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Parametric Estimation of Dispersive Viscoelastic Layered Media with Application to Structural Health Monitoring

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Abstract

We present a sequential Bayesian estimation method to estimate the material parameters that govern the one-dimensional propagation of shear waves through continuous, layered, viscoelastic solids. While the proposed estimation method is generic, namely, can be applied to waveform inversion problems that satisfy the above conditions, we here employ it for system identification of building structures. We approximate the linear-elastic response of building structures subjected to low-amplitude earthquake base excitations by a multilayer dispersive shear beam model with Kelvin-Voigt material subjected to vertically propagating shear waves. Utilizing the proposed sequential Bayesian estimation method, we sequentially update the probability distribution function of the unknown parameters to reduce the discrepancies between the estimated and measured frequency response functions. We next verify and validate the performance of the proposed estimation method and investigate the limitations of the presented structural system identification approach using two case studies. In the first case study, we use the simulated structural response of a three-dimensional 52-story building model subjected to bi-directional low-amplitude ground shakings. We estimate the frequency-dependent phase velocity and damping ratio, as well as the mass distribution along the building height. Then, we verify the structural damage detection and localization capabilities of the presented system identification approach by comparing the wave model parameters estimated from simulated response of undamaged and damaged structural models. In the second case study, we use data measured from a shake table experiment on a full-scale five-story reinforced concrete building specimen, where the estimated wave model parameters capture the progressive structural damage in the test specimen. The validation studies suggest that the sequential Bayesian estimation method based on viscoelastic dispersive wave propagation can be used for system and damage identification of building structures.
1
Parametric Estimation of Dispersive Viscoelastic Layered Media with Application
to Structural Health Monitoring
Hamed Ebrahimian 1, Monica Kohler 2, Anthony Massari 3, Domniki Asimaki 4
1 Postdoctoral Scholar, Department of Mechanical and Civil Engineering, Caltech
2 Research Assistant Professor, Department of Mechanical and Civil Engineering, Caltech
3 Ph.D. Candidate, Department of Mechanical and Civil Engineering, Caltech
4 Professor, Department of Mechanical and Civil Engineering, Caltech
Abstract
We present a sequential Bayesian estimation method to estimate the material parameters that
govern the one-dimensional propagation of shear waves through continuous, layered, viscoelastic
solids. While the proposed estimation method is generic, namely, can be applied to waveform
inversion problems that satisfy the above conditions, we here employ it for system identification
of building structures. We approximate the linear-elastic response of building structures
subjected to low-amplitude earthquake base excitations by a multilayer dispersive shear beam
model with Kelvin-Voigt material subjected to vertically propagating shear waves. Utilizing the
proposed sequential Bayesian estimation method, we sequentially update the probability
distribution function of the unknown parameters to reduce the discrepancies between the
estimated and measured frequency response functions. We next verify and validate the
performance of the proposed estimation method and investigate the limitations of the presented
structural system identification approach using two case studies. In the first case study, we use
the simulated structural response of a three-dimensional 52-story building model subjected to bi-
directional low-amplitude ground shakings. We estimate the frequency-dependent phase velocity
and damping ratio, as well as the mass distribution along the building height. Then, we verify the
structural damage detection and localization capabilities of the presented system identification
approach by comparing the wave model parameters estimated from simulated response of
undamaged and damaged structural models. In the second case study, we use data measured from
a shake table experiment on a full-scale five-story reinforced concrete building specimen, where
the estimated wave model parameters capture the progressive structural damage in the test
specimen. The validation studies suggest that the sequential Bayesian estimation method based
on viscoelastic dispersive wave propagation can be used for system and damage identification of
building structures.
Keywords: Wave inversion, Bayesian estimation, Dispersion, Interferometry, Frequency
domain identification, Structural system identification.
1. Introduction
The linear-elastic response of a building structure subjected to an earthquake base excitation
can be approximated by modeling the propagation of seismic waves through a continuous,
layered, viscoelastic solid. The velocity of shear waves propagating through the building can be
estimated using impulse response functions (IRFs) (e.g., [1], [2], [3], [4], [5], [6]). Since the
shear wave velocity is related to the lateral stiffness of the building structure, several studies
were able to detect structural damage in terms of loss of effective lateral stiffness by comparing
the estimated shear wave velocities from the pre- and post-damage IRFs (e.g., [7], [8], [9]).
2
Among others, Rahmani and Todorovska [10] identified the shear wave velocity profile of an
equivalent multilayer shear beam model, representing the Millikan Library building, using IRFs
estimated from the measured response of the building to a small earthquake. They used a least
squares method to minimize the discrepancies between the main pulses of the estimated and
measured IRFs, and to identify the wave velocity profile along the building height. In another
study, Ebrahimian et al. [11] estimated the frequency-dependent phase velocity of the vertically
propagating shear waves in a building structure. They used an equivalent Timoshenko beam
model and estimated its phase velocities by measuring the wave travel times from IRFs derived
from band-passed-filtered responses at various frequency ranges. This study provided a
nonparametric method for identification of dispersive shear wave propagation in building
