Efficient Probabilistic Back-Analysis of Slope Stability Model Parameters

Transcription

1 Efficient Probabilistic Back-Analysis of Slope Stability Model Parameters J. Zhang 1 ; Wilson H. Tang, Hon.M.ASCE 2 ; and L. M. Zhang, M.ASCE 3 Abstract: Back-analysis of slope failure is often performed to improve one s knowledge on parameters of a slope stability analysis model. In a failed slope, the slip surface may pass through several layers of soil. Therefore, several sets of model parameters need to be back-analyzed. To back-analyze multiple sets of slope stability parameters simultaneously under uncertainty, the back-analysis can be implemented in a probabilistic way, in which uncertain parameters are modeled as random variables, and their distributions are improved based on the observed slope failure information. In this paper, two methods are presented for probabilistic back-analysis of slope failure. For a general slope stability model, its uncertain parameters can be back-analyzed with an optimization procedure that can be implemented in a spreadsheet. When the slope stability model is approximately linear, its parameters can be back-analyzed with sensitivity analysis instead. A feature of these two methods is that they are easy to apply. Two case studies are used to illustrate the proposed methods. The case studies show that the degrees of improvement achieved by the back-analysis are different for different parameters, and that the parameter contributing most to the uncertainty in factor of safety is updated most. DOI: 1.161/ ASCE GT CE Database subject headings: Landslides; Slope stability; Models; Parameters; Failure investigation; Probability; Reliability; Bayesian analysis. Author keywords: Landslides; Slope stability; Failure investigation; Probability methods; Reliability; Bayesian analysis. Introduction 1 Ph.D. Candidate, Dept. of Civil and Environmental Engineering, The Hong Kong Univ. of Science and Technology, Clear Water Bay, Hong Kong. zjxac@ust.hk 2 Chair Professor, Dept. of Civil and Environmental Engineering, The Hong Kong Univ. of Science and Technology, Clear Water Bay, Hong Kong corresponding author. wtang@ust.hk 3 Associate Professor, Dept. of Civil and Environmental Engineering, The Hong Kong Univ. of Science and Technology, Clear Water Bay, Hong Kong. cezhangl@ust.hk Note. This manuscript was submitted on April 2, 28; approved on July 23, 29; published online on December 15, 29. Discussion period open until June 1, 21; separate discussions must be submitted for individual papers. This paper is part of the Journal of Geotechnical and Geoenvironmental Engineering, Vol. 136, No. 1, January 1, 21. ASCE, ISSN /21/ /$25.. A slope failure implies that the factor of safety of the slope at the moment of failure is unity. Based on this information, backanalysis is often carried out to improve knowledge on slope stability parameters. Stability parameters may include both soil strength parameters and pore water pressure parameters at the moment of slope failure. Both deterministic methods e.g., Wesley and Leelaratnam 21; Tiwari et al. 25 and probabilistic methods Luckman et al. 1987; Gilbert et al. 1998; Chowdhury et al. 24 have been used to perform back-analysis. The philosophies behind deterministic and probabilistic back-analysis methods are different. While deterministic back-analysis methods intend to find a set of parameters that would result in the slope failure, probabilistic back-analysis methods recognize that there might be numerous combinations of such parameters, but their relative likelihoods are different, which can be quantified by probability distributions. Major advantages of probabilistic backanalysis methods include: 1 it provides a logical way to incorporate information from other sources in the back-analysis and 2 it is capable of back-analyzing multiple sets of slope stability parameters simultaneously. One possible disadvantage of probabilistic back-analysis methods is that they are usually not as easy to implement compared with deterministic back-analysis methods. The purpose of this paper is to suggest two efficient methods for probabilistic back-analysis of slope failures, i.e., a method based on optimization and a method based on sensitivity analysis. To avoid the potential cumbersome programming work, a spreadsheet template is developed to implement the back-analysis method based on optimization. When the slope stability model is approximately linear, multiple sets of stability parameters may be back-analyzed even without resorting to the optimization process. Two case studies are used to illustrate the suggested methods and implementation ideas. Methods of Probabilistic Back-Analysis Let g,r denote a slope stability model such as a model based on a limit equilibrium method, where vector denoting uncertain input parameters and r vector denoting input parameters without uncertainty. In the following text, r is dropped from the slope stability model for simplicity. The uncertain input parameters may include both soil strength parameters and pore water pressure parameters. In practice, one may already have some prior knowledge on before performing the back-analysis based on either geotechnical tests or engineering experiences. For simplicity, assume that the prior knowledge on can be described by a multivariate normal distribution with a mean of and a covariance matrix of C. The objective of probabilistic back-analysis is JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21 / 99

