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Journal of Earthquake Engineering

ISSN: 1363-2469 (Print) 1559-808X (Online) Journal homepage: http://www.tandfonline.com/loi/ueqe20

Bayesian Parameter Identification and ModelSelection for Normalized Modulus ReductionCurves of Soils

Oluwatosin Victor Akeju, Kostas Senetakis & Yu Wang

To cite this article: Oluwatosin Victor Akeju, Kostas Senetakis & Yu Wang (2017): BayesianParameter Identification and Model Selection for Normalized Modulus Reduction Curves of Soils,Journal of Earthquake Engineering, DOI: 10.1080/13632469.2017.1323051

To link to this article: https://doi.org/10.1080/13632469.2017.1323051

Accepted author version posted online: 19May 2017.Published online: 19 May 2017.

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Bayesian Parameter Identification and Model Selection forNormalized Modulus Reduction Curves of SoilsOluwatosin Victor Akeju, Kostas Senetakis , and Yu Wang

Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong

ABSTACTThis work develops a procedure that involves the use of Bayesianapproach to quantify data scatterness, estimates the optimal valuesof model parameters, and selects the most appropriate model for theconstruction of normalized modulus reduction curves of soils. Theproposed procedure is then demonstrated using real observationdata based on a set of comprehensive resonant column tests oncoarse-grained soils conducted in the study.

ARTICLE HISTORYReceived 20 December 2016Accepted 15 April 2017

KEYWORDSBayesian Approach;Parameter Identification;Model Selection; NormalizedModulus Reduction Curves;Resonant Column Test;Hyperbolic Model

1. Introduction

In geotechnical engineering, there have been numerous studies on the measurement,estimation, and analysis of the reduction (or degradation) of soil shear modulus (G)with strain (γ) using the resonant column (RC) method, the torsional shear apparatus,or monotonic/cyclic triaxial tests [e.g., Ishihara, 1996; Kramer, 1996; Menq, 2003; Zhanget al., 2005; Senetakis et al., 2013a, 2016, among others]. The shear modulus that is used tocharacterize the stiffness of soils is an important parameter required for the design of geo-structures subjected to static and dynamic load patterns. The reduction of shear moduluswith strain is commonly analyzed using the normalized modulus reduction curves (nor-malized with respect to the small-strain value, Gmax), denoted as G=Gmax � log γ curves inthis paper. In geotechnical practice, G=Gmax � log γ curves are important for the analysisof various types of problems including, for example, the prediction of deformations ofearth retaining support structures and foundations [e.g., Atkinson, 1993; Clayton, 2011],and ground seismic response analysis and foundation dynamics due to machine vibrations[e.g., Richart et al., 1970; Seed et al., 1986; Vucetic, 1994; Kramer, 1996]. In computercodes for seismic response analysis such as the codes SHAKE [Schnabel et al., 1972] andEERA [Bardet et al., 2000] for one-dimensional analysis and the code QUAD4M [Hudsonet al., 1994] for two-dimensional analysis, the normalized modulus reduction curves are abasic input parameter.

Over the decades, different families of G=Gmax � log γ curves have been developed forvarious types of soils in the literature. For example, Fig. 1 shows the upper and lowerbounds of G=Gmax � log γ curves for sands reported by Seed and Idriss [1970] withdifferent void ratios (relative densities ranging from 30 to 90%) for an effective vertical

CONTACT Kostas Senetakis ksenetak@cityu.edu.hk Department of Architecture and Civil Engineering, CityUniversity of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong.Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/ueqe.

JOURNAL OF EARTHQUAKE ENGINEERINGhttps://doi.org/10.1080/13632469.2017.1323051

© 2017 Taylor & Francis Group, LLC

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stress of 144 kPa. Figure 1 also shows the upper and lower bounds of G=Gmax � log γcurves for clays reported by Vucetic and Dobry [1991] with a plasticity index valueranging from 0% (lower bound in Fig. 1) to 50% (upper bound in Fig. 1), those reportedby Rollins et al. [1998] for gravels varying from loose to very dense state (confiningpressures between 29 and 490 kPa, and relative densities ranging from 40 to 100%), thoseproposed by Menq [2003] for granular soils with a coefficient of uniformity (Cu) of 1.03 at100 kPa and Cu of 12.5 at 25 kPa, and those recently developed by Oztoprak and Bolton[2013] for sands. These curves have been widely used particularly in problems involvingmoderate strains, i.e., beyond the elastic-linear range of behavior and prior to the large-deformation range where the soil reaches a steady-state behavior. In this range of strains,soil stiffness is highly sensitive to the strain magnitude (or amplitude when cyclic type ofloading is considered).

Nevertheless, in some cases, new test data from given geotechnical projects may not beconsistent with the families of G=Gmax � log γ curves that are already proposed in theliterature as shown in Fig. 1. This mismatch could be the result of data scatterness due toslightly different experimental conditions or most commonly because the literature curveshave been developed on the basis of data for particular types of soils; thus, the reductionmodulus of other types of soils does not necessarily follow the same trend. A typicalexample of this mismatch was reported recently by Senetakis et al. [2013a, 2013b, 2016]who studied different types of volcanic granular soils and concluded that existing reduc-tion modulus curves developed on the basis of quartz sands could not efficiently predictthe medium-strain behavior of the crushed rhyolite and pumice they studied. On the otherhand, the data may belong to a new family of G=Gmax � log γ curves that are significantlydifferent from the pre-existing curves. In such cases, new G=Gmax � log γ curves need tobe developed for such set of data for more accurate geotechnical design and analysis.However, there is no objective procedure to compare the newly developed G=Gmax � log γcurves with various families of pre-existing curves and select the most appropriate one

Figure 1. Upper and lower bounds of G=Gmax � log γ curves proposed for different classes of soil in theliterature.

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with best fitted parameters, even though the research on the reduction of shear moduluswith strain has been extensive since the 1970s.

The development of G=Gmax � log γ curves is usually carried out using analyticalmodels. Several existing models are available in the literature to develop G=Gmax � log γcurves for the analysis of the reduction of shear modulus with strain. However, thehyperbolic models are the most commonly adopted in the development of normalizedmodulus reduction curves because of their simplicity. Examples include the first hyper-bolic model that was proposed by Hardin and Drnevich [1972a, 1972b] and expressed inEq. (1), a modified hyperbolic model proposed by Darendeli [2001] and expressed in Eq.(2) and a further modified hyperbolic model proposed by Oztoprak and Bolton [2013] andexpressed in Eq. (3). These models are particularly applicable for a broad range ofgeotechnical and earthquake engineering problems where the medium-strain range isimportant in design, whereas for problems involving large deformations (failure analysis),plasticity models are preferable [e.g., Atkinson, 1993; Kramer, 1996].

GGmax

� �¼ 1

1þ ðγ=γrÞ(1)

GGmax

� �¼ 1

1þ ðγ=γrÞa(2)

GGmax

� �¼ 1

,1þ γ� γe

γr

� �a� �when γ< γe;G=Gmax ¼ 1 (3)

In Eqs. (1)–(3), γr is the reference shear strain at which G=Gmax = 0.5 [Darendeli, 2001;Menq, 2003], a is a fitting parameter named the coefficient of curvature that controls thecurvature or shape of the G=Gmax � log γ curve, and γe is the elastic threshold strainbeyond which the shear modulus falls below its maximum value. Note that in this study,γe is taken to correspond to the shear strain amplitude at G=Gmax = 0.99 [e.g., Vucetic,1994]. The common approach by practitioners involves using eyeballing and/or the least-squares method to estimate the mean values of the fitting parameters (γrand a) withoutquantifying the scatterness of the observation data. However, the scatterness of theobservation data reflects the degree of uncertainty involved in the prediction of suchdata using the G=Gmax � log γ curve. Thus, there is a need to find a robust method to notonly estimate the optimal (or mean) values of the fitting parameters but also quantify thescatterness of the observation data. Moreover, due to the complexity and addition of morefitting parameters, these hyperbolic models (i.e., Eqs. (1)–(3)) tend to produce differentanalysis and interpretation of the nonlinear behavior of soil even for the same set ofobservation data. Therefore, practitioners also face the task of selecting the most appro-priate hyperbolic model among available ones for a given set of observation data. Notethat traditional methods like eyeballing cannot be used for selecting the most appropriatehyperbolic model in a bid to objectively compare G=Gmax � log γcurves. This study aimsat developing Bayesian probabilistic methods to address these problems. Over the years,probabilistic methods have been successfully applied in geotechnical engineering forsolving problems related to parameter identification [e.g., Yuen, 2010a; Cao and Wang,2013; Wang et al., 2013, 2014; Yan and Yuen, 2015] and model comparison [e.g., Yuen,

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2010b, Cao and Wang, 2014a, 2014b; Wang and Aladejare, 2015]. However, the applica-tion of robust probabilistic approaches in the modeling of the medium-strain behavior ofsoils is less mature in the literature.

