Selecting parameters to optimize ... -...

Computers and Geotechnics 31 (2004) 411–425

www.elsevier.com/locate/compgeo

Selecting parameters to optimize in model calibrationby inverse analysis

Michele Calvello, Richard J. Finno *

Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208, USA

Received 28 May 2003; received in revised form 11 August 2003; accepted 26 March 2004

Available online 25 May 2004

Abstract

A study evaluating the benefits of using inverse analysis techniques to select the appropriate parameters to optimize when

calibrating a soil constitutive model is presented. The factors that affect proper calibration are discussed with reference to the op-

timization of the elasto-plastic Hardening-Soil model for four layers of Chicago glacial clays. The models are initially calibrated using

results from triaxial compression tests performed on specimens from four clay layers and subsequently re-calibrated using incli-

nometer data that recorded the displacements of a supported excavation in these clays. Finite element simulations of both the triaxial

tests and the supported excavation are performed. A parameter optimization algorithm is used to fit the computed results and ob-

served data, expressed in the form of stress–strain curves and inclinometer readings, respectively. A procedure is presented which uses

the results of sensitivity analyses conducted on the soil model parameters for the identification of the relevant and uncorrelated

parameters to calibrate. In both cases the inverse analysis methodology effectively calibrates the soil parameters considered, which

numerically converge to realistic values that minimize the errors between computed responses and experimental observations.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

In a finite element simulation of a geotechnical

problem, calibrations of the models used to reproduce

soil behavior often pose significant challenges. Real soil

is a highly nonlinear material, with both strength and

stiffness depending on stress and strain levels. Numerousconstitutive models have been developed that can cap-

ture many of the important features of soil behavior.

However, developing soil parameters for use in consti-

tutive models is a procedure that involves much judg-

ment and usually is best accomplished by experienced

users of a particular model. An effective and more ob-

jective way to calibrate a soil model employs inverse

analysis techniques to minimize the difference betweenexperimental data (laboratory or field tests) and nu-

merically computed results [1,2].

For some large geotechnical engineering projects, for

example, deep supported excavations in urban envi-

* Corresponding author.

E-mail address: [email protected] (R.J. Finno).

0266-352X/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2004.03.004

ronments, it is usual to record ground movements

developed during construction to evaluate the perfor-

mance of the designed system. In some cases the data are

used to control the construction process and update

predictions of movements given the measured defor-

mations at early stages of constructions. This procedure

is referred to as the ‘‘observational method’’ [3–5]. Thisapproach usually entails the use of pre-construction

analysis and parametric studies coupled with much en-

gineering judgment. Inverse analysis techniques con-

ceptually can be used to enhance the conventional

observational method practice by using the monitoring

data to optimize automatically a numerical model of a

geotechnical project. Recent work in related civil engi-

neering fields (e.g. [6–9]) demonstrate that inversemodeling provides capabilities that help modelers sig-

nificantly, even when the simulated systems are very

complex.

However, there are a number of issues that affect

proper calibration, including the number of parameters

to be optimized, which depends on both the site stra-

tigraphy and number of parameters in the selected

mail to: [email protected]

412 M. Calvello, R.J. Finno / Computers and Geotechnics 31 (2004) 411–425

constitutive model, the interdependence of the model

parameters within the framework of the constitutive

model, the number of observations, and the type of

system under consideration.

In this paper, these factors are discussed and illus-trated by presenting results of inverse analyses used to

optimize the calibration of the Hardening-Soil (H-S)

model [10] for four layers of Chicago glacial clays. The

models are initially calibrated using results from triaxial

compression tests performed on specimens from the four

clay layers and subsequently re-calibrated using incli-

nometer data that recorded the displacements of a

supported excavation in these clays [11]. This paperdescribes the concepts of model calibration by inverse

analysis, summarizes the soil model used to define the

behavior of the clay considered, discusses the factors

that affect proper calibration, presents the results of the

model calibration from triaxial test data and from field

monitoring data and draws conclusions.

2. Model calibration by inverse analysis

In inverse analysis, a given model is calibrated by

iteratively changing input values until the simulated

output values match the observed data (i.e., observa-

tions). Fig. 1 shows a schematic of an inverse analysis

procedure. The input parameters are initially estimated

by conventional means. Much literature exists on thissubject, for example, a number of papers in the McGill

conference (i.e. [12–14]) describe how this first step is

done for a number of constitutive models for soils, i.e.,

a hyperbolic stress–strain model [12,13], and a

bounding surface model [14]. A numerical simulation

Numerical Model

Input parameters

Updated input parameters

Regression(objective function minimization)

Modeloptimized?

Initial input parameters

NO

YES

Computed results

ENDSTART

Optimized input parameters

Observations

Iter

ativ

e pr

oces

s

Fig. 1. Schematic of inverse analysis procedure.

of the problem is conducted and the simulated results

are compared to the available observations. A regres-

sion analysis is performed to minimize an objective

function, which quantifies the fit between computed

results and observations. Its minimization is attainedby the optimization of the input parameters needed to

perform the numerical simulation. If the model fit is

not ‘‘optimal’’, the procedure is repeated until the

model is optimized.