2 then to improve the probability distribution of based on observed slope failure information. One common concern in back-analysis is that the prediction from the theoretical slope stability model g may be subjected to model error. In such a case, even if the calculated factor of safety is unity, the actual factor of safety of the slope may not be unity. To quantify the effect of model imperfection, the model uncertainty can be modeled as a random variable, which is defined as = y g 1 where y=actual factor of safety and =random variable characterizing the modeling uncertainty. For simplicity, assume follows the normal distribution with a mean of and a standard deviation of. With the above assumptions, the improved distribution of can also be approximated by a multivariate normal distribution see Appendix I based on Tarantola 25. Let d and C d denote the improved mean and covariance matrix of, respectively. As a multivariate normal distribution can be fully determined by its mean and covariance matrix, the task in the probabilistic back-analysis is then reduced to determining d and C d. For a general slope stability model g, d is a point that maximizes the chance to observe the slope failure event, and it denotes the most probable combination of parameters that had led to the slope failure event. d can be obtained by minimizing the following misfit function 2S see Appendix I based on Tarantola 25 : 2S = g + 1 T g T C 1 The improved covariance matrix of, C d, which describes the magnitude of uncertainty in each component of as well as the dependence relationships among various components of, can be determined as follows see Appendix I based on Tarantola 25 : C d = GT 1 G 2 + C 1 G = g 4 = d where G=row vector representing the sensitivity of g with respect to at d. The back-analysis based on the above equations is called probabilistic back-analysis method by optimization in this paper. At a first glance, the application of this method seems not easy, as in Eq. 2 the minimization is coupled with the slope stability model g. However, as will be shown later in this paper, the minimization can be automated in a commonly available spreadsheet, thus avoiding the possible troublesome programming work. The expressions in Eqs. 2 4 are indeed quite general, and are applicable no matter g is linear or not. However, when g is approximately linear, d and C d can be determined analytically with the following equations without resorting to the minimization procedure see Appendix II based on Tarantola 25 : 2 3 d = + C H T HC H T g C d = HT 1 H 2 + C 1 H = g 7 = where g =predicted factor of safety calculated at point and H=row vector representing the sensitivity of g with respect to at point. The back-analysis based on Eqs. 5 7 is called simplified probabilistic back-analysis in this paper. Compared with the method by optimization, the simplified method is easier to apply, as it does not involve any minimization procedure. The limitation of the simplified method is that it is applicable only when g is largely linear. Fortunately, based on the authors experience, many stability models are in fact rather linear in spite of the fact that numerical iterations are usually involved in slope stability models, which was also noticed by Mostyn and Li The following comments are made regarding the two methods proposed in this paper for back-analysis of slope failure. 1. In the two methods described above, the emphasis is to improve the knowledge on slope stability parameters through back-analysis. The probability density function of the model uncertainty variable is not updated in the back-analysis. 2. The probabilistic back-analysis procedures described above are based on the assumption that the prior distribution of can be represented or approximated by a multivariate normal distribution. This assumption simplifies the implementation of the back-analysis methods. If this assumption is not satisfied, the prior distribution of can be transformed into a multivariate normal distribution using techniques like Nataf transformation Der Kiureghian and Liu 1986, and the transformed parameters can then be back-analyzed using the above procedures. The transformation would introduce additional computational work into the back-analysis. In the examples presented in this paper, the prior distributions are assumed to be multivariate normal. 3. The credibility of the back-analysis results depends on the accuracy of the identified soil stratigraphy including the accurate location of the slip surface as well as the understanding of the failure mechanism. The back-analysis results may be meaningless if the back-analysis is carried out based on an unrealistic stratigraphy or based on a wrong failure mechanism. It is assumed in this study that both soil stratigraphy relevant to the back-analysis and the location of the slip surface are well defined through postfailure investigation. Moreover, it is assumed that the slope stability model g can correctly reflect the failure mechanism. Back-analysis of slope failures when the stratigraphy is uncertain is out of the scope of this study. 4. Eqs. 2 7 show that the back-analyzed distribution of depends on and C, i.e., the prior knowledge about, as well as the slope failure information. Hence, the backanalyzed distribution is a combination of the prior knowledge about and the knowledge learned from the slope failure event. As will be seen later in this paper, a change in prior distribution will cause changes in the back-analyzed distribution of. This implies that, in a probabilistic backanalysis, information from other sources on the slope stability parameters is also important / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21

