ORIGINAL PAPER

Random field-based regional liquefaction hazard mapping —data inference and model verification using a synthetic digitalsoil field

C. Hsein Juang1,2 & Mengfen Shen2& Chaofeng Wang2 & Qiushi Chen2

Received: 7 February 2017 /Accepted: 10 May 2017# Springer-Verlag Berlin Heidelberg 2017

Abstract Geostatistical tools and random field models havebeen increasingly used in recent liquefaction mapping studies.However, a systematic verification and assessment of randomfield models has yet to be taken, and implications of variousrandom field-based mapping approaches are unknown. In thispaper, an extremely detailed three-dimensional synthetic dig-ital soil field is artificially generated and used as a basis forassessing and verifying various random field-based modelsfor liquefaction mapping. Liquefaction hazard is quantifiedin terms of the liquefaction potential index (LPI), which ismapped over the studied field. A classical CPT-based lique-faction model is adopted to assess liquefaction potential of asoil layer. Different virtual field investigation plans are de-signed to assess the dependency of data inference and modelperformance upon the level of availability of sampling data.Model performances are assessed using three informationtheory-basedmeasures. Results show that when sampling datais sufficient, all random field-based models examined capturefairly well the benchmark liquefaction potentials in the studied

field. As the size of the sampling data decreases, the accuracyof predictions decreases for all models but to different degrees;the three-dimensional random field model gives the best resultin this scenario. All random field-based models examined inthis paper yield a slightly more conservative prediction ofliquefaction potential over the studied field.

Keywords Random field model . Liquefactionmapping .

Liquefaction potential index . Cone penetration test . Datainference .Model verification

Introduction

Liquefaction evaluation is an important issue in the field ofgeotechnical earthquake engineering. Many simplified lique-faction evaluation models have been developed based on col-lected databases of case histories for past earthquakes, follow-ing the framework of the original simplified procedure (Seedand Idriss 1971, 1982). These simplified methods rely on insitu testing, e.g., the standard penetration test (SPT), the conepenetration test (CPT), or the shear wave velocity (Vs) test, asa way to obtain and characterize the strength of the soil toresist liquefaction (e.g., Seed et al. 1985; Robertson andWride 1998; Andrus and Stokoe 2000; Youd et al. 2001;Finn 2002, Juang et al. 2002, 2003; Cetin et al. 2004; Mosset al. 2006; Boulanger and Idriss 2012; Khoshnevisan et al.2015; Shen et al. 2016).

Those simplified methods have been used to evaluate liq-uefaction potential at an in situ test location. To estimate ormap liquefaction potentials over an extended region, the spa-tial variation and dependence of soil properties and liquefac-tion potentials need to be considered. To this end,geostatistical tools and random field models have been in-creasingly used in recent regional liquefaction mapping.

* Qiushi Chenqiushi@clemson.edu

C. Hsein Juanghsein@clemson.edu

Mengfen Shenmengfes@clemson.edu

Chaofeng Wangchaofew@clemson.edu

1 Key Laboratory of Road and Traffic Engineering of the Ministry ofEducation, College of Transportation Engineering, Tongji University,Shanghai 201804, China

2 Glenn Department of Civil Engineering, Clemson University,Clemson, SC 29634, USA

Bull Eng Geol EnvironDOI 10.1007/s10064-017-1071-y

Focusing on how spatial variations and dependence areconsidered and incorporated in the mapping process, threetypes of approaches may be proposed and will be investigatedin this study, i.e., the averaged index approach, the two-dimensional (2D) local soil property approach, and thethree-dimensional (3D) local soil property approach. To illus-trate these approaches, the liquefaction potential at a givenlocation is quantified through an index called the liquefactionpotential index (LPI) (Iwasaki et al. 1978, 1982; Sonmez2003). In the averaged index approach, the spatial dependenceof liquefaction potential (quantified by LPI) at the test loca-tions is characterized and used as input to random field-basedLPI mapping. This approach is widely used in current lique-faction hazard mapping studies (e.g., Holzer et al. 2006a, b; Liet al. 2006; Baise et al. 2006, 2008; Lenz and Baise 2007;Juang et al. 2008; Chen et al. 2015; van Ballegooy et al.2015; Wang et al. 2017). In the 2D and 3D local soil propertyapproaches, the spatial dependence of soil properties, e.g., thecone tip resistance (qc) and the sleeve resistance (fs) obtainedfrom CPT tests, or the corrected blow counts (N1)60 from SPTtests, are characterized and treated as spatially correlated ran-dom variables. Once soil properties of interest are generatedthrough random field models for the entire studied region, thecorresponding liquefaction potential can be calculated. In the2D local soil property approach, only the horizontal correla-tion is considered. The random soil property field is generatedlayer-by-layer considering the horizontal correlation withinthe current layer (e.g., Fenton 1999; Liu and Chen 2006;Baker and Faber 2008; Vivek and Raychowdhury 2014). Incontrast, 3D local soil property approach considers the hori-zontal and vertical correlations simultaneously. There are fewstudies that employ a full 3D soil field in regional liquefactionmapping. Examples include the work by Dawson and Baise(2004), and that by Liu et al. (2016), where the authors applied3D interpolation to evaluate the extent of liquefiable materials.

