Inter-region variability of Robertson and Wride method for liquefaction hazard analysis
Introduction
Due to the difficulties to obtain high quality soil samples, semi-empirical methods based on soil indexes measured with in-situ tests such as cone penetration test (CPT) are widely used to assess the liquefaction potential of soils. Such methods are generally developed based on a database that includes case histories derived from earthquakes in different regions around the world (e.g., Robertson and Wride, 1998, Cetin et al., 2002, Moss et al., 2006, Muduli et al., 2014). The semi-empirical methods are often applied universally around the world. The assumption behind such a practice is that the model uncertainty associated with an empirical model is homogenous for all regions. Such an assumption, however, may not be correct for certain cases. For instance, consider the Robertson and Wride model (Robertson and Wride, 1998, Robertson, 2009), one of the most widely used CPT-based models for evaluating the potential of soil liquefaction. For convenience of presentation, the Robertson and Wride method is referred to hereinafter as the RW model. Table 1 shows the post-earthquake investigation data from 11 major earthquakes, which are adapted from the database compiled in Ku et al. (2012). The factor of safety (FOS) computed based on the RW model is shown in the last column of Table 1. In principle, a soil should liquefy if the FOS against liquefaction is less than unity, and should not liquefy if the FOS is larger than unity. Based on this criterion, there are three out of 11 cases from the 1976 Tangshan earthquake zone in which the liquefaction phenomenon is wrongly predicted by the RW model. For the three wrongly predicted cases, the values of computed FOS are all smaller than unity but liquefaction did not occur, indicating that the RW model may tend to underestimate the actual FOS of soils from the Tangshan earthquake zone. For comparison, among the five cases from the 1980 Mexicali earthquake zone, four cases are wrongly predicted. For these wrongly predicted cases, the values of computed FOS are all larger than unity but soil liquefaction was observed, indicating the RW model may tend to overestimate the actual FOS of soils from the 1980 Mexicali earthquake zone. It seems that the characteristics of model uncertainty associated with the RW model could vary from one region to another, indicating the existence of inter-region variability about the applicability of the RW model.
The existence of inter-region variability in empirical relationships has been previously observed in other geotechnical engineering problems (e.g., Zhang et al., 2004). In foundation engineering, it is often quite typical to develop local empirical relationships to predict the pile capacity based on data in one region to avoid the adverse effect of inter-region variability (e.g., McVay et al., 2003, Reddy and Stuedlein, 2013). For liquefaction problems, however, such an idea is not feasible because post-earthquake soil liquefaction data are scarce and the data in one region are often insufficient to develop a local model. As such, lumping data from different regions for the development of empirical liquefaction potential evaluation models is necessary and the existence of inter-region variability may be unavoidable.
Research studies have been carried out to assess the model uncertainty associated with liquefaction potential evaluation models. Cetin et al. (2002) developed a model to evaluate the liquefaction potential of soils based on the standard penetration test (SPT) data considering the variability in soil parameters, and the model error of the suggested model is characterized during the model development process. Using a similar procedure, Moss et al. (2006) developed a liquefaction potential prediction model based on the CPT data with explicit consideration of model uncertainty. Huang et al. (2012) suggested a Bayesian network for characterizing the model uncertainty associated with a liquefaction model considering the uncertainties in the input soil parameters. A maximum likelihood-based method is employed in Ku et al. (2012) to characterize the model uncertainty associated with the RW model, and a relationship between the computed FOS and liquefaction probability was derived. In these studies it is assumed that the distributions of the model bias factors of different regions are identical, i.e., the model uncertainty is homogenous across regions. The issues of how to model and characterize the inter-region variability and how such variability affects the liquefaction potential evaluation have been seldom addressed in the literature.
The objective of this paper is to extend the work in Ku et al. (2012) and to suggest a method to characterize the inter-region variability of the accuracy of the RW model, through which the effect of inter-region variability of the model bias factor on liquefaction potential evaluation will also be studied. This paper is organized as follows. First, the probabilistic model for modeling the inter-region variability is suggested, followed by a Bayesian method for calibrating the suggested model. Then, the procedure is presented for comparing models based on different assumptions regarding the inter-region variability of the model bias factor. Finally, the inter-region variability of the model bias factor of the RW model is investigated with the proposed approach, and its impact on liquefaction potential evaluation is discussed. This paper provides useful insights on how to address the inter-region variability in the semi-empirical models and how to use such models appropriately for liquefaction potential evaluation. The method suggested in this paper can also be used to characterize the inter-region variability of other semi-empirical methods for liquefaction potential evaluation.
Section snippets
Probabilistic model
Let Fsc denote the FOS calculated by the RW model, and let Fsa denote the actual FOS. Due to the existence of model uncertainty, Fsc may not be exactly Fsa. Suppose Fsc can be related to Fsa via a model bias factor c as follows (e.g., Ang and Tang, 1984)
The model bias factor c can be assumed to follow the lognormal distribution (e.g., Juang et al., 2004, Huang et al., 2012). Let μci and σci denote the mean and the standard deviation of the model bias factor at the ith region. As
Ranking of competing models
As will be seen later in this study, based on the same calibration data, it is possible to construct several probabilistic models depending on whether the inter-region variability is modeled and how it is modeled. Yet, the predictions from these models could be quite different. In such a case, how to compare the validity of different models is important. In this study, different models are compared using Bayes' theorem, where model probability is used to measure the plausibility of different
Calibration database
Ku et al. (2012) compiled a database comprising 165 cases, which are derived from 16 earthquakes around the world based on previous studies of Moss et al. (2006), (2011); Robertson (2009). For the purpose of calibrating the inter-region variability, this database is reviewed and earthquakes with less than 4 cases are excluded for the argument that when the number of cases is small, the observed data provide little information about the model bias factor in that region. The new database in this
Summary and conclusions
The research reported in this paper and findings from this paper can be summarized as follows:
- (1)
A probabilistic model is suggested to model the inter-region variability associated with the model bias factor of the Robertson and Wride (RW) model for liquefaction potential evaluation, in which the mean values of the model bias factor at different regions are assumed different but follow a common distribution. A method based on multivariate Bayesian updating has been suggested for model calibration
Acknowledgment
The first author is grateful to the support from the National 973 Basic Research Program of China (2014CB049100), the Shanghai Rising-star Program (15QA1403800), and the Fundamental Research Funds for Central Universities.
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