bayesianapproachforsequentialprobabilisticbackanalysisof...

Research ArticleBayesian Approach for Sequential Probabilistic Back Analysis ofUncertain Geomechanical Parameters and Reliability Updating ofTunneling-Induced Ground Settlements

Cong Li12 Shui-Hua Jiang 13 Jinhui Li4 and Jinsong Huang1

1School of Civil Engineering and Architecture Nanchang University 999 Xuefu Road Nanchang 330031 China2School of Civil Engineering and Architecture Wuhan Polytechnic University Wuhan 430023 Hubei Province China3State Key Laboratory of Geomechanics and Geotechnical Engineering Institute of Rock and Soil MechanicsChinese Academy of Sciences Wuhan 430071 China4Department of Civil and Environmental Engineering Harbin Institute of Technology (Shenzhen) Shenzhen 518055 China

Correspondence should be addressed to Shui-Hua Jiang sjiangaancueducn

Received 1 September 2019 Revised 20 May 2020 Accepted 25 May 2020 Published 24 June 2020

Academic Editor Paolo Castaldo

Copyright copy 2020 Cong Li et alis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper proposes a new sequential probabilistic back analysis approach for probabilistically determining the uncertaingeomechanical parameters of shield tunnels by using time-series monitoring datae approach is proposed based on the recentlydeveloped Bayesian updating with subset simulation Within the framework of the proposed approach a complex Bayesian backanalysis problem is transformed into an equivalent structural reliability problem based on subset simulation Hermite polynomialchaos expansion-based surrogate models are constructed to improve the computational efficiency of probabilistic back analysise reliability of tunneling-induced ground settlements is updated in the process of sequential back analyses A real shield tunnelproject of No 1 Nanchang Metro Line in China is investigated to assess the effectiveness of the approach e proposed approachis able to infer the posterior distributions of uncertain geomechanical parameters (ie Youngrsquos moduli of surrounding soil layersand ground vehicle load) e reliability of tunneling-induced ground settlements can be updated in a real-time manner by fullyutilizing the time-series monitoring data e results show good agreement with the variation trend of field monitoring data ofground settlement and the post-event investigations

1 Introduction

In order to alleviate the pressure of urban traffic con-structing underground rail transit network has become amain development of urban transportation system Urbanmetro shield tunnel construction often leads to surroundingground movements and further endangers adjacent struc-tures which in turn poses great threat to the tunnel itself(eg [1 2]) To mitigate the adverse impact of shield tunnelconstruction on the surrounding environment the geo-mechanical parameters that can reflect the performance of aspecific construction site and help to analyze the shieldtunneling-induced ground subsidence shall be rationallydetermined (eg [3 4]) Typically only limited amount offield observation data (including geological survey data test

data and monitoring data) can be acquired for a specificconstruction site due to the restrictions on the constructioncost time and project sites (eg [5 6]) erefore how toobtain the geomechanical parameters of the shield tunnelswith low uncertainty efficiently based on the limited fieldobservation data remains an unsolved problem

To make up for the deficiency of field observation datamany researches estimated the geomechanical parameters ofshield tunnels through back analysis For example Chi et al[7] applied an optimization technique for the back analysisof tunneling-induced ground movement based on moni-toring data Zhu et al [8] proposed an artificial bee colonyalgorithm-based displacement back analysis method todetermine the subsoil parameters during shield tunnelingprocess Gao and Ge [4] employed evolutionary neural

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8528304 13 pageshttpsdoiorg10115520208528304

networks to back analyze the soil parameters and the initialgeo-stress of Longtan tunnel and evaluate the stability of thetunnel Although significant advances have been made in theback analysis of shield tunnel parameters the above-men-tioned studies do not account for the influence of the in-herent variability of geomechanical parameters (eg [4 9])It is widely accepted that the geomechanical parameters varyspatially and temporally because of the concealment ofunderground space and complexity of underground envi-ronment (eg [10ndash16]) To obtain the shield tunnel pa-rameters in line with the engineering practice theuncertainties of the geomechanical parameters should betaken into account in the back analysis

Several studies have investigated probabilistic backanalysis problems of uncertain parameters for the shieldtunnels by combining probability theory and statistics Forexample Haas and Einstein [17] employed a Markov chainMonte Carlo (MCMC) method to update the posteriordistributions of surrounding rock mass parameters oftunnels based on the monitoring data Spackova and Straub[18] proposed a dynamic Bayesian network based-tunnelingprocess model to update the probability of tunnel failure byutilizing the observation data from geological survey andconstruction stages Park and Park [19] conducted aprobabilistic back analysis of uncertain parameters fortunnel surrounding rock masses by adopting a responsesurface method Liu et al [20] also employed the MCMCmethod for the probabilistic back analysis of subsoil co-hesion and internal friction angle around a tunnel belowriver Although many investigations on the probabilisticback analysis of shield tunnel parameters have been carriedout the time-series monitoring data that are collectedduring the tunnel advancement were rarely incorporated inthe back analysis To the best of our knowledge only Miroet al [21] applied a MCMCmethod in the probabilistic backanalysis of uncertain parameters for surrounding rockmasses by utilizing the tunnel-induced ground movementsin a sequential manner However the MCMC methodadopted by Miro et al [21] is not effective for high di-mensional problems and the reliability of shield tunnelswith the probabilistic back analysis results was not inves-tigated e updated statistics (ie means standard devia-tions probability distributions) of geomechanicalparameters via the probabilistic back analyses can provide animportant basis for reliability analysis of shield tunnelsBased on the updated probability distributions of the un-certain parameters more realistic reliability assessment ofthe tunnel face stability or serviceability limit state can beachieved

In this paper a Bayesian Updating with Subset simu-lation (BUS)-based sequential probabilistic back analysisapproach is proposed for estimating uncertain geo-mechanical parameters and updating the reliability oftunneling-induced ground settlements To improve com-putational efficiency the proposed approach transforms acomplex Bayesian back analysis problem into an equivalentstructural reliability problem which is then solved by subsetsimulation e Hermite polynomial chaos expansion-basedsurrogate models are constructed In Section 2 the proposed

approach comprising the construction of surrogate modelsand likelihood functions inference of posterior distributionusing BUS approach and reliability updating of tunneling-induced ground settlements is presented In Section 3 theNo 1 NanchangMetro Line tunnel is treated as a typical casefor probabilistically determining the uncertain geo-mechanical parameters based on the time-series monitoringdata of ground settlement Finally the reliability of tun-neling-induced ground settlements is updated on the basis ofthe probabilistic back analysis results

2 BUS-Based Sequential Probabilistic BackAnalysis Approach

Shield tunneling is often fraught with inevitable uncer-tainties that include but not limited to (1) the inherentvariability of soils (2) themeasurement errors in quantifyingthe soil properties by field and laboratory tests and (3) thelimited knowledge on geotechnical conditions and simpli-fied geotechnical models in predicting the ground settle-ments (eg [1 21]) ese uncertainties can be reduced byincorporating the field observation data into a Bayesian backanalysis by estimating the posterior joint probability densityfunction (PDF) of geomechanical parameters (eg [9 22])In this study a BUS-based sequential probabilistic backanalysis approach is proposed to estimate the posterior PDFand its implementation procedure is briefly introduced asfollows

(1) Determine the prior statistics (ie mean coefficientof variation (COV) and distribution) based on theknowledge (eg geological survey reports engi-neering experience and existing data in the litera-ture) about the uncertain input parameters

(2) Construct Hermite polynomial chaos expansion(HPCE)-based surrogate models to facilitate thecalculation of tunneling-induced ground settlementsfor each excavation step

(3) Establish the likelihood function using the moni-toring data of ground settlement obtained from thefirst excavation step

