- Email: [email protected]
Parameter estimation using extended Bayesian method in tunnelling
Computers and Geotechnics 24 (1999) 109±124
Parameter estimation using extended Bayesian method in tunnelling In-Mo Lee a,*, Dong-Hyun Kim b a
Department of Civil Engineering, Korea University, Seoul, 136-701, Korea Geotechnical Engineerring Team, Kolon Construction Co., Ltd., Seoul, 135-100, Korea
b
Received 27 July 1998; received in revised form 30 October 1998; accepted 2 November 1998
Abstract Eort was made in this paper to formulate a parameter estimation technology by systematically combining the ®eld measurements and the prior information of underground structure. The Extended Bayesian Method (EBM) was adopted for the feedback analysis and the ®nite element analysis, which was implemented to predict the ground response. Determined in the present study were various geotechnical parameters including the elastic modulus (E), the initial horizontal stress coecient at rest (Ko), the cohesion (c) and the internal friction angle ( ). The validity of the feedback system proposed herein was demonstrated through an elastoplastic example problem. The proposed method was applied to an actual tunnel site in Pusan, Korea and has shown to be highly eective in actual ®eld problems. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction In spite of drastic development of underground technology, there are still many uncertainties that exist in the designing and constructing of underground structures. This is mainly due to the discrepancy between the laboratory and in-situ tests and limitations of site investigation techniques during the design stage, etc. In order to reduce these uncertainties, ®eld instrumentation results obtained during construction stage are compared with the initially estimated ground properties. The feed back system can be used to estimate the optimum ground properties by minimizing the dierence between the predicted and measured ground motions.
* Corresponding author. Tel.: +82-2-3290-3314; fax: +82-2-928-7656; e-mail: [email protected] 0266-352X/99/$Ðsee front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S026 6-352X(98)0003 1-7
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I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124
The ordinary least squares (OLS) method was widely used, because of its easy application to non-linear geotechnical problems without complex mathematical concepts. However, this method cannot consider the prior information in the process of parameter estimation. Bayesian approach deals with both the prior information and the measurement data [1]. A signi®cant drawback of the Bayesian method is the incommensurate matching between the two components, since the objective function is equally divided by them. To overcome this, Neuman and Yakowitz [2] introduced the adjusting positive scalar term, which adjusts the weights of Jo and Jp : J Jo Jp
1
where, Jo and Jp are objective function of the observed and predicted parameters, respectively. This concept is called extended Bayesian method [3,4]. 2. Background of the proposed feedback system 2.1. Model identi®cation The selection of the most appropriate model among alternative models in the Bayesian approach is possible by introducing the concept of the Akaike Information Criterion [3±5]. The best model among the various alternatives can be identi®ed when the AIC(x) value in Eq. (2) is minimized. 2 AIC x ÿ2 ln fk x j k x 2 dim k where, x is arbitrary input vector, fk x j k is the probability distribution function of the kth alternative model, and dim k is the number of the model parameters for the kth model. The AIC concept is again used in Eq. (3) by the Bayesian approach utilizing the prior and posterior distribution of . AIC ÿ2 ln p x j 2 dim 3 Q where, p xj j f xj d. The ®rst term in Eq. (2) indicates how well the model ®ts with to the observed data, and dim of the second term means the number of model parameters. 2.2. Formulation of extended Bayesian method The extended Bayesian method is a technique to give the optimum match between the observed data and prior information. The observation data vector can be expressed as [3,4], uk uk x j "k
4
I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124
111
where, uk is the ®eld observation at step k, uk is the calculated result vector at step k, "k is the error vector assumed to follow "k N 0; Vu , where Vu is an N N covariance matrix, x is the known input data vector, is the model parameter vector to be estimated, and N is the total number of observation points. The prior information vector is assumed as follows, p
5
where, p is the prior (initially estimated) mean vector of the model parameter vector, is the uncertainty of the prior information assumed to follow N 0; Vp = , where Vp is the prior M M covariance matrix of , is a scalar that adjusts the magnitude of the uncertainty, and M is the number of model parameters. The Bayesian estimator ^ is the one that minimizes the following function with respect to : J j
K X T k uk ÿ uk x j Vÿ1 u ÿ uk x j ÿ pT Vÿ1 u p ÿ p
6
k1
The parameter can be estimated again by the Bayesian theorem maximizing the following function: n o 1 ^ Jp ^ l j u ; p L j u ; p ÿ NK ln Jo 2 9 8 > > > > > > = M j Vÿ1 j 1 < p const ln K > P 2 > > > k T ÿ1 k > > S Vu S Vp j; :j
7
k1
where,
@uk S @T k
^
:
Once we obtain the from Eq. (6), the parameter is then estimated. The modi®ed Box±Kanemasu iteration method is used to estimate the by minimizing Eq. (5) [6]. 2.3. Uncertainty evaluation of model parameters It is a concern to know how to reduce the uncertainties by adopting the proposed technique; the comparison between the uncertainty of initially estimated parameters (the prior estimation) and the uncertainty of the estimated values utilizing the EBM (the posterior estimation). Unfortunately, the posterior distribution is not a simple
