Adaptive sampling strategy for characterizing spatial distribution of soil liquefaction potential using cone penetration test

Zheng Gun, Yu Wng,*, Tengyun Zho

a Department of Architecture and Civil Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China

b School of Human Settlements and Civil Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

Keywords:Liquefaction potential Information entropy Cone penetration test (CPT)Site characterization Compressive sampling

ABSTRACT Characterizing spatial distribution of soil liquefaction potential is critical for assessing liquefactionrelated hazards (e.g. building damages caused by liquefaction-induced differential settlement). However,in engineering practice,soil liquefaction potential is usually measured at limited locations in a specific site using in situ tests,e.g.cone penetration tests(CPTs),due to the restrictions of time,cost and access to subsurface space. In these cases, liquefaction potential of soil at untested locations requires to be interpreted from limited measured data points using proper interpolation method,leading to remarkable statistical uncertainty in liquefaction assessment. This underlines an important question of how to optimize the locations of CPT soundings and determine the minimum number of CPTs for achieving a target reliability level of liquefaction assessment. To tackle this issue, this study proposes a smart sampling strategy for determining the minimum number of CPTs and their optimal locations in a selfadaptive and data-driven manner. The proposed sampling strategy leverages on information entropy and Bayesian compressive sampling(BCS).Both simulated and real CPT data are used to demonstrate the proposed method. Illustrative examples indicate that the proposed method can adaptively and sequentially select the required number and optimal locations of CPTs.

1. Introduction

The occurrence of soil liquefaction may cause severe damage to civil structures during an earthquake,such as foundation failure of buildings,floating of buried tanks,and bridge collapse(e.g.Kramer,1996).In situ tests,e.g.cone penetration tests(CPTs),are commonly used in practice to assess the liquefaction potential of soil,in terms of factor of safety (FS) against liquefaction (e.g. Seed and Idriss,1982; Idriss and Boulanger, 2008). It is widely acknowledged that the damage of structures during an earthquake, such as building foundation failure caused by liquefaction-induced differential settlement, is strongly influenced by spatial distribution of liquefied soils (e.g. Holzer and Bennett, 2007; Cubrinovski et al., 2011; Bray et al., 2014; Bong and Stuedlein, 2018; Guan et al., 2021). Therefore,characterizing spatial distribution of soil liquefaction potential(e.g.FS at each point within a vertical cross-section of a specific site)is critical for assessing liquefaction-related hazards.

However, due to the restrictions of access, time, and cost, CPT soundings are usually sparsely conducted(e.g.one CPT sounding for every tens of meters)at a specific site.The spatial distribution of soil liquefaction potential has to be interpreted from limited CPT soundings,leading to statistical uncertainty in liquefaction assessment(e.g.Dawson and Baise,2005;Guan et al.,2020;Shi and Wang,2021).Such uncertainty might result in significant risk to civil structures when an earthquake occurs. Generally, conducting numerous CPT soundings provides increasingly reliable liquefaction assessment results in a cross-section,but it also dramatically increases the time and expenditure of the liquefaction assessment (e.g. Wang et al., 2019; Guan et al.,2020).A fundamental but challenging question in liquefaction assessment is: how to optimize locations of CPT soundings and determine the minimum number of CPTs for achieving a reliability level of liquefaction assessment results that decision makers of a project want to achieve(i.e.target reliability level).

Currently, no quantitative approach is available in engineering practice for guidance on site investigation for liquefaction assessment.Existing geotechnical manuals and standards only provide some empirical site investigation guidelines. For example, Look (2007)suggested a test spacing of 50-100 m during preliminary design phase,and a spacing of 30-100 m for roads and 20-30 m for bridges and buildings during detail design phase.However,these guidelines only provide the minimum requirements for site investigation. The number and optimal locations of CPTs for a specific site should be determined based on spatial variability of subsurface condition and target reliability level of liquefaction assessment results (e.g. CEN,2007; FHWA-NHI-16-072, 2017; Rix et al., 2018; Guan and Wang,2021). The most challenging issue in site investigation for liquefaction assessment is that much of the information about the subsurface conditions is usually unavailable at the initial stage of site investigation, leading to the difficulty in pre-determining the appropriate number and optimal locations of CPTs(e.g.FHWA-NHI-16-072,2017).