While geostatistical tools and random field models are in-creasingly used in liquefaction mapping studies, a systematicassessment and verification of different approaches to accountfor spatial variation and dependence of soil properties or liq-uefaction potentials are missing and the implications of vari-ous random field-based mapping approaches are unknown.The main challenge is the lack of sufficient data, and thereforelack of knowledge about the soil properties and liquefactionpotentials of the field. Moreover, in situ test data are typicallysampled at selected and sometimes clustered locations,resulting in additional complexities to assess random field-based model performance.

To overcome these challenges, in this paper, an extremelydetailed three-dimensional synthetic digital soil field is artifi-cially generated and used as a basis to assess and verify var-ious random field-based approaches for liquefaction mapping.Soil properties of interest (e.g., the CPT tip resistance) areknown at every location in the synthetic field. The benchmark

liquefaction potential fields can, therefore, be obtained for anygiven hypothetical earthquake event. Moreover, different vir-tual field test plans are designed to assess their effects on datainference and model performances.

Given such an extensive amount of information, this studywill assess and verify various common and uncommon ran-dom field-based liquefaction mapping approaches. In particu-lar, this study will assess: (1) the performance and effective-ness of various approaches in mapping quantities of interest(e.g., soil properties, LPIs) over studied region; (2) the effectof amount of field data on the relative performances of differ-ent approaches, and (3) the optimal random field-based lique-faction model for mapping liquefaction hazards. This studyaims to provide insights on approaches that are commonlyused to account spatial variability and dependence in randomfield-based liquefaction mapping studies.

The remainder of the paper is structured as follows: inSection 2, details of the three random field-based approachesand the adopted random field models are presented; Section 3provides details of the synthetic digital soil field, where theequivalent clean-sand normalized CPT penetration tip resis-tance (qc1N)cs is considered as soil property of interest and thecorresponding benchmark liquefaction potential field is calcu-lated; Section 4 describes the procedure for model verifica-tion, including virtual field testing plans and measures toquantify model performance; Section 5 presents and discussesresults of model verification. For convenient reference, theadopted CPT-based liquefaction model and the calculation ofLPI are briefly summarized in Appendixes 1 and 2,respectively.

Random field-based approaches for liquefactionmapping

In this section, three random field-based approaches for lique-faction mapping are described along with details of the ran-dom field model used in the following simulations.

Mapping liquefaction potentials

In this study, a classical CPT-based liquefaction model pro-posed by Robertson and Wride (1998) and subsequently up-dated by Robertson (2009) is used to evaluate liquefactionpotential of a soil layer. The liquefaction hazard is then quan-tified and mapped over a region in terms of the liquefactionpotential index (LPI) (Iwasaki et al. 1978, 1982). Details ofthe CPT-based liquefactionmodel and the LPI calculations areincluded in Appendixes 1 and 2. Depending on how the spa-tial dependence and variation are integrated in the mappingprocess, three common and uncommon random field-basedapproaches will be assessed and verified: the averaged indexapproach, the two-dimensional (2D) local soil property

C. Hsein Juang et al.

approach and three-dimensional (3D) local soil property ap-proach, which are denoted herein as M1, M2, and M3, respec-tively. A schematic detailing of these three approaches isshown in Fig. 1.

As shown in the figure, the averaged index approach (M1)treats the LPI as the spatially correlated random variable whilethe 2D and 3D local soil property approaches (M2 and M3)treat the soil properties of interest (e.g., tip resistance fromCPT test) as spatially correlated random variables. In the 2Dlocal soil property approach, the random field of soil proper-ties is generated layer-by-layer considering the horizontal cor-relation within each layer. In contrast, the 3D local soil prop-erty approach considers both horizontal and vertical spatialcorrelations. All approaches will rely on the random fieldmodels (described in the next section) and Monte Carlo sim-ulations to generate regional liquefaction potential maps.

Random field models

Spatial structures commonly exist in natural soil deposits asevidenced by the fact that soil properties measured at onelocation are more similar to those at neighboring locationsthan those further away. In this work, spatial structure is de-scribed using a form of covariance known as thesemivariogram γ(h), which is equal to half the variance oftwo variables separated by a vector distance h

γ hð Þ ¼ 1

2Var Z uð Þ−Z uþ hð Þ½ � ð1Þ

where Z(u) and Z(u + h) are the values of the variable underconsideration at locations u and u + h, respectively. A scalarform of the vector distance, denoted as h, is commonly used toaccount for both separation distance and geometric anisotropy

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihxax

� �2

þ hyay

� �2

þ hzaz

� �2s

ð2Þ

where hx, hy and hz are the scalar components of the vectordistance along the principal axes of the field; and ax, ay and azare the corresponding ranges or correlation lengths used tospecify how quickly the spatial dependences decrease alongthose axes.

To generate random field realizations of the variables ofinterest, a conditional sequential Gaussian simulation method(Goovaerts 1997) is implemented, which has been extensivelyused by mining scientists and geostatisticians for natural re-source evaluations and spatial prediction of geohazards. It isworth noting that a multiscale extension of this conditionalsequential Gaussian simulation method has been developedin recent studies (Baker et al. 2011, Chen et al. 2012, 2015,2016, Liu et al. 2017).

