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Journal of Computational Design and Engineering, 2020, 7(1), 107–127

doi: 10.1093/jcde/qwaa011Journal homepage: www.jcde.orgAdvance Access Publication Date: 19 March 2020

RESEARCH ARTICLE

Computer-aided SPT-based reliability model forprobability of liquefaction using hybrid PSO and GAMaral Goharzay1,*, Ali Noorzad1, Ahmadreza Mahboubi Ardakani1 andMostafa Jalal2,*1Faculty of Civil, Water and Environmental Engineering, Shahid Beheshti University, Shahid Shahriari Square,Daneshjou Boulevard, Shahid Chamran Highway, 16589-53571 Tehran, Iran and 2Zachry Department of CivilEngineering, Texas A&M University, College Station, TX 77840, USA

*Corresponding author. E-mail: [email protected], [email protected] (M. Goharzay); [email protected], [email protected](M. Jalal)

AbstractIn this paper, an approach for soil liquefaction evaluation using probabilistic method based on the world-wide SPTdatabases has been presented. In this respect, the parameters’ uncertainties for liquefaction probability have been takeninto account. A calibrated mapping function is developed using Bayes’ theorem in order to capture the failure probabilitiesin the absence of the knowledge of parameter uncertainty. The probability models provide a simple, but also efficientdecision-making tool in engineering design to quantitatively assess the liquefaction triggering thresholds. Within anextended framework of the first-order reliability method considering uncertainties, the reliability indices are determinedthrough a well-performed meta-heuristic optimization algorithm called hybrid particle swarm optimization and geneticalgorithm to find the most accurate liquefaction probabilities. Finally, the effects of the level of parameter uncertainty onliquefaction probability, as well as the quantification of the limit state model uncertainty in order to incorporate the correctmodel uncertainty, are investigated in the context of probabilistic reliability analysis. The results gained from the presentedprobabilistic model and the available models in the literature show the fact that the developed approach can be a robusttool for engineering design and analysis of liquefaction as a natural disaster.

Keywords: liquefaction hazard; probabilistic approach; reliability analysis; optimization algorithm; hybrid PSO-GA

1. Introduction

Several methods for evaluating the liquefaction potential ofsandy soils have been suggested, in which the deterministicanalysis has been widely used among the geotechnical ex-perts due to its simplicity. Although the factor of safety of soilscan be determined utilizing deterministic methods by apply-ing the certain and non-dispersion parameters, some facts suchas anisotropy and non-uniformity of the soils, uncertainties inthe seismic parameters, mismatch between the employed as-sumptions in modelling, and the actual in-situ conditions, aswell as human and instrumental errors, cause unreliable results

for these methods. Thus, in order to overcome these difficul-ties, probabilistic evaluation of liquefaction in which the de-gree of conservatism can be quantified in terms of PL can bea good option. Hence, geotechnical engineers can interpret theprobability of liquefaction occurrence (PL) simplicity in terms oflacking knowledge of the associated uncertainties by calibrat-ing the safety factor (Fs) with the liquefaction field manifesta-tion and by developing the mapping function on the basis of theBayes’ theorem of conditional probability for preliminary calcu-lations of the liquefaction probability (Goharzay, Noorzad, Mah-boubi Ardakani, & Jalal, 2017). However, using reliability assess-ment methods by considering both of the parameter and model

Received: 21 April 2019; Revised: 30 September 2019; Accepted: 30 October 2019

C© The Author(s) 2020. Published by Oxford University Press on behalf of the Society for Computational Design and Engineering. All rights reserved. Forpermissions, please e-mail: [email protected]

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108 Computer-aided SPT-based reliability model for probability of liquefaction using hybrid PSO and GA

uncertainties and utilizing the probability distribution curvesof the parameters instead of only a certain amount for eachone, we can provide the soil liquefaction probability curve. De-sired constitutive modelling of liquefiable soil is very difficult,even with considerable laboratory testing. Hence, cyclic stress-based simplified methods based on in-situ tests such as stan-dard penetration test (SPT), cone penetration test (CPT), andShear wave velocity (Vs) measurements along with the post-liquefaction case histories-calibrated empirical relationshipshave been used widely by the geotechnical engineers to evaluatethe liquefaction potential of soils throughout most part of world(Youd et al., 2001, Cetin et al., 2004; Idriss & Boulanger, 2004;Idriss & Boulanger, 2008). SPT-based method is the most widelyused method among the available in-situ test methods for theevaluation of resistance of soil against the occurrence of lique-faction. Nowadays, CPT is becoming more acceptable as it is con-sistent, repeatable, and able to identify a continuous soil profile.Thus, CPT is being used as a valuable tool for assessing vari-ous soil properties, including liquefaction potential of soil. Themain advantages of the CPT are that it provides a continuousrecord of penetration resistance and is less vulnerable to oper-ator error than the SPT. The main disadvantages of the CPT arethe difficulty in penetrating layers that have gravels or very highpenetration resistance and need to perform companion boringsor soundings to obtain actual soil samples. The use of Vs as anin-situ test index of liquefaction resistance of soil is very wellaccepted because both Vs and cyclic resistance ratio (CRR) aresimilar, but not proportional, and influenced by void ratio, ef-fective confining stresses, stress history, and geologic age. Themain advantages of using Vs for the evaluation of liquefactionpotential are as follows: Vs measurements are possible in soilsthat are difficult to penetrate with SPT and CPT or difficult to ex-tract undisturbed samples; (ii) Vs is a basic mechanical propertyof soil materials, directly related to small-strain shear modulus;and (iii) the small-strain shear modulus is a parameter requiredin analytical procedures for estimating dynamic soil responseand soil-structure interaction analyses. However, the followingare the disadvantages of utilizing Vs for liquefaction resistanceevaluations: (i) seismic wave velocity measurements are madeat small strains, whereas pore-water pressure build-up and theliquefaction triggering are medium- to high-strain phenomena;(ii) seismic testing does not provide samples for the classifica-tion of soils and the identification of non-liquefiable soft clay-rich soils; and (iii) thin, low Vs strata may not be detected if themeasurement interval is too large. Therefore, it is preferred todrill sufficient boreholes and conduct in-situ tests (SPT or CPT)to detect thin liquefiable strata, non-liquefiable clay-rich soils,and silty soils above the ground water table. Whitman (1971)first proposed to use liquefaction case histories to characterizeliquefaction resistance in terms of measured in-situ test param-eters. Seed and Idriss (1971) did a pioneer work in developinga simplified empirical model, using laboratory tests and post-liquefaction field observations in earthquakes for evaluatingliquefaction susceptibility of soils. Based on this procedure, liq-uefaction charts have been developed that correlate soil resis-tance to earthquake-induced stresses. Early efforts to developsuch charts involved using laboratory tests to evaluate stressesneeded for soil liquefaction (Seed & Lee, 1966; Seed & Peacock,1970). Such studies typically quantified soil resistance in termsof relative density. However, it was soon realized that grain dis-turbance during sample extraction from the field significantlyaltered soil resistance, necessitating considerable judgement toestimate in-situ properties from test results (Seed & Idriss, 1982).Iwasaki (1978) proposed the liquefaction potential index (LPI) to

assess the damage potential of liquefaction. Iwasaki et al. (1982)calibrated the values of LPI with the severity of liquefaction-induced damage using data mostly for sandy soils, as providedby 87 SPT borings in liquefied and non-liquefied sites in Japanthat severe liquefaction should be expected if LPI > 15 but notif LPI < 5. This criterion for liquefaction manifestations is re-ferred to as the Iwasaki criterion. The LPI value is inversely pro-portional to the FS and the depth of the saturated layers; thehigher the index, the greater the potential for liquefaction. Dam-age to infrastructure is more likely to result from moderate orsevere liquefaction; the probability of damaging liquefaction atLPI = 5 ranges from 0.05 to 0.17. Thus, the Iwasaki criterion ismore applicable for assessing the damage potential, rather thanthe occurrence of liquefaction. Zhou (1980) first published lique-faction correlation directly based on case history CPT databaseof the 1978 Tangshan earthquake. He presented the critical valueof cone penetration resistance separating liquefiable from non-liquefiable conditions to a depth of 15m. Seed and Idriss (1981)as well as Douglas, Olsen, and Martin (1981) proposed the useof correlations between the SPT and CPT to convert the avail-able SPT-based charts for use with the CPT data. Seed, Idriss,and Arango (1983) extended their previous work in developing amodified model in which they used CSR (τav/σ