(4) Infer the posterior PDFs of uncertain input pa-rameters using the BUS approach en treat theposterior PDFs as the prior PDFs and build a newlikelihood function using the monitoring data ofground settlement obtained from the next excavationstep until the last regarded excavation stage

(5) Update the reliability of tunneling-induced groundsettlements for each excavation step based on theobtained posterior statistics of uncertain inputparameters

e flowchart of the BUS-based sequential probabilisticback analysis approach is presented in Figure 1e involvedconstruction of surrogate models and likelihood functionsinference of posterior distribution using the BUS approachand reliability updating of tunneling-induced ground set-tlements are presented in the following subsections

2 Advances in Civil Engineering

21 Construction of Surrogate Models and LikelihoodFunctions e ground settlements due to shield tunnelingand the corresponding limit state functions are commonlythe nonlinear implicit functions of the geomechanical pa-rameters which need to be evaluated through deterministicfinite element or finite difference analyses Generally tens ofthousands of finite element or finite difference analyses arerequired in a Bayesian back analysis [23] which is com-putationally demanding for a complicated shield tunnelmodel To avoid the high computational burden in theprobabilistic back analysis the polynomial chaos expansionor multivariate adaptive regression splines-based responsesurface method can be adopted to construct the surrogatemodels of the ground settlements (eg [24 25]) e re-sponse surface method can provide a powerful tool to ap-proximate the nonlinear implicit limit state functions andhas many successful applications (eg [26 27]) is studyapplies the HPCE to construct the surrogates of deter-ministic numerical models for facilitating the calculation ofground settlements for each excavation step A surrogatemodel of the ground settlement uj at the jth excavation stepthat involves the uncertain input parameters can be con-structed as follows (eg [11 28 29])

uj a0Γ0 + 1113944n

i11ai1Γ1 ξi1

1113872 1113873 + 1113944n

i111113944

i1

i21ai1i2Γ2 ξi1

ξi21113872 1113873

+ 1113944n

i111113944

i1

i211113944

i2

i31ai1i2i3Γ3 ξi1

ξi2 ξi3

1113872 1113873 + middot middot middot

(1)

where j= 1 2 t t is the total number of excavation stepsn is the number of random variables a0 ai1

ai1 i2 ai1 i2 i3

are the unknown coefficients Γjp(middot) jp = 1 2 3 are

Hermit polynomials with jp degrees of freedom [30] andξ (ξ1 ξ2 ξn)T are a set of independent standardnormal random variables For the nHPCE-th order HPCEthere are a total of M (n + nHPCE)(n times nHPCE) un-known coefficients (ie a0 ai1

ai1 i2 ai1i2 i3

) in equation(1) e unknown coefficients in the HPCE can be evaluatedby solving a series of linear equations given by equation (1)e left side of the linear equations is uj evaluated viadeterministic finite element or finite difference analysisbased on N realizations of the random variables that aregenerated by Latin Hypercube Sampling (LHS) technique(eg [31] in which NgeM)

Having obtained the explicit and approximate expres-sions for calculating the ground settlements at differentshield excavation steps in equation (1) the likelihoodfunctions with the consideration of the uncertainties can beconstructed e likelihood function reflects the model fitwith the field observation data for given input parametersX (X1 X2 Xn)T Typically the difference (ie εmj

measurement error) between a measured and a simu-lated ground settlement for the given values x of randomvariables X can be expressed as

εmj u

mj minus uj(x) (2)

where umj is the measurement of the ground settlement

which is made at the jth excavation step and uj(x) is

Prior distribution 1 and likelihood function L1


Prior distribution n and likelihood function Lt

Prior knowledge 1(prior 1)

Posterior knowledge 1(posterior 1)

Framework of sequential probabilistic back analysis and reliability updating

Prior 2

Prior n

Posterior t

Monitoring data u1m

Surrogate model of u1

MCS or SSBUS operation

Posterior 2MCS or SSPosterior distribution 2

Posterior distribution t

BUS operation

Posterior probabilityof failure Pf1

Monitoring data u2m



Monitoring data utm

Surrogate model of ut

Posterior probabilityof failure Pft

MCS or SSPosterior distribution 1

BUS operation

1st excavation step

2nd excavation step

tth excavation step

g1 (x) = umax ndash u1 (x)


gt (x) = umax ndash ut (x)

Figure 1 Flowchart of BUS-based sequential probabilistic back analysis and reliability updating approach

Advances in Civil Engineering 3

evaluated through the deterministic analysis Note that amathematical transformation between x and ξ in equation(1) can be done by a Nataf transformation procedure [29]Following Miro et al [21] εmj

j= 1 2 t are assumed tobe independent and obey normal distributions with zeromean and constant standard deviations of σεmj

Based onthese the likelihood functions at the jth excavation step canbe established as follows [32]

Lj(x) ϕ um

j minus uj(x)1113960 1113961σεmj1113882 1113883

σεmj

(3)

where ϕ(middot) is the PDF of a standard normal variable Withthe constructed surrogate models using equation (1) thecomputational cost taken on the evaluations of likelihoodfunctions can be substantially reduced and so are the totalcomputational costs of the sequential probabilistic backanalysis

22 Inference of Posterior Distribution BUS approach isadopted herein to infer the posterior distributions of the un-certain input parameters which defines the Bayesian backanalysis problem as an equivalent structural reliability problem[23] Subset simulation (SS) is then employed to solve thestructural reliability problem to obtain samples from fXPrime(x)

(eg [33 34]) e field observation information collected atthe jth excavation step can be described by a likelihood functionLj(x) which is utilized to define a failure domain ΩX in anaugmented outcome space x+= [x p]

ΩX Hj x+( 1113857le 01113966 1113967 (4)

where Hj(x+) is the limit state function which is given by[35]

Hj x+( 1113857 lnp minus ln cLj(x)1113960 1113961 (5)

where p is the realization of a standard uniform randomvariable in [0 1] that is independent with x and c is alikelihood multiplier that satisfies the following inequalityfor all x [23]

cL(x)le 10 (6)

It can be noted that sampling the posterior distributionof x is equivalent to finding the samples generated from theprior distribution of X and falling in the domain ΩX whendetermining the probability of information event Z P(Z)

[36 37] As the subset simulation does the BUS approachcan also express the P(Z) as a product of larger conditionalprobabilities of a series of nested intermediate events

P(Z) P Hj x+( 1113857le 01113960 1113961 P Z1( 1113857 1113945

m

i2Pr Zi

1113868111386811138681113868Ziminus11113872 1113873 (7)

where Z1Z2Zmminus1

Zm

are intermediate events defined as Zi=H(x+)ltgi in which gi i=1 2 m are threshold valuessatisfying g1 gtg2 gt gtgmminus1 gt 0gegm m is the number ofsubset levels required to reach the domain ΩX P(Z1) is the

probability of the first subset level and P(Zi|Ziminus1) is theconditional probability ofZi givenZi minus 1 Specifically amodifiedMetropolis algorithm proposed by Au and Beck [33] is adoptedfor the computation of the conditional probabilities ofP(Zi|Ziminus1) i=2 3 m e threshold values gi i=1 2 m are determined adaptively such that the intermediateconditional probabilities take a target value p0

Once the failure regionΩX is reached the failure samplesxf in the final subset level are extracted and utilized toestimate the posterior statistics of the uncertain input pa-rameters and compute P(Z) e computational effort of theBUS approach decreases significantly with the logarithm ofP(Z) which in turn is proportional to the value of theconstant c To this end the value of c is usually selected aslarge as possible such that equation (6) holds Following Betzet al [35] and Jiang et al [36 37] c is adaptively estimated asthe reciprocal of the maximum of the likelihood functionvalues over the samples at the current subset level ie

ci 1

max cminus1iminus1 L xik1113872 1113873 k 1 2 Nl1113966 11139671113960 1113961

(i 1 2 m)