Following the sequential simulationmethod, the simulationprocess could be briefly described as

ZnjZp ¼ z� �

∼N Σnp⋅Σ−1pp ⋅z;σ

2n−Σnp⋅Σ−1

pp ⋅Σpn

� �ð3Þ

in which the unknown value, Zn, at an unsampled location n isdrawn from the conditional normal distribution with the mean

Σnp⋅Σ−1pp ⋅z

� �and the variance σ2

n−Σnp⋅Σ−1pp ⋅Σpn

� �; Zp is the vector

known data; Σ is the covariance matrix of neighboring mea-surements; the subscription p and n mean Bprevious^ andBnext^, respectively. Once the unknown value Zn is generated,it is inserted into the Bprevious^ vector, i.e., the known datavector Zp, upon which the Bnext^ unknown value at anotherun-sampled location will be generated. The detailed process of

Fig. 1 The approaches forrandom field-based liquefactionmapping

Random field-based regional liquefaction hazard mapping - data

random field modeling may be found in Chen et al. (2012),(2015).

Random field models incorporate the spatial dependence ofthe measured parameter through the covariance matrix. Thecovariance of values at two separated locations could beexpressed as

Σ ¼ COV Zi;Z j ¼ ρZi;Z j

⋅σZi ⋅σZ j ð4Þ

where ρZi;Z jis the correlation between the random variables Zi

and Zj with standard deviations of σZi and σZ j , respectively.

The correlation ρ is used to describe the similarity of spatialmeasurements and is related to the semivariogram γ (h) by

ρ hð Þ ¼ 1−γ hð Þ

COV 0ð Þ ð5Þ.

An exponential semivariogram model can be expressed as

γ hð Þ ¼ 1−e−h ð6Þ

where h can be calculated by Eq.(2).Once the empirical semivariogram γ (h) is characterized, it

will be plugged into the covariance matrix Eq. (4) throughEq. (5). Thus, the unknown value Zn at location n could bedrawn using Eq. (3). The generated value is then assigned tolocation n and treated as known data. This process is repeateduntil all the unsampled locations are assigned with values.

Synthetic digital soil field and benchmarkliquefaction potential field

To verify the random field-based liquefaction models gener-ated by the approaches presented in Section 2.1, a spatially

(a) Three-dimensional view (b) Histogram

(c) Semivariogram in XY plane (d) Semivariogram in YZ plane

Fig. 2 The three-dimensionalview, the histogram, andsemivariograms of the syntheticdigital (qc1N)cs field. The empiri-cal semivariograms (c) and (d)show both the mean values aswell as the error bars (± one stan-dard deviation) from the averag-ing of all layers. (a) Three-dimensional view, (b) Histogram,(c) Semivariogram in XY plane,(d) Semivariogram in YZ plane

Fig. 3 The true LPI field under the hypothetical earthquake (amax = 0.3 gand Mw = 7.0)

Table 1 The classification of the liquefaction potential index (LPI)(Sonmez 2003)

Liquefaction potential index (LPI) Severity class of liquefaction

LPI = 0 I: Non-liquefiable

0 < LPI ≤ 2 II: Low

2 < LPI ≤ 5 III: Moderate

5 < LPI ≤ 15 IV: High

LPI > 15 V: Very high

C. Hsein Juang et al.

correlated synthetic digital soil field is created and its lique-faction potential fields are used as benchmarks.

Spatially correlated synthetic digital soil field

The dimension of the synthetic digital soil field is set as1000 × 1000 × 20 m (width × length × depth) and a soilelement size is correspondingly set as 10 × 10 × 0.05 m.There are a total of 4,010,000 soil elements in the field. Thedepth of the digital field (20 m) corresponds to the integrationdepth in LPI calculation defined in Eq. (B.1). The soil elementis assigned to have a thickness of 0.05 m to match the verticalsampling interval of a typical CPT test.

Within this field, a three-dimensional and spatially corre-lated clean sand equivalent tip resistance (qc1N)cs field is gen-erated and its values are assigned to each soil element asshown in Fig. 2(a). The parameters used to generate the

synthetic field are based on the experience gained throughthe spatial analysis of CPT database in Alameda County ofCalifornia (Chen et al. 2016; USGS 2015).

The (qc1N)cs of the digital soil field is assumed to follow alognormal distribution, and the spatial correlation of the fieldis specified as isotropic in the horizontal plane and anisotropicin the vertical plane. The histogram of the (qc1N)cs is shown inFig. 2(b), with the mean μ and the variance σ2 as 123.98 kPaand 2182.68 kPa, respectively. The semivariogram γ (h) in theXY plane and YZ plane are respectively shown in Fig. 2(c)and (d). The error bars (± one standard deviation σ) representthe variance of empirical semivariogram in the 401 XYplanesand 100 YZ planes. The magenta line is the fitted γ by Eq. (6),and the correlation length ax = ay = 82.59 m, az = 0.915 m.For simplicity, the synthetic digital (qc1N)cs field is denoted asthe Btrue^ (qc1N)cs field for use in subsequent model verifica-tions. It should be noted that the true distribution and spatialstructure of this digital soil field are unknown to random field-based liquefaction modeling and mapping, the same as in thecase of a real soil field. The lognormal and assumptions madeon spatial correlation are for the convenience of generating thedigital field.