′v) instead of peak

ground acceleration (amax) as a measure of seismic action andoverburden pressure-corrected SPT value (N1) instead of relativedensity (Dr) as the site parameter representing its resistance toliquefaction. Robertson and Campanella (1985) developed a CPT-based method for the evaluation of liquefaction potential, whichis a conversion from SPT-based method using empirical corre-lation of SPT–CPT data and follows the same stress-based ap-proach of Seed and Idriss (1971). This method has been revisedand updated by some researchers (Suzuki, Tokimatsu, Taya, &Kubota 1995; Olsen, 1997; Robertson & Wride, 1998). Robertsonand Wride (1998) developed an integrated procedure to evaluatethe liquefaction resistance of sandy soils based solely on CPTdata. Comparison of Robertson and Wride’s CPT-based methodwith SPT-based methods and other CPT-based methods hasdemonstrated that Robertson and Wride’s method is reliable andconvenient (Gilstrap, 1998; Juang, Rosowsky, & Tang, 1999). In ad-dition, Juang et al. (1999) found that the degree of conservatismin the Robertson and Wride method is comparable to that inthe Seed and Idriss (1971, 1982) SPT-based method that has beenwidely used in geotechnical practice around the world for morethan 20 years. Youd et al. (2001) published a summary paper of1996 and 1998 NCEER workshop in which the updates and aug-mentations to the original ‘simplified procedure’ of Seed andIdriss (1971), Seed et al. (1983), and Seed, Tokimatsu, Harder, andChung (1985) for the evaluation of liquefaction potential are rec-ommended using SPT-based methods and are still followed asthe current state of the art on the subject of liquefaction poten-tial evaluation. Cetin et al. (2004) proposed new correlations forthe assessment of liquefaction triggering in soil. Although theabove SPT-based method remains an important tool for evalu-ating liquefaction resistance, it has some drawbacks, primarilydue to the variable nature of the SPT (Robertson & Campanella,1985; Skempton, 1986). Moss (2003) and Moss, Seed, Kayen, Stew-art, and Tokimatsu (2005) presented a CPT-based probabilis-tic model for the evaluation of liquefaction potential using re-liability approach and a Bayesian updating technique. Juang,Yuan, Lee, and Lin (2003) investigated the uncertainties of theRobertson and Wride model (Robertson & Wride, 1998) in orderto assess the liquefaction potential based on the CPT throughthe reliability analysis. Hwang, Yang, and Juang (2004) pro-posed a new practical approach for reliability evaluation of soil

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Journal of Computational Design and Engineering, 2020, 7(1), 107–127 109

liquefaction potential. They calculated the earthquake-inducedcyclic shear stress probability density function utilizing theacceleration attenuation relationships related to Taiwan area(Chi Chi earthquake data in 1999) and applied first-order sec-ond moment method for the combination of cyclic shear stressprobability density function and cyclic shear strength of soil anddeveloped probability models in order to determine the reliabil-ity index (β) and the related liquefaction probability (PL) usingfactor of safety (Fs). Juang, Jiang, and Andrus (2005) studied theuncertainties of the Andrus and Stoke model (Andrus & Stokoe,2000) to assess the soil liquefaction based on the shear wavevelocity by means of the first-order reliability method (FORM).They obtained a probability model to estimate the liquefactionoccurrence. Hanna, Ural, and Saygili (2007) based on Turkey andTaiwan earthquake databases in 1999, taking into account 12 dy-namic and soil parameters, established a neural network modelfor liquefaction potential in sandy soil deposits. Their proposedgeneral regression neural network model (GRNN) propoundeda complex relationship between soil and dynamic parameters.The accuracy of the prediction model was verified by compar-ing of GRNN model results against the results of the simpli-fied methods. Jha and Suzuki (2009) analysed soil liquefactionpotential using analytical approximation methods and MonteCarlo simulation approach based on SPT databases. For thispurpose, they presented a comparative study of the reliabilityapproaches in order to obtain the liquefaction probability re-lated to each factor of safety. Lee, Chi, Juang, and Lee (2010)presented a reliability model to evaluate the liquefaction po-tential of the Yuanlin city of Taiwan by considering the uncer-tainties in seismic forces and soil strength parameters throughthe FORM method. Das and Muduli (2013) used Chi-Chi earth-quake data in order to investigate the soil liquefaction potentialbased on genetic programming (GP). Gene expression program-ing (GEP) was also recently used in earlier study of the authorsto develop deterministic model for liquefaction (Goharzay et al.,2017).

In recent years, artificial intelligence and optimization tech-niques have been successfully applied in various applicationsin civil and mechanical engineering such as civil engineeringmaterials (Ashrafi, Jalal, & Garmsiri, 2010; Jalal & Ramezanian-pour, 2012; Jalal, Ramezanianpour, Pouladkhan, & Tedro, 2013;Fathi, Jalal, & Rostami, 2015; Jalal, 2015), composites (Jodaei,Jalal, & Yas, 2012, 2013; Jalal, Moradi-Dastjerdi, & Bidram, 2019),product manufacturing (Garmsiri & Jalal, 2014; Jalal, Mukhopad-hyay, & Goharzay, 2018; Jalal & Goharzay, 2019; Jalal, Mukhopad-hyay, & Grasley, 2019), etc. Though GEP modelling and optimiza-tion algorithms have been implemented to solve some complexgeotechnical problems, their use in liquefaction potential eval-uation is very limited. In recent years, a lot of work has beendone to assess the liquefaction potential in terms of the proba-bility of liquefaction. A precise estimation of the probability ofliquefaction requires information of both parameter and modeluncertainties. In this research, the issue of the uncertainties hasbeen addressed through rigorous meta-heuristic optimizationalgorithm-based reliability analysis, which is considered to besignificant.

In the present study, an approach for soil liquefaction evalu-ation using probabilistic methods based on the world-wide SPTdatabases as presented in Appendix 1 (Cetin, 2000) has been em-ployed in order to analyse the effects of parameter and modeluncertainty in liquefaction probability on the basis of the ob-tained GEP models (Goharzay et al., 2017) for liquefaction re-sistance and potential evaluation. In this respect, the parame-ters’ and model uncertainties for liquefaction probability have

been taken into account within an extended framework of theFORM. The reliability indexes (β) are determined through a well-performed meta-heuristic optimization algorithm called hybridparticle swarm optimization (PSO) and genetic algorithm (GA)in order to find the most accurate liquefaction probabilities (PL)through the notional failure concept. In addition, the empiricallycalibrated mapping functions with the liquefaction field mani-festation are developed utilizing Bayes’ theorem in order to inferthe relationship between the obtained reliability index (β) andthe related probability of liquefaction (PL). Finally, the effect ofthe level of parameter uncertainty on liquefaction probability(PL), as well as the quantification of the limit state model uncer-tainty in order to incorporate the correct model uncertainty, isinvestigated by comparing the Bayesian mapping functions (β–PL curves) obtained for each uncertainty level in the context ofprobabilistic reliability analysis.

2. Methodology and Analysis2.1. Reliability analysis

Generally, there are several uncertainties in geotechnical engi-neering, especially in liquefaction analysis (e.g. errors associatedwith the dispersion of the measured data, systematic errors, hu-man errors, etc.). Referring to the inability of the determinis-tic methods to take into account the geotechnical uncertainties,They are not applicable to be used in the analysis of the soil liq-uefaction potential. Figure 1 represents the probability densityfunctions of a specified safety factor. The failure probabilities(PL) are shown by the dark areas, which are defined as the zonesof the probability density functions with Fs < 1. According to thefigure, as a curve with a higher mean value has more uncertain-ties than the one with lower mean value, the former curve hasa greater failure probability than the latter. Therefore, it can beseen that the greater Fs does not necessarily mean a less chanceof failure mode. So, the reliability analysis is required in order toconsider both model and parameter uncertainties. The first stepof a reliability analysis is to define a performance function. In theliquefaction potential evaluation, CSR and CRR are representedin the form of the random variables Q and R, respectively. In thisstudy, each variable is assumed to follow a lognormal distribu-tion, which has been shown to provide a good fit to the measuredgeotechnical parameters (Jefferies & Wright, 1988). The safetymargin (Z) is defined as the difference between ‘resistance’ and‘loading’, which also has a lognormal distribution. The perfor-mance function of the liquefaction potential assessment is pre-sented as (Baecher & Christian, 2003)

Z = R − Q. (1)

In terms of Z < 0, the liquefaction is more likely to occur and thefailure mode is expected. When Z = 0, the performance functionis on the limit state boundary, which discriminates between theliquefaction and non-liquefaction cases. If Z > 0, no liquefactionis predicted. The reliability index (β) is defined as the inverse ofthe covariance coefficient, as (Baecher & Christian, 2003)

β = μz

σz= μR − μQ√

σ 2R + σ 2

Q − 2ρRQσRσQ

, (2)

where μR and μQ are the mean values of R and Q, respectively;σ R and σQ are the standard deviations of R and Q, respectively;σ 2

R and σ 2Q are the variances of R and Q, respectively, and ρRQ

is the correlation coefficient between R and Q. The probability ofliquefaction (PL) can be calculated through the notional failure

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Figure 1: Comparison of the failure probabilities corresponding to the two different probability density functions of the factor of safety.

concept, presented as

pf = PL = P [Z ≤ 0] = = �

(−μz

σz

)= � (−β) = 1 − � (β) , (3)

where � is the cumulative distribution function for a standardnormal variable.

Despite the method’s simplicity, it has some significant draw-backs. For example, when the failure function is linearizedaround the average values of the random variables, differentreliability indices can be obtained by choosing different func-tions for the same problem. The above problem is solved by Ha-sofer and Lind (1974), since they proposed a geometric interpre-tation of the reliability index, which is defined as the shortestdistance between the failure surface and the origin of the stan-dard normal coordinate system. They suggested to use a linearapproximation of the failure surface at the design point. Basedon the Hasofer–Lind approach, all the normal random variablesare transformed to their reduced form in standard normal spacewith zero mean and unit standard deviation. Thus, R′ and Q′ asthe standard normal variables and d as the distance of the limitstate line from origin can be expressed as equations (4–6), re-spectively.