(8)

where Nl is the number of samples at each subset levelc1ge c2ge ge cm used in different subset levels shouldguarantee that the intermediate failure domain Zi is entirelycontained in the Ziminus1 i= 2 3 m

23 Reliability Updating of Tunneling-Induced GroundSettlements Once the posterior statistics of the uncertaingeomechanical parameters are obtained via the probabilisticback analysis the reliability of tunneling-induced groundsettlements for each excavation step can be updated elimit state function expressing the maximal ground settle-ment exceeding an admissible threshold can be defined as

gj(x) umax minus uj(x) (9)

where umax is the admissible threshold of ground settlemente posterior probability (Pfj) of tunneling-inducedground collapse can be estimated using direct Monte Carlosimulation (MCS) as follows

Pfj P Fj

11138681113868111386811138681113868Z1113874 1113875 P FjcapZ1113872 1113873

P(Z)

1113936Nf

k1 I umax le uj xkf1113874 11138751113876 1113877

Nf

(10)

where F denotes the ground collapse event Fj x isin ΩFj1113882 1113883

in which ΩFj gj(x)le 01113966 1113967 Nf is the number of posterior

samples xf and I(middot) is the indicator function In most casesthe joint probability P(FjcapZ) is very small thus the estimateof Pfj using the direct MCS will become rather timeconsuming Alternatively Jiang et al [36] proposed toconduct a new SS operation following the BUS operation forcalculating Pfj Interested readers can refer to Jiang et al[36] for detailed procedures for estimating Pfj


3 Project Background

31 ShieldTunnelOverview e shield tunnel project of No1 Nanchang Metro Line is located in Jiangxi provinceChina It is a single-line and double-tunnel structure andabout 28 km in length e shield tunneling interval origi-nates from the Aixi lake west station passes through thefront square of China telecom of Nanchang branch andBeijing east road and finally arrives at the Gaoxin avenuestation e Beijing east road is the main traffic artery inNanchang city e ground traffic volume is large particu-larly on the holiday e underground pipelines are com-plex and thus the tunneling-induced ground subsidence isrelatively easy to occur e shield tunnel in this interval hasa diameter of D 60m and a buried depth of H 140mwhich is constructed with a single-circle shield machineFigure 2 shows the profile of soil layers surrounding theshield tunnel For the concerned tunnel interval the designof five excavation steps is listed in Table 1e total length ofshield tunnel excavation is 112 km

During the shield tunnel construction from the Gaoxinavenue station to the Aixi lake west station a ground col-lapse accident occurred at the 827th ring on October 2 2012Post-event investigations of this accident found the leakageof the underground water pipes in the silty clay layer and thevariation of subsoil property caused by the shield excavationdisturbance were the main reasons of the collapse of the827th ring e Beijing east road above the shield tunnel wasthe main traffic artery so the increase in the vehicle loadsduring the Chinese National day in 2012 was another maincause In addition the shield tunneling construction in-tensively occurred in the gravel layer (see Figure 2) emonitoring data of ground settlement obtained from thepoint Ds826 is utilized for sequential probabilistic backanalysis of uncertain geomechanical parameters and reli-ability updating Note that the point Ds826 is installed on the826th ring and close to the collapsed 827th ring Figure 3presents the monitoring data of ground settlement collectedfrom the point Ds826 at the five excavation steps

32 Numerical Model and Parameters e shield tunnelinginterval between the 821th and 845th rings is selected toestablish the numerical model using the finite differenceprogramme FLAC3D Figure 4 illustrates a three-dimen-sional (3D) finite difference model of the tunnel which has alength of 42m in the Y-axis direction a width of 30 in the X-axis direction and a depth of 35m in the Z-axis directionWith regard to the boundary conditions the normalmovements on the all sides of the 3D model are restrainedwhereas the bottom of the model is not allowed to move inthe three directions e excavation face of the model is freebut the nodes around the excavated tunnel have a fixed radialdisplacement To simulate the influence of the groundmoving vehicle loads on shield tunnel excavation a uniformvehicle load of q 10 kPa is applied to the top of the model inaccordance to Yang et al [38]

e commonly used elastic and perfectly plastic modelbased on the Mohr-Coulomb failure criterion is utilized to

represent the stress-strain behavior of the subsoil massese initial stress is generated by applying gravitational ac-celeration to the model e subsoil masses surrounding thetunnel are modeled using the cylinder elements while therest is modeled using the hexahedral elements e exca-vated tunnel is lined with a linear elastic material with aYoungrsquos modulus of 345GPa a Poissonrsquos ratio of 02 and adensity of 2450 kgm3 e lining segment is prefabricatedwith C50 concrete with a thickness of 03m and a width of12m which is modeled using a shell element A dis-cretization of the model with a total number of 34400 el-ements and 37154 nodes is adopted after a preliminarystudy of the influence of mesh size According to Mollonet al [3] the considered tunnel may result in large groundsettlements since it corresponds to a shallow tunnel with theoverburden depth being about 233 times the outer diameterFor simplicity the groundwater table is not considered inthis study

e geomechanical parameters of different soil layers aredetermined based on the geological survey reports [39] assummarized in Table 2 According to the post-event in-vestigations as mentioned in Section 31 Youngrsquos modulusE1 of the gravel layer Youngrsquos modulus E2 of the silty claylayer and ground vehicle load q that are closely related to theground collapse are identified as random variables after asimple parametric sensitivity studye prior information ofthe three random variables (ie E1 E2 and q) is determinedon the basis of the field observation data and existing data inthe literature (eg [28 38 40ndash42]) e prior statistics ofthree random variables are summarized in Table 3

33 Construction and Validation of Surrogate ModelsTypically 3D deterministic finite difference analysis oftunneling-induced ground settlements suffers from exces-sive computational effort To improve the computationalefficiency of the probabilistic back analysis the 4th orderHPCE-based surrogate models of ground settlements areconstructed for different excavation steps in advance Foreach excavation step the number of unknown coefficients ofthe 4th order HPCE is M 35 N 70 random samples aregenerated according to the prior statistics of three randomvariables by the LHS technique to establish the linearequations and determine the unknown coefficients eexpansion terms and the corresponding coefficients of the4th order HPCE for the 1st excavation step are listed inTable 4

To balance the computational accuracy and efficiency100 direct MCS random samples are generated to verify thesurrogate models Based on these 100 random samples theprobability distributions of the uncertain geomechanicalparameters and ground settlements can be inferred withacceptable accuracy Figures 5(a)ndash5(f) compare the tun-neling-induced ground settlements (ie u1 u2 and u5) andtheir PDFs for three representative excavation steps (ie 1 2and 5) determined from the 4th order HPCE-based surrogatemodels and original deterministic finite difference analysesusing these 100 random samples respectively As observedfrom Figure 5 the ground settlements and their PDFs


obtained from these two methods are in good agreement Itindicates the 4th order HPCE-based surrogate models canwell approximate the 3D numerical models and replace thedeterministic finite difference analyses to accurately calcu-late the uj at each excavation step in this example

34 Sequential Probabilistic Back Analysis Results In thissection the BUS approach is employed to infer the posteriordistributions of E1 E2 and q via the sequential probabilisticback analysis using the time-series monitoring data ofground settlement as shown in Figure 3 Based on the trade-off between the computational accuracy and efficiency thenumber of samples at each subset level Nl 5000 andconditional probability p0 01 are chosen Following Miroet al [21] the standard deviations of measurement errorsσεmj