The main contribution of this paper is to develop an objective procedure for construct-ing suitable G=Gmax � log γ curves for a new set of observation data in a bid to select themost suitable one from various families of pre-existing curves. First, a comprehensive setof high-amplitude RC tests was conducted and presented in the paper on coarse-grainedsands and gravelly soils. Next, a Bayesian approach was developed for parameter identi-fication and quantification of the scatterness of the observation data. In order to select themost appropriate hyperbolic model among available ones, the Bayesian approach was alsodeveloped for selecting the model that provides the best fit with the observation data.Subsequently, the best fit G=Gmax � log γ curve and a reliable confidence interval can beconstructed using the selected hyperbolic model. The proposed procedure was thendemonstrated using the observation data of coarse-grained materials obtained from theRC tests.

2. Resonant Column (RC) Tests

2.1. Materials Used

The samples used in the experimental investigation of the study comprised 15 differentgradings with soils ranging from coarse sand to fine gravel. All the samples were preparedin the laboratory from the same parent soil of sub-angular to angular and low sphericityhard particles adopting typical procedures. The parent soil is a potential fill–backfillmaterial composed of crushed rock [Senetakis and Madhusudhan, 2015]. The parentsoil was washed on sieve no. 200 (0.075 mm opening) in order to remove fine particles,oven-dried at 105°C and sieved in a series of coarse-to-fine grained sieves. Thereafter,samples of target mean grain size (denoted as D50) and coefficient of uniformity (denotedas Cu ¼ D60=D10) were created. The coefficient of curvature (denoted as

Cc ¼ D302 � D10 � D60½ ��1) for all samples had values close to unity. D50 and Cu ranged

from 1.33 to 10.1 mm and 1.03 to 12.5, respectively. Based on the USCS classificationsystem, the 15 laboratory-created samples were classified as SP, SP-SW, and GP [Senetakisand Madhusudhan, 2015].

2.2. Experimental Equipment and Sample Preparation

A fixed-free RC that can accommodate specimens of a solid cross section of about 71 mmin diameter and a ratio diameter:length equal to 1:2 was used in this study [Senetakis,2011; Senetakis and Madhusudhan, 2015]. The apparatus utilizes a system of coils thatsurround the drive mechanism which is fixed on top (active end) of the sample (Fig. 2).Two magnets and one accelerometer are embedded on the drive mechanism for theimplementation of the torsional resonant column mode and the record of sample responseon its top, respectively. A vertically positioned displacement transducer is used for therecord of the axial strain of the specimen.

Samples were prepared in a dry state into a standard split metal mold of appropriatedimensions. For relatively dense solid matrix, samples were prepared in layers of equal

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mass and compacted using light tamping. For relatively loose solid matrix, sampleswere prepared with the hand spooning method. After the preparation of the samples,the system was assembled and vacuum of about 5 kPa was applied in order to supportthe sample before the application of cell pressure. Typically, samples were subjected toincreasing steps of the mean effective confining stress, p0, equal to 25, 50, 100, and 200kPa. Note that p0 corresponded to an isotropic pressure in the study (p0 = σ

03). At each

level of p0, the sample was allowed to reach equilibrium for about 30–60 min withcheck of small-strain stiffness at dense time intervals. In order to determine small-strain stiffness, low-amplitude resonant column tests (LARCTs) in torsional mode ofvibration were performed at p0 equal to 25, 50, and 100 kPa. Thereafter, the sampleswere subjected to the first high-amplitude resonant column test (HARCT) at p0=100kPa with increasing steps of the introduced voltage into the system in order to obtainshear modulus at variable amplitudes of cyclic shear strain. After the completion ofHARCT at p0=100 kPa, samples were subjected to LARCT–HARCT at p0=200 kPafollowed by measurements of small-strain dynamic properties and HARCT at decreas-ing steps of p0.

2.3. RC Testing Program

In total, 26 specimens were prepared and tested in the RC apparatus. Values of initialdry unit weight (γdo) and void ratio (eo) and the levels of p0 at which HARCTs wereconducted are summarized in Table 1. Table 1 also summarizes the grading character-istics of the samples, the content of gravel, and the classification of the soils used inthe experimental investigation. For example, from the poor-graded sand Sample01with Cu = 2.13 and D50 = 1.33 mm, two different specimens were prepared and testedin the RC apparatus, denoted as Sample01-1 prepared at eo = 0.594 and Sample01-2

Figure 2. Close-up view of resonant column with drive mechanism attached on top of the sample (afterSenetakis [2011] and Senetakis and Madhusudhan [2015]).

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prepared at eo=0.820. Complete analysis of the RC tests was conducted by adopting theASTM D4015 [ASTM, 1992] specification. Details of the small-strain dynamic proper-ties of the samples are referred to in Senetakis and Madhusudhan [2015].

2.4. Measurement of Stiffness Degradation and Comparisons with LiteratureCurves

In this study, the normalized shear modulus, G=Gmax, is calculated from the secant shearmodulus G and maximum stiffness, Gmax, which are derived from the RC tests describedin Secs. 2.1–2.3. A typical result of the RC tests consists of two columns of data whichinclude the strain amplitude, γ, and the corresponding normalized shear modulus,G=Gmax. In addition, material damping can be computed from the RC test, but thisproperty was not investigated in the present study. Figure 3 plots all the observationdata obtained from the RC tests for all the coarse-grained materials used in this study. Theupper and lower bounds of G=Gmax � log γ curves reported by Seed and Idriss [1970],Vucetic and Dobry [1991], Rollins et al. [1998], Menq [2003] and Oztoprak and Bolton

Table 1. High-amplitude torsional resonant column testing program on coarse-grained samples.

Code name of specimen D50 (mm) Cu Cc Gravel content (%) USCSγdo

(kN/m3) e0

p0forHARCT(kPa)

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Sample01-1 1.33 2.13 1.01 0 SP 16.43 0.594 25, 100, 200Sample01-2 14.39 0.820 100, 200Sample02-1 1.33 11.8 0.68 20 SP-SW 19.34 0.354 25, 50, 100Sample02-2 17.49 0.498 100, 200Sample03-1 2.00 2.50 1.07 0 SP 16.87 0.553 25, 50, 100Sample03-2 14.80 0.770 25, 50, 100Sample04-1 2.00 5.40 0.50 21 SP 18.19 0.440 100, 200Sample04-2 16.32 0.605 100Sample05-1 2.00 7.30 0.65 25 SP-SW 18.76 0.396 50, 100Sample05-2 16.81 0.558 100, 200Sample06-1 3.07 1.53 0.90 0 SP 16.59 0.579 100, 200Sample06-2 14.27 0.835 50, 100, 200Sample07-1 3.00 2.45 1.10 15 SP 16.26 0.611 50, 100Sample07-2 15.25 0.718 100Sample08-1 3.07 4.24 1.77 20 SP 17.66 0.483 100, 200Sample09-1 2.90 5.95 1.19 30 SP-SW 17.71 0.479 50, 100, 200Sample10-1 3.00 7.85 0.68 40 SP-SW 16.45 0.592 100, 200Sample11-1 3.00 12.5 0.94 40 SP-SW 18.09 0.448 50, 100, 200Sample11-2 16.68 0.570 25, 50, 100Sample12-1 5.50 1.17 0.96 100 GP 15.41 0.700 100, 200Sample12-2 13.95 0.878 50, 100, 200Sample13-2 6.40 2.70 1.19 75 GP 15.12 0.732 50, 100Sample14-1 7.80 1.22 0.94 100 GP 16.13 0.624 50, 100Sample14-2 14.19 0.846 100, 200Sample15-1 10.1 1.03 1.00 100 GP 16.19 0.618 50, 100Sample15-2 14.03 0.867 50, 100

(2)Mean grain size.(3)Coefficient of uniformity.(4)Coefficient of curvature.(5)Percentage of coarse soil retained on No. 4 (4.75 mm) sieve.(6)USCS.(7,8)Initial dry unit weight and void ratio of samples at p′=5 kPa(9)Mean-confining effective pressures where high-amplitude RC tests were performed.SP—Poorly graded sand, SW—Well graded sand, GP—Poorly graded gravel.

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[2013] are also included in Fig. 3. As shown in Fig. 3, the observation data obtained fromthis study are scattered and do not fit perfectly with the pre-existing curves, apart from theupper and lower bounds proposed by Rollins et al. [1998] in which the curves covered theextreme bounds of the RC test data.

Some selected sample data obtained from the RC tests are plotted in Fig. 4. In Fig. 4(a),Sample15-1 at 50 and 100 kPa is indicated with asterisk and open triangle markers,respectively, while Sample15-2 at 50 and 100 kPa is indicated with plus and open squaremarkers, respectively. Theoretical G=Gmax � log γ curves proposed by Menq [2003] forgranular soils are indicated in the plots as dashed and solid lines at 50 and 100 kPa,respectively, for comparison purposes. It can be observed that the dashed and solid linesexhibit more linear behavior than the experimental data from this study. Similar plots areshown in Fig. 4(b), which corresponded to Sample11-1 and Sample11-2 with Cu = 12.5 at50 and 100 kPa. These experimental data exhibit more linear behavior than the dashed andsolid lines at 50 and 100 kPa (i.e., theoretical curves proposed by Menq [2003]). Note thatfor the theoretical curves by Menq (2003) shown in Fig. 4, appropriate values of thecoefficient of uniformity and confining pressure were used so that the model curves arecompared with the experimental results of this study at the same level of pressure andgrading characteristics. Based on the comparative results in Fig. 4, it is apparent that thereis a need to develop new G=Gmax � log γ curves for the coarse-grained materials used inthis study and differentiate it from the pre-existing families of curves. However, there is noobjective procedure for developing new G=Gmax � log γ curves in geotechnical literature.In the next section, the process of developing the Bayesian approach to model thescatterness of the observation data and identify model parameters is discussed.