R′ = R − μR

σR(4)

Q′ = Q − μQ

σQ(5)

d = μR − μQ

σ 2R + σ 2

Q

(6)

As the liquefaction performance function (equation 1) de-pends on multiple variables, it can be represented as Z = R –Q = g(z), in which z is a vector of uncorrelated random variables,z = {(N1)60, FC, σ v, σ ′

v amax, Mw}. Equation (6) can be extendedto six random variables, which are first converted to standardnormal variables (zi

′), as follows:

d =√

z′21 + z′2

2 + . . . + z′26 =

√z′T z′. (7)

Thus, the Hasofer–Lind reliability index is stated as (Der Ki-ureghian, Lin, & Hwang, 1987)

β = min(z′T ρ−1z′)0.5

in terms of g (z) = 0, (8)

where ρ is the correlation matrix of the variables and g(z) = 0 isthe constraint function in the optimization problem.

In order to determine the characteristics of the lognormaldistribution, the mean and standard deviation of equivalent nor-mal variables are calculated as equations (9) and (10), respec-tively.

ξi =√

ln(1 + δ2

zi

)(9)

λi = lnμzi − 0.5ξ2i (10)

in which ξ i is the standard deviation of the equivalent normalvariable, λi is the mean of the equivalent normal variable, μzi isthe mean of the random variable zi, and δzi is the coefficient ofvariation (COV) of zi.

The various numerical methods can be utilized in order tofind out the minimum value of the reliability index (β). A newhybrid evolutionary algorithm combining the PSO algorithm andthe GA referred to as HPSO-GA is presented in order to ex-ploit the advantages and reduce the weaknesses of these twoalgorithms for solving optimization problems. As these mod-ern meta-heuristic optimization algorithms do not deal withthe problem details, they depend only on the quality of theresponse variables. On the Contrary most traditional gradient-based non-linear search and optimization methods (e.g. gradi-ent, conjugate directions, quasi-Newton, and Newton) requireknowledge of derivatives of the objective functions, which maynot be easy to obtain those of the different cost functions forthe sophisticated problems, such as for reliability analysis. Thebenefits of utilizing these approaches provide a fast and easysearch mechanism of the solution space in order to find the op-timal values of the objective functions for the constrained orunconstrained optimization problems by using the computers.

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In the current study, the simulations are performed by develop-ing the hybrid optimization algorithm code in the MATLAB2013asoftware.

2.1.1. Optimization algorithmGAs, first introduced by Holland (1975), were implemented onthe basis of the principles of natural selection as a class of evolu-tionary algorithms (EAs). In the GA, a set of chromosomes, whichare in fact a set of possible answers to the optimization prob-lem, is considered as the initial population. After generating theinitial candidate solutions in the form of strings, the new chro-mosomes known as offspring (child chromosomes) must be cre-ated using the genetic operations (e.g. crossover and mutation).After the creation of children population, by evaluating each ofthe candidate solutions through the selection process, which isbased on the fitness value for each individual in the population,the best chromosomes as the feasible and good quality solutionscan be selected. Therefore, after several generations, the popu-lation will converge towards the optimal solutions to the opti-mization problem. PSO algorithm is a population-based stochas-tic optimization technique developed by Eberhart and Kennedy(1995), originated from the flocking-behaviour simulation of thebirds. In the PSO, each particle represents a possible solution andhas an associated fitness value based on the relevant objectivefunction. Each particle alone has a personal velocity (vi) and po-sition (xi), which can be updated by taking into account the in-fluence of both personal best experience (p best) and the suc-cessful experiences of neighbouring particles or the global bestexperiences (g best) attained by the whole swarm population. So,each particle is pulled towards the best position it has person-ally found in the search space so far, as well as the best posi-tion that any particle has ever found within the entire swarm.The particles adjust their own positions and velocities accord-ing to the previous best solutions as defined in the followingrelations:

vi [t + 1] = wvi [t] + c1r1(xi,best [t] − xi [t]

) + c2r2(xg,best [t] − xi [t]

)(11)

xi [t + 1] = xi [t] + vi [t + 1] , (12)

where i and t are the numbers of particle and iteration, respec-tively, xi and vi are the position and velocity of the particle, re-spectively, xi,best and xg,best are the optimal position of the parti-cle and of the whole particles, respectively, and ω is an inertiaweight, which controls the momentum of the particle and pro-vides a balance between exploration and exploitation within thesearch algorithm. The parameters r1 and r2 are two random val-ues uniformly distributed between 0 and 1, which are used inorder to maintain an appropriate diversity among the feasiblesolutions represented by the particles, and in this way a rapidand comprehensive search is performed over the whole searchspace. The parameters c1 and c2 are the personal and globallearning coefficients, respectively, which quantify the contribu-tions of personal and social experiences. Both GA and PSO algo-rithms begin with an initial random population and the evolu-tionary progression will follow by assessing the fitness of eachcomponent, the information sharing among the particles andgenerating a new population until reaching a stopping criterion.However, the information flow mechanisms in these two algo-rithms are different. In the GA, all chromosomes share infor-mation with each other. Thus, the whole population as a groupmoves towards the optimal region, whereas in the PSO algo-

rithm only the personal best (p best) and global best (g best) giveout information to the other particles. In the course of evolu-tion, the particles’ movement is affected by the best solutions;therefore, the PSO algorithm has a higher convergence rate thanthe GA. Generally, the random search algorithms have two fun-damental objections concerning their performance. First, thesealgorithms are likely to be stuck into the local optima and inthis way, the premature convergence can happen. On the otherhand, although PSO is able to find good solutions much fasterthan many other evolutionary algorithms, the quality improve-ment of the PSO solutions cannot be seen noticeably during thesearch process, due to the large-scale information flow, whichleads to the emergence of similar birds (reducing the varietyof answers), as well as focusing of all the birds at one pointknown as the global best, which is between the personal andsocial optimal positions. Second, these heuristic optimizationalgorithms have an affiliate performance to the problems. Thisdependence is often the result of the way of selecting the algo-rithm parameters, which are chosen individually for each prob-lem. Therefore, selecting different values for them can lead togreat changes in the algorithm performance. These difficultiescan be managed by blending the PSO algorithm with the otherintelligent optimization algorithms such as the GA. The com-bination of these two algorithms was first proposed by Ange-line (1998), which revealed that the performance and efficiencyof the PSO can be significantly improved by adding a selectionprocess similar to that which occurs in the evolutionary algo-rithms. Angeline empirically demonstrated that the selection-based PSO algorithm can enhance the local search ability ofthis algorithm. Thus, a combination of the PSO algorithm withthe concepts of selection, crossover, and mutation from the GAseems to be a logical approach and can provide the better re-sults. Figure 2 illustrates the flowchart of the proposed HPSO-GAalgorithm.

3. Development of SPT-based ReliabilityModels

The liquefaction probability (PL) can be determined for a givenreliability index (β) through a calibration with field manifesta-tions of case history databases on the basis of the Bayesian the-ory as follows (Juang et al., 1999):

P(

L/β)

= fL (β) P (L )fL (β) P (L ) + fNL (β) P (NL )

(13)

in which P(L/β) is the liquefaction probability for a given β, fL(β)and fNL(β) are the probability density functions of β for the lique-fied cases and non-liquefied cases of the database, respectively,and P(L) and P(NL) are the prior probabilities of liquefaction andnon-liquefaction, respectively.

In the absence of the prior information, P(L) can be consid-ered equal to P(NL). Thus, equation (13) can be summarized as

PL = fL (β)fL (β) + fNL (β)

. (14)

The classification of liquefaction likelihood is specified inTable 1.

The assessment of the reliability index (β) associated withsolving the constrained minimization problem, is performed byapplying the proposed optimization algorithm (HPSO-GA) in or-der to determine the optimal reliability index (β) for each casehistory considered in the present study through the FORM in thecase of the lack of knowledge of the model uncertainty.

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Figure 2: Flowchart of the hybrid of particle swarm optimization and genetic algorithm.

The specifications of the hybrid optimization algorithm aregiven as

MaxIt: 500, MaxSubItPSO: 3, MaxSubItGA: 1, nPop: 50,Pcrossover = 0.7, Pmutation = 0.2, and ωinertia weight damping ratio = 0.8.

The parameters c1 and c2 are calculated based on the follow-ing equations (Clerc & Kennedy, 2002):

χ = 2

∅ − 2 + √∅2 − 4∅ (15)

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Table 1: Liquefaction likelihood classification (Das & Muduli, 2013).

ClassLiquefaction

probability (PL) Description of likelihood

1 PL< 0.15 Almost certain that it will not liquefy2 0.15 ≤ PL < 0.35 Unlikely to liquefy3 0.35 ≤ PL < 0.65 Liquefaction and no liquefaction are

equally likely4 0.65 ≤ PL < 0.85 Very likely to liquefy5 PL ≥ 85 Almost certain that it will liquefy

∅1, ∅2 > 0 ∅ �= ∅1 + ∅2 > 4

ω = χ, c1 = χ.∅1, c2 = χ.∅2.

