Zhang et al.,2010, 2012; Wang and Cao, 2013; Juang et al., 2013; Peng et al., 2014).The states of the Markov Chain after it reaches stationary conditionare then used as “equivalent” samples of the random variable with thetarget PDF. MCMCS provides a feasible way to generate samples froman arbitrary PDF, particularly when the PDF is complicated and difficultto express analytically and explicitly. In this study, the Metropolis-Hastings (MH) algorithm (Metropolis et al., 1953, Hastings, 1970, Beckand Au, 2002, Yuen, 2010b) is used in MCMCS to generate a large num-ber, nMCMC, of equivalent samples of r from Eq. (17). Fig. 2 shows a flow-chart for the implementation of MCMCS in this study. Details of theMCMCS-based numerical representation of the PDF of a random vari-able (including the algorithm, advantages, implementation procedure,and computer codes) are referred to Wang and Cao (2013).

After the equivalent samples of r are obtained in each updating level,the corresponding equivalent samples of undrained shear strength Su ,D(i.e., rσv0,D' ) at a given depth D are calculated from the r samples usingthe information onσv0,D' . Then, conventional statistical analyses are per-formed on the equivalent samples of r and Su ,D to estimate their respec-tive statistics (e.g., mean, standard deviation, and quantiles) andprobability distributions (e.g., PDFs). MCMCS can be performed repeat-edly to generate the equivalent samples of r and to estimate the statis-tics and probability distributions of r and Su ,D in each updating level.This provides insights into evolution of the statistics (e.g., mean andstandard deviation) of r and Su ,D as more different types of test resultsare used. Such insights are helpful in inspecting the quality of informa-tion fromdifferent testing procures that are newly incorporated into thesequential updating and their effects on the updated statistics. Then, thetesting procedure that provides consistent and/or informative test re-sults can be identified accordingly, as further discussed and illustratedusing real-life data in the next section.

6. Illustrative example

This section applies the proposed BSU approach to probabilisticallycharacterize Su in a clay layer at a deep excavation site in Taipei (Ouand Liao, 1987; Ou, 2006; Ching et al., 2010). The clay layer extendsfrom about 9.0 m below the surface to the depth of 38.6 m. Multipletypes of test results (including OCR, SPT N60, CPT q data, as shown inTable 1) were obtained from the clay layer through laboratory and in-situ tests during site investigation. Details of site investigation at thesite are referred to Ching et al. (2010). Fig. 3(a)-(c) show the OCR, SPTN60, and CPT q values versus the depth in the clay layer, respectively,which were used by Ching et al. (2010) to illustrate the application ofmultivariate correlations between Su measured by CIUC tests and OCR,SPT N60, CPT q data. These OCR, SPT N60, and CPT q values were inten-tionally chosen by Ching et al. (2010) so that all three types of test re-sults are simultaneously available at the same depth, whichsubsequently enables the use of multivariate correlations at the depth.Note that the pore water pressure behind the cone is not reported inCPT data at the site. Hence, the q values shown in Fig. 3(c) are approxi-mately calculated using the cone tip resistance recorded during CPTwithout consideration of pore water pressure, which somehow leads

Obtain a set of DATA, such as {DATAI}, {DATAI,DATAII}, and {DATAI, DATAII, DATAIII} in updating

levels I-III, respectively

Choose an initial sample for the Markov Chain of r and take the initial sample as the current sample at the first

state of the Markov Chain

No

Estimate statistics (e.g., mean and standard deviation) and probability distributions (e.g., PDFs) of r and that

summarize the information from both DATA and PRIORI

Use MH algorithm to generate a candidate sample of the Markov Chain based on the current sample

Obtain a set of prior knowledge (i.e., PRIORI) on the mean and standard deviation of r

Evaluate the acceptance ratio of the candidate samplebased on DATA and PRIORI, in which Eqs. (7) and (14)-

(17) are used

Accept or not?