3. Bayesian Identification of Model Parameters

Hyperbolic models that are expressed in Eqs. (1)–(3) represent the relationship betweenthe observed (input) and estimated (output) variables for this study. These models can begenerally represented as follows:

+Samples at 25 kPa (This study) Samples at 50 kPa (This study)

Samples at 100 kPa (This study) Samples at 200 kPa (This study)

Figure 3. Experimental data of the coarse-grained materials in this study.

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GGmax

� �i

¼ f γi;ΘkjMk� �

(4)

where GGmax

� iand γi, i ¼ 1; 2; :::; n represent the observable dependent and independent

variables, respectively, which are obtained together as input data pairs from the RC testsdiscussed in Sec. 2, f γi;ΘkjMk

� �represents the right-hand side of each hyperbolic model

Mk, k ¼ 1; 2; :::; nHB (i.e., nHB = 3 for Eqs. (1)–(3)), and Θk represents the unknownmodel parameters for each model Mk that need to be estimated based on the observed(input) data. For example, Eq. (1) is represented as M1 with model parameter Θ1 = ½γr�,Eq. (2) is represented as M2 with model parameter Θ2 = ½γr; a�, and Eq. (3) is representedas M3 with model parameter Θ3 = ½γr; a�. It is important to note that γe in Eq. (3) is anobserved parameter that is obtained “experimentally” as the strain value when G=Gmax =

(a)

(b)

Figure 4. Experimental data of coarse-grained samples fitted with analytical curve from the literaturewhen (a) Cu = 1.03 and (b) Cu = 12.5.

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0.99 as suggested by Vucetic [1994]. These hyperbolic models are imperfect and havevarying degrees of uncertainty. Note that the uncertainties associated with soil samplingand testing equipment are not considered in this study, which could be greater than thoseassociated with the development of an optimal degradation curve in geotechnical engi-neering practice. Nevertheless, the RC method, particularly in testing reconstituted mate-rials, is considered a technique providing relatively highly repeatable results [e.g.,Senetakis, 2011]. This section of the paper seeks to use the Bayesian approach to identifythe best estimate of the values of respective model parameters and also quantify the degreeof uncertainty associated with the model.

First, probabilistic models are formulated with the introduction of the model errorterm, ε, and the general probabilistic model can be expressed as follows:

GGmax

� �i

¼ f γi;ΘkjMk� �þ εi; i ¼ 1; 2; :::; n (5)

where εi is the model error term that represents uncertainties associated with eachhyperbolic model based on the observation data, and εi is considered to follow aGaussian distribution with zero mean and standard deviation σε. Thus, in this study, σεis used to quantify the scatterness of the observation data with respect to the model Mk. σεis an unknown parameter that also has to be estimated in addition to the modelparameters. Due to the introduction of εi, the unknown model parameters for M1, M2,and M3 become Θ1 = ½γr; σε�, Θ2 = ½γr; a; σε� and Θ3 = ½γr; a; σε�, respectively.

Using Bayes’ theorem, the posterior PDF of the model parameters, Θk given theobservation data, Data, and the model Mk, can be expressed as follows [e.g., Ang andTang, 2007; Yuen, 2010a; Cao and Wang, 2013; Wang et al., 2013, 2014]:

P ΘkjData;Mkð Þ ¼ K � P DatajΘk;Mkð Þ � P ΘkjMkð Þ (6)

where K ¼ ð1=P DatajMkð Þ is the normalizing constant, P DatajΘk;Mkð Þ is the likelihoodof observingData from model Mk for given values of Θk, and P ΘkjMkð Þ is the priordistribution of the model parameters for model Mk that reflect prior knowledge about Θk

in the absence of observation data.

Given pairs of Data = γi;G=Gmax

� i

h i; i¼ 1; 2 . . . ;n

n oobtained from the laboratory

investigation of the study (i.e., the RC tests in Sec. 2), the model error

εi ¼ G=Gmax

� i� f γi;ΘkjMk

� �h i; i ¼ 1; 2; :::; n, for each model is taken to be indepen-

dent realizations of the Data; therefore, the general likelihood function for each model(i.e., M1,M2 and M3) is a product of n Gaussian realizations, which is expressed as follows:

P DatajΘk;Mkð Þ ¼Yni¼1

1ffiffiffiffiffiffi2π

pσε

exp � 12σε2

G=Gmax

� i� f γi;ΘkjMk

� �h i2� �(7)

In this study, a noninformative prior distribution is adopted so that the prior isabsorbed into the normalizing constant. Due to this, the Bayesian identification of themodel parameters will depend solely on the likelihood of the observation data. Thus, Eq.(6) can be further expressed as follows:

P ΘkjData;Mkð Þ / P DatajΘk;Mkð Þ (8)

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The optimal model parameters, Θk given a set of observation data, Data and model Mk,can be obtained by maximizing the posterior PDF P ΘkjData;Mkð Þ in Eq. (6). However,based on the noninformative prior distribution and Eq. (8), maximizing the posterior PDFP ΘkjData;Mkð Þ is equivalent to maximizing P DatajΘk;Mkð Þ, or for numerical conveni-ence, minimizing � ln P DatajΘk;Mkð Þ½ � with respect to Θk. This can be solved numeri-cally by using optimization functions in commercial software, such as fmincon function inMATLAB. The model parameters are set to bound constraints by using the likely ranges ofthe model parameters as lower and upper bounds. These likely ranges are basicallyreasonable ranges of the model parameters that are commonly reported in the literature.For example, the likely ranges of γr and a based on experience and those reported in theliterature for this type of soils include [0.001%, 0.1%] for γr and [0.7, 1.3] for a [e.g.,Oztoprak and Bolton, 2013; Senetakis et al., 2013a]. Also, due to the nonnegative physicalmeaning of standard deviation, the minimum value of σε is taken to be 0. In Eq. (5), themaximum possible value of G=Gmax is 1 and ε follows a Gaussian distribution with zeromean and standard deviation σε. Therefore, for this study, the maximum value of σεistaken to be 1.

Furthermore, the Bayesian approach can also be used to quantify the uncertaintyassociated with the estimation of the optimal model parameters Θk. In order to achieve

this, this study makes use of the Hessian matrix, Hk Θk� �

, of the negative logarithm of the

likelihood function in Eq. (7) evaluated at Θk. Hk Θk� �

is calculated as the Hessian matrix

of objective function fobj ¼ � ln P DatajΘk;Mkð Þ½ � with respect to Θk evaluated at Θk.Specifically, the components of Hk for models M1, M2, and M3 are given in Eqs. (9) and(10). Since models M2 and M3 have the same number and type of model parametersi:e:;Θ2 ¼ Θ3 ¼ γr; a; σε

�� �; then H2 ¼ H3 in Eq. (10).

H1 ¼@2fobj@yr2

@2fobj@yr@σε

@2fobj@σε@yr

@2fobj@σε2

24

35 (9)

H2 ¼ H3 ¼

@2fobj@yr2

@2fobj@yr@a

@2fobj@yr@σε

@2fobj@a2

@2fobj@a@σε

sym:@2fobj@σε2

2664

3775 (10)

The components of Hk Θk� �

in Eqs. (9) and (10) can be conveniently computed numericallyusing the finite difference method, which is incorporated in some optimization functions ofcommercial software such as fmincon function inMATLAB. The covariance matrix is given as

the diagonal matrix of the inverse of the Hessian matrix i.e., Hk Θk� � ��1

, and this providesthe associated uncertainty σΘk

of themodel parameters [e.g., Wang et al., 2010; Ng et al., 2016].

4. Selection of the Most Appropriate Hyperbolic Model

The proposed Bayesian identification approach in the previous section provides the optimalvalues as well as the associated uncertainty of the model parameters of all the hyperbolicmodels considered (i.e., M1, M2, and M3). However, only one of the hyperbolic models,

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which is the most appropriate model for a set of observation data, shall be selected todevelop the normalized modulus reduction curve. Therefore, it is important to also developa method to select the most appropriate hyperbolic model among the available ones.