structures. Lastly, in recent studies, Ebrahimian and Todorovska ( [12], [13]) presented a
parametric identification method to identify the frequency dependent phase and group velocities
of vertically propagating waves in building structures. They used a least squares method to
estimate the stiffness parameters of an equivalent multilayer Timoshenko beam model, which
accounts for dispersion due to bending deformation, by minimizing the discrepancies between
the main pulses in the estimated and measured IRFs.
In this paper, we present a sequential Bayesian estimation method for parametric estimation
of wave dispersion in a continuous, multilayer, viscoelastic solid. In contrast to the
aforementioned studies, which used the IRFs to estimate the wave velocity, we formulate our
estimation approach in the frequency domain i.e., using the frequency response functions
(FRFs). It should be noted that the sequential Bayesian estimation method is general, and can be
applied to frequency-domain or time-domain waveform inversion problems. However, we
noticed an improved estimation performance when the presented estimation method is used in
the frequency domain. The estimation method is validated in this study using two building
system and damage identification case studies: the first is based on the numerically simulated
data obtained from linear-elastic structural models, and the second is based on the data measured
from a full-scale shake table experiment. In these case studies, the buildings are identified as
multilayer dispersive shear beam model with Kelvin-Voigt material, in which each layer
represents a story or group of stories. We next propose empirical functions to characterize the
dispersion of phase velocity and damping ratio at each layer. We estimate the parameters
characterizing the frequency-dependent phase velocity, damping ratio, and mass density of each
layer, and quantify their estimation uncertainties, by minimizing the discrepancies between the
measured and predicted FRFs.
The novelty of this study is threefold: (i) we propose a sequential Bayesian estimation
method using FRFs for waveform inversion to estimate the unknown wave propagation
parameters and to quantify their estimation uncertainties; (ii) instead of estimating shear wave
phase velocity and damping ratio for different frequency ranges, we propose closed form
empirical dispersion functions to characterize the phase velocity and damping ratio as a function
of frequency; we then estimate the function parameters through the Bayesian estimation method;
and, (iii) we provide a methodology that can be used for a wide range of waveform inversion
problems, from soil response inversion of downhole geotechnical array recordings to structural
system and damage identification of building structures.
This study has been motivated by the Community Seismic Network (CSN) project ( [14],
[15] [16], [17]), which is a network of low-cost micro-electromechanical systems (MEMS)
accelerometers that have been used to densely instrument buildings and free field locations in the
3
greater Los Angeles area. Most CSN-instrumented buildings are equipped with one or two
triaxial accelerometers per floor. The primary product of the network is the measured structural
response time histories on a floor-by-floor spatial scale, during and after earthquakes. The
complementary product of this network would be robust and computationally efficient methods
for structural damage identification that not only detect but also localize and quantify damage in
the building rapidly after an earthquake. This would provide a proxy indicator of building
damage that can inform emergency response activities mobilized after the event. This study
provides a method that can be used towards this objective.
2. Problem Statement
As mentioned above, we model the building structure as a continuous, non-homogeneous
(along height), viscoelastic solid, subjected to vertically propagating shear waves imposed as
prescribed displacement boundary conditions at the base (Figure 1). The frequency-dependent
phase velocity and attenuation of seismic energy at different wavelengths, together with the
geometric and inertial properties of the building characterize its dynamic response. The objective
of this study is to identify the building system by estimating parameters that characterize the
propagation of shear waves through the system.
To achieve this goal, we estimate the frequency response functions (FRFs) of each floor’s
absolute acceleration response with respect to the base excitation, using a seismic interferometry
approach [18] based on the spectral analysis method [19]. The FRFs estimated from the
measured structural responses (i.e., measured FRFs) are then compared with FRFs predicted
using analytical models for one-dimensional wave propagation in a multilayer dispersive shear
beam model with Kelvin-Voigt material. Through a sequential Bayesian estimation approach,
we estimate the parameters characterizing the phase velocity and damping ratio as a function of
frequency, as well as the homogenized mass density of the building. These parameters will
hereafter be referred to as wave model parameters. These parameters are used for structural
system and damage identification in the ensuing sections of this paper.
Each layer of the multilayer Kelvin-Voigt model is an idealized homogenized story or group
of stories of the building structure. The premise of our study is that estimating the wave model
parameters before and after a damage-inducing earthquake can provide information about the
extent and location of the damage. More specifically, by interpreting the reduction in the phase
velocity as a permanent loss of lateral stiffness at the story level (or group of stories) of the
building structure, we can detect and localize the structural damage (e.g., [7], [8], [9]).
3. Parametric Estimation of Seismic Wave Dispersion in Building Structures
3.1. Multilayered Kelvin-Voigt model for one-dimensional shear wave propagation
The analytical solution for shear wave propagation in a multilayer Kelvin-Voigt shear beam
was first introduced by Gilbert and Backus [20]. This solution is similar to the Thompson [21]
and Haskell [22] propagator matrix approach and is implemented in the computer program,
SHAKE [23]. Although the solution has already been presented in the literature (e.g., [6], [24]
among others), it is briefly reviewed here to ensure the completeness of the discussion.
The solution to the vertically propagating SH-wave in a layered shear beam (Figure 1) for a
harmonic wave of frequency
f
can be expressed as
 
