Bayesian machine learning-based method for prediction of slope failure time

Jie Zhang, Zipeng Wang, Jinzheng Hu,*, Shihao Xiao, Wenyu Shang

a Key Laboratory of Geotechnical and Underground Engineering of the Ministry of Education, Tongji University, Shanghai, 200092, China

b Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, China

c Natural Science College, Michigan State University, MI, 48825, USA

Keywords:Slope failure time (SFT)Bayesian machine learning (BML)Inverse velocity method (INVM)

ABSTRACT The data-driven phenomenological models based on deformation measurements have been widely utilized to predict the slope failure time(SFT).The observational and model uncertainties could lead the predicted SFT calculated from the phenomenological models to deviate from the actual SFT. Currently,very limited study has been conducted on how to evaluate the effect of such uncertainties on SFT prediction.In this paper,a comprehensive slope failure database was compiled.A Bayesian machine learning(BML)-based method was developed to learn the model and observational uncertainties involved in SFT prediction, through which the probabilistic distribution of the SFT can be obtained. This method was illustrated in detail with an example.Verification studies show that the BML-based method is superior to the traditional inverse velocity method(INVM) and the maximum likelihood method for predicting SFT.The proposed method in this study provides an effective tool for SFT prediction.

1. Introduction

Due to the difficulty in precisely obtain the physical models for revealing the effects of complex factors such as external environments, geological conditions and human activities on the slope failure (Crosta and Agliardi, 2002; Intrieri and Gigli, 2016; Kothari and Momayez, 2018; Kardani et al., 2021), data-driven phenomenological models based on deformation measurements have been widely used to predict the slope failure time (SFT) (Petley et al.,2005; Mufundirwa et al., 2010; Federico et al., 2012, 2015; Xue et al., 2014). Two categories of uncertainties may exist in the phenomenological models (Zhang et al., 2020a), i.e. the observational uncertainty caused by factors including measurement error and external disturbance (Mazzanti et al., 2015; Intrieri and Gigli,2016; Carlà et al., 2017), and model uncertainty caused by assumptions associated with the phenomenological models (Carlà et al., 2018; Kothari and Momayez, 2018). Due to the existence of the above uncertainties, the predicted SFT calculated from a datadriven model could deviate from the actual SFT (e.g. Venter et al.,2013; Federico et al., 2015).

Recently, efforts have been made to consider the effect of uncertainties on SFT prediction. (Manconi and Giorden, 2015, 2016)assessed the confidence interval (CI) and forecast reliability of the estimated SFT through a bootstrapping resampling strategy considering the influence of the observational uncertainty. Intrieri and Gigli (2016) evaluated the reliability of the SFT prediction by comparing the predictions from several competing phenomenological models at the same time,through which the influence of the model uncertainty on time-to-failure analysis was considered.Zhang et al. (2020a) suggested a maximum likelihood method in which both the effects of observational and model uncertainties on the SFT were considered based on several simplified assumptions.When such assumptions are not valid, how to explicitly consider both model and observational uncertainties when predicting the SFT remains a challenging task.

The Bayesian machine learning (BML) refers to a data-driven method, which can improve modeling capability and prediction performance based on Bayes’ theorem. As it is flexible in representing uncertainties from different sources and powerful in dealing with complex real-world data, it has been popularly adopted in different fields. The usefulness of the BML has been demonstrated in many studies (Shirzadi et al., 2017; Ching and Phoon, 2019; Contreras and Brown, 2019; Wang, 2020; Ma et al.,2021).

The objective of this paper is to develop a BML-based method for probabilistic prediction of SFT, which can effectively overcome the limitations of the maximum likelihood method as suggested in Zhang et al.(2020a).The proposed method will not only provide a chance to examine the effect of the simplified assumptions in the maximum likelihood method for SFT prediction, but also offer a tool to predict the SFT when the maximum likelihood method is not applicable.

