Evaluation of liquefaction-induced lateral displacement using Bayesian belief networks

Abstract

Liquefaction-induced lateral displacement is responsible for considerable damage to engineered structures during major earthquakes. Therefore, an accurate estimation of lateral displacement in liquefaction-prone regions is an essential task for geotechnical experts for sustainable development. This paper presents a novel probabilistic framework for evaluating liquefaction-induced lateral displacement using the Bayesian belief network (BBN) approach based on an interpretive structural modeling technique. The BBN models are trained and tested using a wide-range case-history records database. The two BBN models are proposed to predict lateral displacements for free-face and sloping ground conditions. The predictive performance results of the proposed BBN models are compared with those of frequently used multiple linear regression and genetic programming models. The results reveal that the BBN models are able to learn complex relationships between lateral displacement and its influencing factors as cause–effect relationships, with reasonable precision. This study also presents a sensitivity analysis to evaluate the impacts of input factors on the lateral displacement.

Keywords

Bayesian belief network / seismically induced soil liquefaction / interpretive structural modeling / lateral displacement

Introduction

Lateral displacement caused by liquefaction, as a potentially damaging effect of seismic soil liquefaction, has drawn the attention of researchers and engineers. Liquefaction-induced lateral displacement generally occurs on gentle slopes founded on loose sand with groundwater table reasonably close to the ground surface; however, open faces such as stream channels are also susceptible to lateral spreading [¹]. This phenomenon can initiate movement of large blocks of soil that can range from a few centimeters to several meters and cause severe damage to overlaying structures. Consequently, the development of appropriate mitigation strategies for estimating displacements as accurately as possible is essential for a safe and economic design of structures. Numerous methods for predicting lateral displacement under different conditions are available in literature. Al-Bawwab [²] categorized these methods into the following four groups: numerical analyses, simplified analytical methods, soft computing techniques, and empirical methods based on statistical analyses or laboratory test data of lateral ground deformation case-history records. A brief overview of these methods [³–²¹] is presented in Table 1.

Soft computing methods have been applied by several researchers in various domains, such as in engineering (Javdanian [¹⁵] and Guo et al. [²²]) and mathematics (Anitescu et al. [²³]), and have achieved comparatively satisfactory results. However, in the case of liquefaction-induced lateral displacement, existing empirical models, analytical, and numerical models have achieved only limited success, according to comparisons of their results with observed lateral displacements in the field. Furthermore, in a majority of cases, because of scarcities and shortages in their databases, some important factors, such as ground intensity measures and fault types, have not been considered. Although a couple of exceptions exist, such as the studies by Raunch [¹⁹] or Bardet and Liu [²⁴], who applied peak ground acceleration and peak ground velocity to address these scarcities and shortages in data, most research studies on the effects of geology and ground motion intensity remain insufficient. At present, even the best of these aforementioned models cannot predict a reasonable and accurate estimate of liquefaction-induced lateral displacements. Therefore, more comprehensive research efforts will have to be conducted to develop more sophisticated and accurate models that capture the essentials of the lateral displacement problem.

This paper proposes a novel probabilistic framework using Bayesian belief networks (BBNs) based on the interpretive structural modeling (ISM) technique to predict lateral displacement due to soil liquefaction. This framework incorporates major causes of lateral displacement, such as the closest horizontal distance of the seismic energy source to the site or nearest fault rupture surface (R, km), earthquake magnitude (M), and average mean grain size within T₁₅ (D50₁₅, mm), and study in the form of a cause–effect relationship. The main significances of this research work are as follows.

1)‚ In the proposed probabilistic graphical models, there is no need to introduce new parameters (e.g., R* in the multiple linear regression, or multilinear regression (MLR), model) or use functional (e.g., logarithmic) values on the parameters; all parameters can be used in the model as they are, without any normalization or calibration.

2) The model is able to learn complex relationships between lateral displacement and its influencing factors in the form of cause–effect relationships with reasonably good precision as compared to most frequently used multiple linear regression and genetic programming models.

3) The proposed probabilistic framework approach for evaluating liquefaction-induced lateral displacement is compatible with the concept of the geological phenomenon and accounts for a wide range of possible ground geometries without any gaps.

4) The main data set is divided into two subsets for the training and testing stages based on statistical properties, instead of random division, to increase the predictive capability and accuracy of the proposed models.

5) In most cases, because of the scarcities and shortages in previously used databases for empirical and soft computing lateral displacement modeling, some aspects of liquefaction-induced lateral displacement phenomena, such as ground intensity measures, have been ignored. Based on the causative fault types of all earthquakes, peak ground acceleration is estimated and added to the present study, and the data set is expanded and becomes more capable and efficient.