3 5. In minimizing Eq. 2, it is possible that the global minimum point cannot be found. To reduce the chance of missing the global minimum point, the minimization of Eq. 2 can be solved starting from various initial points. If indeed several points are found to satisfy the minimization criteria, the differences between posterior means and posterior covariance matrices evaluated at these points would increase when these points are more separated. However, among various points, the point that gives the minimal value of 2S is most plausible because it has the maximum likelihood, and the posterior mean and covariance matrix should be estimated at this point. 6. The optimization method and simplified method for backanalysis are quite general. Both methods are applicable to updating multiple model input parameters, no matter the slope failure occurs in a single soil layer or in layered soils. Both methods are also applicable no matter is correlated or not in its prior distribution. 7. It is known that soil property varies spatially. If spatial variability is to be considered, one should obtain the representative distributions of the spatial averages of the input parameters along the slip surface separately prior to the backanalysis. These prior distributions can then be used in the back-analysis to obtain the updated posterior distributions of the spatial averages of the input parameters. Step-by-Step Implementation In the two suggested methods, the improved joint probability distribution of the uncertain parameters is approximated by the multivariate normal distribution, and the task of back-analysis is then reduced to the determination of the mean d and covariance matrix C d of the multivariate normal distribution. The formulas to calculate d and C d have been summarized in Eqs. 2 4 and Eqs. 5 7, respectively. The probabilistic backanalysis methods described above can be implemented in three steps: 1. Select a stability model, g and identify the uncertain parameters ; 2. Quantify the knowledge on prior to the back-analysis and on the model uncertainty of g. This step is in fact a process of determining, C,, and ; and 3. Improve the probability distribution of considering the slope failure event. In this step, d and C d, which contain the improved knowledge on, are calculated using the formulas presented in the previous section. In the following, these three steps are illustrated in detail using two case studies. Example 1 Shek Kip Mei Slope Failure The slope analyzed here is a cut slope located at Shek Kip Mei, Hong Kong, which failed during a heavy rainstorm on August 25, After the slope failure, FMSW 2 carried out a detailed postfailure investigation into this slope, and the findings reported by FMSW 2 formed the basis of this case study. Fig. 1 shows a representative cross section of this slope as well as the identified slip surface FMSW 2. In this figure, Layers II, III, IV, and V denote slightly decomposed granite SDG, moderately decomposed granite MDG, highly decomposed granite HDG, and completely decomposed granite CDG, respectively. Based Elevation (m) V IV II Groundwater table by FMSW (2) II V/IV II V on GEO 2, while SDG and MDG can be considered as rocks, HDG and CDG are classified as soils. The groundwater table at the moment of slope failure judged by FMSW 2 is also plotted in Fig. 1. Based on this groundwater table, the pore water pressure ratio along the slip surface is in the range of.3, and the average pore pressure ratio is about.13. Slope Stability Analysis Model and Uncertain Parameters The method of Spencer 1967 is used to calculate the factor of safety of this cut slope. Let c and denote the average cohesion and average friction angle along the slip surface, respectively. As c and are uncertain, they are modeled as random variables. In addition to the above two shear strength parameters, the pore water pressures at the moment of slope failure are also unknown. Here the effect of pore water pressure is modeled by a homogeneous equivalent pore water pressure ratio along the slip surface, r u, which is defined as the ratio of pore water pressure to vertical total stress. Although r u is not an accurate representation of a pore pressure distribution along the slip surface, it is an index with effect equivalent to that of the complex and unknown pore pressure El-Ramly et al. 25. Based on the above discussion, the parameters to be back-analyzed for this slope can be denoted by the vector = c,,r u T. Assessment of Uncertainties Fig. 2 shows the saturated triaxial test results from CDG samples retrieved using a Mazier sampler outside the distressed zone of the slope. FMSW 2 judged that the average cohesion and friction angle of the CDG are 8 kpa and 38, respectively. The lower bound cohesion and friction angle inferred from the triaxial tests are 2 kpa and 36, respectively. Assuming that the lower bound implied by the triaxial tests is two standard deviations away from the mean value, the standard deviations of cohesion and friction angle are judged to be 3 kpa and 1, respectively. The above statistics for cohesion and friction angle do not consider the effect of test uncertainty. The existence of test uncertainty would introduce additional uncertainty into the soil strength parameters. On the other hand, the effect of spatial variability of soil properties is also not considered in the previously assessed statistics for II V IV III/II IV/III Horizontal distance (m) Slip surface Fig. 1. A representative cross section of the Shek Kip Mei landslide II, III, IV, and V: slightly, moderately, highly, and completely decomposed granite adapted from FMSW 2 II V II JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21 / 11

