Model uncertainties of SPT, CPT, and VS-based simplified methods for soil liquefaction assessment

Simplified methods for assessing soil liquefaction potential based on the standard penetration test (SPT) are prevalent in practice and widely accepted by several seismic design codes. When encountering sites that have not been investigated using SPT, such as offshore sites or sites with a high level of gravel content, engineers can only substitute the methods based on piezocone penetration test data (CPT-qc methods) or shear wave velocity measurements (VS-based methods); however, these two approaches perform inconsistently with methods based on SPT data (SPT-N methods). As a result, this paper exploits the datasets consisting of SPT, CPTU, and in situ seismic test measurements from 13 alluvium sites in the Taipei Basin to compare the performance of prevalent SPT-N, CPT-qc, and VS-based methods. The discrepancies (uncertainties) of these methods are characterized as Gaussian distribution models, which is believed to be a feasible strategy for predicting equivalent results for SPT-N methods when SPT data are not available. Finally, the application of the proposed models to liquefaction potential index evaluation is demonstrated using a real case study.


Introduction
The investigation of soil liquefaction potential is the cornerstone of modern seismic engineering design. An abundance of approaches for assessing soil liquefaction potential have been developed since the devastating liquefaction damage induced by the 1964 Alaska and Niigata earthquakes (e.g., Idriss 1971, 1982;Ishihara and Li 1972;Kayen and Mitchell 1997;Dobry and Ladd 1980;Seed et al. 1985;Youd et al. 2001;Cetin et al. 2004;Idriss and Boulanger 2010), and the robustness and applicability of these methods have been comprehensively compared and discussed (e.g., National Research Council 2016). It is evident that methods that exploit the measurements of in situ tests, such as the blows of standard penetration tests (SPT-N), the cone resistance of piezocone penetration tests (CPT-q c ), and the shear wave velocity (V S ) for liquefaction potential evaluations, are prevalent in engineering practice. In this paper, these are referred to as "SPT-N methods," "CPT-q c methods," and "V S -based methods," respectively.
SPT-N methods, which are approaches for assessing liquefaction potential based on the number of blows of an SPT and the index properties of split samples, are popular in engineering practice and have been widely accepted in various seismic design codes for civil structures (e.g., Architectural Institute of Japan (AIJ) 2001; American Association of State Highway and Transportation Officials (AASHTO) 2014; Japan Road Association (JRA) 2017; Ministry of Interior (MOI) of Taiwan 2022). SPT-N methods have been diversely developed over the past few decades (e.g., Seed et al. 1985;Youd and Idriss 1997;Juang et al. 2000;Youd et al. 2001;Cetin et al. 2004Cetin et al. , 2018Idriss and Boulanger 2010;Boulanger and Idriss 2014;Hwang et al. 2021) in accordance with the simplified procedure attributed to Seed and Idriss (1971). Among these methods, the one featured in the 1998 NCEER/NSF workshop (Youd et al. 2001) is the earliest procedure on which geotechnical experts and scholars have reached a consensus and also usually appears in relevant comparative research work (e.g., Hwang et al. 2005Hwang et al. , 2021Boulanger and Idriss 2014;Cetin et al. 2018). This method has evidently become the representative SPT-N method in the literature.