The paper is organized into eight sections. Section 2 presents an overview of empirical and soft computing models for the evaluation of lateral displacement. Section 3 outlines the fundamentals of a BBN. Section 4 explains the steps involved in the ISM technique for the development of the BBN. Section 5 presents the development of liquefaction-induced lateral displacement models for free-face and sloping ground conditions. Section 6 describes the evaluation measures, which include metrics of overall accuracy (OA), kappa, user accuracy, and producer accuracy. Section 7 presents a predictive performance and sensitivity analysis of the proposed BBN models and compares them with the multiple linear regression and genetic programming models. The final section presents the conclusions and information on future research. Lastly, Appendix A contains supplementary materials.

Review of empirical and soft computing methods

Hamada et al. [²¹] proposed a preliminary relationship for estimating horizontal ground displacement based on 60 data sets, most of which were collected from Niigata and Noshiro, Japan. This relationship is expressed as follows:

(1)D_{\mathrm{H}}=\mathrm{0.75}{H}^{\mathrm{1}/\mathrm{2}}{\theta }^{\mathrm{1}/\mathrm{3}},

where D_H is the predicted horizontal ground displacement (m), H is the thickness of the liquefied layer (m), and q is the slope of either the ground surface or slope of the base of the liquefied layer (%). This equation is simple and easy to use. The deficiency of this approach, however, is that it has been proposed based on a very limited amount of data. The Hamada et al. [²¹] method suggests only two parameters from the site geometry that affect the value of lateral displacement, and other geotechnical and seismic parameters are not considered. As a result, this equation is not general enough to be used for other sites.

Youd and Perkins [¹⁸] proposed a “liquefaction severity index” (LSI) to predict the maximum horizontal ground displacement induced by an earthquake. LSI is defined by the following equation:

(2)\log \hspace{0.17em}\mathrm{LSI}=-\mathrm{3.49}-\mathrm{1.86}\hspace{0.17em}\log \hspace{0.17em}R+\mathrm{0.98}{M}_{\mathrm{w}}.

The LSI (inches) is estimated based on the distance R (km) to the energy source of the earthquake, and moment magnitude M_w with a maximum limit of 2.5 m. This equation has attracted the attention of engineers at the time. However, this model assumes that if liquefaction occurs and causes lateral displacement, the amount of ground displacement depends only on the seismic parameters (R, M_w). Whereas this equation might have been suitable for estimating lateral displacements in the western part of the United States, it lacked enough generalization and therefore did not become popular worldwide.

Bardet et al. [²⁰] developed Eqs. (3) and (4) to evaluate lateral ground deformation under free-face and sloping ground conditions, respectively, through MLR using data collected by Bartlett and Youd [²⁵,²⁶], which include three groups of input parameters:

1) Seismic parameters: closest horizontal distance of seismic energy source to the site or nearest fault rupture surface (R, km), and earthquake magnitude (M, although M_w is generally used whenever stated).

2) Topographic parameters: gradient of ground surface (S, %) and free-face ratio (W, %).

3) Geotechnical parameters: average mean particle size within T₁₅ (D50₁₅, mm), average fines content included within T₁₅ (F₁₅, %), and cumulative soil layer thickness with corrected standard penetration test (SPT) blow count number less than 15 (T₁₅, m).

(3)\begin{array}{l}\log (D_{\mathrm{H}}+\mathrm{0.01})=-\mathrm{17.372}+\mathrm{1.248}{M}_w-\mathrm{0.923}\log R-\mathrm{0.014}R+\mathrm{0.685}\log W\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{0.3}\log T_{\mathrm{15}}+\mathrm{4.826}\log (\mathrm{100}-F_{\mathrm{15}})-\mathrm{1.091}D{\mathrm{50}}_{\mathrm{15}},\end{array}

(4)\begin{array}{l}\log (D_{\mathrm{H}}+\mathrm{0.01})=-\mathrm{14.152}+\mathrm{0.988}{M}_w-\mathrm{1.049}\log R-\mathrm{0.011}R+\mathrm{0.318}\log S\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{0.619}\log T_{\mathrm{15}}+\mathrm{4.287}\log (\mathrm{100}-F_{\mathrm{15}})-\mathrm{0.705}D{\mathrm{50}}_{\mathrm{15}}.\end{array}