Benchmark liquefaction potential field

To calculate the benchmark liquefaction potential index (LPI)field, the following input data for liquefaction model and for ahypothetical earthquake scenario are used: the moist unitweight of the soil γm is taken as a constant at 15 KN/m3, thesaturated unit weight γsat is 19 KN/m

3, the ground water tableGWT is at 3 m below ground surface, the maximum horizon-tal acceleration at the ground surface amax = 0.3 g and themoment magnitude Mw = 7.0.

The resulting benchmark LPI field is shown in Fig. 3. Itwill be used as the benchmark liquefaction potential field forfurther verification and is denoted as the Btrue^ LPI field.According to the severity class of liquefaction listed inTable 1 (Sonmez 2003), most areas of the field are classifiedas Bhigh^ (IV) or Bvery high^ (V) under the hypotheticalearthquake scenario.

(a) Semivariogram (b) Histogram

Fig. 4 The semivariogram andhistogram of the true LPI field. (a)Semivariogram, (b) Histogram

Fig. 5 The layouts of the virtual site investigation plans (locations A, B,C, D, and E are marked and used subsequently for model verification)

Random field-based regional liquefaction hazard mapping - data

The semivariogram of the true LPI field are shown inFig. 4(a), and the correlation length ax = ay = 114.63 m.Figure 4(b) is the histogram of the true LPI field. The trueLPI approximately follows a lognormal distribution as indi-cated by the magenta line, and the mean μ and variance σ2 are14.20 and 33.66, respectively.

Procedure for model verification

Virtual site investigation plans

As suggested by Webster and Oliver (1992), a sample size of100 should give acceptable confidence to estimate variogramsor semivariograms of soil properties. Two investigation plansare designed in this paper to compare the model performancesunder the scenarios of sufficient and insufficient sample size.As shown in Fig. 5, plan #1 is designed with a total of 225evenly spaced CPT soundings, where the (qc1N)cs is extractedfrom the digital soil field at each sounding location. As acomparison, plan #2 has only 36 evenly spaced CPT sound-ings and is used to gauge the random field model performanceunder insufficient test samples. For the investigation purpose,the element size for investigation plans is designed with20 m × 20 m × 0.05 m, which is identical in the verticaldirection but four times larger in the XY plane than that of

the synthetic digital (qc1N)cs field. Five locations marked in thefigure, A, B, C, D, and E, are used for verification of (qc1N)cprofiles. It should be noted that evenly spaced sampling plansare considered in the current study, which simplifies the infer-ence process for random field model parameters. In the real-world field investigations, unevenly spaced or clustered sam-pling locations are commonly used and the inference of ran-dom field model parameters can be more challenging. A pre-liminary analysis in an ongoing study, however, shows thatthe conclusions reached and presented in this paper are stillvalid with unevenly spaced sampling plans.

Sample (qc1N)cs profiles at selected locations (points A andB in Fig. 5) are shown in Fig. 6(a) and (b), respectively. Thecorresponding profiles of cyclic resistance ratio (CRR), thecyclic stress ratio (CSR), and the factor of safety (FS) arecalculated and plotted. With the FS profile, the LPI value ata specific location can be calculated by integrating the FSalong the soil depth using Eq. (B.1).

Data inference and random field-based liquefactionmodelgeneration

The previously described investigation plans provide Bfielddata^ necessary to infer random field model parameters.For the averaged index approach (M1), the LPI values atfield testing location are needed. For the local soil property

(a) The profiles at location A (b) The profiles at location B

Fig. 6 The profiles for the testsamples at location A and Bmarked in Fig. 5 (the blue dashlines at depth of 3 m represent thegroundwater table; the black dashlines in CSR or CRR subplotsrepresent the CSR and the blacksolid lines represent the CRR). (a)The profiles at location A, (b) Theprofiles at location B

(a) Plan #1 (15 × 15) (b) Plan #2 (6 × 6)

Fig. 7 The histograms of trueand simulated (qc1N)cs fields ofM2 and M3 for both investigationplans. (a) Plan #1 (15 × 15), (b)Plan #2 (6 × 6)

C. Hsein Juang et al.

approaches (M2 and M3), soil property, e.g., (qc1N)csvalues, are needed. Regardless of the mapping approach,statistical parameters (e.g., the statistical distribution, meanμ, and variance σ2) are inferred from field data. Empiricalsemivariograms are calculated, from which analyticals em i v a r i o g r am mod e l s ( e . g . , a n e x pon e n t i a lsemivariogram function) can be fitted. With the parametersfor statistics and semivariogram of the test samples, therandom field-based liquefaction models can be generatedusing procedures discussed in Section 2. For each mappingapproach, 1000 Monte Carlo simulations (MCSs) are per-formed, providing data to estimate not only the expectedvalues of the mapped quantities across the field but also theassociated uncertainties.