For a given set of observation data (i.e., Data), the plausibility of a model Mk,k = 1 ; 2; :::; nHB (i.e., nHB = 3 for Eqs. (1)–(3)) is defined by its occurrenceprobability, P MkjDatað Þ, which is conditional on the observation data (i.e., Data).The most appropriate hyperbolic model, M�

k , is determined by comparing the valuesof P MkjDatað Þ for all the candidate hyperbolic models and selecting the hyperbolicmodel, Mk, with the highest P MkjDatað Þ. P MkjDatað Þ can be obtained using Bayes’theorem, which is expressed as follows [e.g., Beck and Yuen, 2004; Yuen, 2010a; Caoand Wang, 2013, 2014a, 2014b]:

P MkjDatað Þ ¼ P DatajMkð ÞP Mkð Þ=P Datað Þ; k ¼ 1; 2; :::; nHB (11)

where P Mkð Þ is the prior probability of Mk which reflects the prior knowledge of Mk.P Datað Þ ¼ PnHB

k¼1 PðDatajMkÞPðMkÞ is a normalizing constant that is required to make Eq.(11) a proper probability function and it is independent of Mk. P DatajMkð Þ is theconditional probability of Data for a given model Mk. P DatajMkð Þ is frequently calledevidence provided by the Data and it increases with increasing plausibility of Dataconditional on Mk. In the absence of any prevailing prior information about the threeavailable models (i.e., M1, M2, and M3), M1, M2, and M3 would be taken to have the sameprior probability i.e., PðM1Þ ¼ PðM2Þ ¼ PðM3Þ½ � = 1=3. In addition, since P Datað Þ andP Mkð Þ are constants for all the models, P MkjDatað Þ is therefore proportional toP DatajMkð Þ and it is expressed as follows:

P MkjDatað Þ / P DatajMkð Þ (12)

From Eq. (12), selecting the model with the highest plausibility P MkjDatað Þ is thereforeequivalent to selecting the model with the highest value of P DatajMkð Þ. In other words,M�

k can be selected by comparing the values of P DatajMkð Þ and the hyperbolic model thathas the highest value of P DatajMkð Þ is taken as M�

k . The evidence P DatajMkð Þ can beapproximated as follows [e.g., Papadimitriou et al., 1997; Beck and Katafygiotis, 1998;Yuen, 2010a; Cao and Wang, 2013]:

P DatajMkð Þ � P DatajΘk;Mk� �

P ΘkjMk� �

2πð Þnk=2 Hk Θk� ��� ���1=2; k ¼ 1; 2 . . . nHB (13)

where Θk is the optimal value of the model parameters forMk, which is determined in Sec. 3;

P DatajΘk;Mk� �

is the likelihood function of Mk estimated at Θk and is calculated using the

general likelihood function in Eq. (7); P ΘkjMk� �

2πð Þnk=2 Hk Θk� ��� ���1=2 is called the “Ockham

factor,” which can be interpreted as a measure of robustness of Mk and represents a penaltyagainst complicated parameterization [e.g., Gull, 1988; Beck and Yuen; 2004; Yuen, 2010a]; nkis the number of uncertain model parameters ofMk; P ΘkjMk

� �is the prior distribution of Θk

evaluated at Θk. For this study, the respective prior distributions P ΘkjMkð Þ for M1, M2, andM3 are simply taken as joint uniform distribution (i.e., noninformative) of model parametersand they are given in Eqs. (14) and (15), respectively. Note that since ModelsM2 andM3 havethe same number and type of parameters (i.e., Θ2 = Θ3= ½γr; a; σε�), then P Θ2jM2ð Þ =P Θ3jM3ð Þ in Eq. (15).

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P Θ1jM1ð Þ ¼ P γr; σε� � ¼

1γr;max�γr;min

� 1σε;max�σεmin

0

for γr 2 γr;min; γr;max

h i;

σε 2 σε;min; σε;max �

for others

8>>><>>>:

(14)

P Θ2jM2ð Þ ¼ P Θ3jM3ð Þ ¼ P γr; a; σε� � ¼

1γr;max�γr;min

� 1amax�amin

� 1σε;max�σε ;min

0

for γr 2 γr;min; γr;max

h i; a 2 amin; amax½ �;

σε 2 σε;min; σε;max �

for others

8>>><>>>:

(15)

In Eqs. (14) and (15), γr, a, and σε have respective minimum values of γr;min, amin, andσε;min, and respective maximum values of γr;max, amax and σε;max. Note that only the

possible ranges of the model parameters are required to completely define the jointuniform distributions given in Eqs. (14) and (15). The likely ranges of γr, a, and σε stated

in Sec. 3 are taken as their respective prior knowledge in this study. Hk Θk� ��� �� in Eq. (13) is

the determinant of the Hessian matrix of Mk evaluated at Θk. Hk Θk� �

can be calculatedusing Eq. (9) or (10) depending on the type of hyperbolic model Mk.

5. Constructing the Best Fitted G=Gmax � log γ Curve and Its ConfidenceInterval

The development of the best fitted G=Gmax � log γ curve and its confidence interval isbased on the most appropriate hyperbolic model Mk

� evaluated in Sec. 4. The bestfitted G=Gmax � log γ curve and its confidence interval can be developed using theMonte Carlo simulation. The posterior PDFP ΘkjData;Mkð Þ in Eq. (6) can be used togenerate numeric samples of the model parameters for the quantification of themodeling uncertainty. In this study, a large sequence of m samples of the modelparameters Θk is efficiently simulated by the Markov chain Monte Carlo (MCMC)simulation using the posterior PDF P ΘkjData;Mkð Þ in Eq. (6) as the limiting stationarydistribution [e.g., Metropolis et al. 1953; Hastings, 1970; Beck and Au, 2002; Yan andYuen, 2015; Wang et al., 2016; Wang and Akeju, 2016]. The Θk Markov chain startswith an initial state, Θk;1, which is taken to represent the optimal values of the model

parameters Θk evaluated in Sec. 3 depending on Mk�. Note that the use of Θk as the

initial state Θk;1 is to make the MCMC simulation converge quickly to the stationarydistribution in Eq. (6). Subsequently, the candidate sample, Θk;m

� for the mth state(m = 2, 3. . .nmcmc) is generated from a proposal PDF, which is taken as multivariateGaussian PDF, f Θk;m

�jΘk;m�1� �

centered at the previous state, Θk;m�1 depending on thetype of Mk

�. The coefficient of variation of Θk in the proposal PDF is taken as the ratio

of the evaluated uncertainty associated with Θk(i.e., σΘk) to their respective optimal

values Θk (both are evaluated in Sec. 3).

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The chance to accept the candidate sample, Θk;m�, for the mth state depends on the

acceptance ratio, AR, which is evaluated as follows:

AR ¼ P DatajΘk;m�;Mk

� �P DatajΘk;m�1;Mk� �� f Θk;m�1jΘk;m

�� �f Θk;m

�jΘk;m�1� � form ¼ 2; 3; . . . ; nmcmc (16)

where P DatajΘk;m�;Mk

� �and P DatajΘk;m�1;Mk

� �are respective likelihood values of

Θk;m� and Θk;m�1, which can be calculated from Eq. (7); f Θk;m�1jΘk;m

�� �is the joint

conditional PDF of Θk;m�1 given Θk;m�; f Θk;m

�jΘk;m�1� �

is the joint conditional PDF of

Θk;m� given Θk;m�1; f Θk;m�1jΘk;m

�� �and f Θk;m

�jΘk;m�1� �

are both calculated from themultivariate Gaussian PDF with respective mean values of Θk;m

� and depending onMk

� If the calculated AR (i.e., Eq. (16)) is greater than a random number u that isgenerated from a uniform distribution between 0 and 1, then the candidate sampleΘk;m

� is accepted as the mth state of the Θk Markov chain. However, if AR is less thanu, Θk;m

� is rejected and the previous state Θk;m�1 is set as the mth state of the Θk

Markov chain. This procedure is repeated m� 1 times to generate m� 1 samples of

Θk. Together with the initial sample Θk;l, this results in a Markov chain of m samplesof the model parameter Θk.

These numeric samples of Θk generated from the MCMC simulation can be used toobtain the best fitted G=Gmax � log γ curve and its confidence interval at any prescribedpercentile level. In order to achieve this, the m samples of the model parameter Θk areused directly in the probabilistic model (i.e., Eq. (5)) to obtain a large number of mG=Gmax samples for different γ values. The model error term, εi, in Eq. (5) can besimulated using a Gaussian generator with zero mean and standard deviation σε(i.e., theoptimal value of σε obtained from Sec. 3). Subsequently, for each of the γ valuesconsidered, the G=Gmax samples are sorted to identify the corresponding G=Gmax valuesat 50th percentile and the boundary’s desired percentile levels. The percentile values arethen connected at different γ values for each of the percentiles level. The 50th percentilecurve represents the best fitted G=Gmax � log γ curve, while the specified boundary’spercentile (e.g., 2.5th and 97.5th) curves represent the desired confidence interval (e.g.,95% confidence interval) of the best fitted G=Gmax � log γ curve. An important aspect ofthis study is developing not only the best fitted lines of the G=Gmax � log γ curves but alsothe desired confidence interval. The handling of uncertainty for the empirical estimates ofdynamic soil properties in this study will be useful in areas such as probabilistic seismichazard analysis.

6. Implementation Procedure

Figure 5 shows a flowchart for the implementation of the Bayesian approach used in thisstudy. In general, the implementation procedure can be briefly summarized into six stepswhich are listed as follows:

(a) Obtain n pairs of γi and G=Gmaxð Þi data from laboratory tests such as RC tests {i.e.,[γi, G=Gmaxð Þi], i = 1, 2. . . n}.

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Figure 5. Flowchart for the implementation of the approach developed in this study.

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(b) Perform identification of model parameters by minimizing � ln P DatajΘk;Mkð Þ½ �and determine the optimal model parameters Θk and their associated uncertainties,σΘk

.(c) Calculate the evidence P DatajMkð Þ for each model Mk using Eq. (13) and select the

most appropriate hyperbolic model, M�k .

(d) Generate m samples of model parameters Θk(i.e., Θk;m) by the MCMC simulationusing the posterior PDF P ΘkjData;Mkð Þ in Eq. (6) as the limiting stationarydistribution.