The model by Youd et al. [¹⁷], originally proposed in 1992, was derived using the MLR method and was based on the updated results of Bartlett and Youd [²⁵,²⁶] for calculating the estimated lateral ground displacement, D_H (m), under free-face and gently sloping ground conditions, as follows:

(5)\begin{array}{l}\log (D_{\mathrm{H}})=-\mathrm{16.713}+\mathrm{1.532}M-\mathrm{1.406}\log R^{*}-\mathrm{0.012}R+\mathrm{0.592}\log W\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{0.540}\log T_{\mathrm{15}}+\mathrm{3.413}\log (\mathrm{100}-F_{\mathrm{15}})-\mathrm{0.795}\log (D{\mathrm{50}}_{\mathrm{15}}+\mathrm{0.1}\hspace{0.17em}\mathrm{mm}),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array}

(6)\begin{array}{l}\log (D_{\mathrm{H}})=-\mathrm{16.213}+\mathrm{1.532}M-\mathrm{1.406}\log R^{*}-\mathrm{0.012}R+\mathrm{0.338}\log S\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{0.540}\log T_{\mathrm{15}}+\mathrm{3.413}\log (\mathrm{100}-F_{\mathrm{15}})-\mathrm{0.795}\log (D{\mathrm{50}}_{\mathrm{15}}+\mathrm{0.1}\hspace{0.17em}\mathrm{mm}),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array}

where

(7)R^{*}=R_{\mathrm{0}}+{\mathrm{10}}^{\mathrm{0.98}{M}_w-\mathrm{5.64}}.

Because of its use of a wide range of data from different earthquakes, and of the geometry of the site, geotechnical data, and characteristics of earthquakes, this model could find more popularity among geotechnical engineers. However, there are still some limitations in its application. For instance, the free-face equation is used when 5≤W≤20%, whereas the ground slope equation is considered valid when W≤1%. This bordering system for the values of W is discontinuous and does not provide any explanation for cases wherein 1<W<5%. Furthermore, the estimations provided by this model for the Chi-Chi earthquake (Chu et al. [²⁷]) and Kocaeli earthquake in Turkey (Cetin et al. [²⁸] and Youd et al. [²⁹]) are not practically applicable for the observed lateral displacements in the aforementioned sites.

Javadi et al. [¹⁴] estimated the lateral displacement LD (D_H, m) for free-face and gently sloping ground conditions via genetic programming (GP) using the updated case-history records database reported by Youd et al. [¹⁷]. The proposed GP model exhibited advantages over the MLR model. The proposed equations for free-face and gently sloping ground conditions are as follows:

(8)\begin{array}{l}{D}_{\mathrm{H}}=-\mathrm{163.1}{\displaystyle \frac{\mathrm{1}}{M^{\mathrm{2}}}}+\mathrm{57}{\displaystyle \frac{\mathrm{1}}{R\hspace{0.17em}\cdot F_{\mathrm{15}}}}-\mathrm{0.0035}{\displaystyle \frac{T_{\mathrm{15}}^{\mathrm{2}}}{W\hspace{0.17em}\cdot \hspace{0.17em}D{\mathrm{50}}_{\mathrm{15}}^{\mathrm{2}}}}+\mathrm{0.02}{\displaystyle \frac{T_{\mathrm{15}}^{\mathrm{2}}}{F_{\mathrm{15}}\hspace{0.17em}\cdot \hspace{0.17em}D{\mathrm{50}}_{\mathrm{15}}^{\mathrm{2}}}}-\mathrm{0.26}{\displaystyle \frac{T_{\mathrm{15}}^{\mathrm{2}}}{F_{\mathrm{15}}^{\mathrm{2}}}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{0.006}\hspace{0.17em}{T}_{\mathrm{15}}^{\mathrm{2}}-\mathrm{0.0013}{W}^{\mathrm{2}}\hspace{0.17em}+\mathrm{0.0002}{M}^{\mathrm{2}}\hspace{0.17em}\cdot W\cdot \hspace{0.17em}{T}_{\mathrm{15}}+\mathrm{3.7},\end{array}

(9)D_{\mathrm{H}}=-\mathrm{0.8}{\displaystyle \frac{F_{\mathrm{15}}}{M}}+\mathrm{0.0014}{F}_{\mathrm{15}}^{\mathrm{2}}+\mathrm{0.16}{T}_{\mathrm{15}}+\mathrm{0.112}S+\mathrm{0.04}{\displaystyle \frac{S\hspace{0.17em}\cdot \hspace{0.17em}{T}_{\mathrm{15}}}{D{\mathrm{50}}_{\mathrm{15}}}}-\mathrm{0.026}R\hspace{0.17em}\cdot \hspace{0.17em}D{\mathrm{50}}_{\mathrm{15}}+\mathrm{1.14.}