Assessment of model performance

The model performances of random field-based liquefactionmodels are assessed using the Btrue^ data from the syntheticdigital soil field and the benchmark liquefaction potential field(true fields). The models are assessed for two aspects: 1) his-togram assessments to check if the random field models cansimulate the data distribution of the true fields, and 2)semivariogram assessments to verify if the random fieldmodels can capture the spatial variability of the true fields.These assessments are made for both (qc1N)cs field and LPIfield in Section 5.1 and 5.2, respectively. For the (qc1N)cs field,the comparisons of (qc1N)cs profiles at specific locations aremade to verify the random field-based liquefaction model

(a) Plan #1 (15 × 15) in XY plane (b) Plan #1 (15 × 15) in YZ plane

(c) Plan #2 (6 × 6) in XY plane (d) Plan #2 (6 × 6) in YZ plane

Fig. 8 The semivariograms oftrue (qc1N)cs field and simulated(qc1N)cs fields of M2 and M3 forboth investigation plans. (a) Plan#1 (15 × 15) in XYplane, (b) Plan#1 (15 × 15) in YZ plane, (c) Plan#2 (6 × 6) in XYplane, (d) Plan #2(6 × 6) in YZ plane

(a) Plan #1 (15 × 15) (b) Plan #2 (6 × 6)

Fig. 9 The profiles of the trueand simulated (qc1N)cs fields at thesampling locations C, D, and Emarked in Fig. 5 (the black, red,and cyan lines correspond to thetrue (qc1N)cs profiles, thesimulated (qc1N)cs profiles of M2and M3, respectively). (a) Plan #1(15 × 15), (b) Plan #2 (6 × 6)

Random field-based regional liquefaction hazard mapping - data

performances. For the LPI field, the cumulative frequency plotand differences between true and simulated fields are assessedto evaluate the model performance.

In addition, three information theory-based measures areadopted to quantitively assess model performances, i.e., themean absolute percentage error (MAPE), the rootmean squaredeviation (RMSD), and the bias factor (Armstrong andCollopy 1992; Prasomphan and Mase 2013; Kung et al.2007; Juang et al. 2012).

MAPE ¼ 1

n∑n

i¼1

X trueð Þi− X simð ÞiX trueð Þi

�������� ð7Þ

RMSD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n∑n

i¼1X trueð Þi− X simð Þi

2sð8Þ

Bias factor ¼ 1

n∑n

i¼1

X simð ÞiX trueð Þi

ð9Þ

where n is the number of data; i is the ith data; X is the modeloutput value, e.g., the LPI or (qc1N)cs value in this paper; Xtrueis the true LPI or (qc1N)cs value, and Xsim is the simulated orpredicted value.

Smaller MAPE or RMSD value indicates a better modelperformance. For the bias factor, a value of greater than 1means the model overestimates the true field, a value of lessthan 1 means an underestimation, and a value of 1 means anunbiased prediction. As discussed in Section 4.1, the elementsize of the adopted investigation plans is larger than the syn-thetic digital field and the liquefaction potential field in XYplane. Therefore, an average operation is taken when the data

from synthetic digital field and liquefaction potential field areused in the calculation of values of the MAPE, RMSD, andbias factor.

Results and discussions

Following the procedure in Section 4, results of random field-based liquefaction models by the averaged index approach,and the 2D and 3D local soil property approaches (M1, M2,and M3) are assessed and verified in this section. Unless oth-erwise stated, results of the random field-based liquefactionmodels are averaged over 1000 Monte Carlo simulations.

Model assessment and verification: soil property fields

The histograms of the true and simulated (qc1N)cs values forboth investigation plans are plotted in Fig. 7. The blue binsrepresent the true (qc1N)cs histogram, and the red dash linesand cyan dash-dot lines represent histogram fitting curves forsimulated (qc1N)cs values using M2 and M3, respectively. Itcan be seen from Fig. 7(a) that both random field modelspredict the statistical distribution of the true soil property fieldwell, providing that sufficient field data (investigation plan#1) are available to infer model parameters. On the other hand,the model performance deteriorates for the case with insuffi-cient field data (investigation plan #2), as shown in Fig. 7(b).The differences between predictions using 2D (M2) and 3D(M3) local soil property approaches are almost negligible.

The ability of random field models to capture the underly-ing spatial structure of the soil property field is also examined.Empirical semivariograms of the true (qc1N)cs field and thesimulated (qc1N)cs fields are shown in Fig. 8. The red trianglesand cyan circles represent the mean values of the calculatedempirical semivariograms byM2 andM3 in the XYplane andin the YZ plane, respectively. The error bars indicate ± onestandard deviation from themean. It can be seen from the plotsthat for investigation plan #1, both models capture the spatialstructure of the soil property field well. For investigation plan#2, the semivariograms of M2 and M3 deviate from the trend

Table 2 The criteria index for the (qc1N)cs random fields

Index Approach 1 (M1) Approach 2 (M2) Approach 3 (M3)

Plan #1 Plan #2 Plan #1 Plan #2 Plan #1 Plan #2

MAPE NA NA 0.147 0.238 0.136 0.226

RMSD NA NA 24.569 36.480 22.887 35.157

Bias factor NA NA 1.029 1.071 1.024 1.066

Note: NA means not available

(a) Plan #1 (15 × 15) (b) Plan #2 (6 × 6)

Fig. 10 The histograms of thetrue LPI field and simulated LPIfields of M1, M2, and M3 forboth investigation plans. (a) Plan#1 (15 × 15), (b) Plan #2 (6 × 6)

C. Hsein Juang et al.

of the true semivariogram, which is not surprising as insuffi-cient data yield less accurate estimate of model parameters.