Take the candidate sample as the next state

Take the current sample as the next state

equivalent samples

are already generated?

No

Yes

Yes

μ σr r

MCMCn

su,D

Fig. 2. Implementation procedure of MCMCS.

155Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

to underestimation of the corrected cone tip resistance at the site, aspointed out by Ching et al. (2010). This suggests that the informationprovided by q data shown in Fig. 3(c) might not be consistent withthat from OCR and SPT N60 data shown in Fig. 3(a) and (b). Fig.3(d) shows effective vertical stress σv0,D' versus depth in the claylayer. The relationship between the σv0,D' and depth D is approximatedas a linear equation σv0' = 20.84 + 8.38D, where D is the depth belowthe ground surface varying from 11.3 to 26.6 m. Fig. 3(e) shows equiv-alent CIUC Su,D values in the clay layer reported by Ching et al. (2010),which were not directly measured by CIUC tests but were convertedfrom Su values measured by unconsolidated undrained (UU) tests, K0-consolidated undrained (CK0U) tests, and vane shear tests so as to en-able a fair comparison among Su,D values from different testing proce-dures. In general, the Su,D value in the clay layer increases linearly withthe depth, and the variation follows a linear trend Su,D = 5.34D-5.59with a coefficient of determination R2 = 0.802.

All the OCR, SPTN60, and q data shown in Fig. 3(a)-(c) can be used inthe proposed BSU approach to update the knowledge on μrandσr. How-ever, to illustrate the capability of theproposed approach in usingdiffer-ent types of test results measured at different locations and withdifferent quantities, this example only makes use of 7 OCR values and7 SPT N60 values (see solid circles shown in Fig. 3(a) and (b)) belowthe depth of 18.0 m and 6 q values (see solid circles shown in Fig.3(c)) above the depths of 18.0 m as inputs for updating levels I–III, re-spectively. Note that only 6 solid circles can be seen in Fig. 3(a) and(b) because there are two measurements at the depth of 20.2 m (seebold entries in Table 1, indicating the test results used in this example)and they provide virtually identical values of OCR and SPT N60. In theupdating levels II and III, the information on σv0,D' shown in Fig. 3(d) is also used (see Eqs. (15) and (16)). In addition, the prior knowl-edge (i.e., PRIORI) is quantitatively reflected by a joint uniform distribu-tion with a μr ranging from μr ,min = 0.22 to μr ,max = 0.66 and a σr

Table 1In-situ and laboratory test results obtained from a clay site in Taiwan.(After Ou and Liao, 1987; Ou, 2006; Ching et al., 2010).

MeasurementID Depth (m) OCR SPT N60 q (kPa) σv0' (kPa) Su,D (kPa))

1 11.3 1.55 3.96 628.90 115.52 43.102 12.8 1.35 3.00 577.09 127.95 76.453 14.8 1.22 3.34 459.90 144.86 82.354 16.1 1.16 4.00 420.01 155.61 88.675 17.1 1.11 4.00 454.86 164.04 58.366 17.8 1.07 4.00 479.01 169.88 85.817 18.3 1.05 4.00 495.37 174.47 93.838 20.2 1.04 4.00 543.47 189.97 106.159 20.2 1.04 4.00 544.56 190.32 111.2210 20.9 1.03 4.00 561.50 195.79 115.7011 22.7 1.01 4.80 636.74 210.86 101.8812 24.0 1.00 5.58 772.78 221.69 121.3313 26.6 1.00 7.20 1047.25 243.72 139.76

156 Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

ranging from σr ,min = 0.022 to σr ,max = 0.231. The prior knowledge ofμr and σr adopted in this example is consistent with the typical range ofthe undrained shear strength ratio of slightly over-consolidated clay, ofwhich the OCR ranges from 1 to 3 and has a coefficient of variation(COV) from 10 to 35% (e.g., Lacasse and Nadim, 1996; Baecher andChristian, 2003). As shown in Fig. 3(a), the OCR value in the clay layerat the site varies from 1 to 1.55, and the clay at the site is normally con-solidated or slightly over-consolidated. Using the OCR values shown inFig. 3(a), the COV of OCR is calculated as about 14%, which falls withinthe range from 10 to 35%.