Jafarian and Nasri [¹⁶] compiled the most recent LD database, which was based on the uncertainties of distant boreholes, and which outperforms the results by Hamada et al. [²¹], Kanibir [³⁰], Al-Bawwab [²], Bardet et al. [²⁰], Javadi et al. [¹⁴], Youd et al. [¹⁷], and Baziar and Azizkandi [³¹]. The models for free-face and gently sloping ground conditions are expressed as follows:

(10)\begin{array}{l}\log (D_{\mathrm{H}})=-\mathrm{17.95}+\mathrm{1.605}{M}_w-\mathrm{1.8673}{R}^{*}-{(\log (R+\mathrm{20}))}^{-\mathrm{3.3836}}+\mathrm{0.547}\log W\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{0.4431}\log T_{\mathrm{15}}+\mathrm{4.1873}\log (\mathrm{100}-F_{\mathrm{15}})-\mathrm{0.7666}\log (D{\mathrm{50}}_{\mathrm{15}}+\mathrm{0.1}\hspace{0.17em}\mathrm{mm}),\end{array}

(11)\begin{array}{l}\log (D_{\mathrm{H}})=-\mathrm{19.63}+\mathrm{2.0137}{M}_w-\mathrm{2.6124}\hspace{0.17em}\log R^{*}-{(\log (R+\mathrm{20}))}^{-\mathrm{2.7004}}+\mathrm{0.3147}\log S\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{0.6985}\hspace{0.17em}\log T_{\mathrm{15}}+\mathrm{4.1954}\hspace{0.17em}\log (\mathrm{100}-F_{\mathrm{15}})-\mathrm{0.6772}\hspace{0.17em}\log (D{\mathrm{50}}_{\mathrm{15}}+\mathrm{0.1}\hspace{0.17em}\mathrm{mm}).\end{array}

Fundamentals of Bayesian belief network

BBNs were first proposed by Pearl [³²] based on Bayes’ theorem, which is a probabilistic graphical model comprising a directed acyclic graph (DAG) with conditional probability tables. The network consists of nodes, which denote variables, and edges, which represent probabilistic cause–effect relationships, joining the nodes. Nodes and edges represent the qualitative characteristics of the network, whereas conditional probabilities are related to factors or variables denoting the quantitative part, which is determined by the Bayesian formula. The conditional independence rule is as follows:

(12)P(X_i|Y)={\displaystyle \frac{P(Y|X_i)\cdot P(X_i)}{P(Y)}},

(13)P(Y|X_i)={\displaystyle \frac{P(x_{\mathrm{1}},\mathrm{...},\hspace{0.17em}{x}_n|Y)\cdot P(Y)}{P(x_{\mathrm{1}},\mathrm{...},\hspace{0.17em}{x}_n)}},

(14)P(x_{\mathrm{1}},\mathrm{...},\hspace{0.17em}{x}_n)={\displaystyle \prod }_{i=\mathrm{1}}^{n}P(x_i|\pi (x_i)),

where x₁,…, x_n, and Y are random variables; P(X_i|Y) is the posterior probability of variable X_i given evidence Y; P(X_i) and P(Y_i) are the prior probabilities of variables X_i and Y; P(Y|X_i) is the likelihood and is proportional to the conditional probability of observing a particular event given evidence X_i; P(x₁, …, x_n) is the joint probability of variables x₁, …, x_n; and p(x_i) is a set of values for the parents of X_i. Equation (12) corresponds to reverse inference, whereas Eq. (13) denotes sequential inference.

Interpretive structural modeling (ISM)

ISM methodology, as interpretive in judgment, can be used as a systematic means of recognizing the contextual relationships between the elements associated with an issue to be examined (Warfield [³³]). The judgment of expert groups determines the interaction between the elements (Mathiyazhagan et al. [³⁴]). In this study, ISM is used for the development of the qualitative part of the BBN. The steps associated with implementing ISM were proposed by Sushil [³⁵] as follows.

Step 1: Identification of factors related to the problem through a systematic literature review approach.

Step 2: Determination of the contextual relationships within each pair of factors representing whether or not one factor leads to the other.

Step 3: Development of a structural self-interaction matrix (SSIM) based on pairwise comparison among factors.

Step 4: Development of an initial reachability matrix (IRM) from SSIM.