In optimal conditions,

∅1 = ∅2 = 2.05, ω = 0.7298, c1, c2 = 1.4962,

where ω is an inertia weight, ∅1 and ∅2 are the random positivenumbers, and χ is a constriction coefficient.

The penalty function method is a common approach in or-der to transform a constrained optimization problem into anunconstrained one by adding (or multiplying) a certain value tothe original objective functions based on the total amount ofconstraint violation errors to form the adjunct function. Hence,the corresponding objective function can be defined as follows(Yeniay, 2005):

z = z. (1 + αv) (16)

v = max(

CRRCSR7.5

− 1, 0)

, (17)

where z, z., α, and v are the adjunct function, the original ob-jective function (e.g. reliability index), the coefficient of penaltyfunction, and the amount of violation, respectively.

Cyclic stress ratio (CSR7.5) is denoted as (Seed & Idriss, 1971)

C SR7.5 = 0.65(

amax

g

) (σv

σ ′v

)(rd) /MSF.Kσ .Kα, (18)

where CSR7.5 is the cyclic stress ratio adjusted to a benchmarkearthquake of moment magnitude of 7.5 and σ vc

′ = 101 kPa;amax is the peak horizontal ground surface acceleration; g is theground acceleration; σ ′

v and σ v are the effective and total ver-tical stresses at the depth study, respectively, rd is the depth-dependent shear stress reduction coefficient, MSF is the mag-nitude scaling factor, Kσ is the overburden correction factor forCSR, and Kα is the static shear stress correction factor. The valuesof Kσ for all of effective pressures less than 3 atm (1 atm = 101kPa) are obtained according to Fig. 3, where the soil is dividedinto three areas based on Dr and (N1)60 of soil. In Fig. 3, (N1)60 isthe corrected SPT blow count, and Dr is the relative density. Inthis plot, the value of α is equal to the static active shear stressapplied to horizontal plane, which corresponds to the verticaleffective stress in that plane.

Several amendments for rd have been proposed by variousresearchers. In this study, rd is calculated based on the relation-ship presented by Idriss and Boulanger (2010) as follows (Idriss& Boulanger, 2010):

rd = exp [α (z) + β (z) .Mw] (19)

Figure 3: Suggested Kα for effective pressures less than 3 atm (1 tsf = 96 kPa).

α (z) = −1.012 − 1.126sin[5.133 +

( z11.73

)](20)

β (z) = 0.106 + 0.118sin[5.142 +

( z11.28

)], (21)

where Mw is the moment magnitude and z is the depth in me-tres.

MSF can be defined as

MSF = −0.058 + 6.9exp(−Mw

4

)≤ 1.8. (22)

The cyclic loading laboratory test results demonstrate that thesoil susceptibility to the cyclic liquefaction will rise up by in-creasing the overburden effective stress. In order to consider anon-linear relationship between CSR and σ ′

v , the adjustment fac-tor Kσ is calculated as (Seed, 1983)

Kσ = 1 − Cσ ln(

σ ′v

Pa

)≤ 1.1 (23)

Cσ = 1

18.9 − 2.55√

(N1)60,cs

≤ 0.3, (24)

where Pa is the atmosphere pressure (101 kPa).CRR is considered as the upper limit of CSR that a soil can

resist before liquefying (Juang et al., 2000); in the search algo-rithm N1,60, FC and σ ′

v values are kept constant and by changingCSR7.5 (increase/decrease depends on the location of the datapoints), the hypothetical data points on the limit state curve canbe found out. These data points are applied to estimate the limitstate function in the form of CRR = f(N1,60, FC, σ ′

v) by the GEP soft-ware (Goharzay et al., 2017). A brief summary of the databases(Cetin, 2000) used for model development is presented in Table 2.The information includes the maximum and minimum valuesof the mean and the COV of the different variables consideredas inputs and output of the GEP.

For the development of the proposed CRR models, approxi-mately 70% of the databases are randomly selected as trainingand the rest of them as testing data. Among numerous GEP-based CRR models, the best model is chosen on the basis of thestatistical performance [e.g. coefficient of determination (R2) andMSE]. The proposed GEP-based CRR model outputs have a goodconformity with the results obtained based on the Idriss and

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Table 2: Summary of the databases utilized for the development ofthe models in this research.

Modelvariables

Max. meanvalue

Min. meanvalue

Max. COVvalue

Min. COVvalue

d (m) 20.400 1.100Nm 37.000 1.500 0.815 0.007FC (%) 92.000 1 2.000 0.037σv(kPa) 383.930 15.470 0.280 0.031σ

′v(kPa) 198.660 8.140 0.378 0.044

amax (g) 0.693 0.090 0.300 0.011Mw 8.000 5.900 0.025 0

Table 3: Correlation coefficients among the input variables (Lee et al.,2010).

Input parameters

Input parameters (N1)60 FC σ ′v σ v amax Mw

(N1)60 1 0 0.3 0.3 0 0FC 0 1 0 0 0 0σ ′

v 0.3 0 1 0.9 0 0σ v 0.3 0 0.9 1 0 0amax 0 0 0 0 1 0.9Mw 0 0 0 0 0.9 1

Boulanger equations (Idriss & Boulanger, 2010), which demon-strates the competency of the GEP in estimating the compre-hensive predictive models. Equation (25) is found to be the mostappropriate prediction model of CRR through the GEP (Goharzayet al., 2017).

C RR = (N1)6054

σ ′v − F C

+ sin{

4

√sin (σ ′

v) × σ ′v × (N1)60

2

× [(N1)60 − F C

]} + 4√

F C × (N1)603 × [

(N1)60 − σ ′v

]

× 1.190887 +4

√[(N1)60 + 0.37854

]3

(N1)602 + F C + 4.669555

, (25)

where (N1)60 is the corrected SPT blow count and FC is the finecontent percentage.

The correlation coefficients among six input variables aregiven in Table 3 (Juang, Fang, & Li 2008). The reliability indices (β)for the total 94 cases (59 liquefied and 35 non-liquefied cases) ofthe databases, for which the statistical specifications are avail-able, can be determined utilizing the HPSO-GA. The flowchart ofsolving optimal reliability index (β) by the proposed optimiza-tion algorithm is shown in Fig. 4.

4. Results and Discussion4.1. Parameter uncertainty impact on liquefaction

probability without consideration of the modeluncertainty

In the following, corresponding probabilities (PL) are derivedfrom the notional probability concept (equation 3) and theBayesian mapping function (equation 14) without taking into

Enter the mean and COV of the soil and seismic parameters

(Nm, FC, σv, σ′v, Mw, amax)

Enter the max mean and min mean of the parameters

(Nm, FC, σv, σ′v, Mw, amax)

Develop the constrained and objective functions of the optimization

algorithm (Eq. 8)

Enter the HPSO-GA parameters(MaxIt, MaxSubItPSO, MaxSubItGA,

nPop, Pcrossover, Pmutation, φ1 and φ2, wdamp, ωinertia weight damping ratio)

Generate initial population

HPSO-GA main loop

Reliability index (β) calculation

Bayesian probability (Eq. 14) and notional probability (Eq. 3) calculations

Show iteration information(NFE, iteration number and best cost

value)

Start

End

Figure 4: Flowchart of the reliability analysis using HPSO-GA as an optimizationtool.

account the model uncertainty. A difference is observed be-tween the two probability curves depicted in Fig. 5. As the ac-curacy of the calculated notional probability depends on theaccuracy of the reliability index (β), due to the lack of con-sideration of uncertainty associated with the model in thereliability analysis, the calculated β in this case (β in termsof lacking knowledge of model uncertainty is called β1) andthe corresponding notional probability may be either under-estimated or overestimated and cannot be expected to givethe reliable results. While, as the Bayesian probability curveis calibrated empirically with the field observations of thedatabases, the effect of the uncertainty related to the limit statemodel is inherent in the probability calculations, it is supposedto provide a good approximation of the true probability ofliquefaction.