The paper is organized as follows. First, the uncertainties relevant to SFT prediction are explained, and the key assumptions involved in the maximum likelihood method are discussed. Then,Bayesian methods are suggested to learn the model and observational uncertainties, through which the probability distribution of the SFT can be determined. Thereafter, with an example, the proposed method is illustrated in detail.Finally,the proposed method is compared with the traditional method for SFT prediction,i.e.the deterministic method and the maximum likelihood method. The suggested method in this paper provides a versatile tool for SFT prediction considering both the slope-specific information and the information from other slopes.

2. Uncertainties involved in SFT prediction

Many phenomenological models have been suggested for SFT prediction,e.g.Saito(1969)’s method based on the plot of time vs.strain or displacement, the inverse velocity method (INVM) based on the plot of time vs. reciprocal of the velocity (R) (Fukuzono,1985), the slope gradient method via the plot of velocity vs. the value of velocity multiplied by time(Mufundirwa et al.,2010),and the tangential angle method using the transformed plot of displacement vs.time(Xu et al.,2011).A comprehensive review and comparison of methods for SFT prediction have been conducted in Federico et al. (2015). Among these methods, the INVM has been widely used because it is easy to use and the interpretation of results is intuitive.In addition,as it can be expressed in a linear form,efficient machine learning algorithm can be developed based on this method. In this paper, the INVM is considered. The suggested method may also be potentially applicable to other methods such as Saito’s method and the slope gradient method, in which the phenomenological models can also be expressed in a linear form.In the following, the INVM will be briefly described.

It is empirically shown that the plot of the reciprocal of the velocity vs.time often approaches linearity,especially in the critical stage of failure (Rose and Hungr, 2007). Assuming that the reciprocal of the velocity is a linear function of the time at the pre-failure stage of the slope, the relationship between the reciprocal of the velocity and the SFT can be expressed as follows (e.g. Fukuzono,1985; Voight,1988; Rose and Hungr, 2007;Carlà et al., 2017).

where R and t represent the reciprocal of the velocity and the time,respectively; and A and tcare the parameters to be calibrated.

The pre-failure stage is often defined as the stage after the point of onset of acceleration (OOA) (e.g. Dick et al., 2014; Carlà et al.,2017). Fig. 1 illustrates how to determine the SFT by the INVM.Comparing Eq. (1) with Fig. 1, it is shown that tcis indeed the intercept of the fitted R-t linear relationship with the time axis.Assuming that the velocity of the slope movement is infinite when the slope failure occurs, tccan be interpreted as the SFT predicted from the INVM. To derive an efficient algorithm for model uncertainty characterization, Eq. (1) can be rearranged to a linear relationship of unknown parameters as follows:

Fig.1. Schematic of the observational and model uncertainties in the SFT prediction.

where B is a coefficient associated with the slope of the R-t curve.

As shown in Fig. 1, the scattered observational data points around the straight line represent the observational uncertainty.The observational uncertainty can be modeled through a normal random variable εowith the mean and standard deviation (SD) of 0 and σo, respectively:

Due to the simplified modeling assumptions,the intercept of the R-t curve may not be exactly the actual SFT,as shown in Fig.1.The disparity between the calculated SFT and the actual one is called the model uncertainty. Let tarepresent the actual SFT. In order to analyze the effect of model uncertainty, the tarelated to the calculated SFT tcis as follows (Zhang et al., 2020a):

where εmis a normal random variable with the mean and SD of μmand σm, respectively.

To realistically predict the SFT,both the model and observational uncertainties should be taken into consideration. The maximum likelihood method suggested by Zhang et al. (2020a) was used to calibrate the above uncertainties. In the following text, the maximum likelihood method will be briefly reviewed,and the key assumptions involved will be discussed.

3. Review of maximum likelihood method

In Zhang et al. (2020a), the observational uncertainty modeled based on Eq. (1) can be written as follows:

where φ(?) represents the standard normal probability density function (PDF); and ρirepresents the reciprocal of the observed velocity at time ti, i.e. the observed value of R.