Inverse analysis algorithms allow the simultaneous

calibration of multiple input parameters. However,

identifying the important parameters to include in the

inverse analysis can be problematic. Indeed, in mostpractical problems it is not possible to use the regres-

sion analysis to estimate every input parameter of a

given simulation. The number and type of input pa-

rameters that one can expect to estimate simultaneously

depend upon many factors, including the characteristics

of the selected soil model, how the model parameters

are combined within the element stiffness matrix in a

finite element formulation, the site stratigraphy, thenumber and type of observations available, the char-

acteristics of the simulated system, and computational

time issues.

Fig. 2 shows a procedural flowchart used for the

identification of the parameters to optimize in the finite

element simulations of geotechnical problems. Note that

the first step of the procedure refers to the selection of

the model parameters that are relevant to the problemunder study. The last two steps, necessary if multiple soil

layers are calibrated simultaneously, refer to the selec-

tion of the total number of parameters that are opti-

mized in the simulation of the field scale problem. In this

paper, the first step is illustrated with results of simu-

lations of triaxial compression tests and the latter two

steps are illustrated with the simulation of a deep, sup-

ported excavation.In the work described herein, model calibration by

inverse analysis is conducted using UCODE [15], a

computer code designed to allow inverse modeling

posed as a parameter estimation problem. UCODE was

developed for ground-water models, but it can be ef-

fectively used in geotechnical modeling because it works

with any application software that can be executed in a

batch mode. Its model-independency allows the chosennumerical code to be used as a ‘‘closed box’’ in which

modifications only involve model input values. This is

an important feature of UCODE in that it allows one to

develop a procedure that can be easily employed in

practice and in which the engineer will not be asked to

use a particular finite element code or inversion algo-

rithm. Rather, macros can be written in a windows en-

vironment to couple UCODE with any finite elementsoftware. The commercial software PLAXIS 7.11 [16]

was used herein to simulate the soil behavior with the

H-S model [10].

Input.txtPLAXIS Input

PLAXIS Calculation

PLAXIS Output Output

ASCII I/OPLAXIS

Modeloptimized?

START

Initial parameters

NO

YES

Observations

Macro

Regression(multiple PLAXIS runs)

Best-fit parameters

END

Updated parameters

Fig. 3. Inverse analysis with UCODE and PLAXIS.

Fig. 2. Identification of soil parameters to optimize by inverse analysis.

M. Calvello, R.J. Finno / Computers and Geotechnics 31 (2004) 411–425 413

Fig. 3 shows a schematic of the interaction between

UCODE and PLAXIS during the inverse analysis.

PLAXIS, a Windows-based program, does not have any

option to save input or output files in ASCII format.

Therefore, windows-macros were written to convert thePLAXIS I/O into text files. The macros are needed to

produce model changes in PLAXIS-Input from an input

text file, switch between PLAXIS modules, calculate the

simulated model in PLAXIS-Calculation, and generate

an output text file from the PLAXIS-Output. Note that

the procedure needs no user intervention once the

analysis has been started. For more details, see [17].

3. Chicago glacial clays

Much of the subsoil in the Chicago area consists of

fairly distinct strata deposited during the advances and

retreats of a glacier during the Wisconsin Stage. Theadvance and retreat process, marked by terminal mo-

raines, created easily identifiable clay strata. In order of

deposition they are the Valparaiso, Tinley, Park Ridge,

Deerfield, Blodgett, and Highland Park tills [18]. Fig. 4

shows the soil profile at the site of the excavation con-

sidered herein, typical for the downtown area of Chi-

cago. All elevations refer to the Chicago City Datum

-200

-100

0

100

200

0 100 200 300 400

p=(σ1+2σ3)/3

q=σ 1

- σ3

Mohr-Coulombfailure line Shearing

yieldsurfaces

Yield CapSurface

Fig. 5. Hardening-Soil yield surfaces.

-20

-15

-10

-5

0

5

z (m)

Elevation(m CCD)

4.3

0.6-0.3

-4.6

-7.1

-10.8

-15.4

-18.5

Sand / Fill

Clay crust

Soft clay (Layer 1)

Soft-Medium clay (Layer 2)

Medium clay (Layer 3)

Stiff clay (Layer 4)

Very stiff clay

Hard Pan

STRATIGRAPHICUNIT

Blo

dget

tst

ratu

mD

eerf

ield

stra

tum

Park

Rid

gest

ratu

mT

inle

yst

ratu

m

Fig. 4. Subsurface profile.


(CCD), the zero value of which corresponds to the av-

erage level of adjacent Lake Michigan. For the purposes

of the laboratory experimental program, four main

layers were identified in order of increasing depth: the

Upper Blodgett, the Lower Blodgett, the Deerfield, and

the Park Ridge. The laboratory experiments concen-trated on these four normally to lightly overconsolidated

clay layers because they are far more compressible than

the lower Tinley stratum clays, and thus they have the

largest effect on the soil mass response to the excavation.

Note that the Blodgett layer is a supraglacial till and

typically exhibits more variability than the underlying

Table 1

H-S input parameters

Parameter Explanation

Basic parameters

/ Friction angle

c Cohesion

w Dilatancy angle

Eref50 Secant stiffness in standard drained tria

Erefoed Tangent stiffness for primary oedomete

m Power for stress-level dependency of sti

Advanced parameters

Erefur Unloading–reloading stiffness

mur Poisson’s ratio

Rf Failure ratio qf=qak0 k0 value for normally consolidated soil

stratum [18]. In the inverse analysis described herein,

these soil strata are referred to as layers 1, 2, 3, and 4.