Compared with the SPT-N methods, the CPT-q c methods emerged later but piezocone penetration tests (CPTU) have many advantages (National Science Foundation 1994) and have attracted considerable attention in geotechnical engineering (e.g., Campanella et al. 1987;Kulhawy and Mayne 1990;Chen and Mayne 1996;Mayne 2006;Robertson 2009Robertson , 2016 such that CPT-q c methods have gradually evolved in a diverse manner and matured with an increasing amount of investigative data of liquefied and non-liquefied sites. Olsen (1997) utilized the liquefaction data from case histories (Shibata and Teparaska 1988;Stark and Olson 1995;Suzuki et al. 1995) to develop a technique for evaluating liquefaction resistance from CPT soundings along with a soil classification system, which pioneered CPT-q c research methods. Based on these cases, Robertson and Wride (1998) proposed a robust procedure to evaluate cyclic liquefaction potential using CPT soundings, which is also recommended for engineering practice by the 1998 NCEER/NSF workshop (Youd et al. 2001). Hwang et al. (2005) collected CPT data from the 1999 Chi-Chi earthquake history to construct a CPT-q c method for liquefaction assessment that uses hyperbolic functions. They validated that this method performs as well as the methods proposed by Olsen (1997) and Robertson and Wride (1998) using global data from liquefaction case histories. Ku and Juang (2012) developed a unified CPTU-based model for assessing cyclic liquefaction resistance applicable to both sand-like and clay-like soils and examined its performance using data from Adapazari, Turkey, after the 1999 earthquake struck the city. They obtained satisfactory results. Moss et al. (2006) and Boulanger and Idriss (2016) compiled considerably more in situ investigative data from case histories of liquefaction and adopted Bayesian frameworks to develop probabilistic CPTU-based models for evaluating the probability of liquefaction (P L ) of soils subjected to cyclic loading. For deterministic analysis, a model associated with P L = 15% was recommended (Moss et al. 2006;Boulanger and Idriss 2016).
On the other hand, when encountering sites where penetration tests cannot be readily implemented or whose deposits have a high gravel content, in situ seismic tests (e.g., cross-hole tests or P-S suspension logging) are common strategies for site investigation to acquire small-strain soil properties (Sykora 1987;Mayne 2006), which has led to the recent development of V S -based methods for liquefaction potential assessments. There are also deterministic (e.g., Kayen et al. 1992;Robertson et al. 1992;Andrus and Stokoe 2000) and probabilistic (e.g., Kayen et al. 2013) procedures published in the literature. However, the V S -based approaches were developed with fewer case histories of liquefaction, and V S measurements do not provide stratigraphic profiling. V S is also a small-strain property, whereas pore-water pressure buildup and the onset of liquefaction are medium-or large-strain phenomena, such that some concerns about V S -based methods remain (Youd et al. 2001;National Research Council 2016).
Although SPT-N methods have developed more maturely than the other two simplified approaches and are more widely accepted by national seismic design codes and wellknown design principles, their application to sites such as offshore sites or gravel-rich formations is limited. Due to the harsh and rapidly fluctuating oceanographic conditions and the limited time available for investigation at the offshore sites, CPT is the prevalent means for geotechnical investigation in marine engineering owing to its steadiness (Randolph and Gourvenec 2011), which has led it to replace SPT investigation. For gravel-rich formations, penetration tests like SPT or CPTU may not be conducted appropriately due to the large particle size of the deposits, so that in situ seismic tests are adopted to investigate the subsurface conditions. As a result, CPT-q c or V S -based methods for liquefaction potential assessments are commonly substituted for SPT-N methods at these sites.
Even though CPT-q c or V S -based methods can provide liquefaction potential assessments of the aforementioned sites, there are discrepancies between their assessments and those conducted via SPT-N methods, which have been revealed in the literature (e.g., Gilstrap and Youd 1998;Guettaya et al. 2013;Jarushi et al. 2015;Robertson 2015;Hoque et al. 2017), and this has raised concerns about their impact on engineering design and economics among engineers who practice in the areas where SPT-N methods have been prevalently adopted. These discrepancies are attributed to the disparities in the models used for liquefaction resistance evaluation (Baez et al. 2000), and they lead to what is referred to as "model uncertainty" in this paper.