(e) Use the m samples of Θk;m to obtain m samples of G=Gmax for different γ values inEq. (5) and evaluate the corresponding G=Gmax values at 50th percentile andboundary’s desired percentile levels.

(f) Connect the G=Gmax values of the 50th percentile and boundary’s desired percentilelevels separately to obtain the best fitted and confidence interval G=Gmax � log γcurves, respectively.

7. Probabilistic Analysis of the RC Test Results

7.1. Bayesian Parameter Identification Results

The Bayesian parameter identification approach proposed in Sec. 3 is used to estimate theoptimal values (i.e., Θ1 = ½γr; σε�, Θ2 = ½γr; a; σε� and Θ3 = ½γr; a; σε�) and their associateduncertainty (i.e., σΘk

) for models M1, M2, and M3, respectively, using all the 59 data sets

obtained from the RC tests in Sec. 2. Tables 2(a) and 2(b), 3(a) and 3(b), and 4(a) and 4(b)show the estimated optimal values and associated uncertainty of the model parameters formodels M1, M2, and M3, respectively, using all the data presented in this study.

For comparison and validation purposes, the Bayesian identification methodproposed in this study is applied to the observation data (i.e., sandy specimens)presented in Senetakis et al. [2013a]. Tables 5(a) and 5(b) present the resultsobtained from the analysis on representative data previously published by Senetakiset al. [2013a]. Subsequently, the results are compared with the results obtained inSenetakis et al. [2013a] using a traditional approach. Note that unlike in Senetakiset al. [2013a], the method proposed in this study not only evaluates the optimalvalues of the model parameters but also provides associated parameter uncertaintyand model uncertainty to quantify data scatterness. Based on this, only the optimalvalues of model parameters (using Model M2,) obtained in Senetakis et al. [2013a]are compared with those obtained using the method proposed in this study as shownin Tables 6 and 7 for γr and a, respectively. From Table 6, the relative difference ofthe values of γr obtained from both methods ranges from 2.67 to 77.27%, while forthe optimal values of a in Table 7, the relative difference ranges from 0.54 to 19.43%.In addition, the best fitted G=Gmax � log γ curves obtained by directly using theoptimal values of the model parameters from the proposed Bayesian approach andthose from Senetakis et al. [2013a] are shown in Fig. 6 for Sample N1-1 at 200 kPa.Figure 6(a) shows the full view of the two G=Gmax � log γ curves where the shearstrain ranges from 0.0001 to 1%, while Fig. 6(b) shows a snippet view of thecomparison with the shear strain ranging from 0.0001 and 0.1%. Similarly, Fig. 7

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Table 2. Optimal values and associated uncertainty of model parameters (γr and σε) using Model M1 forthe coarse-grained material data.(a) When p0 = 25 and 50 kPa

Code name of specimen

p0 = 25 kPa p0 = 50 kPa

γr(%) σγr (%) σε σσε γr(%) σγr (%) σε σσεSample01-1 0.0176 0.0005 0.0125 0.0027 −NA–Sample01-2 −NA– −NA–Sample02-1 0.0147 0.0004 0.0135 0.0026 0.0207 0.0006 0.0159 0.0029Sample02-2 −NA– −NA–Sample03-1 0.0182 0.0002 0.0081 0.0015 0.0231 0.0005 0.0061 0.0012Sample03-2 0.0228 0.0007 0.0203 0.0036 0.0316 0.0005 0.0079 0.0015Sample04-1 −NA– −NA–Sample04-2 −NA– −NA–Sample05-1 −NA– 0.0199 0.0006 0.0142 0.0026Sample05-2 −NA– −NA–Sample06-1 −NA– −NA–Sample06-2 −NA– 0.0283 0.0008 0.0113 0.0024Sample07-1 −NA– 0.0261 0.0010 0.0084 0.0018Sample07-2 −NA– −NA–Sample08-1 −NA– −NA–Sample09-1 −NA– 0.0215 0.0003 0.0072 0.0014Sample10-1 −NA– −NA–Sample11-1 −NA– 0.0182 0.0005 0.0110 0.0021Sample11-2 0.0227 0.0007 0.0200 0.0036 0.0316 0.0006 0.0090 0.0017Sample12-1 −NA– −NA–Sample12-2 −NA– 0.0296 0.0005 0.0077 0.0015Sample13-2 −NA– 0.0195 0.0005 0.0126 0.0026Sample14-1 −NA– 0.0219 0.0006 0.0094 0.0019Sample14-2 −NA– −NA–Sample15-1 −NA– 0.0220 0.0004 0.0067 0.0013Sample15-2 −NA– 0.0259 0.0007 0.0104 0.0020

(b) When p0 = 100 and 200 kPa

Code name of specimen

p0 = 100 kPa p0 = 200 kPa

γr(%) σγr (%) σε σσε γr(%) σγr (%) σε σσεSample01-1 0.0428 0.0012 0.0074 0.0016 0.0606 0.0015 0.0052 0.0012Sample01-2 0.0489 0.0008 0.0057 0.0011 0.0578 0.0020 0.0090 0.0020Sample02-1 0.0258 0.0006 0.0102 0.0019 −NA–Sample02-2 0.0257 0.0006 0.0103 0.0022 0.0343 0.0007 0.0069 0.0013Sample03-1 0.0381 0.0007 0.0047 0.0009 −NA–Sample03-2 0.0399 0.0009 0.0116 0.0022 −NA–Sample04-1 0.0283 0.0006 0.0083 0.0015 0.0409 0.0004 0.0033 0.0006Sample04-2 0.0296 0.0006 0.0098 0.0019 −NA–Sample05-1 0.0278 0.0006 0.0091 0.0017 −NA–Sample05-2 0.0264 0.0007 0.0105 0.0021 0.0235 0.0014 0.0143 0.0035Sample06-1 0.0383 0.0013 0.0087 0.0018 0.0535 0.0009 0.0039 0.0008Sample06-2 0.0420 0.0010 0.0071 0.0015 0.0604 0.0016 0.0063 0.0013Sample07-1 0.0353 0.0006 0.0054 0.0010 −NA–Sample07-2 0.0416 0.0010 0.0068 0.0013 −NA–Sample08-1 0.0233 0.0010 0.0149 0.0030 0.0286 0.0010 0.0075 0.0017Sample09-1 0.0301 0.0009 0.0123 0.0019 0.0464 0.0016 0.0093 0.0017Sample10-1 0.0287 0.0008 0.0092 0.0021 0.0312 0.0006 0.0058 0.0012Sample11-1 0.0275 0.0005 0.0059 0.0011 0.0372 0.0006 0.0035 0.0006Sample11-2 0.0399 0.0009 0.0116 0.0022 −NA–Sample12-1 0.0367 0.0009 0.0085 0.0016 0.0464 0.0006 0.0035 0.0006Sample12-2 0.0373 0.0005 0.0056 0.0010 0.0487 0.0006 0.0046 0.0008Sample13-2 0.0310 0.0006 0.0065 0.0013 −NA–Sample14-1 0.0310 0.0007 0.0077 0.0016 −NA–Sample14-2 0.0356 0.0007 0.0095 0.0017 0.0534 0.0010 0.0066 0.0012Sample15-1 0.0318 0.0003 0.0043 0.0008 −NA–Sample15-2 0.0315 0.0010 0.0090 0.0019 −NA–

NA—Not available

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Table 3. Optimal values and associated uncertainty of model parameters (γr , a, and σε) using Model M2,for the coarse-grained material data.(a) When p0 = 25 and 50 kPa

Code name ofspecimen

p0 = 25 kPa p0 = 50 kPa

γr(%) σγr (%) a σa σε σσε γr(%) σγr (%) a σa σε σσεSample01-1 0.0180 0.0009 0.9828 0.0391 0.0124 0.0026 −NA–Sample01-2 −NA– −NA–Sample02-1 0.0150 0.0005 0.9725 0.0307 0.0132 0.0025 0.0187 0.0008 1.1048 0.0424 0.0133 0.0024Sample02-2 −NA– −NA–Sample03-1 0.0182 0.0003 0.9996 0.0187 0.0081 0.0015 0.0235 0.0016 0.9893 0.0361 0.0061 0.0012Sample03-2 0.0232 0.0009 0.9604 0.0332 0.0195 0.0033 0.0319 0.0009 0.9894 0.0223 0.0078 0.0015Sample04-1 −NA– −NA–Sample04-2 −NA– −NA–Sample05-1 −NA– 0.0219 0.0010 0.9144 0.0298 0.0115 0.0021Sample05-2 −NA– −NA–Sample06-1 −NA– −NA–Sample06-2 −NA– 0.0265 0.0017 1.0522 0.0486 0.0108 0.0023Sample07-1 −NA– 0.0179 0.0012 1.2184 0.0486 0.0047 0.0010Sample07-2 −NA– −NA–Sample08-1 −NA– −NA–Sample09-1 −NA– 0.0218 0.0006 0.9896 0.0220 0.0072 0.0014Sample10-1 −NA– −NA–Sample11-1 −NA– 0.0206 0.0016 0.9251 0.0392 0.0098 0.0019Sample11-2 0.0233 0.0008 0.9532 0.0324 0.0188 0.0033 0.0317 0.0009 0.9968 0.0243 0.0090 0.0018Sample12-1 −NA– −NA–Sample12-2 −NA– 0.0310 0.0008 0.9552 0.0191 0.0065 0.0012Sample13-2 −NA– 0.0176 0.0006 1.1038 0.0344 0.0093 0.0019Sample14-1 −NA– 0.0217 0.0016 1.0045 0.0463 0.0094 0.0019Sample14-2 −NA– −NA–Sample15-1 −NA– 0.0224 0.0007 0.9861 0.0209 0.0066 0.0012Sample15-2 −NA– 0.0229 0.0012 1.0916 0.0392 0.0086 0.0017