4. The H-S model

The soil model used to simulate the clay behavior is

the H-S model as implemented in PLAXIS 7.11. The H-

S model is an elasto-plastic, multi-yield surface, effective

stress soil model. Failure is defined by the Mohr–Cou-

lomb failure criterion. Two families of yield surfaces are

incorporated in the model to account for both volu-

metric and shear plastic strains. Fig. 5 shows the yieldsurfaces of the model in p–q stress space. A yield cap

surface controls the volumetric plastic strains. On this

cap, the flow rule is associative. On the shearing yield

surfaces, increments of plastic strain are nonassociative

and the plastic potential is defined to assure a hyperbolic

stress–strain response for a triaxial compression loading.

The basic characteristics of the model are a Mohr–

Coulomb failure with input parameters c, / and dilat-ancy angle, w, stress-dependent stiffness according to a

Initial estimates

Slope of faillure line in rn � s stress space

y-axis intercept in rn � s stress space

Function of /peak and /failure

xial test y-axis intercept in logðr3=pref Þ � logðE50Þ spacer loading y-axis intercept in logðrv=pref Þ � logðE50Þ spaceffness Slope of trendline in logðr3=prefÞ � logðE50Þ space

default¼ 3Eref50

default¼ 0.2

default¼ 0.9

conditions default¼ 1� sin/


power law defined by input parameter, m, plastic

straining resulting from primary deviatoric loading with

an input parameter, Eref50 , and plastic straining from

primary compression with an input parameter Erefoed.

Elastic unloading–reloading is defined by input param-eters Eref

ur and mur.Table 1 shows the 10 H-S model input parameters,

their meaning and the conventional way of estimating

them. All parameters can be derived from the results of

drained triaxial compression and standard consolidation

tests. The failure parameters / and c are estimated as-

suming a Mohr–Coulomb failure criterion. The dilat-

ancy angle, w, depends on the volume changecharacteristics of the soil and is equal to zero for nor-

mally consolidated clays. The stiffness parameters Eref50 ,

Erefoed, and m are estimated by assuming the values the

50% secant and oedometric stiffnesses (E50 and Eoed,

respectively) are related to a reference pressure, pref ,usually set equal to 100 stress units. The advanced pa-

rameters are generally set equal to their default values.

Hence to begin a problem, one must estimate six pa-rameters for each soil type.

5. Optimization of laboratory data

Triaxial compression tests were performed on sam-

ples from the four most compressible clay layers on the

site of the Chicago Avenue and State Street subwayrenovation project. The triaxial testing program was

part of an experimental laboratory program conducted

to define the soil properties at the excavation site

[19,20]). The H-S model, used to simulate the behavior

of the clay specimens, was calibrated for the four clay

Table 2

Triaxial experimental program

Layer Depth of

samples (m)

Test type Test name Consolidation

stress (kPa)

1 5–6 CID TXC D1 107

D2 200

D3 400

CIU TXC U1 100

2 10.5–11.5 CID TXC D1 134

D2 220

D3 400

CIU TXC U1 130

3 12–13 CID TXC D1 175

D2 350

D3 450

CIU TXC U1 168

4 16.5–17.5 CID TXC D1 200

D2 350

D3 450

CIU TXC U1 204

layers based on these triaxial results using the inverse

analysis procedure presented in [2].

5.1. Experimental program

Table 2 summarizes the isotropically consolidated,

drained, and undrained triaxial compression tests per-

formed on soil samples from each of the four clay layers

considered. Fig. 6 shows the results of these tests. At

every layer, three drained (CID TXC) and one undrained

(CIU TXC) triaxial compression tests were conducted.

The test samples were first consolidated to different iso-

tropic effective stresses, the in situ vertical effective stress(tests D1 and U1) and two significantly higher stresses

(tests D2 and D3), and then sheared until failure. The

principal stress difference, the axial and volumetric

strains, and/or the excess pore pressures were recorded.

As expected for normally to lightly overconsolidated

clays, results of the drained tests show that the devia-

toric stresses and volumetric strains at failure generally

increase with depth at this site where the upper-strataclay samples are ‘‘weaker’’ than lower-strata ones and,

for a given layer, with increasing consolidation pressure.

Note that trends are not as clear for layers 1 and 2 de-

rived from the Blodgett stratum which is typically more

variable than the lower strata. For the undrained tests,

both the deviatoric stress and the excess pore pressures

at failure increase with increasing depth and, thus, in-

creasing effective consolidation pressure.

5.2. Optimization scheme

The observation points used for the inverse analysis

were selected from the stress–strain curves presented in

Fig. 6. The stress–strain curves of the drained test were

discretized by considering one observation point every

2% axial strain up to a maximum of ea ¼ 12%. Curves forthe undrained test were discretized using eight observa-

tion points per curve; the observation points were se-

lected more frequently at small strains, every 0.15% axial

strain up to ea ¼ 0:9%, so that the pore pressure varia-

tion would be adequately defined. Calvello and Finno [2]

showed that this number of observations points was

sufficient to define the stress–strain responses.