The investigation of the effect of the level of parameter un-certainty on the liquefaction probability can be performed by

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Figure 5: Comparison of the Bayesian probability with the notional probability obtained from the FORM analysis without considering the model uncertainty.

comparing the Bayesian mapping functions derived for each un-certainty level. Figure 6 shows the effect of five levels of the un-certainty (COVs = 0.10, 0.25, 0.40, 0.55, and 0.70) of the parame-ter N1,60 alone on the β1–PL curves (Bayesian mapping function).The results show that within the cases between COV = 0.1 and0.25 the level of N1,60 uncertainty has little effect on the proba-bility curves and the effect of the uncertainty becomes signifi-cant at the greater COV values. In a similar vein, the effect of theuncertainty of the FC parameter for four different COV values(COVs = 0.10, 0.75, 1.25, and 2.00) is investigated. As expected,the FC does not have a remarkable effect on the Bayesian prob-ability curve (Fig. 7). In order to study the effect of the levels ofuncertainty of the parameter σ ′

v , four different scenarios of COVs(0.05, 0.15, 0.25, and 0.35) are performed. Figure 8 shows thatwithin some ranges of COVs (COV = 0.05–0.15 and COV = 0.25–0.35) the level of σ v´ uncertainty has a noticeable effect on the β1–PL relationship, but no significant difference is observed withinthe COV (0.15–0.25). In a similar manner, six different scenariosare studied (COV amax = 0.1 and COV Mw = 0.01, COV amax = 0.1and COV Mw = 0.02, COV amax = 0.2 and COV Mw = 0.01, COVamax = 0.2 and COV Mw = 0.02, COV amax = 0.3 and COV Mw = 0.01,and COV amax = 0.3 and COV Mw = 0.02) for assessing the effectof the levels of uncertainty of the parameters amax and Mw. Asindicated in Fig. 9, in spite of the little influence of the Mw uncer-tainty, the COV of amax has a significant impact on the probabilitycurves. According to the results obtained through the reliabil-ity analysis, it can be concluded that each developed Bayesianprobability function is efficient for the relevant parameter un-certainty level and the suggested Bayesian mapping functionsmay not be applicable if the uncertainty levels of a future casestudy differ remarkably from those of the previous analysis. Asit is not reasonable to develop Bayesian probability functionsfor each level of the parameter uncertainty, a rigorous reliabilityanalysis, which considers both of the associated uncertainties(parameter and the limit state model uncertainties), is highlyrecommended.

4.2. Estimation of the model uncertainty

The accurate determination of the probabilities is not possibleunless the effect of model uncertainty is incorporated within thereliability analysis. In order to evaluate the liquefaction proba-bility of soils, a random variable referred to as model factor (cmf)is considered to quantify the uncertainties related to the limitstate model in the FORM analysis. Since the CRR model is ob-tained on the basis of the CSR model, the uncertainty in the CRRmodel shall be considered alone, which is also recommended asthe appropriate limit state model by Juang et al. (2005) as fol-lows:

g (z) = cmf R − Q = cmfCRR − CSR (26)

in which g(z) is the limit state function in terms of consideringmodel uncertainty and z is a vector of random variables, z = {cmf,(N1)60, FC, σ v, σ ′

v , amax, Mw}.Similar to other variables, the model factor is assumed to

follow a lognormal distribution. The estimation of model un-certainty includes finding out two statistical characteristics ofthe model factor, namely the mean (μcmf) and the COV. As themodel factor has a weak dependence on the other input vari-ables (Phoon & Kulhawy, 2005), in the present study no correla-tion is considered between cmf and other six input parameters.The model uncertainty is correctly estimated if the reliability in-dices (β in terms of considering the model uncertainty is calledβ2) and the related notional probabilities are most compatiblewith those based on the Bayes’ theorem, as well as by plottingthe β2 values against the calibrated probabilities, PL found to be0.5 in the case of β2 = 0, in which the most unbiased evaluationof liquefaction potential of soils can be rest assured. In the cur-rent study, a simple trial-and-error approach is implemented inorder to estimate the model uncertainty within the efficient reli-ability analysis based on a Bayesian framework. Figure 10 showsthe four scenarios of model uncertainty with μcmf = 1, in termsof COVs (0.0, 0.1, 0.2, and 0.3). For each scenario, the reliability

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Figure 6: Investigation of the effect of COV of parameter (N1)60 on the Bayesian mapping function.

Figure 7: Investigation of the effect of COV of parameter FC on the Bayesian mapping function.

indices and the corresponding probabilities are calculated tak-ing into account the effect of model uncertainty for 94 cases ofthe databases through the FORM analysis and Bayesian tech-nique. According to Fig. 10, at μcmf = 1 and COV = 0.2, PL = 0.519at β2 = 0. Thus, the COV component of model uncertainty isconsidered constant at 0.2 for subsequent reliability analyses.Afterwards, the five scenarios of model uncertainty in terms ofCOV = 0.2 and μcmf = (0.90, 0.95, 1.00, 1.05, and 1.10) are investi-gated (Fig. 11). It has been observed that the mapping functionis transmitted from left to right, and the corresponding proba-

bilities increase as the μcmf value increases. In the third scenario(μcmf = 1 and COV = 0.2), β2 and the corresponding PL are 0 and0.5186, respectively, whereas in the second scenario (μcmf = 0.95and COV = 0.2), the PL value is equal to 0.4857 in terms of β2 = 0.Finally, the μcmf value is estimated as 0.97 by using a conjec-tural interpolation. Similar to the first mode, four scenarios ofmodel uncertainty with μcmf = 0.97 and COVs = 0–0.3 are imple-mented through the series of reliability analyses. According toFig. 12, the statistical characteristics of model factor are chosenas μcmf = 0.97 and COV = 0.2, which presents PL = 0.5 at β2 = 0.

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Figure 8: Investigation of the effect of COV of parameter σ ′v on the Bayesian mapping function.

Figure 9: Investigation of the effect of COV of parameters amax and Mw on the Bayesian mapping function.

The effect of the COV component of model uncertainty onthe reliability indices is investigated by plotting the β1 values(β without considering model uncertainty) against the β2 val-ues (β considering model uncertainty) in Fig. 13, which indicatethat the slope of the line passing through the points decreasesas the COV of the model factor increases. Figure 14 shows theeffect of the μcmf component of model uncertainty on the re-liability indices in five scenarios, which illustrate that the in-

tercept of the linear trend line increases by increasing the μcmf

value. Figure 15 demonstrates a comparison between the lique-faction probabilities obtained from the Bayesian theory in termsof lacking knowledge of the model uncertainty (PL1, μcmf = 1,COV = 0), and the probabilities considering the true model un-certainty (PL2, μcmf = 0.97, COV = 0.2). The very good agree-ment (R2 = 0.98839) between both the cases indicates the ro-bustness of the Bayesian mapping approach in estimating the

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Figure 10: Investigation of the effect of COV of the model factor on the liquefaction probability.

Figure 11: Investigation of the effect of μcmf on the liquefaction probability.

liquefaction probabilities (PL) in terms of unknown model un-certainty specifications. A comparison between the calibratedliquefaction probabilities calculated using the Bayesian map-ping function without considering the model uncertainty (PL1,μcmf = 1, COV = 0), and the notional probabilities considering thetrue model uncertainty (PL2, μcmf = 0.97, COV = 0.2) is presentedin Fig. 16, which indicates the strong agreement (R2 = 0.97164)between the two liquefaction probability sets. It can be con-

cluded that the estimated notional probabilities are accurate ifthe true model factor characteristics are incorporated within thereliability analysis.

5. Two Examples

In the following, two cases of the databases are investigated toverify the accuracy of the above study. The first example is the

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Figure 12: Investigation of the effect of COV of model factor on the liquefaction probability in order to find the most unbiased model.

Miyagi-ken-oki earthquake that occurred in Japan on 12 June1978. As a liquefied case, the seismic and soil parameters at thecritical depth (z = 3.7 m) are given below:

Nm = 3.7, CB = 1, CS = 1, CR = 0.77, CE = 1.09, F C = 10%,

σv = 58.83 kPa, σ ′v = 36.28 kPa, amax = 0.2 g, Mw = 7.4.

The COVs of the parameters Nm, FC, σv, σ′v , amax, and Mw are

0.189, 0.2, 0.217, 0.164, 0.2, and 0.1, respectively. By implement-ing the meta-heuristic optimization algorithm (HPSO-GA) andconsidering the estimated true model uncertainty (μcmf = 0.97,COV = 0.2) within the FORM analysis, the reliability index (β2)and the corresponding notional liquefaction probability (PL) us-ing equation (3) are found out to be −1.41 and 0.921, respectively,and Juang et al. (2008) reported the corresponding notional prob-ability for this case as 0.91. The liquefaction probability can alsobe obtained from the Bayesian mapping function (Fs–PL) pro-posed by Goharzay et al. (2017) by considering only the meanvalues of the input variables. The calculated safety factor andthe corresponding Bayesian PL are predicted as 0.573 and 0.943,respectively. These two analyses yield consistent results, whichconfirm the field observation of liquefaction and support the ac-curacy of different estimation approaches used to predict theliquefaction probabilities.

The second example is the San Juan earthquake that oc-curred in Argentina on 23 November 1977. As a non-liquefiedcase, the seismic and soil parameters at the critical depth (z = 2.9m) are as follows:

Nm = 15.2, CB = 1, CS = 1, CR = 0.72, CE = 0.75, F C = 3%,

σv = 45.61 kPa, σ ′v = 38.14 kPa, amax = 0.2g, Mw = 7.4.

The COVs of the parameters Nm, FC, σ v, σ ′v , amax, and Mw are

0.026, 0.333, 0.107, 0.085, 0.075, and 0.1, respectively. In the sim-ilar way, β2 and the corresponding notional probability (PL) interms of considering the true model uncertainty are calculated

as 0.468 and 0.32, respectively. Juang et al. (2008) also investi-gated this case and obtained β2 and the notional PL as 0.533and 0.297, respectively. The liquefaction probability can alsobe obtained from the Bayesian mapping function (Fs–PL) pro-posed by Goharzay et al. (2017) by considering only the meanvalues of the input variables. The calculated safety factor andthe corresponding Bayesian PL are predicted as 1.241 and 0.247,respectively. The results prove the case to be non-liquefied,which indicates the robustness of the methodology used inthis study.