zktfizktfi eBeAtzu *
2
*
2
,
(1)
4
in which
A
,
B
= constants representing the amplitude of the waves travelling in
z
and
z
(upward and downward) directions, respectively, and
is the complex wave number.
The term
iVV 21
*
is the complex shear wave velocity and
Gf

is the damping
ratio, where
denotes the viscosity and
G
denotes the shear modulus of the Kelvin-Voigt
material model [24]. By introducing a local coordinate system for each layer as shown in Figure
1, the displacement and force continuity condition at each layer interface yields the following
two equations.
 
mm
m
h
m
ik
m
m
h
m
ik
mmmm BAeBeAtuthu
1
*1
1
1
*1
111 ,0,
(2)
 
 
mmmm
m
h
m
ik
m
m
h
m
ik
mmmmmm BAkGeBeAkGtth
**
1
*1
1
1
*1
1
*1
*111 ,0,
(3)
where
 
tzum,
and
 
tz
m,
represent the displacement and shear stress of layer m, respectively,
and the shear stress is defined as
z
u
Gm
mm
*
, in which the complex shear modulus is
expressed as
 
mmm iGG
21
*
.
m
h
denotes the thickness of layer m. Equations (2) and (3) can
be solved to find the wave amplitude factors at layer m+1 as a function of those at layer m.
   
   
1
1
1
*1
1
1
*1
1
1
*1
1
1
*1
1
11
11
2
1
m
m
m
h
m
ik
m
m
h
m
ik
m
m
h
m
ik
m
m
h
m
ik
m
m
mB
A
ee
ee
B
A
(4)
in which
*
*11
1mm
mm
mV
V
is the complex impedance ratio between layer m+1 and m,
m
is the
mass density of layer m, and
m
m
mG
V
*
*
.
Figure 1: Multilayer Kelvin-Voigt shear beam model on a rigid base.
By considering a stress-free boundary condition at the top surface of layer n (i.e.,
 
0,0 t
n
),
it can be concluded that
ABA nn
, where
A
is a constant. The recursive solution shown in
equation (4) can be used repeatedly to find the following relationship between the amplitude of
the travelling waves in layer m and layer n, i.e., the top layer.
 