Let θ* represent the maximum value of θ. According to the principle of maximum likelihood, when the number of observations is large, θ can be approximated as a multivariate normal vector with a mean μθ=θ*and a covariance matrix Cθ,where Cθis associated with the Hessen matrix of the logarithm of the likelihood function of θ at point θ*.Note that tcis one of the elements of θ.Provided that the PDF of θ is obtained,the PDF of tc,which is the marginal PDF of θ, can also be obtained.

In Eq.(4),the mean and the SD of εm,i.e.μmand σm,characterize the model uncertainty. Let γ = {μm, σm}. Suppose r slopes are collected to calibrate the model uncertainty. Let μcjand σcirepresent the mean and the SD of tcof the jth slope.Note that the values of μcjand σciare calculated by means of the maximum likelihood procedure introduced previously.Let δjrepresent the observed SFT of the jth slope.Let δ={δ1,δ2,…,δr}.Supposing the SFTs of all the slopes are statistically independent, the likelihood function of μmand σmis written as follows:

According to the principle of maximum likelihood, when the number of slopes is large, the optimal values of μmand σmcan be derived by maximizing Eq. (7).

After the model and observational uncertainties are characterized,the mean and the SD of the SFT can be obtained.Assuming the SFT follows the normal distribution, the cumulative distribution function(CDF)of SFT is computed as follows(Zhang et al.,2020a):

where Φ(?) represents the standard normal CDF.

As mentioned above, the maximum likelihood method is founded on the following assumptions:

(1) Both the model and the observational uncertainties are estimated by means of the maximum likelihood procedure,in which it is assumed that the amount of observed data is far larger than the number of model parameters.

(2) When estimating the observational uncertainty, the distribution of tcis assumed to be normal.

(3) When estimating the model uncertainty, only the best estimate values of the model uncertainty parameters are figured out. The uncertainties associated with μmand σmare not considered.

(4) The distribution of SFT is assumed to be normal.

Therefore, although the above maximum likelihood method suggested in Zhang et al. (2020a) is easy to use, it is necessary to develop methods which can bypass the above assumptions. The availability of such methods will not only provide a chance to examine the effect of the simplified assumptions in the maximum likelihood method for SFT prediction, but also provide a tool to predict the SFT while the maximum likelihood method is not applicable. In the following, we will introduce such a method through BML.As will be shown,the proposed method in the paper requires less assumptions and can provide predictions which accord better with the observed SFT.

4. Machine learning of model uncertainty

4.1. Bayesian network

As the model uncertainty refers to the disparity between model predictions and observed SFT, it can be studied through a systematic analogy between the predicted SFT and the observed one for large amounts of slopes,which involves large amounts of uncertain variables. As the Bayesian network is capable of modeling the complex dependence relationships among a large set of uncertain variables(e.g.Aguilera et al.,2011;Bartlett and Cussens,2017),the machine learning method proposed in this study will be developed through a Bayesian network.

Like the maximum likelihood method,suppose that r slopes are collected to learn the model uncertainty.Fig.2 depicts the structure of the Bayesian network. In this figure, Bj, tcjand σojdenote the unknown parameters when analyzing the jth slope with the INVM as given by Eq.(3).For the jth slope(j=1,2,…,r),suppose there are njdata points for calibration of the INVM. Let tji(i = 1, 2, …, nj)denote the ith observation time corresponding to the observed Rjiof the jth slope.According to Eq.(3)and the normal assumption on εo, the conditional PDF of tjigiven Bj, tcjand σ2ojcan be written as follows:

In a Bayesian network, the nodes at the head and the tail of an arrow are called the child node and the parent node, respectively.The nodes without parents are called the root nodes. In Fig. 2, the root nodes include μm, σ, Bj, tcjand. To perform the Bayesian learning, the prior PDFs of the root nodes should be specified.