Using the prior knowledge and the 7 OCR values, 7 SPT N60 values,and 6 CPT q values shown in Fig. 3(a)-(c), MCMCS is performed foreach updating level to generate 30,000 equivalent samples of r for itsprobabilistic characterization, respectively. For the sake of brevity, thispaper only shows 30,000 equivalent samples of r generated in updatinglevel III, where all the three types of test results are taken into account,and performs the conventional statistical analyses on these equivalentsamples to probabilistically characterize r and the Su ,D profile, asdiscussed in the next subsection.

(a) OCR (b) N60 (c)

10

12

14

16

18

20

22

24

26

28

Dep

th (

m)

OCR

10

13

16

19

22

25

28

Dep

th (

m)

N60

10

13

16

19

22

25

28

Dep

th (

m)

q(kP1 1.5 2 2.5 3 1 4 7 10 0 400

Fig. 3. In-situ and laboratory test data obtained from a clay site in Taiwan (including over-consotical effective stress σv0,D' , and undrained shear strength Su,D) (After Ching et al., 2010).

6.1. Probabilistic characterization of undrained shear strength

Fig. 4 shows a scatter plot for the 30,000 equivalent samples of r inupdating level III. Based on these 30,000 r samples, a histogram of r isconstructed, as shown in Fig. 5, and the 95% confidence interval(i.e., the range from 2.5% quantile to 97.5% quantile) of r is around[0.30, 0.56]. Fig. 5 also shows the respective values (i.e., μr

� = 0.42and σ r� = 0.066) of μr and σr estimated from the 30,000 equivalentsamples. In addition, Fig. 6 shows the PDF of r estimated from the histo-gram (see Fig. 5) of equivalent samples by a dashed line with open tri-angles. The statistics and PDF of r shown in Figs. 5 and 6probabilistically characterize r in the clay layer based on prior knowl-edge and site-specific test data (including OCR, SPTN60, and CPT q data).

For a given depth D, 30,000 equivalent samples of undrained shearstrength Su ,D (e.g., those shown in Fig. 7 for the depth of 20.2m) are cal-culated using the 30,000 equivalent samples of r (see Fig. 4) and the ver-tical effective stress at the depth (see Fig. 3(d)). Based on theseequivalent samples of Su ,D, the statistics (e.g., the mean μSu,D, standarddeviation σSu,D, and 2.5% and 97.5% quantiles) and PDF of Su ,D are esti-mated by conventional statistical analyses, such as those shown in Fig.8 for depths of 11.3 m, 20.2 m and 26.6 m. This leads to probabilisticcharacterization (e.g., mean, standard deviation, and PDF) of the Su ,Dprofile over the depth at the site.

Fig. 9 shows a profile of themeanvalue (i.e., μSu,D) of Su,D versus depthby a solid line with open squares. The μSu,D value increases linearly withthe depth, which are consistentwith the observation obtained from Su,Dvalues shown in Fig. 3(e). Fig. 9 also plots the 2.5% and 97.5% quantilesversus depth by red solid lines, both of which linearly increase withthe depth. The linear variation of the Su,D statistics with depth is mainlyattributed to the effects of σv0,D' on Su ,D, which are well-recognized insoil mechanics (e.g., Terzaghi et al., 1996; Mitchell and Soga, 2005).Such effects ofσv0,D' on Su ,D are incorporated into probabilisticmodelingof undrained shear strength in this study. As indicated by Eqs. (1) and(2), r is modeled by a random variable with spatially constanthyperparameters μr and σr in a soil layer, which forces the mean andstandard deviation of Su,D to linearly increase with depth. However, itis worthwhile to point out that the Su,D statistics may not increasewith depth in a perfectly linear fashion in engineering practice. As thedepth increases from 11.3 m to 26.6 m, the 95% confidence interval

q (d) 'v0 (e) Su,D

a)