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I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124
normal distribution with respect to due to the fact that uk is a non-linear function of , which makes the covariance matrix non-linear. In order to resolve this problem, uk at the estimated values, ^ is linearized. With the linearized uk , it is possible to obtain the posterior covariance matrix based on the conventional Bayesian theory [3,4]: (
K X k ÿ1 Sk T Vÿ1 p u S Vp
)ÿ1 8
k1
As shown in Eq. (8), the covariance matrix of the posterior estimation (p ) is composed of the covariance matrix of the prior estimation (Vp ), and the covariance matrix of measurements (Vu ). It is interesting to observe that, when the ®rst term in Eq. (8) is added to the second term, it makes the variance of the posterior estimation smaller since the covariance matrix of measurements is added to the prior estimation as an inverse matrix form. 2.4. Numerical method for back analysis The existing elasto-plastic ®nite element program, developed by Owen and Hinton [7], was modi®ed to simulate the tunnel excavation and the support system. Mohr± Coulomb failure criterion was used to represent the plastic behavior. The code developed in this study is applied to the example problem and the results are compared with the benchmark analytical solution (Kirsch equation) and the commercial ®nite dierence solution, FLAC-2D. The maximum relative error between the results of analytical solution and the modi®ed FEM analysis is less than 8% in the check points [8]. This relative error is considered acceptable considering many assumptions made in the numerical analysis. These procedures for the proposed system is summarized as a ¯owchart in Fig. 1. 3. Evaluation of the proposed scheme 3.1. Data preparation A comprehensive evaluation of the proposed feedback system was carried out for the elasto-plastic case in a hypothetical site. The geological condition of the site is presented in Fig. 2, and the tunnel is supposed to be located in the residual soil layer. The typical cross section of the tunnel is in a horse shoe shape having a diameter of 10 m, and the primary support system consisting of a 10 cm thick soft shotcrete. The typical values of geotechnical parameters are considered as the initial estimated geotechnical parameters (prior information) for this case. Since the site is a hypothetical one, the observed data are calculated using the ®nite dierence method code, FLAC-2D. The measurement points and displacements are summarized in Fig. 3 and Table 1.
I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124
113
Fig. 1. Framework for the feedback system.
3.2. Sensitivity analysis Sensitivity analysis is done to check the change in the ground response due to perturbations in the parameters. Figs. 4±6 show the results of the sensitivity analysis. During the sensitivity analysis for a chosen parameter, the other parameters are ®xed with initially estimated values. According to these ®gures, it is shown that the ground motion depends on the elastic modulus and the lateral earth pressure coecient during elastic behavior. If plastic deformation occurs, elastic modulus and internal friction angle will be a major in¯uencing factor to the ground motion. It has also been concluded that the ground motion is insigni®cantly aected by cohesion. 3.3. Parameter estimation As shown in Fig. 3, the plastic deformation occurs around the tunnel. Therefore, four parameters are required to be estimated: the cohesion, the internal friction
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Fig. 2. Geological condition of a hypothetical site.
Fig. 3. Location of measuring points.
angle, the elastic modulus, and the initial horizontal stress coecient. Since the tunnel crown is located within the residual soil layer, the parameters only for the residual soil layer are assessed. The coecient of variation of model parameters are assumed to be 0.3 considering large uncertainties are encountered. In order to check the eect of the number of measuring points in feedback system, they are varied during the analysis. The results of the analysis are listed in Table 2, and the prior estimates are also tabulated for comparison. Table 2 shows that at least six measuring points must be installed to obtain a reliable estimations.
I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124 Table 1 Measuring points and displacements No. 1 2 3 4 5 6 7 8 9 10
Distance from tunnel center (m)
Measured (assumed) displacements (mm)
4.5 5.5 6.5 7.5 8.5 4.55 5.7 6.7 7.7 8.7
74.5 66.3 59.6 54.5 50.4 30.8 13.5 4.6 2.0 0.9
Fig. 4. Sensitivity of geotechnical parameters on crown settlement.
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Fig. 5. Sensitivity of geotechnical parameters on sidewall convergence.
4. Case study The application of the proposed method is extended to the subway tunnel in Pusan to illustrate the various applicability of the EBM beyond parameter estimation. Applications in this chapter concentrate on the followings. First, in order to check the most in¯uential geotechnical parameters, the model identi®cation process is performed by the Akaike Information Criterion. Then, parameter estimation process is performed for all the models. Lastly, the reduction of uncertainties by utilizing the EBM is assessed so that the proposed methodology is proved to be a way of systematic consideration of the uncertainties.