Figures 9(a) and (b) plot the (qc1N)cs profiles at selectedlocations (marked as point C, D, and E in Fig. 5) for investi-gation plans #1 and #2, respectively. It can be seen from Fig. 9that with sufficient sampling data (plan #1), the simulated(qc1N)cs profiles of M2 and M3 match the true profile verywell. As the amount of sampling data decreases (plan #2),the (qc1N)cs profiles of M2 and M3 deviates from the true soilprofile, indicating information loss and a reduction in the ac-curacy of predicted soil profiles.

To quantitatively assess model performances, the threemeasures introduced in Section 4.3, i.e., the MAPE, RMSD,and the bias factor, are calculated and summarized in Table 2for the simulated (qc1N)cs fields by the local soil property ap-proaches (M2 and M3). Note that in the averaged index ap-proach (M1), (qc1N)cs field is not needed and therefore, noresult from M1 is presented in Table 2. Smaller MAPE andRMSD values mean better performance, and bias factor closerto one means more accurate model. MAPE and RMSD valuesfor both local soil property approaches (M2 and M3) are rel-atively small compared with the mean (123.98 kPa) and var-iance (2182.68) of the true field, which indicates a relativelygood prediction. M3 performs slightly better than M2. For thebias factors, all model results yield slightly greater than onebias factor, which means the random field-based modelsslightly overpredict. The sampling size also affects the predic-tion accuracy, simulations with sufficient field data (plan #1)

yield better results. For all cases considered, the 3D local soilproperty approach (M3) outperforms the 2D local soil proper-ty approach (M2).

Model assessment and verification: liquefaction potentialfields

In this section, the LPI fields predicted using random field-based approaches are assessed and verified.

Figure 10 plots the histograms of the true and simulatedLPI fields. All of the random field-based liquefaction modelsperform well for investigation plan #1 as the histogram fittingcurves ofM1,M2, andM3 are close to the true LPI histogram.The prediction accuracy decreases with the reduction in sam-ple size, as indicated by Fig. 10(b).

The semivariogram for simulations with investigation plan#1 and plan #2 are shown in Fig. 11(a) and (b), respectively.The blue squares, red triangles, and cyan circles represent themean values of the calculated empirical semivariograms byM1, M2, and M3, respectively. The error bars indicate ±1standard deviation from the mean. It shows that thesemivariograms of M1, M2, and M3 are very close to the trueempirical semivariograms using sufficient samples (investiga-tion plan #1). The variability increased when the distance ofsemivariogram is greater than 800 m as evidenced by longererror bars. However, the use of insufficient samples (plan #2)yields significant differences between the results of the threemodels and true empirical semivariograms.

(a) Plan #1 (15 × 15) (b) Plan #2 (6 × 6)

Fig. 11 The semivariograms ofthe true LPI field and simulatedLPI fields of M1, M2, and M3 forboth investigation plans. (a) Plan#1 (15 × 15), (b) Plan #2 (6 × 6)

(a) Plan #1 (15 × 15) (b) Plan #2 (6 × 6)

Fig. 12 The cumulativefrequency of the true LPI field andsimulated LPI fields of M1, M2,and M3 for both investigationplans. (a) Plan #1 (15 × 15), (b)Plan #2 (6 × 6)

Random field-based regional liquefaction hazard mapping - data

The performances of random field-based liquefactionmodels throughout the studied site are next analyzed withthe cumulative frequencies shown in Fig. 12. From Fig. 12,it can be seen that the cumulative frequencies of M1 and M3are very close to the true ones for both investigation plans. Themodel performance of M2 is worse than M1 and M3, espe-cially under the insufficient test samples, as shown inFig. 12(b). With severity class of liquefaction defined inTable 1, it is possible to estimate the percentage of the studiedsite that may experience a particular level of liquefaction dam-age. For instance, from Fig. 12(a), 96% of the studied site mayexperience a moderate to high liquefaction (LPI > 5) and 37%may experience a very high liquefaction (LPI > 15).

The contours in Fig. 13 are the differences between thesimulated LPI values of M1, M2, and M3 and the true LPIvalues. The red color represents an overestimation, bluecolor represents an underestimation, and green color rep-resents an unbiased prediction. Observations of the

contours clearly reveal that for investigation plan #1, mostof the areas are within the unbiased or little bias region,indicating good model performances of the three randomfield models. Over- and underestimations happen mostlyaround the edges of the field due to a lack of samplingdata. Again, the reduction of the sample size increasesthe bias of the prediction, as indicated by the contourscorresponding to simulations with plan #2 data.

To further examine the model performances, a set of scatterplots for true LPI values (LPItrue) versus simulated LPI values(LPIsim) of M1, M2, and M3 for both investigation plans areshown in Fig. 14. The data for investigation plan #1 are con-centrated around 1:1 line, while many variabilities observed inthe data for investigation plan #2. All three of the random fieldmodels (M1, M2, and M3) overestimate the field at low LPIvalues and underestimation at high LPI values. However, theover or under estimations are mainly located around edges ofthe field, which is similar to the trends observed in Fig. 13.