Fig. 2. Bayesian network for model uncertainty calibration.

4.2. Prior PDF for the Bayesian network

In principle, the prior PDF of the random variables of the root nodes should be settled according to the prior knowledge so that the prior PDF of all variables can be obtained. When the prior knowledge about the root nodes is lacking,non-informative priors can be used(Del Castillo,2007).On the other hand,when conjugate priors are adopted, the involved computational work can be significantly simplified (Del Castillo, 2007). Due to the above considerations, conjugate prior distributions are adopted for the root nodes in this study,which facilitate the development of an efficient Gibbs sampling algorithm to learn the database of SFT.Based on the Bayesian network and the normality of Eqs. (9) and (10), the conjugate prior PDFs for μm, Bj, and tcjare normal (e.g. Gelman et al.,2013):

4.3. Algorithm for machine learning

The implementation of the above algorithm requires the analytical expressions of a series of conditional PDFs as mentioned in Steps (1)-(3), which have been summarized in the Appendix.With the above procedure,the samples of different variables in the Bayesian network can be obtained, including those of μmand σ2m.When drawing samples with MCMCS, one should judge if the Markov chain converges to the equilibrium state. Assessing the convergence of a Markov chain is one of the most challenging problems in MCMCS, and many methods have been suggested to analyze the convergence of the Markov chain (e.g. Cowles and Carlin,1996; Brooks and Roberts,1998; Kass et al.,1998; Sinharay,2003). However, none of these methods can ensure the convergence of a Markov chain within a finite number of samples. A review and comparison of different techniques for convergence checking can be found in Cowles and Carlin(1996).In practice,the convergence is often checked empirically by observing if the Markov chain generates samples with stable statistics such as the median and the correlation(e.g.Geman and Geman,1984).If stable statistics can be obtained, the number of samples in the Markov chain can be considered sufficient (e.g. Gelman et al., 2013; Ching and Phoon, 2019). In this paper, such an empirical method is adopted.For ease of illustration,the dataset of these samples of μmand σ2mis denoted as S-1.

Note in addition to the MCMCS,approaches such as the Bayesian updating with structural reliability method suggested in Straub and Papaioannou (2015) and adaptive Bayesian updating with subset simulation (e.g. Giovanis et al., 2017; Jiang et al., 2020) are also increasingly used for solving high dimensional Bayesian problems.Although MCMCS is used in this study, other approaches may also be potentially useful to solve the model uncertainty characterization problem as described in this study.

5. Machine learning of the observational uncertainty

where X = (1nN×1RN)contains all the observed data; 1nN×1= (1,1,…,1)Tis the vector consists of nN1’s; ^β = (XTX)-1XTtN;and p is the dimension of β which equals 2 in this study. In the literature, X is usually called the design matrix(e.g.Castillo et al.,2015).Based on Eqs.(23)-(26),the posterior samples of β can be obtained,including the samples of tcN. Note that the large sample assumption and the normal assumption about tcNare both not required in the suggested method in this study. The above method is also called Bayesian linear regression in the literature (Smith, 1973; Walter and Augustin, 2010). Starting from k = 1, the samples of tcNcan be drawn based on Eqs. (21) and (22) using the following procedure,i.e. the procedure of the Gibbs sampler.

For ease of illustration,the dataset of the samples of tcNis called S-2 in this paper.

6. Failure time prediction of the new slope

7. Suggested procedure for SFT prediction

To facilitate implementation, the procedure for predicting SFT with the method suggested in this paper is summarized as follows:

(1) Learn the model uncertainty through the Bayesian network based on which the samples of μmand σ2mcan be obtained.In this step, the knowledge from slope failure data is learned.

(2) Learn the observational uncertainty associated with tcNthrough Bayesian regression analysis of the monitoring data at the new slope to be analyzed,for which the samples of tcNcan be obtained. In this step, the knowledge from the monitoring data at the new slope is learned.