4 q' =.5(σ1'-σ3' )(kpa) cohesion and friction angle. As soil property fluctuates spatially along the slip surface, a high value at a certain point will be balanced by lower values at other points. The existence of spatial variability reduces the uncertainty of soil property along the slip surface. Due to lack of information for satisfactory evaluation of test uncertainty and spatial variability, the effects of test uncertainty and spatial variability are assumed to be largely cancelled out. With this assumption, the previously assessed statistics for cohesion and friction angle are used to represent the statistics of cohesion and friction averaged over the slip surface. The coefficient of correlation between c and is not reported in FMSW 2. Previous studies showed that c and may be negatively correlated e.g., Cherubini 2. In this study, the back-analysis will be first carried out assuming c and are statistically independent. A sensitively analysis will be carried out later to check the effect of the correlation between c and on the back-analysis results. There was no measurement on the pore pressures at the moment of slope failure. Therefore, the prior knowledge on average pore pressure ratio is assessed mainly by judgment. In this study, we assume that the average pore pressure ratio r u at the moment of slope failure follows a normal distribution with a mean of.15 and a standard deviation of.1. We further assume that c,, and r u are statistically independent. Therefore, our prior knowledge on c,, and r u can be represented by a multivariate normal distribution with a mean of = 8,38,.15 T, and a covariance matrix as follows: C = p' =.5(σ 1 '+ σ 3 ' )(kpa) Fig. 2. Results of triaxial tests on CDG 1 =major principal effective stress, 3 =minor principal effective stress adapted from FMSW The units for c and in the above and C are kpa and degree, respectively. To be clear, the prior distribution used here is called 1. In the method of Spencer 1967, all the three equilibrium conditions horizontal force equilibrium, vertical force equilibrium, and moment equilibrium are satisfied. Christian et al suggested that the model uncertainty of the simplified method of Bishop 1955 has a mean of.5 and a standard deviation of.7. As the results from Bishop s simplified method are usually close to rigorous methods that satisfy all three equilibrium conditions, it is assumed that the model uncertainty of the method of Spencer 1967 can also be modeled as a normally distributed random variable with a mean of.5 and a standard deviation of.7, i.e., =.5 and =.7. Probabilistic Back-Analysis As mentioned before, since the improved follows a multivariate normal distribution, the task of probabilistic back-analysis is to determine d and C d. For a general slope stability model, d can be found through maximizing the chance to observe the slope failure event, or equivalently, through the minimization of Eq. 2. The work involved in minimizing Eq. 2, however, is unlikely to be trivial because the slope stability model g is coupled with the minimization problem. In this study, a spreadsheet template has been developed to automate the probabilistic back-analysis method by optimization. The layout of the spreadsheet is shown in Fig. 3, which can be accomplished in three steps. 1. Construct the slope stability analysis model g to relate the calculated factor of safety with c,, and r u. The method of Spencer 1967 is constructed in the spreadsheet in this step. In Fig. 3, the slope mass is divided into 2 slices. The factor of safety of the slope can be obtained conveniently through the Solver routine in a spreadsheet. In Fig. 3, the formula in cell N27 is set to be =1* J27. The settings in the Solver are Minimize N27, by changing cell I27 and J27, subjected to M24= and N24=. The implementation of the method of Spencer 1967 in the spreadsheet in this step is similar to that suggested by Low Find the improved mean d by optimization. In the second step, the prior mean, prior covariance matrix C, model uncertainty and, and observed information factor of safety equal to unity are typed into the spreadsheet. Based on the factor of safety calculated from Step 1, the formula for misfit function 2S in Eq. 2 is typed into cell N33. The posterior mean d can be obtained by minimizing cell N33 using the Solver. The setting in the Solver is Minimize N33, by changing cell I27, J27, G25, G26, and G27, subjected to M24= and N24=. 3. Calculate the improved covariance matrix C d. In this step, a sensitivity analysis at d can be performed to obtain the sensitivity vector G as defined in Eq. 4. This sensitivity vector is then substituted into Eq. 3 to calculate C d. To reduce the chance of missing the global minimum point, the minimization of the misfit function is repeated starting from several initial points selected within the possible ranges of the parameters. It is found that all these minimization problems converge to the same minimum point. Therefore, it seems that in this problem, the point that minimizes the misfit function is unique, and this point is then regarded as d. Based on d and C d calculated from the optimization method, the improved distribution for each parameter can be obtained. The prior and the improved probability density functions for c,, and r u are compared in Fig. 4. Although in this study the factor of safety is calculated using the method of Spencer 1967, other slope stability analysis methods can also be used in a similar fashion as long as they can be implemented in a spreadsheet. In fact, many slope stability models have been implemented in spreadsheets, such as Bishop s simplified method Bishop 1955; El-Ramly et al. 22, Janbu s generalized method Janbu 1987; Low and Tang 1997, and Chen- Morgenstern s method Chen and Morgenstern 1983, Low et al Although in this paper it is shown that the probabilistic back-analysis method by optimization can be implemented in a 12 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21