20mm are used Figures 6ndash8 compare the posteriorPDFs of E1 E2 and q estimated from the five differentexcavation steps respectively e prior PDFs of E1 E2 and

q are also plotted in Figures 6ndash8 respectively for com-parison As observed from Figures 6ndash8 the posterior PDFcurves of E1 E2 and q get steeper and narrower as the shieldtunnel advances and are much more peaked than thecorresponding prior PDFse posterior means of E1 and E2become smaller and smaller while that of q becomes largerand larger as the shield tunnel progresses is is consistent

Plain fill

Silty clay

Fine sand

Gravel

Highly weathered silty mudstone

Moderately weathered silty mudstone

60m

54m

14m

1m5m

2m15

m2m

10m

1

2

3

4

5

6

Figure 2 Profile of soil layers surrounding the shield tunnel

Table 1 Excavation design for the concerned shield tunnelinginterval

Step no Excavation time Ring no1 2012930 1500 8362 2012101 0700 8383 2012101 1500 8394 2012102 0700 8405 2012102 1500 841

Mornitoring data from the point Ds826

ndash24

ndash22

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

Mor

nito

ring

data

of g

roun

d m

ovem

ents

(mm

)

2 3 4 51Excavation step

Figure 3 Time-series monitoring data of tunneling-inducedground settlement


with the post-event investigations and the common sensethat the increase in the ground deformation is usually causedby the reduction of soil stuffiness or the increase of externalloads Significant changes can be observed on the posteriorPDFs of E1 E2 and q when the time-series monitoring dataare sequentially incorporated in the probabilistic backanalysis is lies in the fact that the occurrence position ofground collapse is close to the 827th ring and the groundsettlement collected from the monitoring point Ds826sharply increases at the 5th excavation step (see Figure 3) Itindicates the proposed approach not only can make full useof the time-series monitoring data to effectively update thestatistics and reduce the uncertainties of geomechanicalparameters but also can well characterize the realisticchange trends of surrounding subsoil properties

Additionally the COVs of E1 E2 and q decreasesuccessively from the prior COVs as the monitoring dataare sequentially used in the probabilistic back analysis as

shown in Figure 9 e prior COVs of E1 E2 and q are015 015 and 01 respectively which are reduced to 01012 and 0085 at the 3rd excavation step and to 007 011and 008 at the 5th excavation step It is interesting to notethat the uncertainty of E1 is reduced the most whichimplies the gravel layer affects the ground subsidence themost e uncertainties of the geomechanical parametersassociated with the shield tunnel have been significantlyreduced through a Bayesian back analysis in a sequentialmanner

35 Reliability Updating Results of Ground SettlementsBased on the obtained posterior distributions of the un-certain geomechanical parameters for each excavation stepthe reliability of tunneling-induced ground settlements canbe updated using equations (9) and (10) An admissiblethreshold of ground settlement umax 30mm is selected for

XY

Z

Figure 4 3D finite difference model for the No 1 Nanchang Metro Line tunnel

Table 2 Geomechanical parameters for different soil layers

Soil layers Density (kgm3) Youngrsquos modulus (MPa) Poissonrsquos ratio Cohesion (kPa) Friction angle (deg)Plain fill 1813 15 042 5 10Silty clay 1933 15 035 447 193Fine sand 1913 16 04 1 30Gravel 1893 28 039 1 36Highly weathered silty mudstone 2050 120 03 60 37Moderately weathered silty mudstone 2390 450 039 120 32

Table 3 Prior statistics of three random variables

Random variable Mean (MPa) Standard deviation (MPa) COV DistributionYoungrsquos modulus of gravel layer E1 28 42 015 LognormalYoungrsquos modulus of silty clay layer E2 15 225 015 LognormalUniform vehicle load q 10 10 01 Lognormal


3D model constructed at the 1st excavation step

u 1 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)

9

10

11

12

13

14

15

16

17

10 11 12 13 14 15 16 179u1 determined from finite difference analysis (mm)

(a)

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

Finite difference analysis4th HPCE-based surrogate model

000

005

010

015

020

025

030

035

040

10 11 12 13 14 15 16 17 189Ground settlement u1 (mm)

(b)

Figure 5 Continued

Table 4 Expansion terms and coefficients of the 4th order HPCE for the 1st excavation step

No Term Coefficient Value No Term Coefficient Value1 1 a0 1208 19 ξ22ξ3 minus ξ3 a18 minus0032 ξ1 a1 minus138 20 ξ1ξ2ξ3 a19 minus0023 ξ2 a2 minus032 21 ξ41 minus 6ξ21 + 3 a20 0014 ξ3 a3 057 22 ξ42 minus 6ξ22 + 3 a21 minus0055 ξ21 minus 1 a4 027 23 ξ43 minus 6ξ23 + 3 a22 minus0016 ξ22 minus 1 a5 minus016 24 ξ1ξ

32 minus ξ1ξ2 a23 004

7 ξ23 minus 1 a6 001 25 ξ1ξ33 minus ξ1ξ3 a24 0

8 ξ1ξ2 a7 002 26 ξ31ξ2 minus ξ1ξ2 a25 09 ξ1ξ3 a8 minus015 27 ξ2ξ

23 minus ξ2ξ3 a26 minus003

10 ξ2ξ3 a9 minus001 28 ξ31ξ3 minus ξ1ξ3 a27 minus00411 ξ31 minus 3ξ1 a10 minus010 29 ξ32ξ3 minus ξ2ξ3 a28 00412 ξ32 minus 3ξ2 a11 minus001 30 ξ21ξ

22 minus ξ21 minus ξ22 + 1 a29 minus001

13 ξ33 minus 3ξ3 a12 0 31 ξ21ξ23 minus ξ21 minus ξ23 + 1 a30 002

14 ξ1ξ22 minus ξ1 a13 minus001 32 ξ22ξ

23 minus ξ22 minus ξ23 + 1 a31 004

15 ξ1ξ23 minus ξ1 a14 minus001 33 ξ21ξ2ξ3 minus ξ2ξ3 a32 007

16 ξ21ξ2 minus ξ2 a15 006 34 ξ1ξ22ξ3 minus ξ1ξ3 a33 minus003

17 ξ2ξ23 minus ξ2 a16 003 35 ξ1ξ2ξ


18 ξ21ξ3 minus ξ3 a17 009


illustration Figure 10 presents the variation of the posteriorprobability of ground collapse with the excavation step Asseen from Figure 10 the posterior probability of groundcollapse increases continuously as the tunnel starts to ad-vance en it increases dramatically at the 2nd excavationstep and exceeds the prior probability of ground collapse(5152times10minus5) and increases furthermore at the 3rd excava-tion step until reaching 036 at the 5th excavation step evariation trend of the posterior probability indicates a safetycheck and necessary support measures shall be timely takenat the 3rd excavation step to control the monotonous in-crease of ground settlement Otherwise the occurrenceprobability of ground collapse due to the shield tunnelingwill eventually be large and unacceptable Moreover the

variation trend of the posterior probability with the time isconsistent with that of the time-series monitoring data asshown in Figure 3

For the case of the 5th excavation step the posteriorprobability of ground collapse (Pf5) estimated from theproposed approach is 036 To calculate such a probabilitythe proposed approach needs performing 5times 70 runs of 3Ddeterministic finite difference analyses of the tunneling-induced ground settlements to construct five surrogatemodels and additional probabilistic back analysis and reli-ability updating For the same problem the directMCS requires more than 27677 runs of 3D deterministicfinite difference analyses for achieving a target COVPf5below 10 is is because the least number of samples

u 2 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)3D model constructed at the 2nd excavation step

10 11 12 13 14 15 16 17 189u2 determined from finite difference analysis (mm)

9

10

11

12

13

14

15

16

17

18

(c)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(d)

u 5 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)