Fig. 3. Displacement data of the Abbotsford slope (Adapted from Hancox, 2008).

(3) Simulate the samples of the actual SFT with samples of μm,σ2m, and tcNbased on Eq. (27). With these samples, the histogram and empirical CDF of the SFT are obtained.

Notably, the calibration of the model and the observational uncertainty are separate. One might be interested if the two types of uncertainties can be calibrated simultaneously. Such an idea is also tested. However, it turns out that when the two types of uncertainties are calibrated together, the model is very difficult to converge. Hence, the two-stage calibration method is used in this paper.

8. An illustrative example of the abbotsford landslide

8.1. The Abbotsford landslide

The Abbotsford landslide is located in southwest Dunedin,New Zealand (Hancox, 2008). The slope lies on a spur that rises at an angle of 20°-25°for about 100 m and flattens toward the top. A sand quarry modifies the southern end of the slope, and a prehistoric slide lies to the north.The underlying rock of the landslide is comprised of mudstone, overlain by clayey to silty sand. Large cracks were found between Edwards and Mitchell streets to the west side of the slope in early July 1979.Fig.3 shows the cumulative displacement across the crack in Mitchell street determined by the survey monitoring line from June 18 (t = 0) to August 8,1979. In early July,the ground moved about 10 mm per day.Triggered by the rainfall, the ground movements accelerated to about 650 mm per day prior to the final movement.The compression rolls and cracks with a length of 30 m also developed in the prehistoric slide area in mid-July.The final movement of the slope lasted about 30 min from about 9 p.m.on August 8,1979(t=51.88 d).The slope mass with a volume of 5 million m3slid southeast down about 50 m, which destroyed more than 20 houses and dammed Miller Creek.Following the criterion introduced in Dick et al.(2014)and Segalini et al. (2018), the OOA point is identified as t = 39 d. The observed data of the reciprocal of the velocity of this slope after the OOA point are shown in Fig. 4.

Fig.4. Observed data of the reciprocal of the velocity of the Abbotsford slope after the OOA point.

Table 1 The landslide database compiled in this paper.

8.2. Landslide database

To characterize the model uncertainty of the INVM, the displacement data of other 54 landslides are collected. Together with the Abbotsford landslide,a landslide database with 55 slopes is formed. Table 1 summarizes the general information of the 55 landslides. As revealed from Table 1, both rock and soil landslides are contained in the database as both types of slopes can be analyzed through the linear INVM. The majority of the landslides are triggered by water-related factors such as rainfall,groundwater and snow melt, as well as production activities of human being.Next,the monitoring data of the 2nd-55th landslides in Table 1 will be used as the training dataset for model uncertainty characterization, through which the SFT of the Abbotsford slope will be analyzed.

Fig.6. Difference between the calculated SFT in the deterministic INVM and the actual one.

8.3. Calibration of model uncertainty

First,the monitoring data of the 2nd-55th landslides in Table 1 are substituted into the Bayesian network, and the model uncertainty is learned through MCMCS.The length of the Markov chain is 105. It was found that the Markov chain of each variable can soon provide stable statistics with the number of samples increasing.To remove the influence of the initial point, the first 5 ×104samples are discarded.Then the other 5×104samples can be considered in the convergence stage and are collected as the samples of the posterior distributions.Fig.5a and b shows the sample histograms of μmand σ2m, respectively. Computed with these samples, the means of μmand σ2mare 0.36 d and 0.37 d2,respectively;the SDs of μmand σ2mare 0.14 d and 0.15 d2, respectively. As is shown in the above analysis,the uncertainties associated with μmand σ2mare not negligible.