10

13

16

19

22

25

28

Dep

th (

m)

'v0,D (kPa)

20.84+8.38D

10

13

16

19

22

25

28

800 1200 100 150 200 250 0 50 100 150

Dep

th (

m)

Su,D(kPa)

5.34D-5.59R2=0.802

σσ

σ

lidation ratioOCR, standard penetration testN60 value, corrected cone tip resistance q, ver-

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5000 10000 15000 20000 25000 30000

Und

rain

ed s

hear

str

engt

h ra

tio,

r

Number of Markov Chain step, m

Fig. 4. Scatter plot of undrained shear strength ratio r.

0

2

4

6

8

10

0.2 0.3 0.4 0.5 0.6 0.7 0.8

PD

F o

f un

drai

ned

shea

r st

reng

th r

atio

, r

Undrained shear strength ratio, r

Fig. 6. Probability density function (PDF) of undrained shear strength ratio r.

157Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

(i.e., the range from 2.5% quantile to 97.5% quantile) of Su,D becomeswider. This indicates that the variability in Su,D gradually increaseswith the depth. Similar observations on Su,D statistics (e.g., meanvalue, quantiles, and confidence intervals) were also reported in previ-ous studies for clays (e.g., Lumb, 1966; Jaksa, 1995; Cao and Wang,2014).

6.2. Result comparison

For comparison, Fig. 9 also shows the mean values and the 95% con-fidence intervals of Su,D at different depths reported by Ching et al.(2010) using error bars. The solid squares indicate the mean value ofSu,D calculated using the correlations developed by Ching et al. (2010).The 95% confidence interval of Su,D shown by the error bar is determinedas the range of the mean value plus or minus 2 times of standard devi-ation (Ching et al., 2010). In Fig. 9(a), the mean value and standard de-viation of Su,D at the depths from 11.3m to 17.8 m are estimated using qvalues (i.e., see solid circles in Fig. 3(c)) and Eq. (24) in Ching et al.(2010), and those at the depths below 17.8 m are obtained using OCRand N60 values (i.e., see solid circles in Fig. 3(a) and (b)) and Eq. (25)in Ching et al. (2010). As shown in Fig. 9(a), only using q values to cal-culate μSu,D might lead to significant underestimation of μSu,D (see solidsquares above 17.8 m) in this example. This does not mean that thereis a fundamental problem in Eq. (24) given by Ching et al. (2010), butis attributed to the fact that the q values shown in Fig. 3(c) are less

0

2000

4000

6000

8000

0.20 0.30 0.40 0.50 0.60 0.70 0.80

Fre

quen

cy

Undrained shear strength ratio, r

2.5% Quantile= 0.30

95% confidence interval

97.5% Quantile= 0.56

r* = 0.42

r* = 0.066

Fig. 5. Histogram of undrained shear strength ratio r.

than the actual cone tip resistance corrected by pore water pressure be-hind the cone, which is further discussed later in this subsection. Suchunderestimation subsequently results in an unusual profile of μSu ,D,which decreases first as depth increases to about 17.8 m and then in-creases when the depth is greater than 17.8 m. This is inconsistentwith the observation from Su,D values shown in Fig. 3(e). On the otherhand, the proposed BSU approach provides a linear profile of μSu,D, asshown by the solid line with open squares in Fig. 9. The BSU approachreduces negative effects of the low quality CPT q data obtained at depthsabove 17.8 m by incorporating OCR and SPT N60 data measured atdepths below 17.8 m, which provide relatively consistent informationon the site.