Step 5: Checking for transitivity in the IRM (if “x” is related to “y”, and “y” is related to “z”, then “x” is also related to “z”) to obtain the final reachability matrix (FRM).

Step 6: Partitioning of FRM into various levels of ISM hierarchy.

Step 7: Plotting of a directed graph based on the relationship of FRM and removal of transitivity links.

Step 8: Translation of the digraph into an ISM via replacement of the nodes with statements.

Step 9: Checking of the structural model for conceptual inconsistency and implementation of necessary modifications.

The entire process of building the probabilistic framework using a BBN based on the ISM technique for evaluation of lateral displacement is shown in Fig. 1.

Development of liquefaction-induced lateral displacement modeling

Case-history database, predictor variables, and preprocessing

The case-history database used in this study was obtained from three references (Chu et al. [²⁷], Youd et al. [¹⁷], and Cetin et al. [²⁸]) and contains a total of 493 records of lateral displacement from 12 major earthquakes. Among the case-history records, 247 records were associated with free face, whereas the remaining 246 records comprised of sloping ground conditions based on definitions of the topographical parameters free-face ratio, W (H/L), where H is the free- face height, and L is the distance from the toe to the site; and the gradient of surface topography (S), which is the overall inclination of the ground surface (see Fig. 2). The difference between the topography-related descriptive parameters (W and S) is visualized in a diagram of ground surface geometry, shown in Fig. 2.

It has been widely accepted among researchers that the input factors selected by Youd et al. [¹⁷] is a complete and suitable set for controlling the lateral displacement. For this reason, the same factors have been selected as significant factors by many other researchers (e.g., Javadi et al. [¹⁴]; Jafarian and Nasri [¹⁶]; Baziar and Saeedi Azizkandi [³¹]). In the current study, with the addition of ground intensity measure, peak ground acceleration (PGA, a_max) is used to expand the data set, such that it becomes more capable and more efficient in accounting for the mechanisms of earthquakes. Sadigh et al. [³⁶] used the attenuation equation to estimate the PGA (a_max) by considering the causative fault types of all earthquakes. A summary of the case histories and ranges of variation among the factors influencing liquefaction-induced lateral displacement is presented in Table 2.

Previous studies (Tang et al. [³⁷], Hu et al. [³⁸], and Zhang [³⁹]) offer detailed explanations of the discretization of variables and classification, which are beneficial to the present study. The most important reasons for performing data discretization are as follows: (i) well-built capability of BNN to handle discrete variables, (ii) reduction and elucidation of the data set, (iii) quick and easy development of the model, and (iv) production of easily interpretable outputs in the study. Seven significant factors or variables are used for the evaluation of LD in this study: earthquake magnitude (M) FR₁, closest horizontal distance to seismic energy source (R, km) FR₂, peak ground acceleration (a_max, g) FR₃, average fines content (particles<0.075 mm) in T₁₅ (F₁₅, %) FR₄, average mean grain size in T₁₅ (D50₁₅, mm) FR₅, cumulative thickness of saturated layers with corrected SPT number (N₁)₆₀<15 (T₁₅, m) FR₆, free-face ratio (W, %) or slope of surface topography (S, %) FR₇. These factors that influence LD are converted into discrete values prior to the construction of the BBN models, based on possible factor ranges and domain knowledge, as shown in Table 3. The output liquefaction-induced lateral displacement FR₈ was divided into the four grading standards described by Tang et al. [³⁷].

In this study, 80% of the available data (198 cases in free-face and 197 cases in sloping ground conditions) are considered for the training data set. Meanwhile, the test data set is used to assess the predictive ability performance of the proposed models. In this study, data from 49 history records are considered for the testing data set. The data partitioning of the training and testing data sets is based on statistical aspects, such as the mean and standard deviation of the data sets. The statistical consistencies of the training and testing data sets improve the model performance, which helps to evaluate them better. The statistical parameters of the input factors of the training and testing data sets are outlined in Tables 4 and 5 for the free-face and sloping ground conditions, respectively. An important point to emphasize is that because the set of history records includes singular, rare events that may not be reiterated in all data sets, some insignificant differences may subsist among the training and testing sets.