5 1 A B C D E F G H I J K L M N O Step 1: Build the slope stability model 2 Slice# α b h u c tanφ s A B C D Q Q*cos( α - θ ) E Posterior mean, obtained by Q Q*cos(a- θ ) 24 γ H θ,μ minimizing N33 θ d 1E-7 1E c φ β F s F s 27 r u Step 2: Calculate the posterior mean 3 Eq. (2) 31 μ θ C θ μ ε σ ε Observed data 2S( θ) Step 3: Calculate the posterior covariance matrix Posterior covariance, obtained by Eq. (3) 37 Sensitivity vector, Eq. (4) 38 C θ d G Notes: γ and H denote the unit weight of soil and the slope height, respectively; β and F s denote the angle of interslice force and factor of safety of the slope, respectively; α, b, h, and u are the base inclination angle, width, height and pore pressure of a slice, respectively; s, A, B, C, D are five intermediate variables that facilitate coding in the spreadsheet; When Q =and Q*cos(a- θ ) =, force and moment equirlibriums are satisfied, respectively. 49 Fig. 3. Layout of the spreadsheet for probabilistic back-analysis of slope failure spreadsheet, we do not mean that this method must be implemented in the spreadsheet. Other tools can also be used. Fig. 5 shows the relationships between the factor of safety and c,, and r u around. These relationships are rather linear in the likely ranges of these parameters. Therefore, the slope stability parameters may also be back-analyzed using the simplified method. To implement the simplified method, one can perform a sensitivity analysis at point to get g and H first, and then substitute them into Eqs. 5 and 6 to calculate d and C d, respectively. As g and H can be calculated using a conventional slope stability program, the simplified method can be implemented even without resorting to a spreadsheet. To check the accuracy of the simplified method, the updated probability density functions for c,, and r u obtained from this simplified method are also plotted in Fig. 4. The results obtained from the simplified method are almost identical to those from the optimization procedure. The probabilistic back-analysis also provides dependence information among the model parameters. Let c,, c,ru, and,ru denote the correlation coefficients between c and, c and r u, and and r u, respectively. In 1, c,, and r u are assumed statistically independent, so the values of c,, c,ru, and,ru are all zero in the prior distribution. After the back-analysis, the values of c,, c,ru, and,ru become.6,.69, and.2, respectively. The correlations among the variables in the posterior distribution reflect the slope failure event, which imposes a constraint of factor of safety equal to unity at the moment of slope failure. More specifically, in the posterior distribution a higher pore water pressure should be accompanied by higher shear strength parameters such that the calculated factor of safety is close to unity. Since the variance of is very small, the variation in shear strength parameters is dominated by the change in cohesion. This makes the correlation coefficient between c and r u larger than the correlation coefficients among other variables. Information Allocation in Back-Analysis In the probabilistic back-analysis methods, the distributions of all uncertain parameters are modified by the slope failure event simultaneously. A logic question that one may raise is: would the degrees of modification achieved by the back-analysis be the same for all the parameters? This question is answered by Fig. 4, which shows that the changes in the probability distributions for the three parameters are in fact not the same. Among the three parameters, while the probability distribution of r u is changed significantly, the prior and modified distributions for are almost identical. The degree of improvement for a parameter seems to be related to its contribution to the uncertainty in factor of safety. To see this, a simple uncertainty analysis can be performed to calculate the variance of the predicted factor of safety based on the prior knowledge Ang and Tang 27 JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21 / 13

6 (a) Factor of safety (a) c (kpa) c (kpa) (b) (c) φ ( ο ) Factor of safety Factor of safety 1.5 (b) φ ( o ) 1.5 (c) r u r u Fig. 4. Comparison of prior and back-analyzed distributions based on 1 Shek Kip Mei slope 2 var F s g var c c = + g var = + g 2 2 var r u + r 9 u = in which F s =factor of safety. Through Eq. 9, one can identify the contribution of uncertainty from each parameter to the variance of F s, and the results are presented in Fig. 6. The degrees of changes in the probability distributions in Fig. 4 for the three variables are consistent with their contributions to the uncertainty in F s as shown in Fig. 6: while the probability distribution of r u is most updated, the probability distribution of is almost not changed. To further verify the above conclusion, the slope failure is back-analyzed with a different prior distribution on r u, i.e., r u is normally distributed with a mean of.2 and a standard deviation of.7. The mean value of is also increased to 39. The prior mean for in this case is = 8,39,.2 T. The prior covariance matrix in this case is = 32 2 C The above prior distribution is called 2. The posterior distributions of c,, and r u calculated based on 2 are shown in 2 Fig. 5. a Relationship