Although OCR and N60 data obtained above 17.8 m are available inthis study (see open circles in Fig. 3(a) and (b)), they are intentionallynot used in this example to illustrate how the proposed BSU approachdeals with different types of test results measured at different locations.It is worth noting that Eq. (28) in Ching et al. (2010) can also simulta-neously account for different types of test results obtained at the samedepth. Using Eq. (28) in Ching et al. (2010), the mean value and stan-dard deviation of Su,D at the depths from 11.3 m to 17.8 m are re-evaluated based on OCR, N60, and q values, yielding updated 95% confi-dence intervals of Su,D at these depths, as shown in Fig. 9(b). Comparedwith that only using q values, the bias in the estimate of μSu ,D at thedepths from 11.3 m to 17.8 m is generally reduced by incorporatingOCR andN60 values using Eq. (28) in Ching et al. (2010). Such reductionis attributed to incorporating multiple types of test results at the same

40

70

100

130

160

190

0 5000 10000 15000 20000 25000 30000

Und

rain

ed s

hear

str

engt

h, S

u,20

.2 (

kPa)

Number of Markov Chain step, m

Depth = 20.2m

Fig. 7. Scatter plot of undrained shear strength at the depth of 20.2 m.

0

0.02

0.04

0.06

0.08

0.0 50.0 100.0 150.0 200.0

PD

F o

f un

drai

ned

shea

r st

reng

th, S

u,D

Undrained shear strength, Su,D (kPa)

11.3m20.2m26.6m

D (m) 11.3 20.2 26.6

Su,D 48 79 101(kPa)

Su,D (kPa)

quantiles(kPa)

8 13 16

2.5% 34 56 72

97.5% 65 106 136quantiles (kPa)

μ

σ

Fig. 8. Statistics and probability density function (PDF) of undrained shear strength at thedepths of 11.3, 20.2 and 26.6 m.

(a) Comparison with results from Eqs. (24) and(25) in Ching et al. (2010)

(b) Comparison with results from Eqs. (25) and (28) in Ching et al. (2010)

10

13

16

19

22

25

28

0 50 100 150 200

Dep

th (

m)

Statistics of Undrained Shear Strength (kPa)

Mean (This study)

95% Confidence Interval(This study)95% Confidence Interval(Ching et al., 2010)

2.5% Quantile

97.5% Quantile

10

13

16

19

22

25

28

0 50 100 150 200

Dep

th (

m)

Statistics of Undrained Shear Strength (kPa)

Mean (This study)

95% Confidence Interval(This study)95% Confidence Interval(Ching et al., 2010)

2.5% Quantile

97.5% Quantile

Fig. 9. Result comparison.

158 Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

depth. In contrast, the bias reduction is achieved bymaking use of mul-tiple types of test results at different locations (e.g., different depths inthis study) in the proposed BSU approach.

As shown in Fig. 9, the 95% confidence interval of Su,D obtained fromthe BSU approach is narrower than that calculated using the correla-tions developed by Ching et al. (2010). In the proposed BSU approach,hyperparameters μr and σr of r are spatial constants in a soil layer andare updated using combined information from test data obtained at dif-ferent depths. This potentially reduces uncertainties in posterior esti-mates of the Su,D profile. In contrast, multivariate correlations in Chinget al. (2010) were derived by performing Bayesian analysis on the PDFof Su,D, rather than on the PDF of the hyperparameters. These multivar-iate correlations are able to utilizemultiple types of test data obtained atthe same location, or in a close proximity, to estimate the mean andstandard deviation of Su,D at the location. However, information fromdifferent test locations cannot be combined into the estimation usingthe correlations because the Su,D statistics at different locations are esti-mated separately using their corresponding test data. This means thatthe 95% confidence interval of Su,D at a given depth D estimated fromthe correlations only depends on test resultsmeasured at that particulardepth D. On the other hand, the 95% confidence interval of Su,D obtainedfrom the BSU approach relies on all the test data (e.g., OCR, SPT N60, andCPT q data) obtained in the soil layer. In addition, the BSU approach in-tegrates the site-specific test data with prior knowledge and allowssound engineering experience and judgment to be incorporated in theanalysis. It is, hence, not surprising to see that the proposed BSU ap-proach gives a narrower confidence interval. The proposed BSU ap-proach provides rational probabilistic characterization of the Su,Dprofile by integrating both multiple types of test results, which may beobtained at different locations and have different quantities, with priorknowledge. Such probabilistic characterization can be directly used assite-specific statistical estimates of Su in probability-based analysesand designs of geotechnical structures (e.g., Li et al., 2014, 2015, 2016).