6. Conclusions

The following concluding remarks can be obtained from thisstudy:

� Easy implementation, fast computing, and the low mem-ory requirements of the PSO algorithm, making it superiorto the other algorithms. Therefore, HPSO-GA by combiningthe strengths of PSO and GA, as an appropriate optimizationtool for solving complex problems (e.g. reliability analysis),speeds up the convergence rate besides providing the accu-rate solutions.

� The development of the mapping function for each level ofparameter uncertainty to interpret the probability of lique-faction occurrence is not practical. Hence, the reliability anal-ysis, which does not depend only on the mean values of inputparameters and takes into account effects of both uncertain-ties associated with the input parameters and the limit statemodel, is suggested in order to achieve more accurate esti-mation of the right liquefaction probability.

� If the uncertainties related to the limit state model have asignificant impact on the probability assessment, by disre-garding them during the analysis, the reliability indices (β1)and thus, the corresponding notional probabilities (PL) willinclude some errors. Nevertheless, the Bayesian mapping

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(a) (b)

(c) (d)

Figure 13: Investigation of the effect of COV of model factor on the reliability index (β2).

function, which is calibrated empirically with the field obser-vations of the case history databases, covers the effect of theassociated uncertainties of the databases implicitly and con-sequently, provides more accurate predictions in terms oflacking knowledge of model uncertainty.

� Due to obtaining compatible results from the Bayesian map-ping function with and without considering the model uncer-

tainty, it can be concluded that the calibrated Bayesian func-tions can produce excellent predictions of the liquefactionprobability in terms of lacking knowledge of the limit statemodel uncertainties.

� The comparison of the Bayesian probabilities with thosebased on the notional concept in terms of not consideringthe model uncertainty effect and considering the true model

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(a) (b)

(c) (d)

(e)

Figure 14: Investigation of the effect of μcmf on the reliability index (β2).

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Figure 15: Comparison of the liquefaction probabilities obtained from theBayesian mapping function in two different conditions (with/without consid-

eration of model uncertainty).

Figure 16: Comparison of the liquefaction probabilities obtained from theBayesian mapping function without considering the model uncertainty with thenotional probabilities in terms of considering the true model uncertainty.

factor characteristics (μcmf = 0.97, COV = 0.2), respectively,implies the good consistency of the results derived from bothmethods, which demonstrates that the model uncertainty iswell incorporated in the reliability analysis.

Conflict of interest statement

Declarations of interest: none.

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Appendix 1. Post-liquefaction SPT-based Data (Cetin, 2000) Used for Reliability Analysis

Meandepth(m)

Mean σ v

(kPa) COV σ v

Meanσ ′

v (kPa) COV σ ′v

Meanamax (g) COV

MeanNm COV Nm

MeanMw COV Mw

MeanFC(%)

COVFC CR CS CB CE Liquefied?

4.2 69.61 0.149 51.85 0.107 0.24 0.2 6 0.333 7.4 10 0.2 0.8 1 1 1.09 Yes5.2 89.15 0.085 57.29 0.084 0.4 0.1 11.7 0.248 8 2 0.5 0.85 1 1 1.22 Yes3 49.07 0.118 22.22 0.139 0.24 0.2 7 0.229 7.4 12 0.167 0.73 1 1 1 Yes4.5 73.26 0.078 46.37 0.08 0.2 0.075 6 0.017 7.4 50 0.1 0.82 1 1 0.75 Yes3.7 58.83 0.217 36.28 0.164 0.2 0.2 3.7 0.189 7.4 10 0.2 0.77 1 1 1.09 Yes4.7 72.78 0.157 35.43 0.189 0.17 0.265 7.8 0.449 6.5 0.02 40 0.075 0.83 1 1 1.05 No6 102.7 0.123 67.73 0.103 0.24 0.2 9.9 0.424 7.4 10 0.2 0.88 1 1 1.09 No9.9 184.6 0.179 135.57 0.126 0.4 0.1 17.5 0.154 6.9 0.016 20 0.25 1 1 1 1.22 No3.5 53.89 0.057 44.04 0.073 0.26 0.096 13 0.238 7 0.017 3 0.333 0.76 1 1 1 Yes8.5 137.8 0.105 69.39 0.109 0.1 0.2 9.5 0.158 7.3 0.015 5 0.4 0.95 1 1 1 Yes5.5 90 0.095 45.05 0.109 0.095 0.011 4.6 0.087 6.1 13 0.077 0.86 1 1 1.09 No11.4 211 0.054 116.24 0.068 0.18 0.15 24.7 0.121 7.5 0.015 8 0.25 1 1 1 1.21 No6.2 112.6 0.04 93.48 0.044 0.54 0.074 10.3 0.155 6.7 0.019 38 0.605 0.89 1 1 1.13 Yes3.2 52.82 0.133 30.77 0.137 0.09 0.2 2.7 0.556 7.5 0.015 5 0.4 0.74 1 1 1.22 Yes10.7 200.1 0.106 110.44 0.106 0.4 0.1 26 0.096 8 0 0 1 1 1 1.22 No2.4 38.18 0.093 27.02 0.113 0.24 0.2 11.5 0.191 7.4 7 0.143 0.68 1 1 1.12 Yes6.5 117.4 0.036 83.12 0.053 0.22 0.045 13 0.315 7 0.017 5 0.2 0.9 1.1 1 0.92 Yes6 82.08 0.066 57.57 0.072 0.165 0.079 4 0.35 7 0.017 25 0.2 0.88 1.1 1 0.92 Yes8.9 159.8 0.043 142.12 0.049 0.693 0.087 20 0.22 6.7 0.019 43 0.302 0.97 1 1 1.13 Yes7.9 129.6 0.194 66.05 0.169 0.13 0.2 8.4 0.19 7.3 0.015 67 0.104 0.94 1 1 0.83 Yes7.9 143.2 0.137 88.63 0.121 0.24 0.2 19.3 0.13 7.4 5 0.2 0.94 1 1 1.21 No2.7 44.57 0.28 22.52 0.257 0.12 0.2 4.6 0.348 6.7 5 0.2 0.71 1 1 1 Yes5 84.97 0.206 44.91 0.185 0.1 0.2 8.8 0.318 6.7 0 0 0.84 1 1 1.09 No3.7 60.52 0.162 32.09 0.156 0.24 0.2 8.8 0.42 7.4 0 0 0.77 1 1 1 Yes3.7 58.83 0.217 36.28 0.164 0.12 0.2 3.7 0.189 6.7 10 0.2 0.77 1 1 1.09 No5.5 97.24 0.069 52.15 0.089 0.35 0.2 18.1 0.088 8 0.011 20 0.15 0.86 1 1 1 Yes2.5 35.74 0.089 30.84 0.113 0.18 0.111 10.9 0.202 6.7 0.019 30 0.167 0.69 1 1 1.05 No

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Continued

Meandepth(m)

Mean σ v

(kPa) COV σ v

Meanσ ′

v (kPa) COV σ ′v

Meanamax (g) COV

MeanNm COV Nm

MeanMw COV Mw

MeanFC(%)