AfA mm ,θ
and
 
AfB mm ,θ
(5)
Layer 1
Layer m
Layer m+1
Layer nhn
hm+1
hm
h1
nnn
V
,,
111 ,, mmm
V
mmm
V
,,
111 ,,
V
un
zn
um+1
zm+1
um
zm
u1
z1
5
where
 
f
m,θ
and
 
f
m,θ
are scalar valued functions of frequency and the wave model
parameter vector, which is denoted by
θ
. The wave model parameter vector consists of shear
wave velocity, density, and damping ratio of the layers.
The absolute displacement FRF (or transfer function) at the top of layer m with respect to the
displacement input at the base is denoted as
 
fHm,θ
and derived as
   
     
   
1
*
1
1
1
*
1
1
11 ,,
,,
,
,0
,hikhik
mmm
mefef
ff
fhzU fzU
fH
θθ
θθ
θ
(6)
in which
 
fzUm,0
= Fourier transform of the displacement response time history at the top of
layer m (i.e.,
 
tum,0
), and
 
fhzU ,
11
= Fourier transform of the displacement response time
history at the bottom of layer 1 or base (i.e.,
 
thu ,
11
).
Model parameterization
The shear wave propagation in building structures is known to be dispersive, i.e., shear
waves of different wavelength propagate at different velocities. The flexural and flexural-shear
modes of deformation are most likely the main reason for dispersion of shear waves in the
building structure; nevertheless, sudden changes (contrast) in the lateral stiffness of the building
along the height, discontinuity in the lateral load resisting system, reflection of the propagating
waves from geometrical boundaries of the building, and material nonlinearities can also
contribute to the dispersive behavior ( [11], [25]). While using a Timoshenko beam model for
structural system identification can account for the dispersion due to the flexural modes of
deformation [25], other sources of dispersion cannot be characterized by a Timoshenko beam
model. This may lead to biased estimation results, when a Timoshenko beam model is estimated
from the structural response data. To improve the Timoshenko beam model, here we employ an
empirical basis function to characterize the variation of phase velocity as a function of
frequency, as shown in equation (7). This basis function depends on three parameters, which
determine the shape and curvature of the phase velocity function. By defining proper constraints
for the parameters, the basis function can characterize a range of dispersive behaviors, from a
nearly non-dispersive model, to a linearly varying phase velocity as a function of frequency.
 
1
max
2
max
max
ff
c
ff
b
ff
a
fV
V
m
V
m
V
m
m
(7)
In equation (7),
max
f
is the maximum frequency considered in the estimation (i.e.,
max
0ff
).
The function has three degrees of freedom and can be fully identified by determining the slopes
at
0f
and
max
ff
, and the phase velocity at
max
ff
as shown in Figure 2. The shape of the
function is determined by estimating the three unknown parameters, namely
V
m
a
,
V
m
b
, and
V
m
c
for
each layer, from the measurement data fed to the estimation algorithm. To ensure a smooth
positive phase velocity function, monotonically increasing with a negative curvature, the
following constraints are defined for the phase velocity parameters at each layer.
0
2
1
,0,1,0 V
m
V
m
V
m
V
m
V
mcbcba
(8)
6
Figure 2: Schematic representation of the proposed phase velocity function shown in equation
(7).
The phase velocity parameter vector to be estimated for layer m is defined as a (row) vector
 
V
m
V
m
V
m
V
mcba ,,θ
. Consequently, the phase velocity parameter vector for the multilayer model
shown in Figure 1 is defined as
 
V
n
VVV θθθθ ,,, 21
, which is also a row vector.
Likewise, we define the empirical linear function of equation (9) to characterize the damping
ratio as a function of frequency in each layer.
 
mm b
ff
af
max
(9)
where
 
mmm ba ,θ
is the damping parameter (row) vector to be estimated for layer m (
0,
mm ba
) and
 
n
θθθθ ,,, 21
is the damping parameter (row) vector for the multilayer
model. Finally, the layer mass densities are contained in the density (row) vector as
 
n
,,, 21 θ
. Therefore, the wave model parameter vector for the multilayer Kelvin-Voigt
shear beam model is defined as a column vector:
 
T
V
θθθθ ,,
(10)
The successful estimation of all wave model parameters depends on the model
parameterization identifiability. To maintain the model identifiability and eliminate redundancy
in the parameterization, we use the prior knowledge about mass distribution along the height of
building to reduce the number of unknown mass density parameters. We discuss this point in
more detail later in the paper.
Direct differentiation method for computing model response sensitivities
A fundamental step of the Bayesian estimation method is the computation of model response
sensitivities the derivatives of the FRFs with respect to the wave model parameter vector,
θ
,
shown in equation (6). The model response sensitivities can be computed using a finite
difference method, which requires the FRFs to be evaluated at least
 