6.3. Mean and standard deviation of r in updating levels I–III

Fig. 10 shows the estimates (i.e., μr� and σ r�) of mean and standard

deviation, which are calculated from MCMCS samples generated inupdating levels I–III, by solid lines with triangles and circles, respective-ly. When only OCR data is used in updating level I, the values of μr

� andσ r� are 0.44 and 0.088, respectively. The COV of r is then calculated as0.2. As SPT N60 is taken into account in updating level II, μr

� slightly in-creases to 0.46, andσ r� decreases to 0.063. The updated COV of r is thencalculated as around 0.135, which is smaller than the COV (i.e., 0.2) of rin updating level I. Incorporating SPT N60 data leads to uncertainty re-duction in this example. This suggests that the information (includingOCR data, SPT N60 data, and transformation models used to interpretthem) adopted in updating levels I and II is consistent. In updatinglevel III, the values of μr

� and σ r� are further updated to be 0.42 and0.066 by incorporating CPT q data. The resulting COV of r is equal to0.16 in updating level III. As CPT q data is taken into account, the COVof r slightly increases. This indicates that the information used in theupdating level III (including CPT q data and its corresponding transfor-mation model) is inconsistent with that used in the first two updatinglevels, which is true in this example because CPT q values used in thisexample are underestimated. Note that there is no guarantee that theuncertainty is reduced by incorporating more types of test results. Theuncertainty might increase or decrease, depending on the quality of in-formation from newly incorporated test data and the transformationmodel (e.g., Eqs. (8)-(10)) used to interpret the data. The proposed ap-proach not only provides a rational vehicle to systemically combine in-formation from multiple types of site-specific test results forprobabilistic characterization of geotechnical parameters, but also offersadditional insights into evolution of the statistics (e.g., mean and stan-dard deviation) of geotechnical parameters as more different types of

0.06

0.08

0.10

0.12

0.22

0.33

0.44

0.55

0.66

Est

imat

ed s

tand

ard

devi

atio

n of

und

rain

edsh

ear

stre

ngth

rat

io, σσ

r*

Est

imat

ed m

ean

of u

ndra

ined

she

arst

reng

th r

atio

, μr*

Updating Level

Estimated mean value of r

Estimated standard deviation of r

I IIIII

Fig. 10. Evolution of mean and standard deviation of undrained shear strength ratio r.

159Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

test results are taken into account. Such insightsmay assist geotechnicalpractitioners in inspecting thequality of information fromdifferent test-ing procedures (including test results and their corresponding transfor-mationmodels) and identifying testing procedures providing consistentinformation, which leads to reduction of uncertainty.

6.4. Effects of different updating sequences

Although this study uses OCR, SPT N60, and CPT q data as DATAI,DATAII, and DATAIII in updating levels I–III to update the knowledge onthe mean (i.e., μr) and standard deviation (i.e., σr) of r, respectively,the three types of test results can be incorporated into the proposedBSU framework in any other order. In this example, there are six differ-ent updating sequences, denoted by S1-S6, as shown in Table 2. To ex-plore effects of the updating sequence on the performance of theproposed approach, this section applies the proposed BSU approach toestimate μr and σr using all six different updating sequences. For agiven updating sequence, the estimates (i.e., μr� and σ r�) of μr and σrare calculated in each updating level using the proposed BSU approach.