To figure out and quantify the model uncertainties of SPT-N, CPT-q c , and V S -based methods, the datasets of SPT, CPTU, and in situ seismic methods at 13 alluvium sites in the Taipei Basin have been compiled (Wang et al. 2022) as a case study. In this paper, the measurements associated with sandy soils are exploited to first examine the consistencies between the SPT-N methods described in the 1998 NCEER/NSF workshop (hereinafter denoted NCEER-SPT methods) and some popular CPT-q c and V S methods, and to simultaneously calibrate the corresponding model uncertainties. Thereafter, the consistencies of different SPT-N methods are further discussed and characterized with statistical approaches. Finally, an illustration of the application of these calibrated models to the liquefaction potential index (LPI, proposed by Iwasaki et al. 1984) is given using an SPT-CPTU-V S record outside the above database, which is believed to be an available reference for seismic site investigation, geotechnical design, and liquefaction potential map generation.

Seismic testing plan and data acquisition
SPT, CPTU, and cross-hole (XH) seismic tests were carefully planned at 13 selected alluvium sites in the Taipei Basin. Figure 1 is a map of the site locations. The investigated subsurface formation of these sites is the Song-Shan layer, which is the youngest Holocene alluvium in the Taipei Basin and consists of interbedded sands, clays, and silts. The typical layout for each test site and the data acquired are shown in Fig. 2. Two sets of SPT and CPTU were conducted within a distance of approximately 1 m of each other, and the cross-hole seismic tests were implemented in the same boreholes in which SPTs were conducted. It is worth noting that for each site two series of XH measurements were obtained by exchanging receivers and sources thus generating two reverse survey paths at the same depth. In other words, if one series of V S measurements via XH methods was acquired by the receiver in BH-1 with the source in BH-2 (see Fig. 2), then the other was obtained by the receiver in BH-2 with the source in BH-1. These tests were conducted by professionals following the American Society for Testing and Materials (ASTM) standard procedures (ASTM D5778 2012; ASTM D4428 2016; ASTM D1586 2018) at each of the 13 alluvium sites. All these tests reached over 20 m in depth. To reduce potential interference between the seismic tests due to drilling and penetration, the time interval between the drilling and the seismic testing was set to 1 week. To ensure data quality, the test operations were supervised by professional engineers from the National Center for Research on Earthquake Engineering throughout the whole process, and experienced geotechnical experts intermittently inspected the implementation procedures. The procedures for implementing the seismic tests, V S data acquisition, and supervision have been elaborated on by Wang et al. (2022).
SPT-N 60 measurements were at discrete depths with a penetration distance of 30 cm (approximately 12 inches) (left-hand  Fig. 3), continuous CPT-q c profiles were obtained near the borehole in which the SPT was conducted (central profile, Fig. 3), and V S measurements using cross-hole tests with the receiver in the SPT hole were made at 1 m depth intervals (right-hand profile, Fig. 3). The SPT-CPTU-V S data were compiled for each SPT-N measurement as illustrated by the horizontal blue bar in Fig. 3. Taking the SPT-N 60 values obtained in the depth range 13.2 to 13.5 m as an example, the CPT-q c for this depth range was averaged using uniform weights (i.e., the arithmetic mean was calculated). The other CPT measurements (sleeve friction, f s , and pore water pressure, u 2 ) were averaged in the same way. In this way, V S measurements, averaged CPT values, and SPT-N values were paired for each depth. The above procedure is consistent with other relevant research (e.g., Robertson et al. 1983;Gilstrap and Youd 1998;Baez et al. 2000;Stuedlein et al. 2016).
In general, the simplified methods for assessing soil liquefaction potential are applicable to sandy soils only. Thus, this study filtered the SPT-CPTU-V S data associated with sandy soils in accordance with the classification result of each split sample and transformed these data with an overburden correction factor, C N (Andrus and Stokoe 2000; Robertson 2009; Boulanger and Idriss 2016), into normalized parameters as follows. (1) where N 60 is the number of blows of SPT with 60% of standard energy to falling hammer, q t is corrected cone resistance for pore water pressure effects, p a is atmosphere pressure (≈101.3 kPa), and σ v ′ is effective overburden stress (kPa).