5.7 94.84 0.108 38.48 0.153 0.116 0.155 8.9 0.079 7.7 0.013 3 0.333 0.87 1 1 1.22 Yes2.9 45.74 0.238 25.71 0.207 0.2 0.2 5.6 0.143 7.9 20 0.15 0.72 1 1 1.17 Yes8.5 155.1 0.15 120.82 0.102 0.4 0.1 19.7 0.142 6.9 0.016 20 0.25 0.95 1 1 1.22 No7.7 146.5 0.24 94.08 0.192 0.34 0.029 5.7 0.246 6.9 0.016 20 0.25 0.93 1 1 1.22 Yes4.7 84.15 0.096 40.83 0.196 0.19 0.105 21 0.024 6.7 0.019 18 0.167 0.83 1 1 1.05 No3.5 61.66 0.206 38.13 0.171 0.22 0.2 19.7 0.157 8 0.011 5 0.6 0.76 1 1 1 No4.5 72.31 0.118 40.16 0.118 0.13 0.2 9.5 0.263 8 0.011 12 0.25 0.81 1 1 1 Yes4.5 65.59 0.117 41.08 0.102 0.09 0.278 4.5 0.778 6.2 0.023 75 0.133 0.81 1 1 1.05 No7.2 127.4 0.156 78.84 0.126 0.25 0.1 20 0.205 7.6 19 0.105 0.92 1 1 0.65 Yes3.5 53.41 0.065 48.51 0.053 0.28 0.2 11 0.245 7.4 5 0.4 0.76 1 1 1 Yes2 30.93 0.257 16.28 0.256 0.2 0.3 1.8 0.333 8 27 0.111 0.64 1 1 1.17 Yes3 49.72 0.12 29.14 0.126 0.32 0.2 15.5 0.31 7.4 4 0.25 0.73 1 1 1.12 No4.6 68.04 0.099 57.01 0.1 0.14 0.093 4.6 0.717 7 0.017 30 0.233 0.82 1.1 1 0.92 No2.4 41.9 0.223 26.96 0.188 0.225 0.111 14.4 0.153 7.9 5 0.4 0.68 1 1 1.21 No2.6 41.67 0.181 28.44 0.144 0.32 0.2 5.2 0.385 7.4 4 0.25 0.7 1 1 1 Yes5.7 94.84 0.108 38.48 0.153 0.213 0.141 8.9 0.079 7.8 3 0.333 0.87 1 1 1.22 Yes6 87.58 0.066 63.07 0.067 0.13 0.1 4 0.35 7 0.017 25 0.12 0.88 1.1 1 0.92 Yes4.5 75.4 0.089 60.7 0.069 0.5 0.2 29.4 0.139 8 0.011 10 0.2 0.81 1 1 1 No4.2 66.67 0.115 54.66 0.142 0.28 0.2 13.2 0.424 7.4 0 0 0.8 1 1 1.21 No3.6 57.38 0.18 39.24 0.13 0.12 0.2 2 0.4 6.7 60 0.083 0.77 1 1 1 No5.3 81.54 0.073 70.93 0.086 0.45 0.1 3.4 0.265 6.6 55 0.091 0.85 1 1 1.13 Yes6.6 96.57 0.071 87.5 0.085 0.46 0.109 10.2 0.363 7 0.017 27 0.185 0.9 1 1 1.13 Yes6 101.6 0.093 63.9 0.084 0.24 0.2 21 0.09 7.4 0 0.88 1 1 1.12 No3.3 51.47 0.188 33.55 0.142 0.09 0.256 5.9 0.288 5.9 0.025 31 0.097 0.75 1 1 1.05 No4.5 65.59 0.117 41.08 0.102 0.17 0.118 4.5 0.778 5.9 0.025 75 0.133 0.81 1 1 1.05 Yes7.5 123.1 0.226 53.4 0.23 0.227 0.154 7.6 0.224 7.7 0.013 1 1 0.93 1 1 1.22 Yes4.2 66.08 0.131 51.14 0.091 0.13 0.077 6.4 0.516 6.5 0.02 92 0.109 0.8 1 1 1.05 No4.7 72.78 0.147 35.43 0.148 0.1 0.05 7.8 0.449 6.2 0.023 40 0.075 0.83 1 1 1.05 No3.6 57.38 0.18 39.24 0.13 0.24 0.2 2 0.4 7.4 60 0.083 0.77 1 1 1 Yes9.2 158.3 0.056 77.4 0.085 0.15 0.2 13.1 0.229 7.1 0 0 0.97 1 1 1.22 No9.5 174.8 0.17 130.65 0.119 0.4 0.1 30.2 0.235 6.9 0.016 20 0.25 0.98 1 1 1.22 No3 49.07 0.118 22.22 0.139 0.12 0.2 7 0.229 6.7 12 0.167 0.73 1 1 1 No4.5 76.85 0.124 55.93 0.097 0.24 0.2 11.9 0.227 7.4 26 0.192 0.82 1 1 1.09 No2.5 39.27 0.092 34.37 0.078 0.24 0.104 17.9 0.173 7 0.017 1 2 0.69 1 1 1 No3.4 54.08 0.172 37.9 0.225 0.47 0.106 10.6 0.302 6.5 0.02 37 0.135 0.76 1 1 1.13 No4.1 68.41 0.205 45.87 0.15 0.27 0.093 15 0.22 7 0.017 1 2 0.79 1 1 1 No3.5 53.57 0.169 36.9 0.125 0.29 0.086 6.1 0.557 7 0.017 2 1 0.76 1 1 1 Yes5.2 85.16 0.123 42.66 0.146 0.16 0.15 7.9 0.177 7.5 0.015 8 0.25 0.85 1 1 1.09 Yes6 82.08 0.119 57.57 0.084 0.15 0.087 3.9 0.487 7 0.017 50 0.1 0.88 1.1 1 0.92 Yes5.5 104.4 0.155 71.52 0.122 0.225 0.111 25.5 0.067 7.9 5 0.4 0.86 1 1 1.21 No3.5 54.98 0.153 37.83 0.118 0.25 0.22 9.7 0.227 7.7 0.013 1 1 0.76 1 1 1.22 Yes4.5 73.01 0.15 53.9 0.103 0.24 0.104 15.7 0.204 7 0.017 1 2 0.82 1 1 1 Yes3.7 60.52 0.162 32.09 0.156 0.12 0.2 8.8 0.42 6.7 0 0 0.77 1 1 1 No6.5 123.3 0.051 113.51 0.087 0.24 0.1 37 0.038 7 0.017 7 0.286 0.9 1.2 1 0.92 No7.5 122.3 0.124 57.72 0.152 0.16 0.15 9.4 0.16 7.5 0.015 8 0.25 0.93 1 1 1.09 Yes11.4 210.5 0.225 136.98 0.171 0.34 0.118 11.6 0.31 6.9 0.016 20 0.25 1 1 1 1.22 Yes5.2 88.44 0.119 71.78 0.081 0.4 0.125 20.7 0.454 6.9 0.016 18 0.222 0.85 1 1 1.22 No7 117.4 0.108 63.66 0.119 0.2 0.2 12.4 0.056 7.3 0.015 48 0.104 0.91 1 1 1 Yes4.2 66.08 0.142 51.14 0.179 0.19 0.132 6.4 0.516 5.9 0.025 92 0.109 0.8 1 1 1.05 Yes3.5 56 0.199 32.48 0.166 0.22 0.2 5.9 0.136 8 0.011 3 0.667 0.76 1 1 1 Yes4.7 82.08 0.128 35.52 0.165 0.25 0.16 8.5 0.247 6.9 0.016 20 0.35 0.83 1 1 1.22 Yes2.7 44.57 0.28 22.52 0.257 0.32 0.2 4.1 0.39 7.4 5 0.2 0.71 1 1 1 Yes8.2 131.4 0.031 95.58 0.046 0.2 0.075 9 0.233 7.4 20 0.15 0.95 1 1 0.75 Yes7.9 130.5 0.137 61.06 0.161 0.16 0.15 8.8 0.25 7.5 0.015 8 0.25 0.94 1 1 1.09 Yes4.2 66.08 0.131 51.14 0.091 0.174 0.115 6.4 0.516 6.7 0.019 92 0.109 0.8 1 1 1.05 No4.2 69.32 0.26 37.46 0.215 0.2 0.2 4.8 0.583 7.7 0.013 15 0.267 0.8 1 1 1.22 Yes7.5 131.4 0.082 70.63 0.106 0.24 0.2 20.5 0.215 7.4 17 0.176 0.93 1 1 1.12 No

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Continued

Meandepth(m)

Mean σ v

(kPa) COV σ v

Meanσ ′

v (kPa) COV σ ′v

Meanamax (g) COV

MeanNm COV Nm

MeanMw COV Mw

MeanFC(%)