The initial values of the H-S input parameters of thefour clay layers were computed according to conven-

tional calibration procedures briefly summarized in

Table 3. The parameters optimized by inverse analysis

were chosen, according to the procedure shown in

Fig. 2, among the six H-S basic parameters. The four

advanced parameters, as suggested by the PLAXIS

manual, were always set equal to their default values.

The four clay layers were calibrated independently.To evaluate the relative importance of each input

parameter, a sensitivity analysis was conducted for the

six parameters at every layer, using the stress–strain

Fig. 6. Experimental results of triaxial tests.


results of the triaxial tests as observations. The results of

these analyses are used to determine the number ofrelevant and uncorrelated parameters in each layer. The

characteristics of the H-S model as implemented in

PLAXIS, the type of observations, and the stress con-

ditions in the soil samples all influence the sensitivity of

the observations to changes in parameter values.

The relevant parameters are discerned based on the

composite scaled sensitivity, cssj:

cssj ¼XNDj¼1

oy 0iobj

� �bjx

1=2ii

� �2��b

,ND

" #1=2

; ð1Þ

where y 0i is the ith simulated value; bj is the jth esti-

mated parameter; oy0i=obj is the sensitivity of the ithsimulated value with respect to the jth parameter; xjj

is the weight of the ith observation wherein the weight

of every observation is taken as the inverse of its error

variance, and ND is the number of observations. See

[2] for more details concerning x for the laboratory

data and [7] for the inclinometer data. The composite

scaled sensitivities indicate the total amount of infor-

mation provided by the observations for the estima-tion of parameter j and measure the relative

importance of the input parameters being simulta-

neously estimated.

Table 3

Initial and best-fit values of input parameters: laboratory results

Parameter Initial estimate Best-fit value

Layer 1 Layer 2 Layer 3 Layer 4 Layer 1 Layer 2 Layer 3 Layer 4

/a 24.1 27.0 28.9 31.4 23.40 23.50 25.60 32.80

c (kPa)b 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

wb 0 0 0 0 0 0 0 0

Eref50 (kPa)a 2350 3700 4000 11700 4700 7250 6000 8580

Ma 1.00 0.91 1.00 0.94 0.74 0.68 0.85 0.84

The reference pressure is pref ¼ 100 kPa.aOptimized based on regression analysis.b Parameters do not affect predicted results since negligible; values not changed.


The correlated parameters are discerned using the

correlation coefficients, corði; jÞ:

corði; jÞ ¼ covði; jÞvarðiÞ1=2varðjÞ1=2

; ð2Þ

where covði; jÞ equal the off-diagonal elements of the

variance–covariance matrix V ðb0Þ ð¼ s2ðX TxX Þ�1Þ, s2 isthe model error variance, and varðiÞ and varðjÞ refer tothe diagonal elements of V ðb0Þ. The values of corði; jÞindicate the correlation between the ith and jth pa-

rameters. Highly correlated parameters should not be

optimized simultaneously because many combinationscan lead to the same optimized result, but with unre-

alistic values of the optimized parameters. Values close

to )1.0 or 1.0 are indicative of parameters that cannot

be uniquely estimated with the observations used in the

regression.

Fig. 7. H-S input parameters: (a) composite scale

The estimates of parameter Erefoed could not be opti-

mized by inverse analysis because the H-S model im-

plemented in PLAXIS has an internal algorithm that

runs every time a new set of input parameters is speci-

fied. This algorithm considers the deviatoric stress re-sponse of an internally modeled compression test and

‘‘adjusts’’ the values of parameter Erefoed to produce a

hyperbolic curve in a triaxial stress–strain space. The

iterative nonlinear regression method used herein is

based on the value of a sensitivity matrix X ; Xij ¼oyi=obj. Therefore, parameter Eref

oed cannot be included in

the optimization because the regression algorithm would

compute wrong sensitivities every time its value is‘‘corrected’’ by PLAXIS.

Fig. 7 shows that the main parameters that affect the

observations, i.e., those with the higher values of cssj,

are /, Eref50 , and m for all four soil layers. The values of

d sensitivity and (b) correlation coefficients.

CID TXC (D1)

0

100

200

300

400

0 0.05 0.1 0.15

Axial strain

q(k

Pa)

0

0.01

0.02

0.03

0.04

0 0.05 0.1 0.15

Vol

umet

ricst

rain

Experimental Computed (initial) Computed (best-fit)

CID TXC (D2)

0

200

400

600

800

0 0.05 0.1 0.15

Axial strain

q(k

Pa)

0

0.01

0.02

0.03

0.04

0.05

0 0.05 0.1 0.15

Vol

umet

ricst

rain

CIU TXC (U1)

0

50

100

150

200

0 0.01 0.02 0.03 0.04 0.05

Axial strain

q(k

Pa)

020406080

100120

0 0.01 0.02 0.03 0.04 0.05

u exc

ess

(kP

a)CID TXC (D3)

0

200

400

600

800

1000

0 0.05 0.1 0.15

Axial strain

q(k

Pa)

0

0.01

0.02

0.03

0.04

0.05

0 0.05 0.1 0.15

Vol

umet

ricst

rain

Fig. 8. Visual fit between experimental and computed results.


the correlation coefficients, corði; jÞ, for the three pa-

rameters (always lower than 0.75) do not indicate sig-nificant correlation among them. Note that the values of

c and w for a lightly to normally consolidated clay are

expected to be equal to 0, and thus these values would

have no impact on the computed results. The values of cand w are set equal to 0.05 kPa and 0, respectively, and

were not subjected to optimization. As one would ex-

pect, these terms have negligible values of css. The value

of Erefoed was kept constant during the optimization of the

laboratory results, or if outside the valid range, set to the

PLAXIS-suggested value that depends on the values of

parameters Eref50 , E

refur , mur and k0. The results in Fig. 7

suggest that only the failure parameter / and the stiff-

ness parameters, m and Eref50 should be optimized by

inverse analysis.