1
θ
n
times, where
θ
n
is
the size of the wave model parameter vector. Since the finite difference method can be
computationally demanding, the model response sensitivities in this study are computed using a
direct differentiation method to improve accuracy and computational efficiency. Assuming that
 
fHm,θ
as shown in equation (6) is a continuous and differentiable function of
θ
, it follows
that
 
fV
f
max
f
 
max
fV
0f
df
dV
max
ff
df
dV
7
 
 
2
1
*
1
1
1
*
1
1
1
*
1
*
1
11
1
1
*
1
*
1
11
1
1
*
1
1
1
*
1
1
hikhik
hikhik
mm
hikhik
mm
m
ee
e
k
hie
k
hiee
H
θθθθθθ
θ
θ
(11)
in which the dependence of different terms on frequency is dropped for notational convenience.
The terms
θ
m
and
θ
m
are derived using the same recursive approach used to compute
m
and
m
. To find the recursive relation, equation (4) is differentiated with respect to
θ
as
   
   
   
 
θ
θ
θθθθ
θθθθ
θ
θ
1
1
1
*1
1
1
*1
1
1
*1
1
1
*1
1
1
1
1
*1
1
*1
11
1
*1
1
*1
11
1
*1
1
*1
11
1
*1
1
*1
11
11
11
2
1
...
11
11
2
1
m
m
m
h
m
ik
m
m
h
m
ik
m
m
h
m
ik
m
m
h
m
ik
m
m
m
m
h
m
ik
mm
mm
m
h
m
ik
mm
mm
m
h
m
ik
mm
mm
m
h
m
ik
mm
mm
m
m
B
A
ee
ee
B
A
e
k
ihe
k
ih
e
k
ihe
k
ih
B
A
(12)
The terms
θ
*1m
k
and
θ
1m
can be found based on the model parameterization defined in the
previous section. Since
n
A
and
n
B
are constants,
0
θθ
nn BA
; therefore, equation (12) results
in a recursive solution that can be used repeatedly to find
θ
m
A
and
θ
m
B
as
 
Af
Am
m,θα
θ
and
 
Af
Bm
m,θβ
θ
(13)
where
   
θ
θ
θα
f
fm
m,
,
and
   
θ
θ
θβ
f
fm
m,
,
are vector-valued functions.
3.2. Estimation of the frequency response functions from measurements
In seismic interferometry, the transfer functions between output measurements and input
excitations referred to as the FRFs, are usually computed using an empirical transfer function
estimate (ETFE) method (e.g., [4], [5], [6], [26]). An ETFE between the floor absolute
acceleration response measured at level m of a building structure,
 
tam
, and the ground
acceleration input,
 
tag
, can be estimated as
   
 
fA fA
fG
g
m
m
ˆ
(14)
where the numerator and denominator denote the Fourier transforms of the
 
tam
and
 
tag
,
respectively. Assuming that the measurement data are free of noise, the ETFE is an
asymptotically unbiased estimate of the true transfer function (TF). This means that a large
number of data samples in the time domain will ensure the correct estimation of the TFs (or
FRFs herein). Moreover, the ETFE is not a consistent estimator, meaning that the variance of the
estimated TFs will not go to zero even for a large number of data samples in time domain. As a
consequence, the estimated TF fluctuates erratically around the true TF ( [27], [19]). The TF
estimates can be improved by taking advantage of auto/cross spectral estimates as follows
8
   
 
f
f
fG
g
a
g
a
g
a
m
a
m
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(15)
where
 
f
g
a
m
a
ˆ
ˆ
= cross spectral density estimate of
 
tam
and
 
tag
, and
 
f
g
a
g
a
ˆ
ˆ
= auto
spectral density estimate of
 
tag
. The auto/cross spectral density estimates are computed using
Welch's method (e.g., [19]). In order to have an unbiased estimation of the TF values, it is
assumed that the input ground acceleration,
 
tag
, is noiseless and the floor absolute acceleration
response data,
 
tam
, are contaminated with noise that is uncorrelated with the input ground
acceleration.
3.3. Sequential Bayesian estimation of wave model parameters using FRFs
The frequency domain, i.e.,
 
max
,0 f
, is discretized into N sampling points with frequency
spacing
1
max
N
f
f
. At each sampling point, we estimate the FRFs from the analytical
model and measurement data. The predicted FRF from the analytical model at the top of layer m
at the kth frequency sampling point is denoted as
 