8.4. Calibration of observational uncertainty

Substituting the data of Fig. 4 into Eqs. (23)-(26), one can then draw samples for σ2oNand tcNfrom the distribution defined by Eqs.(21) and (22). The histogram of tcNis shown in Fig. 5c. The mean and the SD of the calculated tcNare 50.7 d and 0.98 d,respectively.Note that the value of σmis similar to that of the SD of tcN,indicating that in this example,the observational and the model uncertainties are of similar magnitudes, and hence both types of uncertainties are important for SFT prediction.

Fig. 7. Comparison of results from the maximum likelihood method and BML-based method: (a) Mean of tcN; (b) SD of tcN; (c) μm; (d) σ2m; (e) Mean of taN; and (f) SD of taN.

In the maximum likelihood method,tcNis assumed to follow the normal distribution.After obtaining the samples from the Bayesian method, it is interesting to assess if the normal assumption about tcNis appropriate.Here,the Jarque-Bera test is conducted to verify the normality of tcN, which serves as a normality goodness-of-fit test suitable for large datasets (Jarque and Bera,1987). The critical value for the Jarque-Bera test is 5.97 at the 0.05 significance level.In this example,the Jarque-Bera statistic is 47,which is far larger than the critical value. Therefore, the null hypothesis that tcNis normal should be rejected when the significance level is 0.05.

8.5. SFT prediction

The histogram of taNis shown in Fig.5d.For these samples,the Jarque-Bera statistic is equal to 6.85,slightly larger than the critical value when the significance level is 0.05.Thus,the null hypothesis of normality of taNis rejected.The mean and the SD of taNare 51.12 d and 1.42 d, respectively. Based on the samples of taN, the 95%CI of the SFT is[48.4 d,53.9 d].The observed SFT of the Abbotsford slope,51.88 d,is within the 95%interval of taNderived from the suggested procedure.

For comparison, the predicted SFT is 51.63 d when the traditional deterministic INVM is used. When adopting the maximum likelihood method, the 95% CI of the SFT is [49.77 d, 53.88 d]. The 95% CI predicted via the maximum likelihood method is narrower than that predicted using the BML-based method in this study,indicating that the uncertainty relevant to the SFT is underestimated in the maximum likelihood method. Nevertheless, the SFT is located in the CI determined based on the BML-based method and the maximum likelihood method. In the following, a systematic comparison of the three methods will be conducted.

9. Comparison of methods for SFT prediction

To show the advantage of the suggested method,the SFT of each slope in Table 1 was analyzed using the traditional deterministic method, the maximum likelihood method, and the BML-based method. When analyzing the SFT in Table 1 with the maximum likelihood method or the BML-based method,the rest slopes were used as the training dataset to learn the model uncertainty.

Fig. 8. Jarque-Bera statistics for tcN and taN.

Fig. 9. Comparison of the observed SFT and the 95% CI of the SFT: (a) Maximum likelihood method; and (b) BML-based method.

First, the traditional deterministic method was carried out to forecast the SFT. Let tcN,ddenote the SFT calculated by the traditional deterministic method.Let Δ=ta-tcN,ddenote the difference between the calculated SFT and the actual one. By definition, if Δ>0,the calculated SFT is earlier than the actual one,resulting in false warnings. If Δ < 0, the calculated SFT is later than the actual one, resulting in missing alarms (i.e. warnings that are too late).Fig.6 shows the values of Δ for each slope in Table 1.As can be seen from this figure,there are 33 out of 55 slopes with Δ being positive,and 22 out of 55 cases with Δ being negative.Among the 55 cases,there are 30 cases with the Δ values less than 1 d, indicating that the INVM is reasonable for SFT prediction. Nevertheless, it is also observed that the greatest difference between the actual and the predicted SFTs can be up to about 94 d. Such a difference can be caused by both the observational and the model uncertainties. If such uncertainties are not considered, the reliability of the prediction from the traditional INVM is largely unknown.