Model development using Bayesian belief network based on ISM

The BBN models for lateral displacement are developed using the ISM technique, which was developed by Professor John Warfield to study complex socioeconomic systems. ISM has been effectively utilized in a diverse set of problems, such as identification and benchmarking of significant factors of liquefaction potential (Ahmad et al. [⁴⁰]), risk management in supply chains (Pfohl et al. [⁴¹]), and energy conversion (Sexena et al. [⁴²]). In this study, the step-by-step procedure is applied as proposed by Sushil [³⁵] to construct an ISM-based model. Because ISM proposes the use of domain knowledge (DK) in developing contextual relationships among influencing factors, the contextual relationships between the seven significant liquefaction-induced lateral displacement factors are first developed via literature review and ultimately examined and approved by field experts (Table 6).

In the next step, the SSIM for liquefaction-induced lateral displacement factors is converted into a binary matrix, referred to as the initial reachability matrix (IRM), via replacement of the original symbols with 1 or 0 (see Table 7).

When the IRM is obtained, the transitivity property, which is the fundamental assumption of ISM for the final reachability matrix (FRM), is then checked. The FRM of the aforementioned factors, presented with the values of driving and dependence power, is shown in Table 8.

These factors, collectively with their reachability set (RS), antecedent set (AS), and intersection set (IS), are used for deriving multilevel hierarchy structure levels, which are outlined in Table 9. The results reveal that there are 3 level partitions, as follows:

{{\displaystyle \mathrm{L}}}_{\mathrm{1}}=\{{{\displaystyle \mathrm{F}\mathrm{R}}}_{\mathrm{8}}\};{{\displaystyle \mathrm{L}}}_{\mathrm{2}}=\{{{\displaystyle \mathrm{F}\mathrm{R}}}_{\mathrm{3}},{{\displaystyle \mathrm{F}\mathrm{R}}}_{\mathrm{5}},{{\displaystyle \mathrm{F}\mathrm{R}}}_{\mathrm{6}},{{\displaystyle \mathrm{F}\mathrm{R}}}_{\mathrm{7}}\};{{\displaystyle \mathrm{L}}}_{\mathrm{3}}=\{{{\displaystyle \mathrm{F}\mathrm{R}}}_{\mathrm{1}},{{\displaystyle \mathrm{F}\mathrm{R}}}_{\mathrm{2}},{{\displaystyle \mathrm{F}\mathrm{R}}}_{\mathrm{4}}\}.

A multilevel hierarchy structure of lateral displacement is developed from the FRM. The links between two factors, such as the direct links between earthquake magnitude and liquefaction-induced lateral displacement, are removed because the earthquake magnitude can affect liquefaction-induced lateral displacement via peak ground acceleration, as in the ISM model. Furthermore, there is a restriction of no relations among skipping-level nodes.

In the next step, there is no conceptual discrepancy in the structural model, and thus two ISMs are developed separately for the evaluation of liquefaction-induced lateral displacement in free-face and ground conditions, with the replacement of only one of the topographic parameters (see Fig. 3).

According to the figure, the factors “D50₁₅,” “T₁₅,” “free-face ratio” or “sloping ground condition,” and “peak ground acceleration” in the second level are directly influencing the liquefaction-induced lateral displacement, whereas the parent node of “D50₁₅,” which is “F₁₅,” and of “a_max,” which are “R” and “M,” are in third level of the ISM hierarchy.

The BBN structures are constructed directly in Netica software for parameter learning to obtain the conditional probability distribution of the nodes. Finally, the BBN model is used to evaluate the lateral displacement. A graphical presentation of the results of the proposed liquefaction-induced lateral displacement framework after parameter learning for both conditions is shown in Fig. 4.

Evaluation measures

Because there are no generally recognized performance metrics for multi-class models, the training performance and predictive ability of the proposed lateral displacement models on the case-history records database are measured with the help of four metrics: OA; kappa; producer accuracy, and user accuracy. In this study, these metrics are determined for each class based on the confusion matrix. Let r_ij (i and j = 1,2,...,m) be the joint frequency of observations allotted to class i by prediction and to class j by observation data, r_{i +} be the total frequency of class i as obtained from prediction, and r_+j be the total frequency of class j based on observed data (see Table 10).

The OA is calculated via summation of the diagonal correctly classified instances, followed by division by the total number of instances. The OA is calculated to be

(15)\mathrm{O}\mathrm{A}=\left({\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^m{r}_{ii}}\right)\times \mathrm{100}\%.