5.3. Model calibration

Table 3 shows the initial and best-fit values of the six

H-S input parameters for the four clay layers. The initial

estimates were computed according to conventional

calibration methods. Use of the best-fit values mini-mized the errors between the measured data (Fig. 6) and

the computed results; these errors are expressed by the

regression’s objective function, SðbÞ:

Table 4

Values of the objective functions for initial and calibrated laboratory

results

Objective function

From initial estimates From best-fit estimates

Layer 1 990 277

Layer 2 772 141

Layer 3 744 153

Layer 4 993 95


S bð Þ ¼ y�

� y0 bð Þ�Tx y�

� y 0 bð Þ�¼ eTxe; ð3Þ

where b is a vector containing values of the number of

parameters to be estimated; y is the vector of the ob-

servations being matched by the regression; y0ðbÞ is thevector of the computed values corresponding to the

observations; x is the weight matrix; and e is the vectorsof residuals.

Fig. 8 shows the results of the model calibration for

layer 3. The comparison between the measured data andthe computed results for the initial and best-fit estimates

of the input parameters indicates that the final cali-

brated model (best-fit) fits the data better than the model

calibrated by conventional means (initial), particularly

for the undrained results. The calibration by inverse

analysis of the other clay layers (1, 2, and 4) produced a

similar improvement and the ‘‘optimized’’ models al-

ways fit the experimental data better than the ‘‘initial’’models [17]. These visual fits can be quantified by the

values of the objective functions shown in Table 4. Note

that steps 2 and 3 in Fig. 2 were not needed in this

calibration because of the assumption that the triaxial

tests represented an elemental test and only one soil

layer was optimized in a given analysis.

6. Field application: deep excavation

Inverse analysis becomes more complicated when

multiple soil layers and complicated loading paths areencountered in a problem, as is the case when computing

ground movements associated with deep excavations.

To illustrate the logic behind steps 2 and 3 in Fig. 2, the

calibration of the finite element simulation of a sup-

ported excavation is presented herein. The excavation

made to renovate a subway station in downtown Chi-

cago [11] consisted of removing 12.2 m of soft to me-

dium clay within 2 m of a school supported on shallowfoundations. The support system consisted of a secant

pile wall supported by one level of cross-lot bracing and

two levels of tie-backs. Inclinometers were used to re-

cord ground movements developed during construction.

These monitoring data were used as observations in the

inverse analysis that optimized the numerical model of

the excavation.

PLAXIS was used to compute the response of the soil

around the excavation, and Fig. 9 shows a schematic of

the input. The problem was simulated in plane-strain

conditions. The soil stratigraphy was assumed to be

uniform across the site. The figure shows the centralportion of the finite element mesh and the elevations at

the interfaces of the different soil layers. Eight soil layers

were modeled: a fill layer overlaying a clay crust, a

compressible clay deposit consisting of four distinct clay

layers, and a relatively incompressible deposit consisting

of two clay layers. Dolomite is encountered beneath the

hardpan. The fill layer was modeled as an elastic-per-

fectly plastic Mohr–Coulomb material, whereas all clayslayers were modeled the H-S model. The ends of the

mesh extended 50 m from the closest secant pile wall.

The finite element mesh boundary conditions were set

using horizontal restraints for the left and right bound-

aries and total restraints for the bottom boundary.

Lateral movements of the soil behind the secant pile

wall were recorded using five inclinometers. Observa-

tions from two inclinometers on opposite sides of theexcavation were used to compare computed displace-

ments with the field data. The observation points were

retrieved from the field readings of inclinometers 1

(adjacent to the school) and 4, where movements were

essentially planar [11]. Inclinometer 4 was damaged by

construction activities after stage 3, preventing com-

parisons with computed results for the last two stages of

construction.Table 5 shows the calculation phases and the con-

struction stages used in the finite element simulations.

The tunnel tubes and the school adjacent to the exca-

vation were explicitly modeled in the finite element

simulations to take into account the effect of their con-

struction on the soil surrounding the excavation. Stages

1, 2, 3, 4, and 5 refer to the construction stages for which

the computed results are compared to the field data inthe inverse analysis procedure. PLAXIS employs a

penalty formulation so that undrained conditions can be

explicitly modeled. Construction phases not noted as

‘‘consolidation’’ on Table 5 were modeled as undrained.

Consolidation stages were included in the tunnel,

school, and wall installation calculation phases to per-

mit excess pore water pressures to equilibrate. In

PLAXIS, these stages are computed using Biot’s ap-proach. Secant pile wall installation in the field is a

three-dimensional process. To simulate this construction

in the plane strain analysis, elements representing the

wall were excavated and a hydrostatic pressure equiva-

lent to a water level located at the ground surface was

applied to the face of the resulting trench (calculation

phase 13 in Table 5). After computing the movements

associated with this process, the excavated elementswere replaced by elements with the properties of the

secant pile wall (calculation phase 14). More details

about the definition of the finite element problem, the

Fig. 9. Schematic of PLAXIS input.