Extensive studies have been carried out to develop CPT-based methods for assessing soil liquefaction potential (e.g. Stark and Olson, 1995; Moss et al., 2006; Robertson, 2010). More recently,several studies have been performed to investigate the influences of soil spatial variability on liquefaction evaluation(e.g.Assimaki et al.,2003; Lenz and Baise, 2007; Montgomery and Boulanger, 2017;Wang et al.,2017a;Bong and Stuedlein,2018).Popescu et al.(1997)indicated that soil spatial variability greatly influences the liquefaction behavior of soils.Chen et al.(2016)proposed a random fieldbased method for evaluating liquefaction potential considering soil spatial variability. However, no quantitative and rational methods are available for optimizing CPT locations and determining the minimum number of CPTs for liquefaction assessment with consideration of spatial variability of soil properties.

Totacklethisissue,thisstudyproposesasmartsamplingstrategyfor determining the minimum number of CPTsoundings and their optimal locations in a self-adaptive and data-driven manner.Using information entropytheoryandBayesiancompressivesampling(BCS),theproposed sampling strategy sequentially increases the number of CPTs at the optimal location until the target reliability level of the interpreted liquefaction assessment results is achieved. The rest of this paper is organizedas follows.The proposedCPTsampling strategy frameworkis firstly introduced,followed by its detailed implementation procedure.Then,an illustration example of simulated CPT data is provided.Finally,real CPT data from New Zealand Geotechnical Database are used to demonstrate the proposed sampling strategy.

2. The proposed smart sampling strategy

In geotechnical engineering practice, it is usually difficult to conduct a comprehensive site investigation for spatial liquefaction assessment,particularly during the initial stage of site investigation due to a lack of information about subsurface conditions.However,as the subsurface information is gradually acquired during site investigation, engineers’ knowledge on the subsurface condition accumulates and improves,and the effectiveness of site investigation can also be improved using the accumulated knowledge of the site.Therefore,for example, FHWA-NHI-16-072 (2017) recommends a phased site investigation plan for a poorly understood site. In the phased site investigation, preliminary investigation points (e.g. CPTs) might be performed at a relatively large spacing.Then,the preliminary interpretation of subsurface condition from the initial investigation point measurementsisevaluatedbasedonthedesiredreliabilitylevel.If the preliminary interpretation results do not meet the requirement,additional investigationpoints are performed to improve reliability of the interpreted subsurface conditions. It should be noted that geomorphology plays an important role in geotechnical site investigation, and careful consideration of geology and geomorphology can help to reduce uncertainty in planning explorations.

Key idea of the proposed method is consistent with the phased site investigation plan. To deal with the challenge in unknown subsurface condition,the proposed sampling strategy conducts the CPT soundings in a self-adaptive manner.It continuously evaluates the FS in the cross-section interpreted from the already performed CPT soundings and decides whether or not adaptation(i.e.addition CPT soundings) is required. In other words, site investigation for liquefaction assessment is considered as a sequential process, in which determining the k-th CPT sounding is based on the information obtained from the previously conducted (k-1) CPT soundings, and self-adaptive process stops when reliability of the interpreted FS in the cross-section achieves the target level. Thus,even without sufficient prior information about the subsurface conditions, the minimum number and optimal locations of CPTs can be determined automatically.

In this study, the interpretation of FS at each point of a crosssection is performed using two-dimensional (2D) BCS and limited CPT soundings. The 2D BCS is adopted in the proposed sampling strategy,because it provides not only a point(single)estimate of FS in a cross-section but also the full marginal probability distribution of the estimated FS,which quantify uncertainty and offers error bars(e.g. mean, μ ± standard deviation (SD), σ) on the interpretation results. These error bars or coefficient of variation, COV = σ/μ,reflect confidence (or reliability) level on interpreted results. In addition,BCS has unique capability of dealing with non-stationary spatial data and is directly applicable to CPT data in subsurface conditions with multiple, but unknown, number of soil layers without pre-stratification of subsurface space(e.g.Zhao et al.,2020).

The framework of the proposed method mainly contains five steps, as illustrated in Fig.1. In Steps 1 and 2, the target reliability level in terms of target coefficient of variation, COVT, and preliminary number of CPT soundings are determined.Then,in Step 3,for a given earthquake,the reliability level,COVM,of FS in the crosssection interpreted from preliminary number of CPTs is evaluated.