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Fig. 6. Variation of objective function by the variation of the geotechnical parameters.
4.1. Ground condition The ground at the site consists of ®ll layer (cohesionless soil), residual soil and highly weathered rock from the ground surface, as shown in Fig. 7. Below the overburden, it is composed of Bulkuksa granite. The tunnel was constructed through the highly weathered rock, which is classi®ed as `Poor Rock' with a RMR description. The typical cross section of the tunnel is shown in Fig. 8. 4.2. Measurement The crown subsidence and sidewall convergence measured by tape extensometer are used as the observed data. The location of the measuring points is shown in Fig. 9. The characteristic lines are obtained from the extrapolation of the measured
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Table 2 Optimized parameters with changing of the number of measuring points Number of measuring points Assumed values 2 4 6 8 10
E (t/m2)
Ko
c (t/m2)
(deg)
3000 3182.1 3008.8 3013.9 3013.7 3013.6
0.50 0.51 0.52 0.47 0.46 0.46
3.00 3.00 3.01 3.02 3.01 3.02
30.0 31.5 26.2 27.6 27.6 28.0
Fig. 7. Idealized ground condition of the subway tunnel in Pusan.
data. Fig. 10 and Table 3 show the obtained characteristic lines and absolute displacement. In this case, the ratio of deformation which occurred ahead of the tunnel face was 45% of the absolute displacement [9]. The two stations located within close proximity are chosen for the tunnel convergence measurements; however, for the application of the EBM, two stations are combined by endowing each section with time steps 1 and 2. 4.3. Selection of model parameters The zone in which the ground shows plastic deformation due to tunnel excavation is assessed by comparing the stresses obtained from the elastic ®nite element calculation and the strength obtained from the Hoek±Brown model. As shown in Fig. 11, the ground motion is almost elastic except at the invert corner zones. Therefore, the elastic analysis appears appropriate for this site. Since the residual soil with granite origin is mostly cohesionless, it is almost impossible to obtain undisturbed samples with typical sampling techniques. The
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Fig. 8. Typical tunnel cross section.
Fig. 9. Location of measuring points.
elastic modulus and the strength parameters can be empirically estimated from correlations with SPT N-value. Estimation of Ko values is even more dicult. Therefore, some typical values shown in Fig. 7 are used as the prior information in this study. The coecient of variation of SPT N-value is about to be 0.26 [10]. The coecient of variation of the E and K0 are selected as 0.3 including uncertainty of the SPT N-value.
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Fig. 10. Characteristic lines.
4.4. Model identi®cation 4.4.1. Description of alternative models In order to assess the best model, three models are introduced. The model I-1 is adopted directly from the site investigation (see Fig. 7). Since the elastic modulus of residual soil selected in the model I-1 seems too small, the model I-2 adopted the larger elastic modulus for the residual soil layer. The estimated elastic modulus of granitic soil for model I-2 was about 10,000 ton/m2, which was obtained from triaxial test results by Kim [11]. In the model II, residual soils and weathered rocks are considered as a single `weathered zone'.
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Table 3 Results of regression Locations
Station I Station II
Side Wall Crown Surface Sidewall Crown Surface
Functions
Displacements from regression analysis (mm)
Absolute displacementsa (mm)
u x 2:593 1 ÿ eÿ0:098x u x 7:316 1 ÿ eÿ0:126x ± u x 2:833 1 ÿ eÿ0:165x u x 2:593 1 ÿ eÿ0:122x ±
3.56 7.38 3.41 3.48 7.77 3.41
6.47 13.42 6.20 6.32 14.12 6.20
a Absolute displacement is the sum of the displacement occurred ahead of the tunnel face and the displacement from regression analysis.
Fig. 11. Plastic zone assessed by Hoek±Brown's criterion.
4.4.2. Selection of the best model for back analysis The measurement error, Vu , might then be signi®cant by ®eld conditions; however, since the relative magnitude of Vu to Vp can be adjusted by applying the , Vu is assumed to be a unit matrix for the comparison under the same condition. The comparison of the model I-1, model I-2 and model II by the AIC is shown in Fig. 12. It is found from this ®gure that the model I-2 gives the minimum AIC among the three models, and the model II also looks small, even though slightly larger than model I-2. 4.5. Parameter estimation Using the values obtained in the previous stage, the optimized parameters are obtained. As shown in Table 4, the objective functions of the model I-2 and model II are quite small compared with the function of the model I-1.
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Fig. 12. The hyperparameter vs AIC value in each model.