The filtered normalized datasets (i.e., (N 1 ) 60 -q t1 -V S1 ) are shown in Fig. 4. Figure 4 shows that most of the (N 1 ) 60 values are between 5 and 15, the q t1 values are mainly in the interval of 15-60, and the range of V S1 is 150-210 m/s. The number of these datasets is 102, and the sample mean, sample coefficient of variation, and 10%-and 90%-quantile values of these datasets are listed in Table 1. It can be seen from Table 1 that over 90% of the (N 1 ) 60 values are smaller than 15, which indicates most of the sandy soils in these studied sites were classified as loose to medium sand (Terzaghi et al. 1996).

Comparison of performance of selected simplified methods
This section aims to compare the performance of selected CPT-q c and V S -based methods with the representative SPT-N method, NCEER-SPT, using the datasets from these alluvium sites. For CPT-q c -based approaches, there were six prevalent methods selected in this study: the Olsen (1997) method (denoted as OS97), the Robertson and Wride (1998) method, the Hwang et al. (2005) Robertson and Wride (1998), and the updated procedure was adopted and denoted as RB09 in this study for comparison. For V S -based procedures, those proposed by Andrus and Stokoe (2000) and Kayen et al. (2013) were chosen in this study, and these two methods are respectively denoted as AN00 and KY13 in this paper. This section compares the factor of safety against cyclic liquefaction (FS) values assessed via these simplified methods to examine the consistency of their performance. FS values obtained via such simplified methods can be evaluated as follows.
where CRR is the cyclic resistance ratio of the soil, CSR is the averaged cyclic stress ratio caused by the earthquake, CRR 7.5 is the cyclic resistance ratio of equivalent clean sand corresponding to an earthquake magnitude M w = 7.5, MSF is the magnitude scaling factor, PGA is the peak ground acceleration (g), r d is the shear stress reduction factor, σ v is the total overburden stress (kPa), and σ v ′ is the effective overburden stress (kPa). Except for PGA, σ v , and σ v ′, the parameters in Eq. (5) are dimensionless. It is worth noting that the recommended formulas for CRR 7.5 , MSF, and r d for each simplified method are distinct. That is, the evaluations of these parameters are the characteristics of the simplified methods. Fig. 4 Scatter plots of q t1 -(N 1 ) 60 -V S1 data The evaluated FS values of the aforementioned datasets subjected to the design earthquake (475-year return period) in the Taipei Basin in accordance with the seismic design codes for ordinary buildings (MOI 2022) by the selected CPTq c , V S -based, and NCEER-SPT methods are shown in Fig. 5. This figure shows that most of the FS values assessed by the CPT-q c methods are lower than those assessed by the NCEER-SPT method, which indicates that these CPT-q c methods are more conservative than the NCEER-SPT method. Further observation shows that OS97 is less biased with the NCEER-SPT method (its FS evaluations concentrate more around the 45° line), whereas it is more uncertain (its FS evaluations are more scattered) than the other CPT-q c methods. The FS values evaluated by KJ12 are evidently below the 45° line, which suggests that KJ12 was the most conservative among the CPTq c methods. As stated above, the assessment of liquefaction potential using the CPT-q c methods is inconsistent with that using the NCEER-SPT method at the studied sites.
It is also observed that the two selected V S -based methods performed in an inconsistent manner compared with the NCEEER-SPT method. The FS values evaluated using the V S -based methods are significantly lower than those evaluated using the NCEER-SPT method (as shown in Fig. 5), which suggests that these two V S -based methods are more conservative than the NCEER-SPT method. Comparing these two V S -based methods, it is evident that AN00 is more conservative than KY13 since the FS values assessed by AN00 are significantly below the 45° line.