Cohen’s kappa coefficient calculates the proportion of units that are correctly classified in units proportion after the probability of chance agreement, which is a robust index that considers the probability of an event classification by chance [⁴³], is eliminated. Kappa can be calculated using the following formula:

(16)\mathrm{k}\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{a}={\displaystyle \frac{n{\displaystyle {\sum }_{i=\mathrm{1}}^m{r}_{ii}-{\displaystyle {\sum }_{i=\mathrm{1}}^m(r_{i+}\cdot r_{+i})}}}{n^{\mathrm{2}}-{\displaystyle {\sum }_{i=\mathrm{1}}^m(r_{i+}\cdot r_{+i})}}},

where n is the number of examples, m is the number of class values, r_ii is the cell count in the main diagonal, and r_{i +} and r_+i are the row and column total counts, respectively.

Landis and Koch [⁴⁴] suggested a scale to explain the degree of concordance (see Table 11). A kappa value lower than 0.4 indicates poor agreement, whereas a value of at least 0.4 indicates good agreement (Sakiyama et al. [⁴⁵]; Landis and Koch [⁴⁴]). According to Congalton and Green [⁴⁶], the producer’s accuracy of class i (PA_i) can be calculated as

(17){\mathrm{PA}}_i=\left({\displaystyle \frac{r_{ii}}{r_{+m}}}\right)\times \mathrm{100} \hspace{0.17em}\%=\left({\displaystyle \frac{r_{ii}}{{\displaystyle {\sum }_{i=\mathrm{1}}^m{r}_{im}}}}\right)\times \mathrm{100}\hspace{0.17em}\%,

and user’s accuracy of class i (UA_i) can be calculated as

(18){\mathrm{UA}}_i=\left({\displaystyle \frac{r_{ii}}{r_{m+}}}\right)\times \mathrm{100} \hspace{0.17em}\%=\left({\displaystyle \frac{r_{ii}}{{\displaystyle {\sum }_{i=\mathrm{1}}^m{r}_{mj}}}}\right)\times \mathrm{100}\hspace{0.17em}\%.

Results and discussion

Training performance of BBN models

Because the OA, kappa statistics, user accuracy, and producer accuracy that are determined for each class can intuitively and comprehensively present the classification performance of the models, these four metrics can be used to determine the learning ability of the two BBN models. Tables 12 and 13 show clearly that for the BBN–W model, OA= 97.98%, kappa= 0.771 (substantial), PA= 0%–100%, and UA= 80%–100%, whereas for the BBN–S model, OA= 99.492%, kappa= 0.664 (substantial), PA= 50%–100%, and UA= 99.49%–100%. Except for two instances (the BBN–W model has the lowest value of PA in the “none” class of lateral displacement, whereas the BBN–S model has a moderate value, i.e., 50%, for PA in the “small” class of lateral displacement), these parameters are all very prominent. These high values indicate that both BBN models have good learning ability on the training data sets.

Predictive performance comparison with existing models

To quantify the performance measures of the classified instances of lateral displacement, such as “none,” “small,” “medium,” and “large,” accuracy evaluation based on a confusion matrix is performed using a testing data set consisting of 49 case-history records for free-face and sloping ground conditions.

For the free-face condition, the BBN–W model achieved the highest OA (85.714%), whereas the MLR model by Jafarian and Nasri [¹⁶] and the GP model by Javadi et al. [¹⁴] exhibited at-par OA (83.673%), and the MLR model by Bardet et al. [²⁰] exhibited a relatively low OA, which is 73.469%. On the other hand, the kappa of the BBN–W model was calculated to be in moderate-strength agreement, whereas the MLR model by Jafarian and Nasri [¹⁶] and the GP model by Javadi et al. [¹⁴] exhibited fair to slight agreements, respectively, based on the concordance scale proposed by Landis and Koch [⁴⁴]. The PA and UA values indicate that some liquefaction-induced lateral displacement classes are better classified using the BBN–W model than with others, as shown in Table 14. The “medium” LD (PA: 50% and UA: 100%) for the BBN–W model indicates a relatively better performance than those of the MLR model by Jafarian and Nasri [¹⁶] and the GP model by Javadi et al. [¹⁴], which both exhibited at-par measures, i.e., (PA: 16.67% and UA: 100%), and of the MLR model by Bardet et al. [²⁰] (0% for PA and UA). Because of the fewer number of samples in free-face liquefaction-induced lateral displacement classes “small” and “none,” i.e., 2.02% and 1.62%, respectively, the low PA and UA effects may be inherited in the BBN–W model.