4.6. Uncertainty evaluation The error ratio of mechanical instrumentations is generally within 4% [12]. However, gross errors may include reading errors, computational errors, incorrect installation, missing the installation time, etc. Therefore, it is likely to be a larger value in the ®eld. The measurement errors of 4, 20, and 30% are considered in this study to evaluate the in¯uence of the measurement accuracy on the magnitude of uncertainties. Table 5 shows the reduction of uncertainties from the prior to the posterior with the variation of measurement errors in model I-2. The elastic modulus of the weathered rock layer is more sensitive by the changing of measurement accuracy.
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Table 4 Results of parameter estimation Ground condition
Model I-1 Model I-2 Model II a
Elastic modulus (t/m2)
Initial horizontal stress coecient
Prior information
Optimized value
Prior information
Optimized value
3000 20,000 10,000 20,000 20,000
4610 22,630 11,990 19,620 18,990
0.50 0.50 0.50 0.50 0.50
0.49 0.50 0.50 0.505 0.49
R.S. W.R. R.S. W.R. W.Z.a
Objective function
28.60 0.26 0.76
W.Z., weathered zone.
Table 5 Coecient of variation of geotechnical parameters
E
Prior error Posterior error
a
a
Measurement error 0:3 Measurement error 0:2 Measurement error cew 0:04
Ko
R.S.
W.R.
R.S.
W.R.
0.3
0.3
0.3
0.3
0.261 0.215 0.052
0.237 0.189 0.045
0.295 0.252 0.064
0.292 0.248 0.062
(.), coecient of variation.
5. Conclusions Developed in the present study is a new feedback system in which the ®eld measurements and the prior information are systematically combined together. The Extended Bayesian Method (EBM) was adopted for the back analysis. The main advantage of the Extended Bayesian Method is the introduction of the hyperparameter . The is in¯uenced by sensitivity of the parameters and uncertainties existing in both the prior information and the measurements. The small value of implies that the measurements possess more weight than the initially estimated information, and vice versa. Therefore, by observing the value of , the relative importance in feedback analysis can easily be seen. The conclusions of this study could be summarized as follows: 1. The four parameters needed to evaluate the ground motion in elasto-plastic condition are; E, Ko , c, and . It is shown from the example problem that estimating the four parameters is extremely time consuming and sometimes dicult in achieving convergence with the limited number of observation data. For the given example study, six measuring points appear to be the minimum requirement for the analysis of the elasto-plastic ground behavior.
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2. The model identi®cation is carried out between two initial models based on the site investigation and the simpli®ed model in the subway tunnel. From the model identi®cation, the model I-2 was selected as the optimized model in this site. And the model II was also found to be a good choice showing a low AIC value and a very small objective function (even though larger than model I-2). And the AIC value of model I-1 was obtained to be much higher. This result shows that the appropriate selection of initially estimated parameter is a very important task in the feedback system, and sometimes the simpli®ed models can give a good result. 3. Table 5 shows the trend of reduction in the prior information errors by applying the EBM with the observed data. The more accurate the ®eld instrument is, the more reduction in the uncertainty of the prior information can be accomplished. References [1] Cividini A, Maier G, Nappi A. Parameter estimation of a static geotechnical model using a Bayes's approach. Int J Rock Mech Min Sci & Geomech Abstr 1983;20(5):215±26. [2] Neuman SP, Yakowitz S. A statistical approach to the inverse problem of aaquifer hydrologyÐ1. Theory, Water Resources Research 1979;15(4):845±60. [3] Honjo Y, Wen-Tsung L, Sakajo S. Application of Akaike information criterion statistics to geotechnical inverse analysis: the extended Bayesian method. Structural Safety 1994;14:5±29. [4] Honjo Y, Wen-Tsung L, Guha S. Inverse analysis of an embankment on soft clay by extended Bayesian method. Int J Numer Anal Methods Geomech 1994;18:709±34. [5] Akaike H. Information theory and an extension of the maximum likelihood principle. 2nd Int. Symp. on Information Theory, Akademiai Kiado, 1973, pp. 267±281. [6] Beck JV, Arnold KJ. Parameter estimation in engineering and science. New York: John Wiley & Sons, 1977. [7] Owen DRJ, Hinton E. Finite elements in plasticity: theory and practice. Swansea: Pineridge Press, 1980. [8] Kim DH. Geotechnical approach on geotechnical parameter estimation for underground structures. Ph.D. thesis, Department of Civil Engineering, Korea University, 1996. [9] Lee IM, Kim DH, Choi HS, Choi SI. The 3-dimensional tunnel analysis considering stress concentration; load-distribution ratio. J of Korean Geotech Soc 1996;12(1):87±108. [10] Harr ME. Reliability-based design in civil engineering, New York: McGraw-Hill, 1987. [11] Kim YJ. Constitutive characteristics of decomposed Korean granites, Ph.D. thesis, Department of Civil Engineering, Korea University, 1994. [12] Hanna TH. Field instrumentation in geotechnical engineering. Rockport, MA: Trans Tech, 1985.