3D model constructed at the 5th excavation step

12

13

14

15

16

17

18

19

13 14 15 16 17 18 1912u5 determined from finite difference analysis (mm)

(e)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040

13 14 15 16 17 18 1912Ground settlement u5 (mm)

(f )

Figure 5 Validation of the surrogate models underlying three representative excavation steps (a) Comparison of u1 (b) Comparison of thePDF of u1 (c) Comparison of u2 (d) Comparison of the PDF of u2 (e) Comparison of u5 (f ) Comparison of the PDF of u5


required for the MCS to estimate Pf5 is calculated byNsim ge (1 minus Pf5)(Pf5(COVPf5

)2) [22] e computationaltime required for one run of 3D deterministic finite dif-ference analysis is 800 seconds when the computations are

performed on a desktop with 8GB RAM and one Intel Corei7-4790 CPU clocked at 36GHz e computational timetaken on the probabilistic back analysis and reliabilityupdating with the constructed surrogate models equals 18seconds which is only 144 of that required for one run of3D deterministic finite difference analysis Based on theseabout 6150 hours will be required for the direct MCS while5times 800 + 18 seconds (11 hours) are required for the pro-posed approach to calculate the posterior probability of

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

15 20 25 30 35 40 45 50 5510Youngrsquos modulus of gravel layer E1 (MPa)

Prior PDFPosterior PDFs

The 1st excavation stepThe 2nd excavation stepThe 3rd excavation stepThe 4th excavation stepThe 5th excavation step

Figure 6 Comparison of the posterior PDFs of Youngrsquos modulusof gravel layer for different excavation steps

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

10 15 20 25 305Youngrsquos modulus of silty clay layer E2 (MPa)



Figure 7 Comparison of the posterior PDFs of Youngrsquos modulusof silty clay layer for different excavation steps



00

10 times 10ndash4

20 times 10ndash4

30 times 10ndash4

40 times 10ndash4

50 times 10ndash4

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

8 9 10 11 12 13 14 157Ground vehicle load q (KPa)

Figure 8 Comparison of the posterior PDFs of ground vehicle loadfor different excavation steps

Prior COV

Prior COV

COVE1COVE2COVq

006

008

010

012

014

016

Coef

ficie

nt o

f var

iatio

n (C

OV

)

2 31 54Excavation step

Figure 9 Variation of the coefficients of variation of input pa-rameters with the excavation step


ground collapse at the 5th excavation stepis confirms thatthe proposed approach is much more efficient in theprobabilistic back analysis of the uncertain geomechanicalparameters and the reliability updating Such high efficiencywill greatly facilitate the applications of the proposed ap-proach in geotechnical engineering

4 Conclusions

A BUS-based sequential probabilistic back analysis is proposedto estimate the uncertain geomechanical parameters and up-date the reliability of tunneling-induced ground settlementse shield tunnel project of No 1 Nanchang Metro Line inChina is investigated to assess the effectiveness of the proposedapproach Several conclusions can be drawn from this study

(1) e proposed approach can well infer the posteriordistributions of uncertain geomechanical parametersby fully utilizing the time-series monitoring datae reliability of tunneling-induced ground settle-ments is updated in a real-time manner e com-putational efficiency has been improved throughtransforming the Bayesian back analysis probleminto an equivalent structural reliability problem andconstructing the surrogate models of the outputresponses of shield tunnels by the Hermite poly-nomial chaos expansion

(2) By employing the proposed approach the variationtrends of the means of uncertain geomechanicalparameters and the posterior probability of groundcollapse match well with those of time-series mon-itoring data and the post-event investigations eprobability distributions of geomechanical parame-ters gradually converge to the target distribution andthe uncertainties of geomechanical parameters arereduced successively after updating ese demon-strate the effectiveness of the proposed approach

(3) e sequential probabilistic back analysis and reli-ability updating results can provide an importantreference for the reduction of the uncertainties ofgeomechanical parameters during shield tunnelexcavation and consequently the mitigation of thepotential risk of ground collapse For the consideredreal example the probability of ground collapseincreases markedly from October 1 2012 700 toOctober 1 2012 1500 which can provide valuableinformation for the practitioners to formulate earlywarning measures to prevent the occurrence ofground collapse accident

Data Availability

Some or all data models or code generated or used duringthis study are available to the readers upon request eitems are listed as follows

(1) Time-series monitoring data of tunneling-inducedground settlement

(2) Hermite polynomial chaos expansion code that isused for constructing the surrogate models of theoutput responses of shield tunnels

(3) BUS code that is used for inferring the posteriordistribution of geomechanical parameters and esti-mating the posterior probability of ground collapse

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Project nos 41867036 and 41972280)Jiangxi Provincial Natural Science Foundation (Project nos2018ACB21017 20181ACB20008 and 20192BBG70078) andOpen Research Fund of State Key Laboratory of Geo-mechanics and Geotechnical Engineering (Project noZ019019) e financial support is gratefully acknowledged

References

[1] C Camos O Spackova D Straub and C Molins ldquoProba-bilistic approach to assessing and monitoring settlementscaused by tunnelingrdquo Tunnelling and Underground SpaceTechnology vol 51 pp 313ndash325 2016

[2] V H Franco G D F N Gitirana and A P De AssisldquoProbabilistic assessment of tunneling-induced buildingdamagerdquo Computers and Geotechnics vol 113 Article ID103097 2019

[3] G Mollon D Dias and A-H Soubra ldquoProbabilistic analysesof tunneling-induced ground movementsrdquo Acta Geotechnicavol 8 no 2 pp 181ndash199 2013

[4] W Gao and M Ge ldquoBack analysis of rock mass parametersand initial stress for the longtan tunnel in Chinardquo Engineeringwith Computers vol 32 no 3 pp 497ndash515 2016

[5] W Gong C H Juang J R Martin H Tang Q Wang andH Huang ldquoProbabilistic analysis of tunnel longitudinal

Prior probability of ground collapsePosterior probability of ground collapse

1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse


Figure 10 Comparison of prior and posterior probabilities ofground collapse induced by shield tunneling


performance based upon conditional random field simulationof soil propertiesrdquo Tunnelling and Underground SpaceTechnology vol 73 pp 1ndash14 2018

[6] J Hu W Liu Y Pan and H Zeng ldquoSite measurement andstudy of vertical freezing wall temperatures of a large-di-ameter shield tunnelrdquo Advances in Civil Engineeringvol 2019 Article ID 8231458 11 pages 2019

[7] S-Y Chi J-C Chern and C-C Lin ldquoOptimized back-analysis for tunneling-induced ground movement usingequivalent ground loss modelrdquo Tunnelling and UndergroundSpace Technology vol 16 no 3 pp 159ndash165 2001

[8] C Zhu H Zhao and M Zhao ldquoBack analysis of geo-mechanical parameters in underground engineering usingartificial bee colonyrdquo De Scientific World Journal vol 2014Article ID 693812 13 pages 2014

[9] Y Sun J Huang W Jin S W Sloan and Q Jiang ldquoBayesianupdating for progressive excavation of high rock slopes usingmulti-type monitoring datardquo Engineering Geology vol 252pp 1ndash13 2019

[10] K-K Phoon and F H Kulhawy ldquoCharacterization of geo-technical variabilityrdquo Canadian Geotechnical Journal vol 36no 4 pp 612ndash624 1999

[11] D-Q Li S-H Jiang Y-F Chen and C-B Zhou ldquoReliabilityanalysis of serviceability performance for an undergroundcavern using a non-intrusive stochastic methodrdquo Environ-mental Earth Sciences vol 71 no 3 pp 1169ndash1182 2014

[12] X M Li ldquoStudy on ground subsidence induced by earthpressure balanced shield tunnelingrdquo PhD esis NanjingUniversity Nanjing China 2014