Fig. 11(a) and (b) show the evolution ofμr� andσ r� in three updating

levels for each updating sequence, respectively. As shown in Fig. 11(a),the values (see the dashed and solid lineswith circles) ofμr� in updatinglevel I of S5 and S6, where CPT q data is used, is considerably less thanthose obtained in updating level I of S1-S4, where OCR and SPT N60data are used. This is, again, attributed to the underestimation of qvalues at the site. As OCR and SPT N60 data are incorporated into S5and S6 in updating levels II and III, the values of μr� in the two updatingsequences increase, and the bias on μr

� resulted from underestimationof CPT q data is reduced by considering information from OCR and SPTN60 data. Finally, the values of μr� obtained using all six updating se-quences converge to the same value (i.e., 0.42) in updating level III.

Table 2Summary of data used in different updating levels of six updating sequences.

Sequence ID

Data used in different updating levels

DATAI DATAII DATAIII

S1 OCR SPT N60 CPT qS2 OCR CPT q SPT N60S3 SPT N60 OCR CPT qS4 SPT N60 CPT q OCRS5 CPT q OCR SPT N60S6 CPT q SPT N60 OCR

This is intuitively reasonable because the information used in the finalupdating level (i.e., the information from PRIORI, OCR data, SPT N60data, CPT q data, and transformation models) is the same regardless ofthe updating sequence. Similarly, the values of σ r� converge to thesame value (i.e., 0.066) for all six updating sequences, as shown in Fig.11(b). The final value of σ r� is smaller than those (i.e., 0.088, 0.106,and 0.114, respectively) obtained only using OCR, SPT N60, or CPT qdata in updating level I of S1-S6. Compared with using a single type oftest results, incorporating all the three types of test results into theBSU framework leads to reduction of uncertainty in this example al-though such uncertainty reduction is not guaranteed, as discussedpreviously.

Moreover, it is also worth pointing out that the difference betweenthe statistics (i.e., μr� and σ r�) of r obtained using a single type of testresults (e.g., OCR data for S1) in updating level I and those estimatedusing all three types of test results in updating level III indicates how sig-nificantly the probability distribution of r is updated by taking into ac-count two more types of test data (e.g., SPT N60, and CPT q data forS1). If the difference is relatively small, the probability distribution of robtained using a single type of test results in updating level I is closeto the final probability distribution of r considering all the three typesof test results. This means that the test results used in the updatinglevel I dominate the probabilistic characterization of r, and are, hence,relatively informative and valuable. Based on this rational, the most in-formative test results can be identified by comparing the statistics andprobability distributions obtained using one type of test results withthose obtained using all the three types of test results. For example,when compared with only using SPT N60 data (see updating level I ofS3 and S4) or CPT q data (see updating level I of S5 and S6), the differ-ence between the statistics (i.e., μr� and σ r�) of r from only using OCRdata (see updating level I of S1and S2) and those obtained using allthe three types of test results is relatively small, indicating that theOCR data used in this example are more informative than SPT N60 andCPT q data. In other words, the OCR data are the most informative andvaluable information source among the three types of test results usedin this example.

7. Summary and conclusions

This paper developed a Bayesian sequential updating (BSU) frame-work for probabilistic characterization of geotechnical parameters byintegrating information from multiple sources, including prior knowl-edge that is available prior to the project and different types of site-

(a) Effects on the estimated mean of r

(b) Effects on the estimated standard deviation of r

0.22

0.33

0.44

0.55

0.66

Est

imat

ed m

ean

of u

ndra

ined

shea

r st

reng

thra

tio, μ μ

r*

Updating Level

S1: OCR data-SPT data-CPT dataS2: OCR data-CPT data-SPT dataS3: SPT data-OCR data-CPT dataS4: SPT data-CPT data-OCR dataS5: CPT data-OCR data-SPT dataS6: CPT data-SPT data-OCR data

I II III

0.00

0.06

0.12

0.17

0.23

Est

imat

ed st

anda

rd d

evia

tion

of u

ndra

ined

shea

r st

reng

th r

atio

, σr*

Updating Level

S1: OCR data-SPT data-CPT dataS2: OCR data-CPT data-SPT dataS3: SPT data-OCR data-CPT dataS4: SPT data-CPT data-OCR dataS5: CPT data-OCR data-SPT dataS6: CPT data-SPT data-OCR data

III III

Fig. 11. Effects of the updating sequence.