Table 5

Construction stages for updating model predictions

Calculation

phase

Construction stage

0 Initial conditions – at-rest

conditions

1–4 Tunnel construction (1940)

5 Consolidation stage

6–10 School construction (1960)

11 Consolidation stage

Wall installation 12 Reset displacements (1999)

13 Excavate secant-pile wall

14 Place concrete in wall> stage 1

15 Consolidation stage (20 days)

Excavation 16 Excavate [þ2:75 m] and install

strut> stage 2

17 Excavate [�0:9 m]

18 Prestress first tiebacks> stage 3

19 Excavate [�4:6 m]

20 Prestress second tie-

backs> stage 4

21 Excavate [�7:9 m]> stage 5


calculation phases, and the model parameters used inthe simulation described herein can be found in [17].

6.1. Definition of inverse analysis problem

According to Hill [21], a well-posed regression

problem is one that will converge to an optimal set of

parameter values given reasonable initial values. Given

commonly available data for supported excavations, therequirement of maintaining a well-posed regression will

produce rather simple models with relatively few esti-

mated parameters. Often, however, it is this simple level

of model complexity that can be supported by the data

based on regression methods. Thus, determining the

greatest possible level of model complexity while main-

taining a well-posed regression can be thought of as an

objective analysis of the information provided by thedata. Composite scaled sensitivities and parameter cor-

relation coefficients can be used to define parameters

and to decide which parameters to estimate using re-

gression. To obtain to a well-posed regression one mustuse both (i) objective values such as composite scaled

sensitivities and parameter correlation coefficients and

(ii) good engineering judgment, i.e., whether or not the

optimized parameters are consistent with those expected

based on the geotechnical meaning of the parameters.

The finite element simulation of the excavation was

re-calibrated by inverse analysis at the five construction

stages indicated in Table 5. The soil strata calibrated byinverse analysis are the same ones for which the triaxial

tests were performed. The initial estimates of the input

parameters in this excavation problem for layers 1–4 are

based on the results of the calibration conducted using

the triaxial data (i.e., best-fit values in Table 3). The

procedure described in Fig. 2 was used to determine the

number of parameters to optimize in the inverse analysis

of the excavation.Because four soil layers with five parameters each

may need to be simultaneously optimized in this com-

plicated finite element simulation of the excavation

system, the ‘‘principle of parsimony’’ [21] was applied.

The calibration was conducted by estimating as fewest

number of simulation parameters that together repre-

sented the main features of interest for the problem

studied. Consequently, the problem of ‘‘spreading thedata too thin’’ [8], which occurs when the ratio between

the number of observations available and the number of

parameters estimated is too low, was avoided. For in-

stance, at the first optimization stage, 24 observations

were used in the regression analysis (inclinometer read-

ings on both sides of the excavation at various eleva-

tions). Obviously, the maximum number of parameters

that a regression can estimate is equal to the number ofobservations used. In practice, the number of estimated

parameters had to be significantly lower than the num-

ber of observations because many of the observations

used were redundant (they ‘‘carried’’ the same infor-

mation).


The observations, soil movements, and the many

types of loading paths associated with the excavation

simulation are very different from the stress–strain data

used as observations in the triaxial compression tests

used previously to calibrate the H-S model. Yet, theresults of a sensitivity analysis performed on the H-S

basic parameters indicated that the parameters that are

most relevant to the excavation problem are Eref50 , m, and

/, the same as those found in the results of the cali-

bration based on laboratory data. As before, the values

of c and w had negligible influence on the results, and

Erefoed cannot be optimized in PLAXIS since it may be

internally adjusted.Fig. 10 shows the composite scaled sensitivities of the

three relevant parameters for layers 1–4. The bar chart

refers to sensitivities computed using all the observa-

tions, and the line charts refer to sensitivities computed

from the observations of the different layers. From a

0 10 20 30 40 50

Composite scaled sensitivity

Obs from all layers

Obs from layer 1

Obs from layer 2

Obs from layer 3

Obs from layer 4

Laye

r1

E50

m

Laye

r2

Laye

r3

Laye

r4

E50

m

E50

m

E50

m

Fig. 10. Composite scaled sensitivities of parameter Eref50 , m and / for

layers 1 to 4.

Table 6

Values of correlation coefficients: field study

Correlation coefficients

Between parameters Value Between para

Layer 1 mð1Þ and Eref50ð1Þ �0:70 /ð1Þ and Eref

50ð1ÞLayer 2 mð2Þ and Eref

50ð2Þ �0:85 /ð2Þ and Eref50ð2Þ

Layer 3 mð3Þ and Eref50ð3Þ �0:87 /ð3Þ and Eref

50ð3ÞLayer 4 mð4Þ and Eref

50ð4Þ �0:99 /ð4Þ and Eref50ð4Þ

simulation perspective, results show that the parameters

that most influence the simulation are the ones relative

to layers 1, 3, and 4. Layer 1 is the softest soil layer, thus

its major influence on the displacement results is ex-

pected. Layer 3 is the stratum wherein the excavationbottoms out. Layer 4 is the stiff clay layer below the

bottom of the excavation into which the secant pile wall

is tipped. The high sensitivity values of this stratum in-

dicate that the strength and the stiffness of the clay be-

low the excavation have significant impact on

movements, as one would expect. Fig. 10 also shows

that the observations relative to a soil layer are mainly

influenced by changes in that soil layer’s parameters.For instance, the values of the sensitivities from layers 3

and 4 show a clear ‘‘peak’’, respectively, for the layer 3

and layer 4 input parameters.