[13] H Huang W Gong S Khoshnevisan C H Juang D Zhangand LWang ldquoSimplified procedure for finite element analysisof the longitudinal performance of shield tunnels consideringspatial soil variability in longitudinal directionrdquo Computersand Geotechnics vol 64 pp 132ndash145 2015

[14] S-H Jiang and J-S Huang ldquoEfficient slope reliability analysisat low-probability levels in spatially variable soilsrdquo Computersand Geotechnics vol 75 pp 18ndash27 2016

[15] S-H Jiang J Huang C Yao and J Yang ldquoQuantitative riskassessment of slope failure in 2-D spatially variable soils bylimit equilibrium methodrdquo Applied Mathematical Modellingvol 47 pp 710ndash725 2017

[16] H Cheng J Chen R Chen J Huang and J Li ldquoree-di-mensional analysis of tunnel face stability in spatially variablesoilsrdquo Computers and Geotechnics vol 111 pp 76ndash88 2019

[17] C Haas and H H Einstein ldquoUpdating the decision aids fortunnelingrdquo Journal of Construction Engineering and Man-agement vol 128 no 1 pp 40ndash48 2002

[18] O Spackova and D Straub ldquoProbabilistic assessment oftunnel construction performance based on datardquo Tunnellingand Underground Space Technology vol 37 pp 62ndash78 2013

[19] D Park and E-S Park ldquoInverse parameter fitting of tunnelsusing a response surface approachrdquo International Journal ofRock Mechanics and Mining Sciences vol 77 pp 11ndash18 2015

[20] W Liu X Luo J Huang L Hu and M Fu ldquoProbabilisticanalysis of tunnel face stability below river using BayesianframeworkrdquoMathematical Problems in Engineering vol 2018Article ID 1450683 8 pages 2018

[21] S Miro M Konig D Hartmann and T Schanz ldquoA prob-abilistic analysis of subsoil parameters uncertainty impacts ontunnel-induced ground movements with a back-analysisstudyrdquo Computers and Geotechnics vol 68 pp 38ndash53 2015

[22] H S Ang and W H Tang Probability Concepts in Engi-neering Emphasis on Applications to Civil and Environmental

Engineering John Wiley amp Sons New York City NY USA 2edition 2007

[23] D Straub and I Papaioannou ldquoBayesian updating withstructural reliability methodsrdquo Journal of Engineering Me-chanics vol 141 no 3 Article ID 04014134 2015

[24] W G Zhang and A T C Goh ldquoMultivariate adaptive re-gression splines for analysis of geotechnical engineeringsystemsrdquoComputers and Geotechnics vol 48 pp 82ndash95 2013

[25] D-Q Li D Zheng Z-J Cao X-S Tang and K-K PhoonldquoResponse surface methods for slope reliability analysis re-view and comparisonrdquo Engineering Geology vol 203 pp 3ndash14 2016

[26] W Zhang and A T C Goh ldquoMultivariate adaptive regressionsplines and neural network models for prediction of piledrivabilityrdquoGeoscience Frontiers vol 7 no 1 pp 45ndash52 2016

[27] X Liu D-Q Li Z-J Cao and Y Wang ldquoAdaptive montecarlo simulationmethod for system reliability analysis of slopestability based on limit equilibrium methodsrdquo EngineeringGeology vol 264 Article ID 105384 2020

[28] G Mollon D Dias and A-H Soubra ldquoprobabilistic analysisof circular tunnels in homogeneous soil using responsesurface methodologyrdquo Journal of Geotechnical and Geo-environmental Engineering vol 135 no 9 pp 1314ndash13252009

[29] D Li Y Chen W Lu and C Zhou ldquoStochastic responsesurface method for reliability analysis of rock slopes involvingcorrelated non-normal variablesrdquo Computers and Geo-technics vol 38 no 1 pp 58ndash68 2011

[30] R G Ghanem and P D Spanos Stochastic Finite Element ASpectral ApproachmdashRevised Version Dover PublicationMineola NY USA 2003

[31] S K Choi R A Canfield and R V Grandhi ldquoEstimation ofstructural reliability for gaussian random fieldsrdquo Structureand Infrastructure Engineering vol 2 no 3-4 pp 161ndash1732006

[32] I Papaioannou and D Straub ldquoReliability updating in geo-technical engineering including spatial variability of soilrdquoComputers and Geotechnics vol 42 pp 44ndash51 2012

[33] S-K Au and J L Beck ldquoEstimation of small failure proba-bilities in high dimensions by subset simulationrdquo ProbabilisticEngineering Mechanics vol 16 no 4 pp 263ndash277 2001

[34] J Huang G Fenton D V Griffiths D Li and C Zhou ldquoOnthe efficient estimation of small failure probability in slopesrdquoLandslides vol 14 no 2 pp 491ndash498 2017

[35] W Betz I Papaioannou J L Beck and D Straub ldquoBayesianinference with subset simulation strategies and improve-mentsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 331 pp 72ndash93 2018

[36] S-H Jiang I Papaioannou and D Straub ldquoBayesianupdating of slope reliability in spatially variable soils with in-situ measurementsrdquo Engineering Geology vol 239 pp 310ndash320 2018

[37] S-H Jiang J Huang X-H Qi and C-B Zhou ldquoEfficientprobabilistic back analysis of spatially varying soil parametersfor slope reliability assessmentrdquo Engineering Geology vol 271Article ID 105597 2020

[38] D Yang H Huang and J Zhang ldquoStudy on probabilitydistribution of vehicle load and its load effectrdquo China Journalof Guangzhou University vol 13 no 5 pp 56ndash60 2014

[39] Jiangxi Survey and Design Institute Geotechnical Investiga-tion Nanchang Metro Line Nanchang China 2009

[40] J Bauer and W Puła ldquoReliability with respect to settlementlimit-states of shallow foundations on linearly-deformable


subsoilrdquo Computers and Geotechnics vol 26 no 3-4pp 281ndash308 2000

[41] G B Baecher and J T Christian Reliability and Statistics inGeotechnical Engineering JohnWiley amp Sons New York CityNY USA 2003

[42] Y Li L Tang Z Liu and Y Liu ldquoStatistics and probabilityanalysis of vehicle overloads on a rigid frame bridge fromlong-term monitored strainsrdquo Smart Structures and Systemsvol 9 no 3 pp 287ndash301 2012


networks to back analyze the soil parameters and the initialgeo-stress of Longtan tunnel and evaluate the stability of thetunnel Although significant advances have been made in theback analysis of shield tunnel parameters the above-men-tioned studies do not account for the influence of the in-herent variability of geomechanical parameters (eg [4 9])It is widely accepted that the geomechanical parameters varyspatially and temporally because of the concealment ofunderground space and complexity of underground envi-ronment (eg [10ndash16]) To obtain the shield tunnel pa-rameters in line with the engineering practice theuncertainties of the geomechanical parameters should betaken into account in the back analysis