Parameter estimation using extended Bayesian method in tunnelling In-Mo Lee a,*, Dong-Hyun Kim b a
Department of Civil Engineering, Korea University, Seoul, 136-701, Korea Geotechnical Engineerring Team, Kolon Construction Co., Ltd., Seoul, 135-100, Korea
b
Received 27 July 1998; received in revised form 30 October 1998; accepted 2 November 1998
Abstract Eort was made in this paper to formulate a parameter estimation technology by systematically combining the ®eld measurements and the prior information of underground structure. The Extended Bayesian Method (EBM) was adopted for the feedback analysis and the ®nite element analysis, which was implemented to predict the ground response. Determined in the present study were various geotechnical parameters including the elastic modulus (E), the initial horizontal stress coecient at rest (Ko), the cohesion (c) and the internal friction angle ( ). The validity of the feedback system proposed herein was demonstrated through an elastoplastic example problem. The proposed method was applied to an actual tunnel site in Pusan, Korea and has shown to be highly eective in actual ®eld problems. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction In spite of drastic development of underground technology, there are still many uncertainties that exist in the designing and constructing of underground structures. This is mainly due to the discrepancy between the laboratory and in-situ tests and limitations of site investigation techniques during the design stage, etc. In order to reduce these uncertainties, ®eld instrumentation results obtained during construction stage are compared with the initially estimated ground properties. The feed back system can be used to estimate the optimum ground properties by minimizing the dierence between the predicted and measured ground motions.
* Corresponding author. Tel.: +82-2-3290-3314; fax: +82-2-928-7656; e-mail: [email protected] 0266-352X/99/$Ðsee front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S026 6-352X(98)0003 1-7
110
I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124
The ordinary least squares (OLS) method was widely used, because of its easy application to non-linear geotechnical problems without complex mathematical concepts. However, this method cannot consider the prior information in the process of parameter estimation. Bayesian approach deals with both the prior information and the measurement data [1]. A signi®cant drawback of the Bayesian method is the incommensurate matching between the two components, since the objective function is equally divided by them. To overcome this, Neuman and Yakowitz [2] introduced the adjusting positive scalar term, which adjusts the weights of Jo and Jp : J Jo Jp
1
where, Jo and Jp are objective function of the observed and predicted parameters, respectively. This concept is called extended Bayesian method [3,4]. 2. Background of the proposed feedback system 2.1. Model identi®cation The selection of the most appropriate model among alternative models in the Bayesian approach is possible by introducing the concept of the Akaike Information Criterion [3±5]. The best model among the various alternatives can be identi®ed when the AIC(x) value in Eq. (2) is minimized. 2 AIC x ÿ2 ln fk x j k x 2 dim k where, x is arbitrary input vector, fk x j k is the probability distribution function of the kth alternative model, and dim k is the number of the model parameters for the kth model. The AIC concept is again used in Eq. (3) by the Bayesian approach utilizing the prior and posterior distribution of . AIC ÿ2 ln p x j 2 dim 3 Q where, p xj j f xj d. The ®rst term in Eq. (2) indicates how well the model ®ts with to the observed data, and dim of the second term means the number of model parameters. 2.2. Formulation of extended Bayesian method The extended Bayesian method is a technique to give the optimum match between the observed data and prior information. The observation data vector can be expressed as [3,4], uk uk x j "k
4
I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124
111
where, uk is the ®eld observation at step k, uk is the calculated result vector at step k, "k is the error vector assumed to follow "k N 0; Vu , where Vu is an N N covariance matrix, x is the known input data vector, is the model parameter vector to be estimated, and N is the total number of observation points. The prior information vector is assumed as follows, p
5
where, p is the prior (initially estimated) mean vector of the model parameter vector, is the uncertainty of the prior information assumed to follow N 0; Vp = , where Vp is the prior M M covariance matrix of , is a scalar that adjusts the magnitude of the uncertainty, and M is the number of model parameters. The Bayesian estimator ^ is the one that minimizes the following function with respect to : J j
K X T k uk ÿ uk x j Vÿ1 u ÿ uk x j ÿ pT Vÿ1 u p ÿ p
6
k1
The parameter can be estimated again by the Bayesian theorem maximizing the following function: n o 1 ^ Jp ^ l j u ; p L j u ; p ÿ NK ln Jo 2 9 8 > > > > > > = M j Vÿ1 j 1 < p const ln K > P 2 > > > k T ÿ1 k > > S Vu S Vp j; :j
7
k1
where,
@uk S @T k
^
:
Once we obtain the from Eq. (6), the parameter is then estimated. The modi®ed Box±Kanemasu iteration method is used to estimate the by minimizing Eq. (5) [6]. 2.3. Uncertainty evaluation of model parameters It is a concern to know how to reduce the uncertainties by adopting the proposed technique; the comparison between the uncertainty of initially estimated parameters (the prior estimation) and the uncertainty of the estimated values utilizing the EBM (the posterior estimation). Unfortunately, the posterior distribution is not a simple