(b) When p0 = 100 and 200 kPa

Code name ofspecimen

p0 = 100 kPa p0 = 200 kPa

γr(%) σγr (%) a σa σε σσε γr(%) σγr (%) a σa σε σσεSample01-1 0.0438 0.0034 0.9871 0.0418 0.0074 0.0016 0.0477 0.0023 1.1290 0.0287 0.0029 0.0007Sample01-2 0.0506 0.0022 0.9779 0.0254 0.0055 0.0011 0.0405 0.0013 1.2279 0.0249 0.0029 0.0006Sample02-1 0.0226 0.0004 1.1247 0.0190 0.0048 0.0009 −NA–Sample02-2 0.0226 0.0005 1.1245 0.0212 0.0048 0.0010 0.0297 0.0007 1.1066 0.0178 0.0034 0.0007Sample03-1 0.0324 0.0014 1.0905 0.0255 0.0033 0.0006 −NA–Sample03-2 0.0384 0.0014 1.0382 0.0316 0.0111 0.0019 −NA–Sample04-1 0.0258 0.0006 1.0775 0.0206 0.0060 0.0010 0.0404 0.0012 1.0066 0.0170 0.0033 0.0006Sample04-2 0.0269 0.0010 1.0825 0.0309 0.0078 0.0015 −NA–Sample05-1 0.0263 0.0011 1.0456 0.0304 0.0085 0.0015 −NA–Sample05-2 0.0229 0.0006 1.1233 0.0257 0.0061 0.0013 0.0205 0.0030 1.0803 0.0907 0.0137 0.0032Sample06-1 0.0270 0.0014 1.2097 0.0374 0.0043 0.0009 0.0469 0.0024 1.0670 0.0269 0.0031 0.0007Sample06-2 0.0355 0.0012 1.1157 0.0253 0.0040 0.0009 0.0456 0.0016 1.1591 0.0238 0.0028 0.0006Sample07-1 0.0307 0.0009 1.0831 0.0190 0.0037 0.0006 −NA–Sample07-2 0.0430 0.0035 0.9825 0.0405 0.0068 0.0013 −NA–Sample08-1 0.0163 0.0005 1.2826 0.0324 0.0047 0.0010 0.0211 0.0017 1.1661 0.0509 0.0051 0.0012Sample09-1 0.0250 0.0009 1.1611 0.0356 0.0079 0.0014 0.0307 0.0005 1.2652 0.0136 0.0017 0.0003Sample10-1 0.0247 0.0011 1.1126 0.0346 0.0063 0.0014 0.0319 0.0017 0.9865 0.0282 0.0058 0.0012Sample11-1 0.0244 0.0012 1.0692 0.0299 0.0050 0.0009 0.0296 0.0011 1.1086 0.0197 0.0020 0.0004Sample11-2 0.0384 0.0014 1.0382 0.0316 0.0111 0.0019 −NA–Sample12-1 0.0356 0.0022 1.0217 0.0392 0.0084 0.0016 0.0428 0.0013 1.0485 0.0181 0.0028 0.0005Sample12-2 0.0385 0.0009 0.9733 0.0162 0.0052 0.0010 0.0531 0.0016 0.9456 0.0163 0.0035 0.0006Sample13-2 0.0274 0.0010 1.0831 0.0263 0.0047 0.0009 −NA–Sample14-1 0.0275 0.0015 1.0802 0.0370 0.0064 0.0013 −NA–Sample14-2 0.0353 0.0014 1.0069 0.0301 0.0095 0.0019 0.0585 0.0033 0.9469 0.0289 0.0060 0.0011Sample15-1 0.0301 0.0007 1.0386 0.0169 0.0037 0.0007 −NA–Sample15-2 0.0285 0.0024 1.0595 0.0510 0.0085 0.0018 −NA–

NA—Not available

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Table 4. Optimal values and associated uncertainty of model parameters (γr , a, and σε) using Model M3,for the coarse-grained material data.(a) When p0 = 25 and 50 kPa

Code name ofspecimen

p0 = 25 kPa p0 = 50 kPa

γr(%) σγr (%) a σa σε σσε γr(%) σγr (%) a σa σε σσεSample01-1 0.0200 0.0002 0.8034 0.0047 0.0015 0.0003 −NA–Sample01-2 −NA– −NA–Sample02-1 0.0158 0.0004 0.8035 0.0165 0.0073 0.0014 0.0195 0.0012 0.9648 0.0510 0.0160 0.0030Sample02-2 −NA– −NA–Sample03-1 0.0182 0.0003 0.9154 0.0162 0.0072 0.0013 0.0367 0.0031 0.7401 0.0249 0.0043 0.0009Sample03-2 0.0229 0.0006 0.8205 0.0202 0.0118 0.0020 0.0338 0.0009 0.8378 0.0174 0.0066 0.0013Sample04-1 −NA– −NA–Sample04-2 −NA– −NA–Sample05-1 −NA– 0.0243 0.0007 0.7760 0.0147 0.0059 0.0011Sample05-2 −NA– −NA–Sample06-1 −NA– −NA–Sample06-2 −NA– 0.0323 0.0019 0.8350 0.0287 0.0065 0.0014Sample07-1 −NA– 0.0297 0.0038 0.8525 0.0454 0.0047 0.0010Sample07-2 −NA– −NA–Sample08-1 −NA– −NA–Sample09-1 −NA– 0.0234 0.0010 0.8678 0.0265 0.0078 0.0014Sample10-1 −NA– −NA–Sample11-1 −NA– 0.0295 0.0018 0.7014 0.0186 0.0049 0.0009Sample11-2 0.0229 0.0006 0.8205 0.0210 0.0121 0.0022 0.0338 0.0009 0.8378 0.0182 0.0064 0.0012Sample12-1 −NA– −NA–Sample12-2 −NA– 0.0334 0.0011 0.8291 0.0205 0.0061 0.0012Sample13-2 −NA– 0.0189 0.0009 0.9252 0.0336 0.0093 0.0019Sample14-1 −NA– 0.0288 0.0020 0.7874 0.0275 0.0058 0.0012Sample14-2 −NA– −NA–Sample15-1 −NA– 0.0250 0.0010 0.8295 0.0220 0.0065 0.0012Sample15-2 −NA– 0.0294 0.0017 0.8214 0.0253 0.0062 0.0012

(b) When p0 = 100 and 200 kPa

Code name ofspecimen

p0 = 100 kPa p0 = 200 kPa

γr(%) σγr (%) a σa σε σσε γr(%) σγr (%) a σa σε σσεSample01-1 0.0644 0.0032 0.7526 0.0157 0.0029 0.0006 0.0713 0.0073 0.8540 0.0355 0.0039 0.0009Sample01-2 0.0633 0.0038 0.8028 0.0249 0.0055 0.0011 0.0548 0.0053 0.9264 0.0443 0.0056 0.0012Sample02-1 0.0242 0.0007 0.9679 0.0219 0.0059 0.0011 −NA–Sample02-2 0.0243 0.0011 0.9461 0.0324 0.0073 0.0016 0.0352 0.0021 0.8996 0.0331 0.0066 0.0013Sample03-1 0.0488 0.0052 0.8222 0.0365 0.0050 0.0010 −NA–Sample03-2 0.0416 0.0011 0.8638 0.0159 0.0055 0.0010 −NA–Sample04-1 0.0275 0.0010 0.9445 0.0228 0.0064 0.0011 0.0471 0.0024 0.8743 0.0229 0.0045 0.0009Sample04-2 0.0302 0.0010 0.8995 0.0202 0.0051 0.0010 −NA–Sample05-1 0.0294 0.0020 0.8780 0.0371 0.0104 0.0019 −NA–Sample05-2 0.0260 0.0011 0.9030 0.0257 0.0061 0.0013 0.0361 0.2289 0.7000 1.8493 0.0052 0.0555Sample06-1 0.0447 0.0052 0.8243 0.0396 0.0050 0.0010 0.0716 0.0079 0.8148 0.0347 0.0043 0.0009Sample06-2 0.0438 0.0027 0.8951 0.0306 0.0052 0.0011 0.0702 0.0094 0.8436 0.0472 0.0058 0.0012Sample07-1 0.0429 0.0032 0.8223 0.0273 0.0055 0.0010 −NA–Sample07-2 0.0679 0.0077 0.7436 0.0339 0.0058 0.0011 −NA–Sample08-1 0.0193 0.0018 1.0004 0.0586 0.0092 0.0019 0.0322 0.0029 0.8655 0.0313 0.0036 0.0008Sample09-1 0.0279 0.0007 0.9769 0.0176 0.0041 0.0007 0.0363 0.0024 1.0467 0.0376 0.0048 0.0009Sample10-1 0.0304 0.0017 0.8724 0.0278 0.0053 0.0011 0.0437 0.0043 0.7838 0.0339 0.0060 0.0013Sample11-1 0.0352 0.0024 0.8141 0.0246 0.0042 0.0008 0.0553 0.0097 0.7863 0.0466 0.0051 0.0009Sample11-2 0.0416 0.0011 0.8638 0.0159 0.0055 0.0010 −NA–Sample12-1 0.0471 0.0024 0.7875 0.0205 0.0047 0.0009 0.0540 0.0053 0.8593 0.0406 0.0066 0.0012Sample12-2 0.0415 0.0012 0.8590 0.0174 0.0055 0.0010 0.0632 0.0026 0.8205 0.0176 0.0038 0.0007Sample13-2 0.0333 0.0018 0.8859 0.0259 0.0047 0.0010 −NA–Sample14-1 0.0341 0.0021 0.8705 0.0275 0.0052 0.0011 −NA–Sample14-2 0.0387 0.0010 0.8651 0.0156 0.0048 0.0009 0.0734 0.0036 0.7982 0.0179 0.0037 0.0007Sample15-1 0.0342 0.0018 0.9007 0.0277 0.0060 0.0012 −NA–Sample15-2 0.0465 0.0038 0.7436 0.0251 0.0045 0.0009 −NA–