Several studies have investigated probabilistic backanalysis problems of uncertain parameters for the shieldtunnels by combining probability theory and statistics Forexample Haas and Einstein [17] employed a Markov chainMonte Carlo (MCMC) method to update the posteriordistributions of surrounding rock mass parameters oftunnels based on the monitoring data Spackova and Straub[18] proposed a dynamic Bayesian network based-tunnelingprocess model to update the probability of tunnel failure byutilizing the observation data from geological survey andconstruction stages Park and Park [19] conducted aprobabilistic back analysis of uncertain parameters fortunnel surrounding rock masses by adopting a responsesurface method Liu et al [20] also employed the MCMCmethod for the probabilistic back analysis of subsoil co-hesion and internal friction angle around a tunnel belowriver Although many investigations on the probabilisticback analysis of shield tunnel parameters have been carriedout the time-series monitoring data that are collectedduring the tunnel advancement were rarely incorporated inthe back analysis To the best of our knowledge only Miroet al [21] applied a MCMCmethod in the probabilistic backanalysis of uncertain parameters for surrounding rockmasses by utilizing the tunnel-induced ground movementsin a sequential manner However the MCMC methodadopted by Miro et al [21] is not effective for high di-mensional problems and the reliability of shield tunnelswith the probabilistic back analysis results was not inves-tigated e updated statistics (ie means standard devia-tions probability distributions) of geomechanicalparameters via the probabilistic back analyses can provide animportant basis for reliability analysis of shield tunnelsBased on the updated probability distributions of the un-certain parameters more realistic reliability assessment ofthe tunnel face stability or serviceability limit state can beachieved

In this paper a Bayesian Updating with Subset simu-lation (BUS)-based sequential probabilistic back analysisapproach is proposed for estimating uncertain geo-mechanical parameters and updating the reliability oftunneling-induced ground settlements To improve com-putational efficiency the proposed approach transforms acomplex Bayesian back analysis problem into an equivalentstructural reliability problem which is then solved by subsetsimulation e Hermite polynomial chaos expansion-basedsurrogate models are constructed In Section 2 the proposed

approach comprising the construction of surrogate modelsand likelihood functions inference of posterior distributionusing BUS approach and reliability updating of tunneling-induced ground settlements is presented In Section 3 theNo 1 NanchangMetro Line tunnel is treated as a typical casefor probabilistically determining the uncertain geo-mechanical parameters based on the time-series monitoringdata of ground settlement Finally the reliability of tun-neling-induced ground settlements is updated on the basis ofthe probabilistic back analysis results

2 BUS-Based Sequential Probabilistic BackAnalysis Approach

Shield tunneling is often fraught with inevitable uncer-tainties that include but not limited to (1) the inherentvariability of soils (2) themeasurement errors in quantifyingthe soil properties by field and laboratory tests and (3) thelimited knowledge on geotechnical conditions and simpli-fied geotechnical models in predicting the ground settle-ments (eg [1 21]) ese uncertainties can be reduced byincorporating the field observation data into a Bayesian backanalysis by estimating the posterior joint probability densityfunction (PDF) of geomechanical parameters (eg [9 22])In this study a BUS-based sequential probabilistic backanalysis approach is proposed to estimate the posterior PDFand its implementation procedure is briefly introduced asfollows

(1) Determine the prior statistics (ie mean coefficientof variation (COV) and distribution) based on theknowledge (eg geological survey reports engi-neering experience and existing data in the litera-ture) about the uncertain input parameters

(2) Construct Hermite polynomial chaos expansion(HPCE)-based surrogate models to facilitate thecalculation of tunneling-induced ground settlementsfor each excavation step

(3) Establish the likelihood function using the moni-toring data of ground settlement obtained from thefirst excavation step

(4) Infer the posterior PDFs of uncertain input pa-rameters using the BUS approach en treat theposterior PDFs as the prior PDFs and build a newlikelihood function using the monitoring data ofground settlement obtained from the next excavationstep until the last regarded excavation stage

(5) Update the reliability of tunneling-induced groundsettlements for each excavation step based on theobtained posterior statistics of uncertain inputparameters

e flowchart of the BUS-based sequential probabilisticback analysis approach is presented in Figure 1e involvedconstruction of surrogate models and likelihood functionsinference of posterior distribution using the BUS approachand reliability updating of tunneling-induced ground set-tlements are presented in the following subsections



uj a0Γ0 + 1113944n

i11ai1Γ1 ξi1

1113872 1113873 + 1113944n

i111113944

i1

i21ai1i2Γ2 ξi1

ξi21113872 1113873

+ 1113944n

i111113944

i1

i211113944

i2

i31ai1i2i3Γ3 ξi1

ξi2 ξi3


(1)


ai1 i2 ai1 i2 i3



ai1 i2 ai1i2 i3




εmj u

mj minus uj(x) (2)









Prior 2

Prior n

Posterior t

Monitoring data u1m





BUS operation


Monitoring data u2m



Monitoring data utm




BUS operation

1st excavation step

2nd excavation step

tth excavation step









Lj(x) ϕ um

j minus uj(x)1113960 1113961σεmj1113882 1113883

σεmj

(3)




ΩX Hj x+( 1113857le 01113966 1113967 (4)




cL(x)le 10 (6)



P(Z) P Hj x+( 1113857le 01113960 1113961 P Z1( 1113857 1113945

m

i2Pr Zi

1113868111386811138681113868Ziminus11113872 1113873 (7)

where Z1Z2Zmminus1

Zm




ci 1


(i 1 2 m)

(8)





Pfj P Fj

11138681113868111386811138681113868Z1113874 1113875 P FjcapZ1113872 1113873

P(Z)

1113936Nf

k1 I umax le uj xkf1113874 11138751113876 1113877

Nf

(10)



















Plain fill

Silty clay

Fine sand

Gravel



60m

54m

14m

1m5m

2m15

m2m

10m

1

2

3

4

5

6





ndash24

ndash22

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

Mor

nito

ring

data

of g

roun

d m

ovem

ents

(mm

)








XY

Z








u 1 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)

9

10

11

12

13

14

15

16

17


(a)

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)


000

005

010

015

020

025

030

035

040


(b)

Figure 5 Continued



32 minus ξ1ξ2 a23 004








23 minus ξ22 minus ξ23 + 1 a31 004



17 ξ2ξ23 minus ξ2 a16 003 35 ξ1ξ2ξ


18 ξ21ξ3 minus ξ3 a17 009





u 2 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm



9

10

11

12

13

14

15

16

17

18

(c)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(d)

u 5 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)


12

13

14

15

16

17

18

19


(e)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(f )






Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)







00

10 times 10ndash4

20 times 10ndash4

30 times 10ndash4

40 times 10ndash4

50 times 10ndash4

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)



Prior COV

Prior COV

COVE1COVE2COVq

006

008

010

012

014

016

Coef

ficie

nt o

f var

iatio

n (C

OV

)





4 Conclusions





Data Availability







Acknowledgments


References







1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse















































uj a0Γ0 + 1113944n

i11ai1Γ1 ξi1

1113872 1113873 + 1113944n

i111113944

i1

i21ai1i2Γ2 ξi1

ξi21113872 1113873

+ 1113944n

i111113944

i1

i211113944

i2

i31ai1i2i3Γ3 ξi1

ξi2 ξi3


(1)


ai1 i2 ai1 i2 i3



ai1 i2 ai1i2 i3




εmj u

mj minus uj(x) (2)









Prior 2

Prior n

Posterior t

Monitoring data u1m





BUS operation


Monitoring data u2m



Monitoring data utm




BUS operation

1st excavation step

2nd excavation step

tth excavation step









Lj(x) ϕ um

j minus uj(x)1113960 1113961σεmj1113882 1113883

σεmj

(3)




ΩX Hj x+( 1113857le 01113966 1113967 (4)




cL(x)le 10 (6)



P(Z) P Hj x+( 1113857le 01113960 1113961 P Z1( 1113857 1113945

m

i2Pr Zi

1113868111386811138681113868Ziminus11113872 1113873 (7)

where Z1Z2Zmminus1

Zm




ci 1


(i 1 2 m)

(8)





Pfj P Fj

11138681113868111386811138681113868Z1113874 1113875 P FjcapZ1113872 1113873

P(Z)

1113936Nf

k1 I umax le uj xkf1113874 11138751113876 1113877

Nf

(10)



