If COVM> COVT, reliability of the interpreted FS in the crosssection requires improvement, and hence, in Step 4, the adaptive process is triggered to reduce COVMsequentially by performing an additional CPT sounding at its optimal location.Note that,only one additional CPT sounding is conducted at each adaptation.The selfadaptive process stops when COVM≤COVT. After that, the minimum number and optimal locations of CPT soundings are determined automatically for achieving the target reliability level of liquefaction assessment results in Step 5. The approaches and procedure of each step are described in detail below.

Fig.1. Framework of proposed CPT sampling strategy for liquefaction assessment.

2.1. Steps 1 and 2: determining target reliability level and preliminary CPT soundings

In Step 1,COVTfor the interpreted FS in the cross-section is first determined. The selection of COVTshould be based on soil spatial variability at a specific site, importance of the structures and risks associated with the potential earthquake.For example,if a structure is extremely sensitive to differential settlement, a stringent target reliability level should be adopted.The preliminary number of CPT soundings is selected depending on the local engineering experience or the recommendations provided by geotechnical design codes and guidelines in Step 2 (e.g. FHWA-NHI-16-072, 2017). As suggested by most geotechnical design codes and manuals, the investigation points should be evenly distributed within a site when no pre-existing information about subsurface condition is available(e.g.CEN,2007;FHWA-NHI-16-072,2017;Zhao and Wang,2019).In addition,based on the information entropy theory,Zhao et al.(2021)showed that when no pre-existing CPT data are available, equal spacing along with the horizontal direction is an efficient sampling strategy for CPT. Therefore, an equally spaced sampling strategy is adopted for preliminary CPT soundings in this study.

2.2. Step 3: Evaluation of the interpreted FS in the cross-section

In Step 3,measured data points from preliminary CPT soundings are used to interpret FS in a cross-section. There are two key modules in Step 3: (1) CPT-based simplified procedure for liquefaction assessment, and (2) BCS for interpretation of high spatial resolution FS in the cross-section from limited CPT measurements.

2.2.1. Simplified procedure for liquefaction assessment using CPTs

Liquefaction triggering analysis is usually performed following the simplified procedure (e.g. Seed and Idriss, 1971; Cetin et al.,2004).In the simplified procedure,FS is used to quantify the liquefaction potential of soil,which is computed as(e.g.Youd et al.,2001):

where CRR is the cyclic resistance ratio, which indicates soil liquefaction resistance; CSR is the cyclic stress ratio, which represents the earthquake demand on soil; CRR7.5is the CRR when the earthquake magnitude M = 7.5; MSF = 174/M2.56represents the magnitude scaling factor;and M represents earthquake magnitude.If the calculated FS < 1, liquefaction is estimated to occur.

Based on case histories, Robertson and Wride (1998) proposed the following equations to estimate CRR7.5directly from CPT measurement data:

Using the simplified procedure, FS can be estimated at the locations with CPT measurements.To obtain FS over the whole crosssection,(qc1N)cswithin a cross-section is interpolated from limited measured CPT data using 2D BCS.

2.2.2. BCS interpretation of FS at each point of a cross-section

The 2D BCS is an innovative interpolation method that can reconstruct a high spatial resolution cross-section of spatially correlated properties(e.g.(qc1N)cs)from limited measurement data(e.g.Ji et al.,2008,2009;Babacan et al.,2009;Zhao et al.,2018).It not only provides an estimate of(qc1N)csat untested locations from limited measurements, but also quantifies the confidence level of the estimated(qc1N)cs.Note that BCS is a non-parametric and datadriven method which can deal with soil properties with non-Gaussian marginal distribution functions, such as lognormal distribution, gamma distribution, exponential distribution and even unknown distribution functions (e.g. Wang et al., 2021). Mathematically, (qc1N)csat each point within a cross-section can be represented by an Nx1×Nx2data matrix F.Mcrepresents the number of CPT soundings. (qc1N)csmeasurements from McCPT soundings conducted within the site(i.e. the corresponding CPT data from F)are denoted by a matrix Y with a dimension of Nx1× Mc. In the context of 2D BCS,F can be written as a linear summation of Nx1×Nx22D basis functions(e.g.Candes et al.,2006;Candès and Wakin,2008; Zhao et al., 2020):

where ^Xi,jrepresents the entry in the i-th row and j-th column of the reconstructed cross-section of ^X; COV^Xi,jrepresents the coefficient of variation of ^Xi,j;and μ^Xi,jand σ^Xi,jrepresent the mean and SD of ^Xi,j, respectively. Using Eqs. (1)-(10), reliability level of the interpreted FS in the cross-section from limited CPT soundings is estimated.