between factor of safety and c =38, r u =.15 ; b relationship between factor of safety and c =8 kpa, r u =.15 ; and c relationship between factor of safety and r u c=8 kpa, =38 Shek Kip Mei slope Fig. 7. For comparison, the prior distributions of c,, and r u are also plotted in this figure. Both the optimization method and simplified method are used to calculate the posterior distributions. The two methods yield practically the same results, as shown in Fig. 7. Among the three variables, while c and r u are more updated, the distribution of is almost not changed. Fig. 8 shows the contribution of uncertainty from each variable to the variance Contribution to variance of Fs (%) cc' f' φ ru r u eε Fig. 6. Contributions to the variance of factor of safety F s from different parameters based on 1 Shek Kip Mei slope 14 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21

7 (a) μc d = 8.4 kpa σc d = 2.57 kpa () (a) μc d = 7.71 kpa σc d = 2.8 kpa () c (kpa) c (kpa) (b) μ φ d = 39. o σ φ d =.99 o () (b) μ φ d = 38.2 o σ φ d = 1. o () φ ( ο ) φ ( ο ) (c) μru d =.2 σru d =.5 () (c) μru d =.17 σru d =.5 () Fig. 7. Comparison of prior and back-analyzed distributions based on 2 Shek Kip Mei slope i d and i d denote backanalyzed mean and standard deviation of i, respectively of F s. The degree of updating for each variable is also consistent to its contribution to the variance of F s. In Eq. 9 it is assumed that the model parameters are statistically independent. When model input parameters are correlated, the correlation among the input parameters could also affect the contribution of a parameter to the variance of F s e.g., Ang and Tang 27. In such a case, the individual contribution of a parameter to variance of F s is not directly obvious. Nevertheless, the Contribution to variance of Fs (%) r u cc' f' φ ru r u eε Fig. 8. Contributions to the variance of factor of safety F s from different parameters based on 2 Shek Kip Mei slope Fig. 9. Comparison of prior and back-analyzed distributions based on 3 Shek Kip Mei slope information allocation analysis for an uncorrelated prior distribution in this section still gives useful insights on how information is allocated in the back-analysis. Effect of Correlation between Model Input Parameters To examine the effect of the correlation between c and on the back-analysis of a slope failure, 1 is modified by assuming c and are negatively correlated with a coefficient of correlation of.5. The resultant prior distribution is called 3 here. Based on the above assumption, the mean of in 3 is = 8,38,.15 T, and the covariance matrix of 3 is = 32 C The slope failure is then back-analyzed again using 3, and a comparison between prior and posterior distributions of c,, and r u are shown in Fig. 9. Both the optimization method and simplified method are used to calculate the posterior distribution, and the two methods again yield practically identical results, as shown in Fig. 9. Through the back-analysis, the values of c,, c,ru, and,ru are changed from.5,, and to.51,.62, and.12, respectively. The negative correlation coefficient between and r u is caused by the combined effects of the positive correlation between c and r u and the negative correlation between c and. For comparison, the posterior values of c,, c,ru, and,ru r u JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21 / 15

8 Elevation (m) 2 1 Silty sand Clay 1 Clay 2 Clay 3 Water table Slip surface Horizontal distance (m) Fig. 1. A representative cross section of Congress Street Cut adapted from Ireland 1954 are.6,.69, and.2, respectively, if the back-analysis is started with 1. This shows that prior distribution has an important effect on the back-analyzed distribution. The backanalyzed distribution is a combination of prior knowledge and observed slope failure information. It is important to obtain highquality prior knowledge on model input parameters before the back-analysis is carried out. Example 2 In many cases the purpose of back-analysis is to improve one s knowledge on the shear strength parameters. The Shek Kip Mei landslide example shows that the knowledge on pore water pressure has important effects on the updating of shear strength parameters. When the knowledge on pore water pressure is poor, the uncertainty caused by pore pressure may dominate the uncertainty in F s. Therefore, the information collected from the back-analysis is mainly allocated to reduce the uncertainty in pore water pressure rather than to reduce the uncertainty in the shear strength parameters. However, if pore water pressure uncertainty is small or pore water pressure is not involved in the back-analysis, the probability distributions of shear strength parameters can be improved more significantly. To illustrate this, the failure of Congress Street Cut Ireland 1954 is analyzed as follows. Congress Street Cut Fig. 1 was located mainly in saturated clays and failed in an undrained manner in 1952 during the construction of Congress Street in Chicago Ireland The slip surface passed through four different layers of soil, i.e., a sandy fill overlaying three saturated clay layers. The cohesion of the sandy fill was zero Oka and Wu 199. In the back-analysis, the factor of safety of this slope is calculated using Bishop s simplified method Let s u1, s u2, and s u3 denote the average undrained shear strengths of the three clay layers, and denote the average friction angle of the sandy fill. The parameters to be back-analyzed in this case are = s u1,s u2,s u3, T. For this slope, sampling was carried out using 51 mm diameter Shelby tubes with samples taken at least 1 m apart Tang et al