Ikm
Rkmmkm HiHfkHH 1,θθ
, where
Rkm
H
and
Ikm
H
are the real and imaginary parts of
km
H
, respectively. The values of the estimated
FRFs can be written in compact form as follows:
 
   
T
I
nk
R
nk
I
n
R
n
I
n
R
n
Ik
R
k
IRIR
kHHHHHHHHHHHH 22111112121111
ˆθy
(16)
in which
 
12
ˆ
kn
ky
(
Nk 1
) is the predicted FRF vector, which contains the real and
imaginary parts of the analytical FRF from first to kth frequency sampling point, and n depicts
the number of layers. Similarly, the measured FRFs are estimated at the same frequency
sampling points and contained in the measured FRF vector,
k
y
, as
T
I
nk
R
nk
I
n
R
n
I
n
R
n
Ik
R
k
IRIR
kGGGGGGGGGGGG
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ22111112121111 y
(17)
The objective of the parameter estimation process is to infer information about the wave
model parameters from the measurements. Due to various sources of estimation uncertainties, the
unknown wave model parameter vector,
θ
, is modeled as a random vector (denoted by
Θ
). The
uncertainties in
Θ
are characterized by a probability distribution function (pdf). The prior
knowledge about the wave model parameter vector is modeled as a prior pdf, which is updated to
posterior pdf by comparing the discrepancies between the predicted and measured FRFs through
the Bayes’ rule as
 
 
k
k
kppp
py
θθy
yθ|
|
(18)
in which
 
k
pyθ|
= posterior pdf of
Θ
,
 
θy|
k
p
= likelihood function,
 
θp
= prior pdf of
Θ
,
and
 
θθθyy dppp kk
|
is a normalizing constant.
9
In order to solve the estimation problem, we next postulate assumptions on the source of
discrepancies between the predicted and measured FRFs. The discrepancies between the
predicted and measured FRFs is expressed as
   
θγθyy kkk ˆ
(19)
in which
k
γ
denotes the simulation error vector and represents the misfit between the predicted
and measured FRFs.
The simulation error vector accounts for uncertainties in the wave model parameters,
estimation uncertainties in the measured FRF vector
k
y
, and also model uncertainties. For
example, the multilayer dispersive shear beam model with Kelvin-Voigt material is an
idealization of the dynamic response behavior of a building structure and the predicted FRFs
would be inherently different from the measured FRFs, no matter how well the parameters are
tuned. This effect is referred to as model uncertainties. If the physics of the idealized model
differ significantly from the physics that characterize the measurements, model uncertainties
would be non-negligible and could result in biased estimation of wave model parameters [28]. In
this case, even though the predictions obtained from the estimated models may match well the
measurements, the estimated model may not be a physically accurate representation of the
building structure. Nevertheless, it is assumed here that the effects of model uncertainties are
negligible. The simulation error vector is therefore modeled as an independent and identically
distributed Gaussian white noise. Consequently, the likelihood function in equation (18) will be
Gaussian, i.e.,
 
kk pp γθy|
. It is furthermore assumed that the prior pdf of the wave model
parameter vector is Gaussian. Thus, equation (18) leads to
 
 
 
 
 
   
2/1
2/
ˆ
1
ˆ
ˆ
2
1
2/1
1
2
1
ˆ
2
2
1
|
P
R
y
yθ
θ
θθPθθθγRθγ
n
T
k
kn
kk
T
k
k
kee
p
p
(20)
where
   
knkn
k22
R
= the covariance matrix of the simulation error vector
k
γ
;
θ
ˆ
is the prior
mean vector and
θθ
Pnn
ˆ
is the prior covariance matrix of the wave model parameter vector;
and,
X
denotes the determinant of matrix
X
. Since
k
γ
is modeled as independent and
identically distributed Gaussian white noise,
k
R
will be a diagonal matrix (refer to [29] for more
details).
Equation (20) results in a non-Gaussian distribution for the posterior pdf of the wave model
parameter vector. To simplify the mathematical derivations, the posterior pdf can be
approximated as Gaussian using a first-order approximation. The first-order Taylor series
expansion of
 
θγk
with respect to
θ
at
θ
ˆ
can be expressed as
   
 
θθCθyyθyyθγ ˆˆ
ˆˆ kkkkk
(21)
in which
 
θθ
θ
θy
C
ˆ
ˆk
is the model response sensitivity matrix and will be computed using the
direct differentiation method described in Section 3.1. By substitution of equation (21) into
equation (20), the Gaussian posterior pdf of
Θ
is approximated as (see Appendix A1 for proof)
 