2.3. Steps 4 and 5: Improvement of the liquefaction assessment results from additional CPTs

When the preliminary interpreted liquefaction assessment results indicate that reliability level obtained from the preliminary investigation results does not reach the target reliability level,additional CPT soundings are required to obtain further measurement data and enhance reliability level of the interpreted liquefaction assessment results until the target reliability level is achieved. Therefore, when COVMof interpreted FS in the crosssection is larger than COVT, additional CPTs are adaptively and sequentially performed at their optimal locations.In each adaptive process, one additional CPT sounding is conducted at its optimal location and used together with the previously obtained measured data points to re-interpret the FS in the cross-section. The key element in Step 4 is to determine the optimal location for an additional CPT sounding based on the information from the previously conducted CPTs, which is introduced below.

In this study, the optimal location for an additional CPT sounding is selected using an information entropy-based method (e.g.Zhao et al., 2021). Information entropy is able to evaluate the uncertainty in a random process(e.g.Goodfellow et al.,2016).Spatial variation of FS in a cross-section may be considered as a random process,and thus information entropy can be used to measure the uncertainty in FS in the cross-section interpreted from limited CPT measurements. Note that, for a random process modeling of a soil property,autocorrelation function is used to model autocorrelation of soil property among different points(e.g.Baecher and Christian,2005).Generally,as the number of measured data points increases,the information entropy or uncertainty of the interpreted crosssection of FS decreases. When FS at each point of the whole cross-section is measured completely, information entropy of the interpreted FS in the cross-section decreases to zero, i.e. no uncertainty.

3. Numerical example

where Δx1and Δx2are the vertical and horizontal distances between two data points, respectively; and λhand λvare the horizontal and vertical correlation lengths taken as 60 m and 1.5 m,respectively,which are within the typical ranges of λvand λhfor CPT data (e.g. Cao et al., 2016). After the random field of (qc1N)csis specified, a random field generator, such as circulant embedding algorithm (e.g. Dietrich and Newsam,1993), is used to generate a cross-section of(qc1N)cs.Fig.2 shows the generated cross-section of(qc1N)cswith a depth of 15 m and width of 250 m. The vertical resolution of the cross-section of (qc1N)csis 0.1 m and the horizontal resolution is 1 m. In total, the cross-section has 151 ×251 =37,901 data points.

For illustration purpose, the designed ground motion intensity measures are taken as M = 7 and amax= 0.15g. Saturated unit weight of soils is taken as 18 kN/m3, and the groundwater table is assumed to be located at ground surface. In Step 1, COVTof interpreted liquefaction assessment results is taken as 10%. Then, the preliminary number of CPTs,i.e.the starting point of the proposed sampling strategy,is first selected.Based on the recommendations from FHWA-NHI-16-072 (2017), CPT soundings with a spacing of 50 m might be adopted for preliminary interpretation.The width of the cross-section is 250 m, and thus Mc= 250/50 + 1 = 6 CPT soundings should be conducted, as illustrated in Fig. 2 by dashed lines.

Fig. 2. Simulated equivalent cone penetration resistance in clean sand, (qc1N)cs in a vertical cross-section.

In Step 3, measurement data of the six CPT soundings are obtained from the generated cross-section of (qc1N)csin Fig. 2 and used to interpret FS in the cross-section. To mimic the process of performing six CPTs (i.e. C-1, C-2, …, C-6) in site investigation,6 × 151 = 906 data points of (qc1N)csat these CPT locations are taken from the original simulated (qc1N)csdata, as illustrated in Fig. 3. Using 2D BCS method and MCMC, Nb= 500 samples of(qc1N)cscross-section are generated from these measurement data of(qc1N)cs.Nbrepresents the size of generated sample realizations.The mean and SD of generated (qc1N)cssamples are illustrated in Fig.4.The 2D discrete cosine transform(DCT)basis function is used in this illustration example, because DCT is widely used in image and signal processing (e.g. Candès and Wakin, 2008). Note that class selection framework of Bayesian model can be used to select the most appropriate basis function in engineering practice (e.g.Wang et al.,2017b).Spatial resolutions of the generated samples ofare set as the same as the original one.