Calibration of the model uncertainties of the simplified methods
The discrepancies in the assessments by the different simplified methods as observed above are characterized using statistical approaches in this section. These discrepancies are referred to in this paper as the "model uncertainties" of the simplified methods. The model uncertainties are quantified as the ratio of the factors of safety (RFS). For instance, the RFS of the FS value obtained using CPT-q c methods (FS CPT ) to the value obtained using SPT-N methods (FS SPT ) can be expressed according to Eq. (5) as: Equation (6) shows that RFS only consists of the parameters evaluated by the distinctions of each method and without site-specific parameters such as the ground response. That is, RFS represents the relationship between different models and its variability reflects the model uncertainty of the simplified methods. To characterize this variability, the maximum likelihood estimation (MLE), which was developed based on frequentist principles, was adopted because it is known that it has many positive statistical properties, including being unbiased and having asymptotic normal distributions, asymptotic minimum variance, and consistency (DeGroot and Baecher 1993). Applications of the characterization of various geotechnical material properties can be found in the literature (e.g., Ang and Tang 2007;Ching and Phoon 2012;Juang et al. 2013). RFS cannot be negative, evidently, and so the logarithm of RFS, ln(RFS), is adopted to be characterized using MLE. For a Gaussian distribution, the likelihood function is expressed as follows.
where N is the number of data points and μ and σ are parameters of the Gaussian distribution representing mean value and standard deviation, respectively.
After these parameters are estimated using MLE, the goodness of fit of the probabilistic model needs to be determined. The goodness-of-fit test can be viewed as the normality test because a Gaussian distribution has been adopted as the probabilistic model in this study. The normality of the estimation can be validated using the Kolmogorov-Smirnov (K-S) test (Conover 1999). In general, the goodness-of-fit test is considered as passed when the p-value associated with the K-S test is equal to or greater than 5%.

Model uncertainties of the CPT-SPT methods
The ln(RFS) values of the six selected CPT-q c methods with respect to the representative SPT-N method (NCEER-SPT) were evaluated with the datasets compiled in this study, and their histograms and the corresponding characterized probabilistic models from MLE are shown in Fig. 6. Figure 6 shows that Gaussian distribution models with the estimated parameters listed in Table 2 can reasonably fit the distribution of each ln(RFS) of the CPT-SPT methods. Furthermore, the p-values associated with the K-S test of these characterized probabilistic models are listed in Table 2. They are all higher than 5%, which indicates that a Gaussian distribution is suitable for the model uncertainties of these six selected CPT-q c methods with respect to the NCEER-SPT method.

Model uncertainties of the VS-SPT methods
To calibrate the model uncertainties of the V S -SPT methods, the ln(RFS) values of the two selected V S -based methods with respect to the representative SPT-N method (NCEER-SPT) were also evaluated and are shown as histograms in Fig. 7. The calibrated Gaussian distribution models of these ln(RFS) are plotted as blue curves in Fig. 7 as well. It is evident that these Gaussian distribution models fit the distributions of the ln(RFS) histograms reasonably well. The goodness of fit of these Gaussian distribution models is also indicated by the p-values associated with their K-S tests (see Table 2). A Gaussian distribution is thus a suitable probabilistic model for the model uncertainties of these two selected V S -based methods with regard to the NCEER-SPT method.

Consistency issues for SPT-N methods
The NCEER-SPT method, the SPT-N method described in the 1998 NCEER/NSF workshop, was adopted as the representative SPT-N method being the basis for comparison in this study, but it does not always perform in a manner consistent with other SPT-N methods. This means that the model uncertainties of the characterized CPT-SPT and V S -SPT methods are not definitely compatible with other SPT-N methods. Although the SPT-N methods that have appeared over the past few decades in the literature were developed in accordance with the framework proposed by Seed and Idriss (1972), and although their success rates at predicting whether or not soil is liquefied do not differ significantly (Hwang et al. 2005;Chang et al. 2011), some studies have indicated that their assessments for the cyclic resistance and FS of soils with different normalized SPT penetration resistance are still distinct (e.g., Cetin et al. 2018;Hwang et al. 2021;Cetin and Bilge 2022).