Based on reported triaxial tests, the mean values of s u1, s u2, and s u3 are 43.14, 32.65, and kpa, respectively, and the standard deviations for these variables are 1.22, 5.47, and 6.77 kpa, respectively Yucemen et al. 1973; Tang et al Variance reduction due to spatial variability was considered in deriving the statistics for s u1, s u2, and s u3 Yucemen et al. 1973; Tang et al The mean of is assumed to be 3 based on Oka and Wu 199, and the coefficient of variation of is assumed to be.2. Assuming further s u1, s u2, s u3, and are statistically independent, the prior knowledge on can be described by a multivariate normal distribution with a mean of = 43.14,32.65, 37.35,3 T and a covariance matrix of 5.47 C = Based on Christian et al. 1994, the model uncertainty of the method of Bishop 1955 is modeled as a normal random variable with a mean of.5 and a standard deviation of.7, i.e., =.5 and =.7. Sensitivity analysis shows that the slope stability model in this case is approximately linear, so the simplified method is applicable. Fig. 11 compares the prior and back-analyzed distributions for s u1, s u2, s u3, and. The distribution of s u3 is modified most significantly by the observed slope failure. This example illustrates that when pore water pressure uncertainty is not involved in the back-analysis, the probability distributions of shear strength parameters may be improved more significantly. The distributions of other three parameters have also been improved by the backanalysis, but the degrees of improvement for them are not as significant as that for s u3. This is because the uncertainty in s u3 has contributed most to the uncertainty in the factor of safety, as shown in Fig. 12. Fig. 11 shows the marginal posterior distributions of the strength parameters. The posterior correlation matrix d, which reflects the dependence relationships among the four strength parameters, is = d It shows that the strength parameters are negatively correlated as expected to satisfy the constraint that the calculated factor of safety based on the combination of posterior parameters should be close to unity as a reflection of the slope failure event. Among the four uncertain model input parameters, the correlation between s u1 and s u3 as well as the correlation between s u2 and s u3 are relatively stronger. The marginal distributions of s u1, s u2, and s u3 are more updated so these variables have more freedom to adjust their correlation coefficients. Summary and Conclusions The research reported in this paper and findings from the paper are summarized as follows: 1. Back-analysis of slope failure is often carried out to improve one s knowledge on the parameters of a slope stability model. To back-analyze multiple sets of slope stability parameters simultaneously under uncertainty, the back-analysis can be implemented in a probabilistic way, in which uncertain parameters are modeled as random variables, and their distributions are improved using observed slope failure information. 16 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21

9 (a) μ su1 d = 4.38 kpa σ su1 d = 9.74 kpa Back analysis s u1 (kpa) Contribution to variance of Fs (%) s c1 c2 c3 f m u1 s u2 s u3 φ ε Back analysis s u2 (kpa) (b) (c) μ su2 d = kpa σ su2 d = 5.24 kpa Back analysis μ su3 d = kpa σ su3 d = 4.27 kpa s u3 (kpa) Fig. 12. Contributions to the variance of factor of safety F s from different parameters Congress Street Cut failure caused by rainfall infiltration. The stability model of this case is largely linear, and the back-analysis results from the two methods are indeed similar. 4. Case studies in this paper show that the degree of change in the probability distribution of a parameter achieved by the back-analysis depends on its contribution to the uncertainty in the factor of safety: the probability distribution of the parameter that contributes the greatest uncertainty to the factor of safety will be changed most. This conclusion is valid if the parameters to be updated are not correlated in the prior distribution. For a slope failure in which the knowledge in pore water pressure is poor, the uncertainty in pore water pressure may dominate the back-analysis. In such a case, the knowledge on shear strength parameters may not be effectively improved by probabilistic back-analysis. Acknowledgements (d) Back analysis μ φ d = o σ φ d =6. o This research was substantially supported by the Research Grants Council RGC of the Hong Kong SAR Project Nos and 6294/4E. Appendix I. Theory of Probabilistic Back-Analysis for a General Slope Stability Model φ ( ο ) Fig. 11. Comparison of prior and back-analyzed distributions Congress Street Cut 2. Two methods are presented for probabilistic back-analysis of slope failure, i.e., a method based on optimization for a general slope stability model, and a simplified method based on sensitivity analysis. While the simplified method is easier to apply in practice, it is only applicable when the slope stability model is approximately linear. 3. For efficient analysis, a spreadsheet template has been developed to implement the back-analysis method based on optimization. Also, the simplified method involves simply a sensitivity analysis together with a conventional slope stability program. The results from the optimization method and the simplified method are compared in a case study of slope Basic Theory The general probabilistic back-analysis theory is first introduced here. Let g,r be a general model to predict the response of a system, where and r are vectors denoting unknown and known parameters, respectively. In the following text, r is dropped from the prediction model for simplicity. Let d be a vector representing the actual system response. The objective of the probabilistic back-analysis is to improve the probability distribution of based on some observed system response d obs through the prediction model g. Before the probabilistic back-analysis, one may already have some prior knowledge on from various sources. Assume the prior knowledge on can be denoted by a multivariate normal distribution f = k 1 exp 1 2 T C 1 14 where k 1 =normalization constant to make the probability density function valid i.e., making the integration of probability density JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21 / 17