After that,for each CRR sample together with estimated CSR,the cross-section of FS is calculated using Eq.(1),leading to Nb=500 FS in the cross-section samples.Fig.7a and b shows the mean and SD of the Nb=500 FS samples generated from six CPT soundings.The reliability level of FS in the cross-section interpreted from six CPT soundings is estimated as COVM= 14.2% using Eq. (10), and it is greater than COVT= 10%. This suggests that the reliability of the interpretation results should be further improved.Hence,the selfadaptive process is triggered to perform additional CPT soundings at their optimal locations.

Fig. 3. Equivalent cone penetration resistance in clean sand, (qc1N)cs measurement data of six CPT soundings: (a) C-1, (b) C-2, (c) C-3, (d) C-4, (e) C-5, and (f) C-6.

Fig. 4. Interpreted cross-section of equivalent cone penetration resistance in clean sand, (qc1N)cs, using six preliminary CPT soundings: (a) Mean and (b) SD.

Fig. 5. High-resolution cyclic resistance ratio, CRR, cross-section interpreted from six CPTs: (a) Mean and (b) SD.

In Step 4,the target reliability level of interpreted FS in the crosssection is reached in a self-adaptive manner. One additional CPT sounding is conducted at its optimal location for each adaptation based on the information obtained from performed CPT soundings,and the new (qc1N)csmeasurement from the additional CPT sounding together with the previously measured data points are used to re-interpret FS in the cross-section. Once the target reliability level is reached, the self-adaptive process stops. As introduced in previous section, the optimal CPT location is the one where the summation of variance or SD of the interpreted FS along the depth is the largest. Fig. 7c shows the summation of SD of the interpreted FS along the depth at different locations.It is observed from that the summation of SD at around 175 m is larger than that at around 75 m. This is reasonable because the uncertainty of the soil properties interpolated from the proposed method depends not only on the distance from the sampled points, but also on the local variation of measured data points.Fig.3 shows that the overall variability of(qc1N)csfrom C-4 and C-5 conducted at the horizontal coordinate of 150 m and 200 m is larger than that from C-2 and C-3 conducted at the horizontal coordinate of 50 m and 100 m,leading to the relatively large summation of SD at around 175 m. Such results are consistent with the geotechnical engineering experience and judgment that subsurface conditions are expected to be highly variable when the nearby measured soil properties exhibit significant variability. Fig. 7c indicates that the maximum summation of SD along the depth is 60 which is located at the horizontal coordinate of 125 m. This indicates that an additional CPT sounding shall be conducted at this location for the first adaptation.

Fig. 6. CSR estimated from maximum acceleration, amax = 0.15g.

Fig.7. Interpreted high-resolution FS in the cross-section using six CPTs:(a)Mean and(b) SD of FS, and (c) Summation of SD of interpreted FS along the depth.

To mimic the process of conducting additional CPT,(qc1N)csdata points along the depth at the horizontal coordinate of 125 m are taken from the simulated (qc1N)cscross-section in Fig. 2 and used together with the previous 906(qc1N)csdata points to re-interpret FS in the cross-section. In total, 7 × 151 =1057 (qc1N)csmeasurement data obtained from Mc=6+1=7 CPT soundings are used to interpret the FS in the cross-section.Fig.8a and b shows the mean and SD of the FS in the cross-section interpreted from seven CPT soundings. Fig. 8c shows the summation of SD of interpreted FS along the depth.

A comparison between Figs. 7c and 8c shows that SD of the interpreted FS close to the location (i.e. 125 m in the horizontal direction)of the additional performed CPT is significantly reduced.The COVMis calculated to be 13.3%.This suggests that the reliability level of the interpreted liquefaction assessment results is enhanced after conducting an additional CPT. However, COVM= 13.3% is still greater than COVT= 10%, and therefore, the self-adaptive process continues to further improve reliability level of the interpreted results.It is observed from Fig.8c that the maximum summation of SD is located at the horizontal coordinate of 29 m,and thus the next CPT sounding is conducted at this location.Fig.9a and b shows the mean and SD of the FS in the cross-section interpreted from Mc=6+2=8 CPT soundings.The COVMis further reduced to 12.1%.

The Board Meeting had come to an end. Bob started to stand up and jostled（，） the table, spilling his coffee over his notes. How embarrassing. I am getting so clumsy in my old age.