112
I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124
normal distribution with respect to due to the fact that uk is a non-linear function of , which makes the covariance matrix non-linear. In order to resolve this problem, uk at the estimated values, ^ is linearized. With the linearized uk , it is possible to obtain the posterior covariance matrix based on the conventional Bayesian theory [3,4]: (
K X k ÿ1 Sk T Vÿ1 p u S Vp
)ÿ1 8
k1
As shown in Eq. (8), the covariance matrix of the posterior estimation (p ) is composed of the covariance matrix of the prior estimation (Vp ), and the covariance matrix of measurements (Vu ). It is interesting to observe that, when the ®rst term in Eq. (8) is added to the second term, it makes the variance of the posterior estimation smaller since the covariance matrix of measurements is added to the prior estimation as an inverse matrix form. 2.4. Numerical method for back analysis The existing elasto-plastic ®nite element program, developed by Owen and Hinton [7], was modi®ed to simulate the tunnel excavation and the support system. Mohr± Coulomb failure criterion was used to represent the plastic behavior. The code developed in this study is applied to the example problem and the results are compared with the benchmark analytical solution (Kirsch equation) and the commercial ®nite dierence solution, FLAC-2D. The maximum relative error between the results of analytical solution and the modi®ed FEM analysis is less than 8% in the check points [8]. This relative error is considered acceptable considering many assumptions made in the numerical analysis. These procedures for the proposed system is summarized as a ¯owchart in Fig. 1. 3. Evaluation of the proposed scheme 3.1. Data preparation A comprehensive evaluation of the proposed feedback system was carried out for the elasto-plastic case in a hypothetical site. The geological condition of the site is presented in Fig. 2, and the tunnel is supposed to be located in the residual soil layer. The typical cross section of the tunnel is in a horse shoe shape having a diameter of 10 m, and the primary support system consisting of a 10 cm thick soft shotcrete. The typical values of geotechnical parameters are considered as the initial estimated geotechnical parameters (prior information) for this case. Since the site is a hypothetical one, the observed data are calculated using the ®nite dierence method code, FLAC-2D. The measurement points and displacements are summarized in Fig. 3 and Table 1.
I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124
113
Fig. 1. Framework for the feedback system.
3.2. Sensitivity analysis Sensitivity analysis is done to check the change in the ground response due to perturbations in the parameters. Figs. 4±6 show the results of the sensitivity analysis. During the sensitivity analysis for a chosen parameter, the other parameters are ®xed with initially estimated values. According to these ®gures, it is shown that the ground motion depends on the elastic modulus and the lateral earth pressure coecient during elastic behavior. If plastic deformation occurs, elastic modulus and internal friction angle will be a major in¯uencing factor to the ground motion. It has also been concluded that the ground motion is insigni®cantly aected by cohesion. 3.3. Parameter estimation As shown in Fig. 3, the plastic deformation occurs around the tunnel. Therefore, four parameters are required to be estimated: the cohesion, the internal friction
114
I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124
Fig. 2. Geological condition of a hypothetical site.
Fig. 3. Location of measuring points.
angle, the elastic modulus, and the initial horizontal stress coecient. Since the tunnel crown is located within the residual soil layer, the parameters only for the residual soil layer are assessed. The coecient of variation of model parameters are assumed to be 0.3 considering large uncertainties are encountered. In order to check the eect of the number of measuring points in feedback system, they are varied during the analysis. The results of the analysis are listed in Table 2, and the prior estimates are also tabulated for comparison. Table 2 shows that at least six measuring points must be installed to obtain a reliable estimations.
I.-M. Lee, D.-H. Kim / Computers and Geotechnics 24 (1999) 109±124 Table 1 Measuring points and displacements No. 1 2 3 4 5 6 7 8 9 10
Distance from tunnel center (m)
Measured (assumed) displacements (mm)
4.5 5.5 6.5 7.5 8.5 4.55 5.7 6.7 7.7 8.7
74.5 66.3 59.6 54.5 50.4 30.8 13.5 4.6 2.0 0.9
Fig. 4. Sensitivity of geotechnical parameters on crown settlement.
115
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Fig. 5. Sensitivity of geotechnical parameters on sidewall convergence.
4. Case study The application of the proposed method is extended to the subway tunnel in Pusan to illustrate the various applicability of the EBM beyond parameter estimation. Applications in this chapter concentrate on the followings. First, in order to check the most in¯uential geotechnical parameters, the model identi®cation process is performed by the Akaike Information Criterion. Then, parameter estimation process is performed for all the models. Lastly, the reduction of uncertainties by utilizing the EBM is assessed so that the proposed methodology is proved to be a way of systematic consideration of the uncertainties.