Thus, the consistency issue for NCEER-SPT is discussed in the current section. The model uncertainties are further calibrated, which allows the characterized models of the CPT-SPT and V S -SPT model uncertainties to be applicable to other SPT-N methods. Four SPT-N methods validated by global liquefied and non-liquefied case histories were selected to examine this consistency issue. They were proposed by Cetin et al. (2004) (denoted as CE04), Boulanger and Idriss (2014) (denoted as BI14), Cetin et al. (2018) (denoted as CE18), and Hwang et al. (2021) (denoted as HBF21). Note that CE04, BI14, and CE18 are probabilistic and deterministic versions, whereas only deterministic versions were adopted in this study.
The model uncertainties of these four SPT-N methods were quantified as ln(RFS) and characterized using statistical approaches identical to the procedure for the calibration of the model uncertainties of the CPT-SPT and V S -SPT methods. The characterization results are shown in Fig. 8 and listed in Table 3. Figure 8 and the p-values associated with the K-S test in Table 3 show that the Gaussian distribution is also suitable for the model uncertainties of these SPT-N methods. The model uncertainties of these SPT-N methods (σ s ) are reasonably lower than those of the CPT-SPT or V S -SPT methods (σ c or σ v ).

Prediction of FS equivalent to the SPT-N methods
As long as the model uncertainties of CPT-SPT and V S -SPT methods have been characterized, their assessments of liquefaction potential equivalent to the SPT-N methods can be sensibly inferred based on the assessments using CPT-q c or V S -based methods, even at those sites lacking SPT data. Given a set of factors of safety against cyclic liquefaction assessed using CPT-qc methods, FS CPT , the corresponding FS values assessed using the NCEER-SPT method, FS NCEER-SPT , can be expressed as a log-normal random variable as follows.
where μ c and σ c are calibrated parameters that can be found in Table 2. Therefore, FS NCEER-SPT can be inferred according to Eq. (8). For instance, the median value of FS NCEER-SPT is exp[ln(FS CPT )-μ c ], and the 95% confidence intervals can be estimated using the following equation: On the other hand, the assessments using other SPT-N methods can also be reasonably inferred with the calibrated models in this study. Consider that the FS values evaluated from HBF21 (FS HBF21 ) are of interest. Given a set of FS CPT , the corresponding FS HBF21 values can be expressed as a lognormal random variable as well: where μ s and σ s are calibrated parameters that can be found in Table 3, and FS HBF21 can be inferred according to Eq. (10). For instance, the median value of FS NCEER-SPT is exp[ln(FS CPT )-(μ c-μ s )], and the 95% confidence intervals can be estimated using the following equation:  The approach to predicting FS values assessed using SPT-N methods given FS VS is similar to the procedures based on FS CPT . FS NCEER-SPT predicted based on FS VS can also be expressed as a log-normal random variable, whose median value is exp[ln(FS VS )-(μ v )], and the 95% confidence interval is shown below: where μ v and σ v are calibrated parameters that can be found in Table 2.

Illustrated applications of the calibrated models
This section demonstrates the application of the calibrated models to the prediction of FS SPT by CPT-q c and V S -based methods and the corresponding liquefaction potential index (LPI) using a set of SPT-CPTU-V S records excluded from the datasets used for the calibrations above. LPI is an index proposed by Iwasaki et al. (1984) to quantify the severity of ground manifestation or the damage to low-rise buildings due to liquefaction; it is commonly adopted for liquefaction hazard map generation as well (e.g., Power and Holzer v 1996;Rix and Romero-Hudock 2006;Cabalar et al. 2019). In practice, the LPI value is regularly evaluated following the formula proposed by Iwasaki et al. (1984) with the FS values generated using SPT-N methods on SPT records to assess the liquefaction potential of the site of interest. However, at sites lacking SPT data, CPT-q c or V S -based methods are substituted for SPT-N methods, and their variability may impact the LPI evaluation and other such assessments of liquefaction hazards. This study calibrated probabilistic models for the model uncertainties of CPT-SPT, or V S -SPT methods to provide strategies for reasonable estimations of LPI values using SPT-N methods, even when SPT data are lacking.