For the sloping ground condition, the BBN–S model achieved a classification accuracy of 100%. Out of 49 testing data records, all were correctly classified (Table 15). In terms of OA for the testing data set, BBN–S and the MLR model by Jafarian and Nasri [¹⁶] achieved the highest OA (100%), followed by the at-par OA rates of the GP model by Javadi et al. [¹⁴] and the MLR model by Bardet et al. [²⁰] (95.918%). The kappa values of BBN–S and the MLR model by Jafarian and Nasri [¹⁶] are both equal to 1, corresponding to almost perfect, whereas those of the GP model by Javadi et al. [¹⁴] and the MLR model by Bardet et al. [²⁰] for the testing data set exhibited fair values, based on the concordance scale proposed by Landis and Koch [⁴⁴]. The PA and UA of BBN–S and the MLR model by Jafarian and Nasri [¹⁶] are 100% for “large” lateral displacement, whereas those of the GP model by Javadi et al. [¹⁴] and the MLR model by Bardet et al. [²⁰] are at par, i.e., 97.92% for PA, and 100% for UA. As demonstrated in Table 13, the “small” class of lateral displacement was predicted wrongly by the GP model by Javadi et al. [¹⁴] and the MLR model by Bardet et al. [²⁰].

It is obvious, judging from the comparison, that the results predicted by the proposed BBN models show an improvement over the MLR models, except for two instances (the BBN–S and MLR model by Jafarian and Nasri [¹⁶] for sloping ground conditions have at-par values for OA and kappa), and a more significant improvement over the GP model in terms of overall accuracy and kappa. Unfortunately, the compiled database for the free-face condition did not contain a sufficient portion of “small” and “none” classes. By contrast, the database for sloping ground conditions included “small,” “medium,” and “none” classes. Therefore, additional new data sets to cater displacement data for the aforementioned classes are necessary to further update and improve the predictive performance results of the current BBN models.

Sensitivity analysis

Because of natural variability of and insufficient knowledge on soil properties, site conditions, and earthquake loads, the resolution of geotechnical problems, particularly for highly nonlinear conditions such as soil liquefaction and its induced lateral displacement, encounters many uncertainties. Various techniques have been used in previous studies to quantify the influence of uncertain input factors on uncertain model outputs. For example, Vu-Bac et al. [⁴⁷] proposed a unified framework that links different steps, from generating samples, to constructing a surrogate model, to implementing a sensitivity analysis method, to determine the key input parameters of an output of interest.

In this study, to govern and quantify these uncertainties, the sensitivity analysis function of the Netica software was used to determine the nodes that have greater impact on the liquefaction-induced lateral displacement hazard. The key objective of a sensitivity analysis is to determine significant factors that have a higher impact on the outcome of lateral displacement with the intention of designing countermeasures against this hazard, to decrease its effects. Mutual information between two nodes can reveal the dependency of the nodes on each other, and if any, how close their relationship is (Cheng et al. [⁴⁸]). The sensitivity analysis results for the BBN models are presented in Table 16, where it is evident that for the BBN–W and BBN–S models, the “peak ground acceleration,” having mutual info values of 0.07718 and 0.12110, respectively, and “D50₁₅,” having mutual info values of 0.02282 and 0.05334, respectively, are the most significant factors. Therefore, to counteract the consequences of lateral displacement, soil improvements might be a more efficient solution for more highly seismic-prone zones, for controlling the impact of on-site PGA.

Conclusions and future research

In this paper, a novel probabilistic framework for predicting lateral displacement using the BBN approach based on an ISM technique is presented. Two BBN models were trained and tested using case-history records for free-face and sloping ground conditions. A comparison of predictive results revealed that the proposed BBN models provide relatively better accuracies than those of commonly used MLR and GP models. Furthermore, it is no longer necessary for any normalization or calibration of parameters prior to the introduction of new parameters, such as R* in the MLR model, or the use of logarithmic functional values for affecting parameters; all parameters can be fed into the model as they are, without any normalization or calibration. It is evident from the results that the BBN is able to learn complex relationships between lateral displacement and its influencing factors as cause–effect relationships, with reasonable precision. With the results of the sensitivity analysis, it is concluded that the “peak ground acceleration” and “D50₁₅” are the two most significant factors in both BBN models.

The basic emphasis of this study was to introduce a novel probabilistic framework using BBN to evaluate the lateral displacement caused by liquefaction. One of the most important abilities of BBN models is to update the conditional probability based on updated data sets, which will improve its prediction performance and overthrow previous results. Therefore, for future research, the database used in this study should be expanded to a wider range of input factors, because 60.65% of the records are related to the Niigata 1964 earthquake, wherein M= 7.5 and R= 21 km. Moreover, new data sets of almost balanced displacement sites are necessary to further evaluate and update the current BBN models.

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