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Fig. 6. Variation of objective function by the variation of the geotechnical parameters.
4.1. Ground condition The ground at the site consists of ®ll layer (cohesionless soil), residual soil and highly weathered rock from the ground surface, as shown in Fig. 7. Below the overburden, it is composed of Bulkuksa granite. The tunnel was constructed through the highly weathered rock, which is classi®ed as `Poor Rock' with a RMR description. The typical cross section of the tunnel is shown in Fig. 8. 4.2. Measurement The crown subsidence and sidewall convergence measured by tape extensometer are used as the observed data. The location of the measuring points is shown in Fig. 9. The characteristic lines are obtained from the extrapolation of the measured
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Table 2 Optimized parameters with changing of the number of measuring points Number of measuring points Assumed values 2 4 6 8 10
E (t/m2)
Ko
c (t/m2)
(deg)
3000 3182.1 3008.8 3013.9 3013.7 3013.6
0.50 0.51 0.52 0.47 0.46 0.46
3.00 3.00 3.01 3.02 3.01 3.02
30.0 31.5 26.2 27.6 27.6 28.0
Fig. 7. Idealized ground condition of the subway tunnel in Pusan.
data. Fig. 10 and Table 3 show the obtained characteristic lines and absolute displacement. In this case, the ratio of deformation which occurred ahead of the tunnel face was 45% of the absolute displacement [9]. The two stations located within close proximity are chosen for the tunnel convergence measurements; however, for the application of the EBM, two stations are combined by endowing each section with time steps 1 and 2. 4.3. Selection of model parameters The zone in which the ground shows plastic deformation due to tunnel excavation is assessed by comparing the stresses obtained from the elastic ®nite element calculation and the strength obtained from the Hoek±Brown model. As shown in Fig. 11, the ground motion is almost elastic except at the invert corner zones. Therefore, the elastic analysis appears appropriate for this site. Since the residual soil with granite origin is mostly cohesionless, it is almost impossible to obtain undisturbed samples with typical sampling techniques. The
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Fig. 8. Typical tunnel cross section.
Fig. 9. Location of measuring points.
elastic modulus and the strength parameters can be empirically estimated from correlations with SPT N-value. Estimation of Ko values is even more dicult. Therefore, some typical values shown in Fig. 7 are used as the prior information in this study. The coecient of variation of SPT N-value is about to be 0.26 [10]. The coecient of variation of the E and K0 are selected as 0.3 including uncertainty of the SPT N-value.
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Fig. 10. Characteristic lines.
4.4. Model identi®cation 4.4.1. Description of alternative models In order to assess the best model, three models are introduced. The model I-1 is adopted directly from the site investigation (see Fig. 7). Since the elastic modulus of residual soil selected in the model I-1 seems too small, the model I-2 adopted the larger elastic modulus for the residual soil layer. The estimated elastic modulus of granitic soil for model I-2 was about 10,000 ton/m2, which was obtained from triaxial test results by Kim [11]. In the model II, residual soils and weathered rocks are considered as a single `weathered zone'.
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Table 3 Results of regression Locations
Station I Station II
Side Wall Crown Surface Sidewall Crown Surface
Functions
Displacements from regression analysis (mm)
Absolute displacementsa (mm)
u x 2:593 1 ÿ eÿ0:098x u x 7:316 1 ÿ eÿ0:126x ± u x 2:833 1 ÿ eÿ0:165x u x 2:593 1 ÿ eÿ0:122x ±
3.56 7.38 3.41 3.48 7.77 3.41
6.47 13.42 6.20 6.32 14.12 6.20
a Absolute displacement is the sum of the displacement occurred ahead of the tunnel face and the displacement from regression analysis.
Fig. 11. Plastic zone assessed by Hoek±Brown's criterion.
4.4.2. Selection of the best model for back analysis The measurement error, Vu , might then be signi®cant by ®eld conditions; however, since the relative magnitude of Vu to Vp can be adjusted by applying the , Vu is assumed to be a unit matrix for the comparison under the same condition. The comparison of the model I-1, model I-2 and model II by the AIC is shown in Fig. 12. It is found from this ®gure that the model I-2 gives the minimum AIC among the three models, and the model II also looks small, even though slightly larger than model I-2. 4.5. Parameter estimation Using the values obtained in the previous stage, the optimized parameters are obtained. As shown in Table 4, the objective functions of the model I-2 and model II are quite small compared with the function of the model I-1.
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Fig. 12. The hyperparameter vs AIC value in each model.