Self-adaptive process continues until COVM≤COVT. The evolution of COVMwith the number of adaptations is shown in Fig.10.It is found that COVMdecreases from 14.2% to 9.5% as Mcincreases from 6 to 12. The reliability level of interpreted FS in the crosssection is monitored at each adaptation, and the FS in the crosssection is re-interpreted with the newly conducted CPT sounding,which in turn is used for decision on further CPT soundings, if needed.When Mcincreases to 12,COVM=9.5%

Fig. 8. Interpreted high-resolution FS in the cross-section using previously conducted six CPTs together with one additional CPT: (a) Mean and (b) SD of FS in the crosssection, and (c) Summation of SD of interpreted FS along the depth.

4. Real CPT data example

In this section,the proposed sampling strategy is demonstrated using a set of real CPT data from a site located at the central business district of Christchurch, New Zealand with a latitude of 43.522°south and a longitude of 172.655°east. Based on site investigation reports,sediments of this site consist of 5-6 m thick layer of silty sand and sandy silt with a saturated unit weight of 18 kN/m3.The water table is about 1.2 m below the ground surface.To evaluate the performance of the proposed method,it is applied to evaluating liquefaction hazard at this site under the 2011 Christchurch Earthquake with M=6.2.Based on recordings from a nearby strong motion station,amax=0.52g is used in this example(e.g. Bradley and Cubrinovski, 2011).

Fig. 10. COVM of interpreted FS in the cross-section versus the number of additional CPT soundings (i.e. number of adaptation).

The proposed sampling strategy is performed for cross-section A-A′, as illustrated in Fig.12 by a solid line. Seven CPT soundings includingCPT_385,CPT_55692,CPT_55682,CPT_55684,CPT_55680, CPT_55685 and CPT_55683 conducted by Lankelma Ltd. and Tonkin & Taylor Ltd. in 2012 are used to demonstrate the proposed method, as shown in Fig.12 by solid triangles and solid circles,respectively.In Step 1,a target reliability level,COVT=5%is used for illustration. Then, in Step 2, five CPT soundings (i.e.CPT_385,CPT_55692,CPT_55682,CPT_55684 and CPT_55680)with a horizontal spacing of 40-50 m are adopted for preliminary interpretation of soil liquefaction potential, as illustrated in Fig.12 by solid triangles. Fig. 13 shows the CPT measurement data (qcand fs) of these CPT soundings which are downloaded from New Zealand Geotechnical Database(NZGD)(NZGD,2021).For each set of qcand fs, (qc1N)csis calculated using Eqs. (3) and (4).

In Step 3,measured data points from the selected CPT soundings for the preliminary analyses are used to interpret the FS in the cross-section. Using measured (qc1N)csdata from these five CPT soundings, Nb= 500 samples of (qc1N)cscross-section are generated using 2D BCS and MCMC. The mean and SD of generated(qc1N)cssamples are shown in Fig.14. The vertical and horizontal resolutions of the samples of(qc1N)cscross-section are 0.01 m and 1 m, respectively. Then, Nb= 500 CRR cross-section samples are computed from the samples of(qc1N)csusing Eqs.(1)and(2).Fig.15 shows the mean and SD of the generated CRR samples.For the given earthquake (i.e. amax= 0.52g), the CSR within cross-section A-A′is computed using Eqs. (5) and (6), as illustrated in Fig.16.

Fig.11. Interpreted high-resolution FS in the cross-section using 12 CPTs:(a)Mean and(b) SD.

Using the generated CRR samples together with estimated CSR,Nb= 500 FS samples in the cross-section are calculated using Eq.(1). Fig. 17a and b shows the mean and SD of the FS samples generated from five CPT soundings. The reliability level of interpreted FS in the cross-section is calculated to be COVM=11.4%from Eq.(10),which is greater than COVT=5%.Therefore,additional CPT soundings are conducted,one by one,at their optimal locations for improving reliability of the interpretation results in Step 4.Fig.17c shows the summation of SD of the interpreted FS along the depth at different locations. It is observed from Fig.17c that the maximum summation of SD along the depth is 34.7 and occurs at the horizontal coordinate of 112 m.To mimic the process of conducting CPT at this location,CPT_55683,which was approximately conducted at the horizontal coordinate of 112 m and obtained from NZGD, is used together with previous five CPT soundings to re-interpret FS in the cross-section. CPT measurement data (qcand fs) from CPT_55683 are shown in Fig.18a. The interpretation results from six CPT soundings are illustrated in Fig.19.The COVMof the FS in the cross-section interpreted from six CPT is computed to be 8%, and the self-adaptive process continues. As shown in Fig. 19c, the optimal location for next CPT sounding is 158 m in the horizontal direction.