9.9 183.8 0.139 115.18 0.115 0.34 0.118 9.5 0.137 6.9 0.016 20 0.25 1 1 1 1.22 Yes1.1 15.47 0.243 8.14 0.378 0.16 0.281 5.8 0.241 6.5 0.02 80 0.125 0.5 1 1 1.05 Yes3.3 51.47 0.188 33.55 0.142 0.51 0.098 5.9 0.288 6.5 0.02 31 0.097 0.75 1 1 1.05 Yes6 108.3 0.066 61.46 0.077 0.18 0.15 30 0.097 7.5 0.015 0 0 0.88 1 1 1.21 No4.9 85.81 0.181 59.83 0.13 0.25 0.1 34.2 0.216 7.6 19 0.105 0.84 1 1 0.65 No4.2 69.61 0.149 51.85 0.107 0.14 0.171 6 0.483 6.7 10 0.2 0.8 1 1 1.09 No2.9 49.88 0.163 39.1 0.169 0.156 0.128 26.4 0.394 6.7 0.019 25 0.16 0.72 1 1 1.13 No4.2 71.87 0.067 40.01 0.095 0.2 0.2 10.4 0.538 8 0.011 5 0.6 0.8 1 1 1 Yes1.1 15.47 0.243 8.14 0.378 0.19 0.105 5.8 0.241 6.7 0.019 80 0.125 0.5 1 1 1.05 No6.2 117.8 0.048 85.96 0.057 0.25 0.04 6 0.65 7 0.017 8 0.375 0.89 1.1 1 0.92 Yes3.7 61.59 0.214 24.99 0.231 0.16 0.15 3.9 0.179 7.5 0.015 10 0.3 0.77 1 1 1.09 Yes7.9 141.6 0.053 98 0.054 0.4 0.125 25.9 0.193 6.9 0.016 2 0.5 0.94 1 1 1.22 No4.7 72.78 0.148 35.43 0.249 0.23 0.087 7.8 0.449 5.9 0.025 40 0.075 0.83 1 1 1.05 Yes5.9 91.33 0.119 74.9 0.114 0.14 0.171 9 0.2 6.7 5 0.6 0.88 1 1 1 No12.1 165.6 0.187 78.98 0.155 0.135 0.111 13 0.269 7.5 3 0.333 1 1 1 0.75 No4.5 73.95 0.153 33.27 0.176 0.12 0.183 7.7 0.091 7.1 0 0 0.81 1 1 1.22 Yes7 115.4 0.084 92.84 0.062 0.41 0.122 20 0.165 7 0.017 13 0.154 0.91 1 1 1.13 Yes9.1 129 0.238 54.35 0.213 0.135 0.111 4.3 0.302 7.5 3 0.333 0.97 1 1 0.75 Yes6.5 98.26 0.086 82.58 0.063 0.46 0.109 9 0.1 7 0.017 15 0.133 0.9 1 1 1.13 Yes4.5 65.59 0.117 41.08 0.102 0.2 0.2 4.5 0.778 6.6 0.02 75 0.133 0.81 1 1 1.05 No3.2 52.33 0.089 44.37 0.08 0.14 0.2 9 0.233 6.7 20 0.15 0.74 1 1 1.09 No7.5 135.6 0.119 80.04 0.108 0.18 0.15 17.5 0.063 7.5 0.015 8 0.25 0.93 1 1 1.21 No4.7 80.68 0.066 37.36 0.105 0.17 0.118 21 0.024 5.9 0.025 18 0.167 0.83 1 1 1.05 No9.2 172.4 0.045 120 0.048 0.51 0.118 24.4 0.123 6.7 0.019 25 0.2 0.97 1 1 1.13 Yes4.2 133.4 0.122 105.94 0.072 0.2 0.2 23.3 0.094 7.4 10 0.2 0.8 1 1 1.21 No4.2 69.32 0.26 37.46 0.215 0.15 0.2 4.8 0.583 7.1 15 0.267 0.8 1 1 1.22 No4 65.91 0.183 32.15 0.211 0.225 0.111 4.5 0.156 7.9 5 0.4 0.79 1 1 1.09 Yes3 47.6 0.144 34.39 0.115 0.24 0.2 4.3 0.581 7.4 10 0.2 0.73 1 1 1 Yes4.5 65.59 0.118 41.08 0.104 0.18 0.106 4.5 0.778 6.5 0.02 75 0.133 0.81 1 1 1.05 Yes3.3 53.17 0.224 32.21 0.175 0.4 0.3 3.6 0.278 7.3 0 0 0.75 1 1 1.17 Yes4.7 72.78 0.147 35.43 0.148 0.18 0.028 7.8 0.449 6.6 0.02 40 0.075 0.83 1 1 1.05 Yes3.5 53.06 0.157 38.2 0.123 0.2 0.3 6 0.333 8 13 0.077 0.76 1 1 1.17 Yes1.7 26.7 0.15 19.35 0.166 0.25 0.1 6.5 0.062 7 0.017 35 0.143 0.61 1 1 1 Yes6 86.4 0.12 66.79 0.078 0.42 0.119 8 0.45 7 0.017 22 0.136 0.88 1 1 1.13 Yes4.2 65.12 0.154 47.19 0.114 0.2 0.3 1.5 0.4 8 25 0.12 0.8 1 1 1.17 Yes3.9 61.38 0.144 46.19 0.183 0.24 0.104 10.2 0.039 7 0.017 3 0.333 0.79 1 1 1 Yes6 86.01 0.091 71.3 0.145 0.37 0.135 9 0.244 7 0.017 8 0.25 0.88 1 1 1.13 Yes5 81.53 0.076 38.4 0.114 0.16 0.15 4.4 0.364 7.5 0.015 0 0 0.84 1 1 1.09 Yes20.4 383.9 0.04 198.66 0.065 0.4 0.1 8.5 0.259 8 − 10 0.3 1 1 1 1.22 Yes7.5 140 0.071 95.86 0.068 0.25 0.04 12 0.425 7 0.017 10 0.1 0.93 1.1 1 0.92 Yes7.5 137.9 0.168 103.54 0.117 0.4 0.125 14.5 0.407 6.9 0.016 25 0.2 0.93 1 1 1.22 Yes3.4 54.08 0.179 37.9 0.235 0.13 0.154 10.6 0.302 6.7 0.019 37 0.135 0.76 1 1 1.13 No14.4 238.5 0.044 105.28 0.074 0.095 0.011 3.6 0.167 6.1 27 0.037 1 1 1 1.09 No4.5 73.95 0.153 33.27 0.176 0.278 0.144 7.7 0.091 7.7 0.013 0 0 0.81 1 1 1.22 Yes8.7 152.6 0.056 101.11 0.058 0.4 0.125 11 0.236 6.9 0.016 2 0.5 0.96 1 1 1.22 Yes2.5 35.74 0.075 30.84 0.099 0.16 0.125 10.9 0.183 5.9 0.025 30 0.167 0.69 1 1 1.05 No6.5 106.4 0.142 67.22 0.119 0.18 0.15 12 0.25 7.5 0.015 0 0 0.9 1 1 1.09 No4.2 71.34 0.158 53.42 0.112 0.24 0.2 9.6 0.448 7.4 3 0.333 0.8 1 1 1.09 Yes3 47.6 0.144 34.39 0.115 0.12 0.2 4.3 0.581 6.7 10 0.2 0.73 1 1 1 No2.5 35.74 0.079 30.84 0.102 0.16 0.119 10.9 0.202 6.5 0.02 30 0.167 0.69 1 1 1.05 No2.4 39.26 0.182 27.31 0.144 0.2 0.075 13.9 0.007 7.4 4 0.375 0.68 1 1 0.75 No3.2 47.09 0.157 32.88 0.266 0.47 0.106 2.7 0.815 6.5 0.02 29 0.155 0.74 1 1 1.13 Yes4.2 71.34 0.158 53.42 0.112 0.14 0.2 9.6 0.448 6.7 3 0.333 0.8 1 1 1.09 No6.2 94.92 0.072 84.47 0.077 0.45 0.1 7.2 0.333 6.6 50 0.1 0.89 1 1 1.13 Yes5 84.97 0.206 44.91 0.185 0.2 0.2 8.8 0.318 7.4 0 0 0.84 1 1 1.09 Yes3.2 47.09 0.166 32.88 0.276 0.15 0.133 2.7 0.815 6.7 0.019 29 0.155 0.74 1 1 1.13 No11.6 181.8 0.052 133.22 0.05 0.2 0.075 11.4 0.14 7.4 20 0.15 1 1 1 0.75 Yes7.9 132.2 0.197 90.83 0.143 0.35 0.3 16.1 0.155 7.3 4 0.25 0.94 1 1 1.3 Yes8.5 124.5 0.068 70.58 0.077 0.41 0.122 18.5 0.151 7 0.017 20 0.15 0.95 1 1 1.13 Yes

Dow




Continued

Meandepth(m)

Mean σ v

(kPa) COV σ v

Meanσ ′

v (kPa) COV σ ′v

Meanamax (g) COV

MeanNm COV Nm

MeanMw COV Mw

MeanFC(%)


8.3 104.3 0.186 63.97 0.121 0.135 0.111 14.9 0.154 7.5 3 0.333 0.95 1 1 0.75 No4.7 83.3 0.095 47.19 0.101 0.2 0.2 9.8 0.204 8 0.011 12 0.25 0.83 1 1 1 Yes3.5 53.41 0.065 48.51 0.053 0.14 0.2 11 0.209 6.7 5 0.4 0.76 1 1 1 No2.9 45.61 0.107 38.14 0.085 0.2 0.075 15.2 0.026 7.4 3 0.333 0.72 1 1 0.75 No4.2 68.93 0.221 41.97 0.202 0.18 0.056 6 0.55 7 0.017 20 0.2 0.8 1.1 1 1.13 Yes5.9 91.33 0.119 74.9 0.114 0.24 0.2 9 0.444 7.4 5 0.6 0.88 1 1 1 Yes2.4 38.18 0.093 27.02 0.113 0.12 0.2 11.5 0.191 6.7 7 0.143 0.68 1 1 1.12 No1.1 15.47 0.243 8.14 0.378 0.17 0.118 5.8 0.241 5.9 0.025 80 0.125 0.5 1 1 1.05 No3.2 52.33 0.089 44.37 0.08 0.24 0.2 9 0.256 7.4 20 0.15 0.74 1 1 1.09 Yes13.5 240.2 0.045 141.19 0.058 0.4 0.125 5.8 0.483 6.9 0.016 18 0.222 1 1 1 1.22 Yes6.5 119.9 0.037 88.49 0.045 0.4 0.1 9.5 0.074 6.7 0.019 37 0.135 0.9 1 1 1.13 Yes6.1 111.1 0.043 80.23 0.085 0.22 0.045 12.6 0.246 7 0.017 3 0.333 0.88 1.1 1 0.92 Yes7.5 121.3 0.14 62.7 0.13 0.2 0.2 10.3 0.32 7.3 0.015 5 0.4 0.93 1 1 1 Yes3.4 54.08 0.17 37.9 0.124 0.26 0.096 10.5 0.067 7 0.017 2 1 0.76 1 1 1 Yes2.9 49.88 0.163 39.1 0.169 0.47 0.106 26.4 0.394 6.5 0.02 25 0.16 0.72 1 1 1.13 No2.6 41.67 0.181 28.44 0.144 0.12 0.2 5.2 0.385 6.7 4 0.25 0.7 1 1 1 No3.3 51.47 0.188 33.55 0.142 0.16 0.125 5.9 0.288 6.7 0.019 31 0.097 0.75 1 1 1.05 No

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