NA—Not available

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illustrates the relative differences between the two G=Gmax � log γ curves, which arepresented in Fig. 6. Other samples were also compared, but the results are notpresented in this paper because of page limit. The significant relative differences inthe optimal values of the model parameters (i.e., γr in Table 6 and a in Table 7) as

Table 5. Optimal values and associated uncertainty of model parameters (γr , a, and σε) for sandyspecimen data in the work by Senetakis et al. [2013a].(a) When p0 = 25 and 50 kPa

Code name ofspecimen

p0 = 25 kPa p0 = 50 kPa

γr(%) σγr (%) a σa σε σσε γr(%) σγr (%) a σa σε σσεSample N1-1 −NA– 0.0584 0.0039 0.9680 0.0326 0.0058 0.0011Sample N2-3 −NA– 0.0521 0.0016 0.9832 0.0197 0.0052 0.0009Sample Q1-1 −NA– 0.0332 0.0016 0.9571 0.0301 0.0080 0.0016Sample Q2-1 0.0214 0.0009 0.9711 0.0311 0.0115 0.0022 −NA–Sample Q3-1 0.0180 0.0009 0.9828 0.0391 0.0124 0.0026 −NA–

(b) When p0 = 100 and 200 kPa

Code name ofspecimen

p0 = 100 kPa p0 = 200 kPa

γr(%) σγr (%) a σa σε σσε γr(%) σγr (%) a σa σε σσεSample N1-1 0.0800 0.0046 1.0026 0.0301 0.0052 0.0009 0.0677 0.0059 1.1791 0.0404 0.0018 0.0004Sample N2-3 0.0429 0.0024 1.0630 0.0354 0.0073 0.0014 0.0954 0.0049 0.9786 0.0235 0.0036 0.0006Sample Q1-1 0.0670 0.0028 0.9954 0.0245 0.0052 0.0010 0.0865 0.0037 1.0555 0.0215 0.0025 0.0005Sample Q2-1 0.0488 0.0023 0.9553 0.0237 0.0047 0.0008 0.0488 0.0014 1.0985 0.0188 0.0026 0.0005Sample Q3-1 0.0438 0.0034 0.9871 0.0418 0.0074 0.0016 0.0477 0.0023 1.1290 0.0287 0.0029 0.0007

NA—Not available

Table 6. Comparison of the values of γr obtained from the proposed Bayesian approach and traditionalapproach in the work by Senetakis et al. [2013a].

Code name ofspecimen

p0 = 25 kPa p0 = 50 kPa p0 = 100 kPa p0 = 200 kPa

γr(BA)(%)

γr (TA)(%)

Rel.Diff.(%)

γr(BA)(%)

γr(TA)(%)

Rel.Diff.(%)

γr(BA)(%)

γr(TA)(%)

Rel.Diff.(%)

γr(BA)(%)

γr (TA)(%)

Rel.Diff. (%)

Sample N1-1 −NA– 0.0584 0.0620 6.16 0.0800 0.0910 13.69 0.0677 0.1200 77.27Sample N2-3 −NA– 0.0521 0.0560 7.48 0.0429 0.0525 22.27 0.0954 0.1100 15.34Sample Q1-1 −NA– 0.0332 0.0370 11.56 0.0670 0.0690 2.94 0.0865 0.0940 8.67Sample Q2-1 0.0214 0.0220 2.67 −NA– 0.0488 0.0435 10.83 0.0488 0.0570 16.85Sample Q3-1 0.0180 0.0185 2.94 −NA– 0.0438 0.0450 2.85 0.0477 0.0600 25.83

BA—Bayesian Approach developed in this study, TA—Traditional Approach used in Senetakis et al. [2013a], NA—Notavailable.

Table 7. Comparison of the values of a obtained from the proposed Bayesian approach and traditionalapproach in the work by Senetakis et al. [2013a].

Code name ofspecimen

p0 = 25 kPa p0 = 50 kPa p0 = 100 kPa p0 = 200 kPa

a (BA) a (TA)

Rel.Diff.(%) a (BA) a (TA)

Rel.Diff.(%) a (BA) a (TA)

Rel.Diff.(%) a (BA) a (TA)

Rel.Diff.(%)

Sample N1-1 −NA– 0.9680 0.9500 1.86 1.0026 0.9500 5.24 1.1791 0.9500 19.43Sample N2-3 −NA– 0.9832 0.9700 1.35 1.0630 0.9700 8.75 0.9786 0.9500 2.92Sample Q1-1 −NA– 0.9571 0.9200 3.88 0.9954 0.9900 0.54 1.0555 1.0300 2.41Sample Q2-1 0.9711 0.9500 2.18 −NA– 0.9553 1.0200 6.77 1.0985 1.0200 7.14Sample Q3-1 0.9828 1.0000 1.75 −NA– 0.9871 1.0000 1.31 1.1290 1.0200 9.66

BA—Bayesian Approach developed in this study, TA—Traditional Approach used in Senetakis et al. [2013a], NA—Notavailable.

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well as in Figs. 6 and 7 reflect the importance of using an objective method such asthe Bayesian approach proposed in this study for parameter identification. In addi-tion, the evaluation of model uncertainty (i.e., σε) and associated parameter uncer-tainty (i.e., σΘk

) by the proposed approach has provided a platform through which

the scatterness of observation data obtained from RC tests can be objectively andexplicitly quantified. The quantification of the scatterness of observation data isuseful in probabilistic site response analysis for the safe design of geotechnicalstructures involving dynamic problems and seismic hazard studies.

(a)

(b)

Figure 6. Comparison of best fitted curves obtained from the Bayesian approach and Senetakis et al.[2013a] for Sample N1-1 at 200 kPa. (a) Full view and (b) snippet view.

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7.2. Model Selection Results

The model selection method discussed in Sec. 4 is applied to select the most appropriatehyperbolic model among three candidate hyperbolic models (i.e., M1, M2, and M3) usingthe data of the coarse-grained materials presented in Sec. 2 of this study (Fig. 3). Thoughthe Bayesian parameter identification method has been applied to obtain the optimalvalues and associated uncertainty of model parameters based on M1, M2, and M3, for allthe 59 sets of data obtained from the RC tests in Sec. 6.1, only selected sets of data will beused to illustrate the model selection method. The three sets of data include resultsobtained from specimen Sample03-1 at 25, 50, and 100 kPa. Sample03-1 is selected

(a)

(b)

Figure 7. Relative differences between the best fitted curves obtained from the Bayesian approach andSenetakis et al. [2013a] for Sample N1-1 at 200 kPa. (a) Full view and (b) snippet view.

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because it is one of the few samples tested at the three initial mean-confining pressures(i.e., 25, 50, and 100 kPa). Table 8 shows the three selected sets of data, which includeresults obtained from the RC test for Sample03-1 at 25, 50, and 100 kPa. The values of γefor each data set that are indicated in the bottom row of Table 7 were obtained “experi-mentally,” where G=Gmax= 0.99 as suggested by Vucetic [1994].

By using each of the data sets presented in Table 8 and the corresponding optimal valuesof the model parameters obtained in Sec. 6.1, the respective values of the logarithm ofevidence ln P DatajMkð Þ½ � for models M1, M2, and M3 are calculated using Eq. (11). Thehyperbolic model with the highest value of ln P DatajMkð Þ½ � is taken as the most appropriatehyperbolic model. From Table 9, the highest value of ln P DatajMkð Þ½ � at 25 kPa is obtainedby using M1 (i.e., 36.96) for Sample03-1. Also, at 50 kPa, the highest value ofln P DatajMkð Þ½ � is obtained by using M3 (i.e., 35.91), while by using M2 (i.e., 46.60), thehighest value of ln P DatajMkð Þ½ � is obtained at 100 kPa for Sample03-1. This indicates thatfor Sample03-1, M1 is the most appropriate model at 25 kPa, M3 is the most appropriatemodel at 50 kPa, and M2 is the most appropriate model at 100 kPa. Table 10 shows asummary of the selected hyperbolic model for all the 59 data sets obtained from the RCtests. Model M1 was selected for 12 data sets, M2 was selected for 16 data sets, and M3 wasselected for 31 data sets. From Table 10, it can be seen that for most samples, different typesof models were selected by the proposed approach at different mean-confining pressures(e.g., Sample03-1 at 25, 50 and 100 kPa). This shows the importance of applying an objectivemodel selection method to choose the most appropriate model from available ones.