As an illustration of the application of these calibrated probabilistic models, one set of records from an alluvium site in Taipei City was examined. The records consist of SPT, CPTU, and down-hole seismic test data. The distance between the SPT borehole and CPTU was approximately 2 m, and the down-hole seismic test was implemented at the same location as the CPTU. The normalized data (i.e., (N 1 ) 60 , q t1 , V S1 ) are shown in Fig. 9. This figure shows that these three normalized datasets have reasonably consistent profiles when plotted against depth, which indicates that the horizontal spatial variability of this site is not significant. As such, so that it is ignored hereinafter. Based on these records, the FS NCEER-SPT profile subjected to the design earthquake with a 475-year return period (MOI 2022) is shown as a series of yellow circles in Fig. 10, whereas the results of predicted profiles of FS NCEER-SPT using the calibrated models conditioning the six FS CPT and two FS VS models are plotted as solid blue lines (median) and dashed lines (95% confidence interval). It can be seen that the median profiles of the predicted FS NCEER-SPT values based on the FS CPT or FS VS profiles are reasonably consistent with the actual FS NCEER-SPT , and the corresponding 95% confidence intervals include most of the actual FS NCEER-SPT values. This suggests that the statistical consistencies of the characterized models for the model uncertainties of the CPT-SPT and V S -SPT methods are acceptable. After further evaluation, the LPI values assessed based on FS NCEER-SPT , FS CPT , and FS VS are shown in Fig. 11 as red dashed lines and black squares. The median values and 95% confidence intervals of the LPI value predictions using the NCEER-SPT methods conditioning FS CPT and FS VS are also shown in Fig. 11(a) and (b) as blue circles and grey lines. Figure 11 shows that the medians of the predicted LPI are much closer to the actual LPI NCEER-SPT value, and the 95% confidence intervals also cover the actual LPI NCEER-SPT values. This also suggests that the calibrated models in this study perform reasonably consistently at LPI prediction.

Concluding remarks
This paper examined the inconsistencies of some prevalent simplified methods for liquefaction potential assessment, including SPT-N, CPT-q c , and V S -based procedures, and adopted the ratio of factors of safety (RFS) to quantify the variability of their performance, which were referred to as "model uncertainties" in this study. These model uncertainties were calibrated as probabilistic models using statistical approaches based on the SPT-CPTU-V S datasets from 13 alluvium sites in the Taipei Basin. These models demonstrated that they can provide reasonable estimations of the factors of safety against liquefaction (FS) evaluated via one simplified method inferred from the FS values by another method. The applications of these models to the prediction of liquefaction potential indices evaluated using other methods were illustrated with a real case in the final section. These calibrated models are believed to provide reasonable strategies for consideration of the discrepancies of different methods applied to geotechnical design against liquefaction and thus yield more reliable results.
Some other observations are enumerated below: 1. The six selected CPT-q c methods in this study were all found to be more conservative than the NCEER-SPT method. Among them, the method proposed by Ku and Juang (2012) was the most conservative, whereas the method developed by Olsen (1997) was the least conservative. The procedure developed by Olsen (1997) was less biased compared with the NCEER-SPT method than the other CPT-q c methods, but its variability was the highest when compared with the NCEER-SPT method. 2. For V S -based methods, the two selected methods were more conservative than the NCEER-SPT method. Furthermore, it was evident that the method developed by Andrus and Stokoe (2000) was significantly more conservative, and its variability compared with the NCEER-SPT method was higher than the other V S -based method. 3. This study selected four SPT-N methods that had been validated with global case histories of liquefaction in order to discuss their consistencies with the NCEER-SPT method. The results showed that they all performed more consistently compared with the NCEER-SPT method than the CPT-q c or V S -based methods. The procedure proposed by Hwang et al. (2021) was the least biased, and the uncertainty was lower, which indicates that it performed the most consistently compared with the NCEER-SPT method.
It should be noted that the probabilistic models calibrated in this study were established based on data from Taipei Holocene alluvium deposits only. Their feasibility for other geological environments or stratigraphic formations of different ages requires further validation.