10 function in Eq. 14 over the entire domain of equal to unity. Due to the existence of observational uncertainty, the observed data d obs may not be the same as the actual system response d. Assuming that the observational uncertainty can be described by a multivariate normal distribution with a mean vector of zero and a covariance matrix of C d, the probability density function of d given d obs is f d d obs = k 2 exp 1 2 d d obs T C d 1 d d obs 15 where k 2 is another normalization constant to make the probability density function valid. As any prediction model is only an abstraction of the real world, model uncertainty always exists. The existence of model uncertainty makes the predicted response g different from the actual response d even when is exactly known. Assuming the prediction model is unbiased and that the model uncertainty can be described by a multivariate normal distribution with a mean vector of zero and a covariance matrix of C m, the probability density function of d given is f d = k 3 exp 1 2 d g T C 1 m d g 16 where k 3 is again a normalization constant to make the probability density function valid. With the above assumptions, the improved knowledge of considering both f and d obs can be described by its posterior probability distribution Tarantola 25 f d obs = k 4 exp g d obs T C 1 D g d obs + T C 1 17 where k 4 =normalization constant to make the probability density function valid; f d obs =posterior distribution of given d obs ; =prior means vector of ; C =prior covariance matrix of and C D =C d +C m. The posterior density function of may not have a closedform expression except when g is linear. Let be the point where f d obs is maximized. If the prediction model g is nonlinear but is linearizable in the neighborhood of, f d obs can be approximated by a multivariate normal distribution with a mean of d and a covariance matrix of C d, where d and C d are Tarantola 25 d = 18 Extension to a Biased Prediction Model In the literature, prediction models are often assumed unbiased. In geotechnical engineering, however, prediction models could be biased. For example, a two-dimensional slope stability model might be biased towards the conservative side as it neglects threedimensional effects Christian et al Assume that the model uncertainty of a biased model follows a multivariate normal distribution with a mean of m and a covariance matrix of C m. The magnitude of m measures the degree of bias in the model prediction. For example, m = means the model is unbiased, and in this case the model uncertainty is described by Eq. 16. Inthe case where m, although the prediction from g is biased, the prediction from g =g + m is unbiased. Therefore, when g is biased, the above formulations also apply if g is substituted with g, i.e., g + m. For example, the misfit function for a biased model is 2S = g + m d obs T C 1 D g + m d obs + T C 1 22 Application to Back-Analysis of Slope Failure In back-analysis of slope failure, the observation data is that the factor of safety at the moment of slope failure is unity, which is a scalar. In such a case, the model uncertainty, the observation data, and the observational uncertainty are all scalars. Let 2 d,, and 2 denote C d, m, and C m in the back-analysis of slope failure, respectively, where d and are the standard deviations of observation and model uncertainty variables, respectively. Note that in back-analysis of slope failure d obs =1. Assume the observational uncertainty is negligible, d =. In such a case, Eqs. 22, 19, and 2 become Eqs. 2 4, respectively. Appendix II. Probabilistic Back-Analysis Theory for an Approximately Linear Slope Stability Model For a general linear prediction model g =H, the posterior mean and covariance of can be determined using the following relationships without resorting to the optimization procedure Tarantola 25 d = + C H T HC H T + C D 1 d obs H 23 C d = G T C D 1 G + C 1 1 G = g = * 19 2 C d = H T C D 1 H + C If g =H is biased with the mean model uncertainty of m, one can view H + m as a unbiased prediction model, and the posterior mean d can then be determined as follows: can be found by maximizing Eq. 17, or equivalently, minimizing the misfit function 2S, which is defined as follows Tarantola 25 : 2S = g d obs T C 1 D g d obs + T C 1 21 The general probabilistic back-analysis method described above is also known as theory of system identification e.g., Schweppe d = + C H T HC H T + C D 1 d obs H m 25 If the slope stability model is largely linear, Eqs. 25 and 24 may be used to determine the improved mean and covariance matrix of approximately, respectively. As mentioned before, in the back-analysis of slope failure, the observed information is a scalar. The scalar form of Eqs. 25 and 24 are Eqs. 5 and 6, respectively. The accuracy of this simplified method has been checked in the case study. 18 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 21

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