Table 6 shows the correlation coefficients between the

three parameters at every layer. The rather high corre-

lation between Eref50 and m indicate that these parameters

are not likely to be simultaneously and uniquely opti-

mized, even though the results of the analysis are sen-sitive to both. Parameter Eref

50 , rather than parameter m,

was chosen to ‘‘represent’’ the stiffness of the H-S model

because changes in Eref50 values also produce changes in

the values of parameters Erefoed (equal to 0.7 times Eref

50 )

and Erefur , thus its calibration can be considered as

‘‘representative’’ of the calibration of all H-S stiffness

parameters.

The results of the sensitivity analysis seem to indicatethat the total number of relevant parameters is 8 (i.e.,

Eref50 and / for layers 1–4). However a final reduction of

the parameters to optimize was necessary to establish a

‘‘well-posed’’ problem where the solution converged.

The stiffness parameters (Eref50 ) were chosen over the

failure parameters (/Þ because the excavation-induced

stress conditions in the soil around this excavation were,

for the most part, far from failure, and the laboratoryestimated values of u are judged to be more accurate

than Eref50 since sampling disturbance affects stiffness

much more than u. Thus, the variations in stiffness pa-

rameters were perceived to be more relevant to the

simulated problem. Note that, when the stiffness and

failure parameters are optimized simultaneously or only

the failure parameters are calibrated, the regression

analysis never converged to geotechnically reasonable

meters Value Between parameters Value

)0.42 mð1Þ and /ð1Þ 0.33

)0.59 mð2Þ and /ð2Þ 0.41

)0.58 mð3Þ and /ð3Þ 0.25

)0.07 mð4Þ and /ð4Þ �0.14


values [17]. This emphasizes the point that convergence

does not necessarily ensure that reasonable results are

attained when optimizing a nonlinear problem such as a

supported excavation in soil. In the spirit of the ‘‘prin-

cipal of parsimony’’, layers 1 and 2 were combined be-cause layer 2 had a much lower impact on the computed

results, as indicated by the low values of composite scaled

sensitivities for this layer in Fig. 10. Further justification

for combining layers 1 and 2 come from the geotechnical

perspective that there was not always a consistent trend

in the stress–strain responses in Fig. 6 and the two layers

are derived from the same geologic stratum.

7. Results

Table 7 shows the parameters that were re-calibrated

at every construction stage. Two parameters per layer,

Eref50 and Eref

oed, were updated by inverse analysis at each

stage. The value of parameter Erefoed, which cannot be

independently estimated by the regression, is related tothe value of parameter Eref

50 .

The simplest way to evaluate the difference between

the results of the simulations based on the initial guess

and the recalibrated parameters is to compare the in-

clinometer data with the computed horizontal dis-

placements for each case. Fig. 11 shows the visual fit

between the observations and the results computed be-

fore (i.e., initial) and after (i.e., best-fit) the calibrationby inverse analysis. The comparison shows that the

initial simulation computes displacements significantly

larger than the measured ones at every construction

stage (the maximum computed displacements at stage 5

are about two times the measured ones) and the com-

puted displacement profiles result in significant and

unrealistic movements in the lower clay layers. When the

model is calibrated by inverse analysis, the fit betweenthe computed and measured response is quite good. At

the end of the construction the maximum computed

displacement exceeds the measured data by less than

10% and the distributions of lateral deformations are

consistent throughout the excavation.

Note that all observations (i.e., stages 1–5) were used

to calibrate the finite element model of the excavation

that produced the good fit shown in Fig. 11. The sim-ulation was calibrated starting at stage 1 and re-cali-

brated at every subsequent construction stage using the

inclinometer data available up to that stage.

Table 7

Parameters recalibrated by inverse analysis: field study

Layers Parameters optimized Related parameters

1 and 2 E1=2 ¼ Eref50 Eref

oed ¼ 0:7Eref50

3 E3 ¼ Eref50 Eref

oed ¼ 0:7Eref50

4 E4 ¼ Eref50 Eref

oed ¼ 0:7Eref50

Table 8 shows the initial and optimized values of the

Eref50 values at the different optimization stages. The

maximum changes in parameter values occur at stage 1

when the observations relative to the installation of the

secant-pile wall are used. Subsequent calibrations re-sulted in parameters that only changed slightly at later

stages. By the end of stage 3 the model essentially is

calibrated. Indeed, the values of the optimized input

parameters do not change after that stage, indicating

that the observations at stages 4 and 5 ‘‘match’’ the

computed results of the model calibrated at stage 3.

Further improvement in the fit of the model is neither

possible nor necessary. In fact, most improvement oc-curred after the first optimization stage after the secant

pile wall was installed. While as much as 10 mm of

lateral movement in the clays occurred during this stage

(Fig. 11), one must remember that the tunnel and school

construction were also simulated so as to obtain rea-

sonable estimates of the ‘‘initial’’ stresses in the ground

before the wall was installed. Because of the proximity of

the tunnel, one would not expect at-rest in situ stresses,and hence the simulation included steps for tunnel and

school construction as noted in Table 5.