Then,the maximum likelihood method and BML-based method are used to analyze the SFT of each slope.Fig.7a and b compares the mean and the SD of tcNcalculated through the maximum likelihood method and the BML-based method, respectively. As is shown in these two figures, the mean values of tcNpredicted based on the two methods are quite close. However, the SD predicted based on the maximum likelihood method are generally smaller than those calculated using the BML-based method, indicating that the observational uncertainty is underestimated in the maximum likelihood method.

Fig.7c and d compares the values of μmand σmestimated by the maximum likelihood method and the BML-based method respectively. As mentioned previously, the uncertainties of μmand σmin the maximum likelihood method are not considered. Hence, the values of μmand σmpredicted by the maximum likelihood method are represented as points in these two figures. For comparison, in the BML-based method, the uncertainties associated with μmand σmare considered.The 95%CI of μmand σmestimated by the BMLbased method is shown in Fig. 7c and d. It is meaningful to notice that the values of μmand σmestimated by the maximum likelihood method are well within the 95%CI estimated based on the Bayesian method. On the other hand, the uncertainties associated with μmand σmare not negligible.Hence,the maximum likelihood method also underestimates the model uncertainty.

Fig.7e and f compares the mean and the SD of taNcalculated by the maximum likelihood method and the BML-based method. As can be seen in Fig. 7e, the mean values of taNcalculated using the two methods are largely consistent. The values of SD calculated with the maximum likelihood method are generally smaller than those calculated by the BML-based method in this paper. This is reasonable, since the model uncertainty is underestimated by the maximum likelihood method as well the observational uncertainty.

To check the normality assumptions about tcNand taNmade in the maximum likelihood method, Fig. 8 shows the Jarque-Bera statistics for tcNand taNfor each slope in Table 1. As mentioned above,the critical value for the Jarque-Bera test for normality test is 5.97 at the significance level of 0.05.As can be seen from Fig.8,the Jarque-Bera statistics for tcNand taNfor the slopes in Table 1 are generally greater than 5.97. Therefore, the null hypothesis of normality should be rejected.

Let taLand taUdenote the lower and upper bounds of the 95%CI of the predicted SFT, respectively. To facilitate verification, a normalized actual SFT, ta,norm, is defined as follows:

By definition,if ta,normis between-1 and 1,it indicates that the 95%CI of the predicted SFT covers the actual SFT.If ta,normis smaller than-1,the actual SFT is earlier than the lower bound of the 95%CI.If ta,normis greater than 1, the actual SFT is later than the upper bound of the 95%CI of the SFT.Fig.9a shows the normalized actual SFT of all the 55 slopes when each of them is considered as the verification example for the case of maximum likelihood method.There are 6 slopes out of 55 slopes with the 95%CI of the predicted SFT not covering the observed SFT. Fig. 9b compares the observed SFT and the 95%CI of the SFT calculated by the suggested method in this study.In this case,the actual SFT of all the 55 slopes falls within their predicted 95% CI, respectively. Overall, while the maximum likelihood method can lead to reasonable prediction of the SFT, it could be unconservative due to the underestimation of the model and observational uncertainties.On the other hand,the BML-based method suggested in this paper is based on less assumptions and can provide more reliable predictions on the SFT.

10. Concluding remarks

Due to model and observational uncertainties, accurate SFT prediction is challenging. In this paper, a BML-based method is proposed to predict the SFT, in which both the model and observational uncertainties are considered. A comprehensive slope database is compiled so that the model uncertainty can be learned through the INVM. Compared with the BML-based method, the previous maximum likelihood method underestimates both the model and observational uncertainties, and hence also underestimates the uncertainty in the actual failure time. A comprehensive comparison among predictions from different methods shows that the prediction from the BML-based method accords better with the observations of slope failure phenomenon.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was substantially supported by the Shuguang Program from Shanghai Education Development Foundation and Shanghai Municipal Education Commission, China (Grant No.19SG19), National Natural Science Foundation of China (Grant No.42072302), and Fundamental Research Funds for the Central Universities, China.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2021.09.010.