Table 8. Three sets of γ and G=Gmaxdata obtained from resonant column test.Sample03-1(25 kPa)

Sample03-1(50 kPa)

Sample03-1(100 kPa)

γ(%) G=Gmax γ(%) G=Gmax γ(%) G=Gmax

1 0.00025 1.00000 0.00025 1.00000 0.00032 1.000002 0.00036 0.99346 0.00033 0.99444 0.00039 0.995753 0.00053 0.98046 0.00045 0.98614 0.00048 0.992934 0.00076 0.96753 0.00060 0.97925 0.00057 0.992935 0.00110 0.94513 0.00078 0.96964 0.00071 0.988696 0.00159 0.91670 0.00104 0.95463 0.00095 0.980257 0.00220 0.88717 0.00139 0.93974 0.00130 0.970468 0.00288 0.85661 0.00194 0.91696 0.00180 0.957939 0.00385 0.81768 0.00271 0.88789 0.00229 0.9441110 0.00585 0.75244 0.00375 0.85672 0.00300 0.9290211 0.00869 0.68177 0.00475 0.82737 0.00402 0.9032412 0.01296 0.59676 0.00606 0.80101 0.00528 0.8751713 0.01860 0.49875 NA NA 0.00659 0.8527814 0.02213 0.44263 NA NA 0.00815 0.82037

For Sample03-1 at 25 kPa, γe(%) = 0.00036; at 50 kPa,γe(%) = 0.00045 and at 100 kPa, γe(%) = 0.00071, NA – Not available.

Table 9. Results of the model selection for Sample03-1 at 25, 50, and 100 kPa.

Hyperbolic models

ln P DatajMkð Þ½ �Sample03-1 (25 kPa) Sample03-1 (50 kPa) Sample03-1 (100 kPa)

M1 [Hardin and Drnevich, 1972a, 1972b] 36.96 34.00 45.03M2 [Darendeli, 2001] 34.48 32.09 46.60M3 [Oztoprak and Bolton, 2013] 35.83 35.91 42.89

For Sample03-1 at 25 kPa, γe(%) = 0.00036; at 50 kPa,γe(%) = 0.00045 and at 100 kPa, γe(%) = 0.00071, NA – Not available.

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7.3. Best Fitted G=Gmax � log γ Curves and Their Confidence Intervals

Based on the posterior PDF in Eq. (6), the MCMC simulation approach discussed in Sec. 5is used to generate 100,000 samples (i.e., m = 100,000) of model parameters depending onthe type of selected model (Secs. 4 and 6.2). Using the method described in Sec. 5, the bestfitted curves and the 95% (i.e., 2.5th and 97.5th percentile levels) confidence intervalcurves for Sample03-1 at 25, 50, and 100 kPa are constructed in Fig. 8(a)–(c), respectively,for Sample10-1 at 100 and 200 kPa in Fig. 9(a) and (b), respectively, as well as forSample14-1 at 50 kPa and 100 kPa in Fig. 10(a) and (b), respectively. Note that in Figs.8–10, the best fitted curves are represented with the solid lines. These solid lines representthe average G=Gmax � log γcurves for the observation data and can be directly used inpredicting the normalized modulus reduction. Furthermore, the 95% confidence intervalcurves represented as dashed lines are developed for each set of observation data, i.e.,Sample03-1 at 25, 50, and 100 kPa; Sample10-1 at 100 and 200 kPa; and Sample14-1 at 50and 100 kPa.

The 95% confidence interval (i.e., dashed lines in Figs. 8–10) reflects the scatterness ofthe observation data related to the selected hyperbolic model. They represent the band ofscatter around the best fitted curves where the actual observation data are located. Thetraditional method commonly employed in the literature [e.g., Menq, 2003; Senetakiset al., 2013a, 2016] for optimizing the fitting parameters of normalized modulus curvesdoes not involve the quantification of the scatterness of the observation data, which iscarried out by the proposed method in this study. Note that the Bayesian approach may

Table 10. Summary of the selected model for all the data from the resonant column tests.

Code name of specimen

p0 = 25 kPa p0 = 50 kPa p0 = 100 kPa p0 = 200 kPa

Selected Model Selected Model Selected Model Selected Model

Sample01-1 M3 NA M3 M2Sample01-2 NA NA M1 M2Sample02-1 M3 M2 M2 NASample02-2 NA NA M2 M2Sample03-1 M1 M3 M2 NASample03-2 M3 M1 M3 NASample04-1 NA NA M2 M1Sample04-2 NA NA M3 NASample05-1 NA M3 M1 NASample05-2 NA NA M3 M3Sample06-1 NA NA M2 M1Sample06-2 NA M3 M2 M2Sample07-1 NA M3 M2 NASample07-2 NA NA M3 NASample08-1 NA NA M2 M3Sample09-1 NA M1 M3 M2Sample10-1 NA NA M3 M1Sample11-1 NA M3 M3 M2Sample11-2 M3 M3 M3 NASample12-1 NA NA M3 M1Sample12-2 NA M3 M1 M2Sample13-2 NA M3 M3 NASample14-1 NA M3 M3 NASample14-2 NA NA M3 M3Sample15-1 NA M1 M1 NASample15-2 NA M3 M3 NA

NA—Not available

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(a)

(b)

(c)

Figure 8. Normalized modulus reduction curves of coarse-grained samples in this study. (a) Best fittedcurve and its 95% confidence interval for Sample03-1 at 25 kPa, (b) Best fitted curve and its 95%confidence interval for Sample03-1 at 50 kPa, and (c) Best fitted curve and its 95% confidence intervalfor Sample03-1 at 100 kPa.

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suffer from local maxima and degeneracy problem, which may affect MCMC performancein Sec. 5 [e.g., Doucet et al., 2000; Robert and Casella, 2004; Dias and Wedel, 2004; Ferozand Hobson, 2008; Koblents and Miguez, 2015]. However, the result from this studyshows that the proposed Bayesian approach performs satisfactorily and the problem oflocal maxima and degeneracy in MCMC is not profound for this application. Both the bestfitted curves and the 95% confidence interval curves can be jointly used in predicting thenormalized modulus reduction for probabilistic site response analysis in geotechnicalengineering.

(a)

(b)

Figure 9. Normalized modulus reduction curves of coarse-grained samples in this study. (a) Best fittedcurve and its 95% confidence interval for Sample10-1 at 100 kPa and (b) Best fitted curve and its 95%confidence interval for Sample10-1 at 200 kPa.

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8. Summary and Conclusion

This paper developed a procedure for constructing suitable G=Gmax � log γcurves for thenew set of observation data to differentiate them from pre-existing G=Gmax � log γ curves.Though several families of G=Gmax � log γ curves have been developed in the literature,not all new geotechnical data are consistent with them due to data scatterness. Moreover,numerous analytical models are available in the literature for constructing G=Gmax � log γcurves, making it difficult for practitioners to select the most appropriate model fordeveloping G=Gmax � log γ curves. The proposed procedure involves the use of Bayesian

(a)

(b)

Figure 10. Normalized modulus reduction curves of coarse-grained samples in this study. (a) Best fittedcurve and its 95% confidence interval for Sample14-1 at 50 kPa and (b) Best fitted curve and its 95%confidence interval for Sample14-1 at 100 kPa.

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approach to quantify data scatterness, estimate the optimal values of the model para-meters, and select the most appropriate model for the construction of the normalizedmodulus reduction curves of soil.

Moreover, this study also presents a laboratory investigation of the dynamic propertiesof coarse-grained materials of a potential fill–backfill material of hard particles usingtorsional RC tests. The observation data obtained from these RC tests were then used todemonstrate the proposed procedure for developing suitable G=Gmax � log γcurves.Unlike the traditional method commonly used by practitioners, the Bayesian parameteridentification method proposed in this study not only evaluates the optimal values ofmodel parameters but also objectively provides associated parameter uncertainty andmodel uncertainty to quantify the scatterness of observation data. Such quantification ofuncertainty and scatterness of data is particularly important to probability-based designand analysis in geotechnical engineering. The model selection method also offers a meansto objectively select the most appropriate model from available ones which range fromsimple to complex analytical models. The different types of models selected by theprocedure for most samples of the observation data even at different mean-confiningpressures show the significance of applying an objective model selection approach. Thesubsequent confidence intervals developed reflect the degree of uncertainty involved in theprediction of the data using the selected analytical models. In addition to the best fittedcurves, they can be jointly utilized in predicting the normalized modulus reduction forprobabilistic site response analysis. The optimal values of the model parameters and theconstructed G=Gmax � log γ curves using the proposed procedure in this study can be usedto differentiate the nonlinear behavior of a given soil from those of other soils or pre-existing curves.

Funding

The work described in this paper was supported by grants from the Research Grants Council of theHong Kong Special Administrative Region, China (Project No. 9042172 (CityU 11200115) andProject No. 1141880 (CityU 112813). The financial support is gratefully acknowledged.

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