To illustrate the impact these activities had on the

ground conditions, contours of equivalent shear strain

computed after the wall was installed (calculation phase

14 in Table 5) are presented in Fig. 12. The shear strains

were as large as 4% in all four of the clay layers that

were optimized. These strains were large enough to‘‘exercise’’ the model so that the parameters could be

optimized. Recall that both stiffness, reflected by Eref50 ,

and failure, reflected by u, parameters were shown in

Fig. 10 to be important at these relatively large strain

levels. Note that Calvello and Finno [22] showed that

when the tunnel and school construction were not in-

cluded in the simulation, while the computed displace-

ments matched the observed values after optimization,the optimized Eref

50 values did not make increase with

depth and hence did not make sense from a geotechnical

viewpoint. These results emphasize the important point

that one must consider all factors that affect the stresses

in the soil to have a meaningful result based on inverse

analysis.

Results in Table 8 also show that the initial estimates

of the stiffness parameters based on the laboratory dataare significantly lower than the optimized values based

on the field performance data. At stage 1, the changes in

parameters for layers 1 and 2 are smaller than those for

layers 3 and 4. Layers 1 and 2 are subjected to larger

displacements, and hence strains, than the deeper layers

(see Fig. 11). This trend suggests that the estimation of

the parameters affecting the smaller strain responses

applicable to layers 3 and 4 is more difficult than esti-mation of the parameters at larger strains applicable to

layers 1 and 2, at least with the H-S model. The observed

trends in parameter changes reflect that fact that accu-

EA

ST

sid

eW

ES

Tsi

de

Stage 4 Stage 5Stage 1 Stage 2 Stage 3

-20

-15

-10

-5

0

0 20 40 60displacement (mm)

Ele

vatio

n(m

CC

D) 1

Layers

2

3

4

-20

-15

-10

-5

0


-20

-15

-10

-5

0


-20

-15

-10

-5

0


-20

-15

-10

-5

0

0 20 40 60diplacement (mm)

-20

-15

-10

-5

0

0 20 40 60

Ele

vatio

n(m

CC

D)

-20

-15

-10

-5

0

0 20 40 60

-20

-15

-10

-5

0

0 20 40 60

Field data

Computed displacements (initial)

Computed displacements (best-fit)

1

2

3

4

Layers

Fig. 11. Measured vs. computed horizontal displacements for initial and best-fit estimates of parameters.

M.Calvello

,R.J.Finno/Computers

andGeotech

nics

31(2004)411–425

423

Table 8

Best-fit values of reference moduli at various optimization stages: field

study

E1=2 (kPa) E3 (kPa) E4 (kPa)

Initial 288 288 413

Stage 1 306 862 2031

Stage 2 327 841 2112

Stage 3 362 752 2573

Stage 4 362 752 2573

Stage 5 362 752 2573

5/Initial 1.3 2.6 6.2

The reference pressure is pref ¼ 100 psf¼ 4.8 kPa.


rately estimating stiffness at small strains via laboratory

testing is not possible when defining response via global

measures of strains in a conventional triaxial apparatus,

but better results can be achieved at larger strains.Sample disturbance would have a larger effect on the

secant moduli at smaller strains than at larger strains,

the same trend that is exhibited in the results. Further-

more, as the excavation deepens, ratio of the length of

the wall to the depth of the cut becomes smaller and

departs from plane strain conditions. The stiffening ef-

fect of the corner of the excavation cannot be accom-

modated in a plane strain simulation, except byincreasing the stiffness of the soil [7]. Both these factors

contributed to the amount of change observed in the

optimized parameters.

Fig. 12. Shear strain distributio

8. Conclusions

The inversemodeling procedure described in this paper

combines a finite element analysis and a parameter opti-

mization algorithm to efficiently calibrate a soil model byminimizing the errors between experimental observations

and computed responses. Based on the results presented

herein, the following conclusions can be drawn:

1. Inverse analysis can be effectively used to calibrate a

numerical soil model based on triaxial experimental

results. When the relevant and uncorrelated H-S in-

put parameters were updated using an automated re-

gression algorithm, the computed results fit theexperimental data (relative to four different clay lay-

ers) better than the initially calibrated models.

2. A good understanding of the problem is necessary to

define an adequate optimization scheme when dealing

with a finite element simulation of a geotechnical pro-

ject (e.g., supported excavation), which involves the

calibration of multiple soil layers. Important factors

include the number of parameters to be optimized,equal to the number of model parameters times the

number of soil layers, number of observations avail-

able, and the type of system being analyzed. Statistics

that are generated by inverse analysis techniques,

such as composite scaled sensitivity and correlation

coefficients, are helpful in reducing the number of rel-

evant parameters to manageable levels.

n after wall installation.


Of course, if the constitutive model does not ade-

quately represent the important stress–strain responses

in the field problem, then convergence to realistic soil

parameters is not possible.

Acknowledgements

This work was supported by funds from Grant CMS-

0115213 from the National Science Foundation. The

support of Dr. Richard Fragaszy, the cognizant pro-

gram manager, is greatly appreciated.

References

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[18] Chung CK, Finno RJ. Influence of depositional processes on the

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