(a) M1, plan #1 (b) M1, plan #2

(c) M2, plan #1 (d) M2, plan #2

(e) M3, plan #1 (f) M3, plan #2

Fig. 13 Contours of thesimulated LPI values (LPIsim) ofM1, M2, and M3 minus true LPIvalues (LPItrue) for bothinvestigation plans. (a) M1, plan#1, (b) M1, plan #2, (c) M2, plan#1, (d) M2, plan #2, (e) M3, plan#1, (f) M3, plan #2

C. Hsein Juang et al.

The model performances of LPI field are also quantitative-ly assessed with the MAPE, RMSD, and bias factor. The cal-culated indices for M1, M2, and M3 are summarized inTable 3. All three models predict relatively accurate LPIvalues over the entire field when the data from plan #1 areused. TheM3 outperformsM1, and the latter outperformsM2.All three approaches slightly overestimate the LPI field as the

bias factors are all greater than one. When the number of testsamples are insufficient (plan #2), the model performancesbased on MAPE and bias factor are M3 (best), M2 (second),and M1 (worst). By RMSD, however, M2 (3.876) is slightlybetter thanM3 (3.897) and better thanM1 (3.965). It indicatesthat the local soil property approach (M2 and M3) is superiorto the averaged index approach (M1) in predicting the lique-faction potential field when the sampling data is insufficient.

The computational efficiency of a model is also of concernwhen evaluating the model performance. The computationaltime required for obtaining the 1000 LPI random fields basedon investigation plan #2 are 4.5 mins, 1090.6 mins, and5237.4 mins for M1, M2, and M3, respectively. These num-bers are meaningful only on a relative basis, as they depend onthe computer used in the computation. The averaged indexapproach (M1) dominates in terms of computational efficien-cy and would be the clear choice when computational cost isof major concern.

(a) M1, plan #1 (b) M1, plan #2

(c) M2, plan #1 (d) M2, plan #2

(e) M3, plan #1 (f) M3, plan #2

Fig. 14 The true LPI values(LPItrue) versus simulated LPIvalues (LPIsim) of M1, M2, andM3 for both investigation plans.(a) M1, plan #1, (b) M1, plan #2,(c) M2, plan #1, (d) M2, plan #2,(e) M3, plan #1, (f) M3, plan #2

Table 3 The criteria index for the LPI random fields

Index Approach 1 (M1) Approach 2 (M2) Approach 3 (M3)

Plan #1 Plan #2 Plan #1 Plan #2 Plan #1 Plan #2

MAPE 0.146 0.300 0.150 0.293 0.130 0.285

RMSD 2.030 3.965 2.047 3.876 1.878 3.897

Bias factor 1.085 1.151 1.088 1.138 1.070 1.120

Random field-based regional liquefaction hazard mapping - data

Based on the comparisons of the accuracy criteria and thecomputational efficiency, M3 outperforms M2 in predictingthe local soil property field, but M2 is more efficient than M3in both scenarios of sufficient and insufficient test samples.For the liquefaction potential field, M3 performs better thanM1 and M2 given the sufficient test samples. However, theperformance ofM1 is similar toM3when the comparisons arebased on histogram, semivariogram, and cumulative frequen-cy. In addition, M1 is the most efficient random field model.Therefore, under the scenario of sufficient test samples, M1 isrecommended for constructing the liquefaction potential field.Under the scenario of insufficient test samples, however, M3is recommended as it is more accurate than M2, and offersmore information than M1.

Discussions

In this work, data inference and model verification are carriedout based on a synthetic digital soil field. The synthetic fieldaffords us extremely detailed information on soil propertiesand a benchmark field for liquefaction potential. The focusof this work is on understanding and verifying different ran-dom field-based approaches for liquefaction potential map-ping, and use of synthetic digital field in this fundamentalstudy has distinctive advantages over any real-world site in-vestigation data.

On the other hand, it is important to note the assumptionsand limitations of the synthetic field and the associated modelverification process when drawing conclusions from the anal-ysis. For instance, in preparing the synthetic field and in gen-erating random field models, stationarity of the random field isassumed. Soil properties are assumed to be isotropic on ahorizontal plane and anisotropic on a vertical plane. In reality,non-stationary variations of soil properties are quite common.In addition, only evenly spaced virtual field investigationplans are considered in this study, which simplifies the infer-ence of random field model parameters. In real-world fieldinvestigations, unevenly spaced and/or clustered sampling lo-cations are common in engineering practice. Further study toconsider the effect of unevenly spaced and/or clustered sam-pling plans on the data inference and model verification pro-cesses and outcomes for random field-based liquefaction haz-ard mapping is warranted.