4.6. Uncertainty evaluation The error ratio of mechanical instrumentations is generally within 4% [12]. However, gross errors may include reading errors, computational errors, incorrect installation, missing the installation time, etc. Therefore, it is likely to be a larger value in the ®eld. The measurement errors of 4, 20, and 30% are considered in this study to evaluate the in¯uence of the measurement accuracy on the magnitude of uncertainties. Table 5 shows the reduction of uncertainties from the prior to the posterior with the variation of measurement errors in model I-2. The elastic modulus of the weathered rock layer is more sensitive by the changing of measurement accuracy.
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Table 4 Results of parameter estimation Ground condition
Model I-1 Model I-2 Model II a
Elastic modulus (t/m2)
Initial horizontal stress coecient
Prior information
Optimized value
Prior information
Optimized value
3000 20,000 10,000 20,000 20,000
4610 22,630 11,990 19,620 18,990
0.50 0.50 0.50 0.50 0.50
0.49 0.50 0.50 0.505 0.49
R.S. W.R. R.S. W.R. W.Z.a
Objective function
28.60 0.26 0.76
W.Z., weathered zone.
Table 5 Coecient of variation of geotechnical parameters
E
Prior error Posterior error
a
a
Measurement error 0:3 Measurement error 0:2 Measurement error cew 0:04
Ko
R.S.
W.R.
R.S.
W.R.
0.3
0.3
0.3
0.3
0.261 0.215 0.052
0.237 0.189 0.045
0.295 0.252 0.064
0.292 0.248 0.062
(.), coecient of variation.
5. Conclusions Developed in the present study is a new feedback system in which the ®eld measurements and the prior information are systematically combined together. The Extended Bayesian Method (EBM) was adopted for the back analysis. The main advantage of the Extended Bayesian Method is the introduction of the hyperparameter . The is in¯uenced by sensitivity of the parameters and uncertainties existing in both the prior information and the measurements. The small value of implies that the measurements possess more weight than the initially estimated information, and vice versa. Therefore, by observing the value of , the relative importance in feedback analysis can easily be seen. The conclusions of this study could be summarized as follows: 1. The four parameters needed to evaluate the ground motion in elasto-plastic condition are; E, Ko , c, and . It is shown from the example problem that estimating the four parameters is extremely time consuming and sometimes dicult in achieving convergence with the limited number of observation data. For the given example study, six measuring points appear to be the minimum requirement for the analysis of the elasto-plastic ground behavior.
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2. The model identi®cation is carried out between two initial models based on the site investigation and the simpli®ed model in the subway tunnel. From the model identi®cation, the model I-2 was selected as the optimized model in this site. And the model II was also found to be a good choice showing a low AIC value and a very small objective function (even though larger than model I-2). And the AIC value of model I-1 was obtained to be much higher. This result shows that the appropriate selection of initially estimated parameter is a very important task in the feedback system, and sometimes the simpli®ed models can give a good result. 3. Table 5 shows the trend of reduction in the prior information errors by applying the EBM with the observed data. The more accurate the ®eld instrument is, the more reduction in the uncertainty of the prior information can be accomplished. References [1] Cividini A, Maier G, Nappi A. Parameter estimation of a static geotechnical model using a Bayes's approach. Int J Rock Mech Min Sci & Geomech Abstr 1983;20(5):215±26. [2] Neuman SP, Yakowitz S. A statistical approach to the inverse problem of aaquifer hydrologyÐ1. Theory, Water Resources Research 1979;15(4):845±60. [3] Honjo Y, Wen-Tsung L, Sakajo S. Application of Akaike information criterion statistics to geotechnical inverse analysis: the extended Bayesian method. Structural Safety 1994;14:5±29. [4] Honjo Y, Wen-Tsung L, Guha S. Inverse analysis of an embankment on soft clay by extended Bayesian method. Int J Numer Anal Methods Geomech 1994;18:709±34. [5] Akaike H. Information theory and an extension of the maximum likelihood principle. 2nd Int. Symp. on Information Theory, Akademiai Kiado, 1973, pp. 267±281. [6] Beck JV, Arnold KJ. Parameter estimation in engineering and science. New York: John Wiley & Sons, 1977. [7] Owen DRJ, Hinton E. Finite elements in plasticity: theory and practice. Swansea: Pineridge Press, 1980. [8] Kim DH. Geotechnical approach on geotechnical parameter estimation for underground structures. Ph.D. thesis, Department of Civil Engineering, Korea University, 1996. [9] Lee IM, Kim DH, Choi HS, Choi SI. The 3-dimensional tunnel analysis considering stress concentration; load-distribution ratio. J of Korean Geotech Soc 1996;12(1):87±108. [10] Harr ME. Reliability-based design in civil engineering, New York: McGraw-Hill, 1987. [11] Kim YJ. Constitutive characteristics of decomposed Korean granites, Ph.D. thesis, Department of Civil Engineering, Korea University, 1994. [12] Hanna TH. Field instrumentation in geotechnical engineering. Rockport, MA: Trans Tech, 1985.