Fig. 12. Layout of five preliminary CPT soundings at a site in Christchurch, New Zealand.

Fig.13. Measurement data (cone tip penetration resistance, qc and sleeve friction, fs)from preliminary CPT soundings at a site in Christchurch, New Zealand: (a) CPT_385,(b) CPT_55692, (c) CPT_55682, (d) CPT_55684, and (e) CPT_55680.

Fig.14. Cross-section of equivalent cone penetration resistance in clean sand,(qc1N)cs,interpreted from preliminary CPTs: (a) Mean and (b) SD.

Fig.15. High-resolution cyclic resistance ratio,CRR,cross-section interpreted from five preliminary CPTs: (a) Mean and (b) SD.

CPT_55685 was approximately conducted at the horizontal coordinate of 158 m, and thus it is obtained from NZGD and used together with previous six CPT soundings to re-interpret FS in the cross-section. CPT measurement data (qcand fs) from CPT_55685 are shown in Fig.18b. Fig. 20 shows the FS interpreted from seven CPT soundings. It is found from Fig. 20b that the overall SD of interpreted FS is significantly reduced after performing two additional CPT soundings. The reliability level of interpretation results(COVM) is improved from 11.4% to 4.9% achieving the target reliability level, COVT= 5%. The layout of additional CPT soundings is illustrated in Fig.12 by solid circles. Using the proposed sampling strategy, seven CPT soundings and their locations are determined for achieving COVT= 5%.

Fig.16. CSR estimated from maximum acceleration, amax = 0.52g.

Fig.17. Interpreted high-resolution FS in the cross-section using five preliminary CPTs:(a) Mean and (b) SD, and (c) Summation of SD of interpreted FS along the depth.

It is observed from Fig. 20 that most of soils within the crosssection A-A′are likely to be liquefied during the earthquake, and thus significant liquefaction-induced building damages are expected. Such results are consistent with the field observations.Based on the field observations reported by Tasiopoulou et al.(2011), although liquefaction-induced surface ejection was not observed at this site, numerous nearby buildings were damaged during the earthquake.Note that,in this real data example,it takes 150-170 s to perform the analysis using a personal computer with Intel?Core?i7-6700 3.4 GHz(4 Cores)CPU and 16.0 GB RAM and determine the optimal location of the subsequent CPT.Therefore,it is a nearly real-time method. Once the CPT data are collected, the proposed method can provide the liquefaction assessment results and corresponding uncertainty within minutes for decisionmaking of further CPT soundings.

Fig.18. Measurement data (cone tip penetration resistance, qc and sleeve friction, fs)from additional CPT soundings at a site in Christchurch, New Zealand: (a) CPT_55683 and (b) CPT_55685.

Fig. 19. Interpreted high-resolution FS in the cross-section using six CPTs: (a) Mean and (b) SD, and (c) Summation of SD of interpreted FS along the depth.

5. Conclusions

Fig.20. Interpreted high-resolution FS in the cross-section using seven CPTs:(a)Mean and (b) SD.

A smart sampling strategy was developed in this study for characterizing spatial distribution of soil liquefaction potential in a vertical cross-section using CPTs.A high-resolution CPT data crosssection is interpolated from limited measurements using 2D BCS,while the location for an additional CPT is optimized using the information entropy-based method. When reliability level of the interpreted liquefaction assessment results does not meet the target and requires improvement, a self-adaptive process is triggered to perform additional CPT soundings at optimal locations(i.e.the locations with the maximum entropy reduction). Once the target reliability level is reached, the self-adaptive process stops.Using the proposed method, the minimum number and optimal locations of CPT soundings are automatically determined for achieving a target reliability level of the liquefaction assessment results. Note that more study will be conducted to consider uncertainties in liquefaction triggering procedure and that associated with seismic hazard in the model. In addition, the proposed method has not considered the liquefaction-induced damages and consequences (e.g. building damage caused by liquefactioninduced differential settlement) which will be addressed in the future study. The proposed method can be readily extended to three-dimensional (3D) space, while demanding intensive computation.Thus,an efficient computational method is required.The proposed sampling strategy for liquefaction assessment was illustrated using both simulated and field CPT data. The results show that the proposed sampling strategy is able to provide the optimal locations and minimum number of required CPT soundings.

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

The work described in this paper was supported by grants from the Research Grant Council of Hong Kong Special Administrative Region, China (Project Nos. CityU 11202121 and CityU 11213119).The financial support is gratefully acknowledged.