Plain fill

Silty clay

Fine sand

Gravel



60m

54m

14m

1m5m

2m15

m2m

10m

1

2

3

4

5

6





ndash24

ndash22

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

Mor

nito

ring

data

of g

roun

d m

ovem

ents

(mm

)








XY

Z








u 1 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)

9

10

11

12

13

14

15

16

17


(a)

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)


000

005

010

015

020

025

030

035

040


(b)

Figure 5 Continued



32 minus ξ1ξ2 a23 004








23 minus ξ22 minus ξ23 + 1 a31 004



17 ξ2ξ23 minus ξ2 a16 003 35 ξ1ξ2ξ


18 ξ21ξ3 minus ξ3 a17 009





u 2 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm



9

10

11

12

13

14

15

16

17

18

(c)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(d)

u 5 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)


12

13

14

15

16

17

18

19


(e)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(f )






Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)







00

10 times 10ndash4

20 times 10ndash4

30 times 10ndash4

40 times 10ndash4

50 times 10ndash4

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)



Prior COV

Prior COV

COVE1COVE2COVq

006

008

010

012

014

016

Coef

ficie

nt o

f var

iatio

n (C

OV

)





4 Conclusions





Data Availability







Acknowledgments


References







1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse

















































Lj(x) ϕ um

j minus uj(x)1113960 1113961σεmj1113882 1113883

σεmj

(3)




ΩX Hj x+( 1113857le 01113966 1113967 (4)




cL(x)le 10 (6)



P(Z) P Hj x+( 1113857le 01113960 1113961 P Z1( 1113857 1113945

m

i2Pr Zi

1113868111386811138681113868Ziminus11113872 1113873 (7)

where Z1Z2Zmminus1

Zm




ci 1


(i 1 2 m)

(8)





Pfj P Fj

11138681113868111386811138681113868Z1113874 1113875 P FjcapZ1113872 1113873

P(Z)

1113936Nf

k1 I umax le uj xkf1113874 11138751113876 1113877

Nf

(10)



















Plain fill

Silty clay

Fine sand

Gravel



60m

54m

14m

1m5m

2m15

m2m

10m

1

2

3

4

5

6





ndash24

ndash22

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

Mor

nito

ring

data

of g

roun

d m

ovem

ents

(mm

)








XY

Z








u 1 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)

9

10

11

12

13

14

15

16

17


(a)

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)


000

005

010

015

020

025

030

035

040


(b)

Figure 5 Continued



32 minus ξ1ξ2 a23 004








23 minus ξ22 minus ξ23 + 1 a31 004



17 ξ2ξ23 minus ξ2 a16 003 35 ξ1ξ2ξ


18 ξ21ξ3 minus ξ3 a17 009





u 2 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm



9

10

11

12

13

14

15

16

17

18

(c)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(d)

u 5 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)


12

13

14

15

16

17

18

19


(e)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(f )






Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)







00

10 times 10ndash4

20 times 10ndash4

30 times 10ndash4

40 times 10ndash4

50 times 10ndash4

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)



Prior COV

Prior COV

COVE1COVE2COVq

006

008

010

012

014

016

Coef

ficie

nt o

f var

iatio

n (C

OV

)





4 Conclusions





Data Availability







Acknowledgments


References







1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse




























































Plain fill

Silty clay

Fine sand

Gravel



60m

54m

14m

1m5m

2m15

m2m

10m

1

2

3

4

5

6





ndash24

ndash22

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

Mor

nito

ring

data

of g

roun

d m

ovem

ents

(mm

)








XY

Z








u 1 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)

9

10

11

12

13

14

15

16

17


(a)

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)


000

005

010

015

020

025

030

035

040


(b)

Figure 5 Continued



32 minus ξ1ξ2 a23 004








23 minus ξ22 minus ξ23 + 1 a31 004



17 ξ2ξ23 minus ξ2 a16 003 35 ξ1ξ2ξ


18 ξ21ξ3 minus ξ3 a17 009





u 2 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm



9

10

11

12

13

14

15

16

17

18

(c)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(d)

u 5 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)


12

13

14

15

16

17

18

19


(e)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(f )






Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)







00

10 times 10ndash4

20 times 10ndash4

30 times 10ndash4

40 times 10ndash4

50 times 10ndash4

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)



Prior COV

Prior COV

COVE1COVE2COVq

006

008

010

012

014

016

Coef

ficie

nt o

f var

iatio

n (C

OV

)





4 Conclusions





Data Availability







Acknowledgments


References







1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse


















































XY

Z








u 1 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)

9

10

11

12

13

14

15

16

17


(a)

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)


000

005

010

015

020

025

030

035

040


(b)

Figure 5 Continued



32 minus ξ1ξ2 a23 004








23 minus ξ22 minus ξ23 + 1 a31 004



17 ξ2ξ23 minus ξ2 a16 003 35 ξ1ξ2ξ


18 ξ21ξ3 minus ξ3 a17 009





u 2 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm



9

10

11

12

13

14

15

16

17

18

(c)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(d)

u 5 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)


12

13

14

15

16

17

18

19


(e)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(f )






Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)







00

10 times 10ndash4

20 times 10ndash4

30 times 10ndash4

40 times 10ndash4

50 times 10ndash4

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)



Prior COV

Prior COV

COVE1COVE2COVq

006

008

010

012

014

016

Coef

ficie

nt o

f var

iatio

n (C

OV

)





4 Conclusions





Data Availability







Acknowledgments


References







1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse















































u 1 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)

9

10

11

12

13

14

15

16

17


(a)

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)


000

005

010

015

020

025

030

035

040


(b)

Figure 5 Continued



32 minus ξ1ξ2 a23 004








23 minus ξ22 minus ξ23 + 1 a31 004



17 ξ2ξ23 minus ξ2 a16 003 35 ξ1ξ2ξ


18 ξ21ξ3 minus ξ3 a17 009





u 2 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm



9

10

11

12

13

14

15

16

17

18

(c)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(d)

u 5 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)


12

13

14

15

16

17

18

19


(e)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(f )






Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)







00

10 times 10ndash4

20 times 10ndash4

30 times 10ndash4

40 times 10ndash4

50 times 10ndash4

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)



Prior COV

Prior COV

COVE1COVE2COVq

006

008

010

012

014

016

Coef

ficie

nt o

f var

iatio

n (C

OV

)





4 Conclusions





Data Availability







Acknowledgments


References







1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse

















































u 2 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm



9

10

11

12

13

14

15

16

17

18

(c)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(d)

u 5 d

eter

min

ed fr

om 4

th H

PCE-

base

dsu

rrog

ate m

odel

(mm

)


12

13

14

15

16

17

18

19


(e)


Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

000

005

010

015

020

025

030

035

040


(f )






Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)







00

10 times 10ndash4

20 times 10ndash4

30 times 10ndash4

40 times 10ndash4

50 times 10ndash4

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)



Prior COV

Prior COV

COVE1COVE2COVq

006

008

010

012

014

016

Coef

ficie

nt o

f var

iatio

n (C

OV

)





4 Conclusions





Data Availability







Acknowledgments


References







1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse

















































Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)

00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

20 times 10ndash7

25 times 10ndash7

30 times 10ndash7

35 times 10ndash7

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)







00

10 times 10ndash4

20 times 10ndash4

30 times 10ndash4

40 times 10ndash4

50 times 10ndash4

Prob

abili

ty d

ensit

y fu

nctio

n (P

DF)



Prior COV

Prior COV

COVE1COVE2COVq

006

008

010

012

014

016

Coef

ficie

nt o

f var

iatio

n (C

OV

)





4 Conclusions





Data Availability







Acknowledgments


References







1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse















































4 Conclusions





Data Availability







Acknowledgments


References







1E ndash 5

1E ndash 4

1E ndash 3

001

Prob

abili

ty o

f gro

und

colla

pse














































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