Conclusions

In this paper, a three-dimensional synthetic digital soil field isartificially generated and used as a basis to assess and verifyvarious random field-based models for liquefaction mapping.The liquefaction potential is assessed using a classical CPT-based liquefaction model, and the result is expressed in termsof the liquefaction potential index. Three random field-based

liquefaction models are assessed and verified, namely, theaveraged index approach (M1), the two-dimension local soilproperty approach (M2), and the three-dimension local soilproperty approach (M3). Two virtual field testing plans aredesigned. Here, performances of the three models are evalu-ated in terms of resulting sample histograms, empiricalsemivariograms and are compared using three informationtheory-based criteria, i.e., the mean absolute percentage error(MAPE), the root mean square deviation (RMSD), and thebias factor. Main finds are summarized as:

(1) All three random field models examined can closely cap-ture the statistical distribution and spatial structure of thetrue (qc1N)cs and LPI fields, provided that the amount offield test data for model parameter inference is sufficient.The model performances deteriorate with the reductionof test samples as expected.

(2) All random field models are found to overestimate slight-ly liquefaction potentials over the studied area, comparedto the benchmark liquefaction potential fields.

(3) When there is sufficient amount of field data for modelparameter inference, the 3D local soil property approach(M3) slightly outperforms the averaged index approach(M1) and the 2D local soil property approach (M2) interms of the accuracy in predicting the liquefaction po-tentials, while M1 is significantly more efficient thanM2and M3.

(4) When there are sufficient field test data to infer modelparameters, it is recommended that the averaged indexapproach (M1) be used for liquefaction mapping consid-ering a tradeoff between efficiency (in terms of compu-tational effort) and accuracy. On the other hand, underthe scenario of insufficient data, the 3D local soil prop-erty approach (M3) is recommended for its highest accu-racy among the three models examined.

It should be noted that the above conclusions were reachedusing a synthetic digital field with the assumptions of station-arity of the random field and evenly spaced virtual field inves-tigation plans. Thus, these conclusions should be viewed withcaution and further study to quantify the effect of these as-sumptions is warranted.

Acknowledgments The first author wishes to thank Shanghai SubmitDiscipline Development Program for the support during the course of thisstudy. The second author wishes to thank the support of ChinaScholarship Council (CSC). The third author wishes to acknowledgethe support of the Shrikhande Family Foundation through theShrikhande Graduate Fellowship. This study is supported in part by theU.S. Geological Survey (Grant No. G17AP00044). Clemson Universityis acknowledged for the allotment of computer time on the Palmetto highperformance computing facility.

C. Hsein Juang et al.

Appendix 1: CPT-based liquefaction model

The CPT-based liquefactionmodel proposed in Robertson andWride (1998) and subsequently updated by Robertson (2009)is adopted in this study. The factor of safety against liquefac-tion is defined as the ratio of cyclic resistance (CRR) andcyclic stress (CSR).

FS ¼ CRR

CSRðA:1Þ

The CRR provides soil resistances and it is defined inEq.(A.2), which is a function of the equivalent clean sandnormalized CPT penetration tip resistance (qc1N)cs.

CRR ¼ 0:8333 qc1Nð Þcs=1000 þ 0:05 if qc1Nð Þcs < 50

93 qc1Nð Þcs=1000 3 þ 0:08 if 50 < qc1Nð Þcs < 160

(

ðA:2Þ

The CSR represents the earthquake loadings, and the fol-lowing adjusted form is adopted (Youd et al. 2001)

CSR ¼ 0:65amax

g

� �σvo

σ0vo

� �rdð Þ 1

MSF

� �1

Kσ

� �ðA:3Þ

where amax is the maximum horizontal acceleration at theground surface; g is the gravitational acceleration and equalto 9.81 m/s2, σvo and σ′vo are the respective total and effectivevertical overburden stresses.

The stress reduction factor rd is a function of depth z anddefined below (Youd et al. 2001)

rd ¼ 1:000−0:4113z0:5 þ 0:04052zþ 0:001753z1:5

1:000−0:4177z0:5 þ 0:05729z−0:006205z1:5 þ 0:001210z2

ðA:4Þ

The MSF is the magnitude scaling factor and related to themoment magnitude Mw as (Youd et al. 2001)

MSF ¼ 102:24

M 2:56w

ðA:5Þ

The Kσ in Eq.(A.6) is the overburden correction factor forCSR (Youd et al. 2001). The correction is applied when the σ′

vo greater than 100 kPa.

Kσ ¼ σ0vo

Pa

� � f −1ð ÞðA:6Þ

where Pa is the atmospheric pressure; f is an exponent andrecommended as: f = 0.7–0.8 for relative densities between40 and 60%; f = 0.6–0.7 for relative densities between 60 and80%.

Appendix 2: Liquefaction potential index

The liquefaction potential index (LPI) was developed byIwasaki et al. (1978, 1982). It is based on the assumption thatthe liquefaction potential is related to the thickness of theliquefied layer and the factor of safety against liquefaction,and its equation is expressed as follows:

LPI ¼ ∫200 ω zð ÞFLdz ðB:1Þ

where z is the soil depth in meters and it is commonlyevaluated for the top 20 m of soil profile; ω(z) is the functionof soil depth and FL is a function of factor of safety (FS)against liquefaction listed as follows:

ω zð Þ ¼ 10−0:5z ðB:2Þ

FL ¼0 FS≥1:21−FS FS≤0:952� 106e−18:427FS 0:95 < FS < 1:2

8<: ðB:3Þ

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