160 Z.-J. Cao et al. / Engineering Geology 211 (2016) 150–161

specific test results, which might be obtained at different locations in asoil layer. For illustration, the proposed BSU framework was formulatedto probabilistically characterize the undrained shear strength Su of clayusing OCR, SPT, and CPT data. The information provided by OCR, SPT,and CPT data is incorporated into the proposed Bayesian framework ina sequential manner and is systematically combined with the priorknowledge. The combined information is then transformed into alarge number of equivalent samples of the Su profile using MarkovChain Monte Carlo simulation (MCMCS) for determination of its statis-tics (e.g., mean, standard deviation, and quantiles) and probability dis-tributions (e.g., PDF) at a particular site, providing necessary input forprobability-based geotechnical analyses and designs at the site.

Equationswere derived for the proposed BSU approach, and the pro-posed approach was illustrated using real-life data at a clay site inTaiwan. The results showed that the proposed approach rationallycombines prior knowledge with the information from different typesof site-specific test results obtained at different depths to probabilisti-cally characterize Su at the site. It is also found that although incorporat-ing multiple types of test results might reduce the bias and uncertaintyin probabilistic characterization of geotechnical parameters, this is notguaranteed but depends on the information from newly incorporateddata and the transformation model used to interpret the data. The pro-posed BSU approach not only provides a rational vehicle to systemicallyaccumulate information from different types of test results obtainedduring geotechnical site investigation in a sequential manner, but also

offers additional insights into evolution of the statistics of geotechnicalparameters asmore types of test results are taken into account. Such in-sights may assist geotechnical practitioners in inspecting the quality ofinformation from different testing procedures and identifying the onesproviding consistent test results, which lead to reduction of uncertainty.

Effects of the updating sequence were also explored. It is found thatthe updating sequence has no effect on the final results of probabilisticcharacterization of geotechnical parameters when the same informa-tion sources (e.g., prior knowledge, test results, and transformationmodels) are used in different updating sequences. Moreover, the mostinformative test results can be identified by comparing the statistics(e.g., mean and standard deviation) obtained using a single type oftest results with those estimated from multiple types of test results.The smaller the difference between the two sets of results, the more in-formative the single type of test results (e.g., OCR data in the illustrativeexample) is. This information is valuable for interpretation of site inves-tigation data and planning of site investigation, such as choosing testingprocedures during site investigation.

Acknowledgments

The authors would like to thank the reviewers for their constructiveand valuable comments, which helped to improve the quality of thepaper. The work described in this paper was supported by grants fromthe Research Grants Council of the Hong Kong Special AdministrativeRegion, China (Project No. CityU 11200115 and Project No. T22-603/15N), National Natural Science Foundation of China (Project Nos.51409196, 51579190, 51528901), National Science Fund for Distin-guished Young Scholars (Project No. 51225903), and an open fundfrom State Key Laboratory Hydraulics and Mountain River Engineering,Sichuan University (Project No. SKHL1318). The financial supports aregratefully acknowledged.

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Site-specific characterization of soil properties using multiple measurements from different test procedures at different ...1. Introduction2. Bayesian sequential updating (BSU) framework for geotechnical site investigation2.1. Probabilistic modeling of undrain shear strength2.2. BSU of model parameters of undrained shear strength ratio

3. Prior distribution4. Likelihood functions5. Probabilistic characterization of undrained shear strength using equivalent samples6. Illustrative example6.1. Probabilistic characterization of undrained shear strength6.2. Result comparison6.3. Mean and standard deviation of r in updating levels I–III6.4. Effects of different updating sequences

7. Summary and conclusionsAcknowledgmentsReferences