θθPθθ
yθ
ˆ
1
ˆ
ˆ
2
1
|T
kecp
(22)
10
in which
c
is a constant normalizing factor. The posterior mean vector and covariance matrix of
Θ
are respectively defined as
 
θyyKθθ ˆ
ˆ
ˆˆ kk
and
 
1
1ˆˆ
PCRCP k
T
(23)
where “–” and “+” superscripts indicate the prior and posterior estimate, respectively. The matrix
K is known as the Kalman gain matrix [30], which can be expressed as
 
1
1
1
1ˆ
k
T
k
TRCPCRCK
(24)
Constrained estimation
The model parameterization described in the previous section requires a series of linear
constraints on the wave model parameters, which the posterior estimates obtained from equation
(23) should satisfy. To this end, first the posterior Gaussian pdf that resulted from equation (23)
is truncated at the constraint edges. The constrained posterior mean vector and covariance matrix
are then calculated using the truncated pdf. However, truncating the multi-dimensional pdf of
Θ
with constraints that are linear combinations of wave model parameters could be very
complicated; so, in order to simplify the problem, the wave model parameters are transformed to
a new space to decouple the constraints, so that each constraint will involve only one parameter.
Then, the pdf of the transformed single parameter is truncated to correct for the constraints, and
the corrected mean and variance obtained from the truncated pdf will be transformed back to the
wave model parameter space. This method is borrowed from [30] and [31]. More details are
provided in Appendix A2.
Sequential Bayesian estimation algorithm
We now summarize the Sequential Bayesian estimation algorithm. First, a prior mean vector
and covariance matrix for the wave model parameters are postulated. The prior information is
sequentially updated by comparing the predicted FRF vector,
 
θyˆ
ˆk
, with the measured FRF
vector
k
y
for
Nk ,,2,1
. The posterior mean vector and covariance matrix are corrected to
satisfy the constraints. The corrected mean vector is used in the next step as the prior mean
vector. To improve the convergence of the sequential estimation procedure, a constant diagonal
matrix, depicted by
Q
(known as the process noise covariance matrix in the Kalman filter
estimation method [32]) is added to the posterior covariance matrix to yield the prior covariance
matrix for the next step, i.e.,
kk θθ ˆˆ 1
,
QPP
kk ˆˆ 1
(25)
Table 1 recapitulates the algorithm for the proposed sequential Bayesian method for estimation
of the wave model parameters.
Table 1: Algorithm for sequential Bayesian estimation of wave model parameters.
1. Estimate the measured FRF vector,
N
y
.
2. Postulate a prior mean vector and covariance matrix,
c,0
ˆ
θ
and
c,0
ˆ
P
, respectively.
3. Postulate the diagonal matrix
Q
and constant
r
.
4. For
Nk 1
:
4.1.
ckk ,1
ˆˆ θθ
,
QPP
ckk ,1
ˆˆ
.
4.2. Compute the predicted FRF vector form the n-layer Kelvin-Voigt shear beam model,
 
kk θyˆ
ˆ
.
11
4.3. Compute the model response sensitivity matrix,
 
k
k
θθ
θ
θy
C
ˆ
ˆ
.
4.4. Set
   
knknkr22
IR
.
4.5. Find
 
k
T
kk
TRCPCRCK 1
1
1ˆ
.
4.6. Find
 
kkkkk θyyKθθ ˆ
ˆ
ˆˆ
and
 
1
1ˆˆ
kk
T
kPCRCP
.
4.7. For
s
ns 1
, where
s
n
is the number of constraints, check if the sth constraint (
s
T
ss ba θΦ
) is
violated. If yes, correct for the constraint (refer to appendix A2 for definition of notations):
4.7.1. Find the eigendecomposition
T
kTWTP
ˆ
.
4.7.2. Find the orthogonal matrix
ψ
using the Gram-Schmidt method (see equation (A. 10)).
4.7.3. Find the mean,
1
, and variance,
2
1
, of a standard normal distribution truncated between
 
2/12/1 ˆ
ˆ
,
ˆ
ˆ
sk
T
s
k
T
ss
sk
T
s
k
T
ss ba
ΦPΦ
θΦ
ΦPΦ
θΦ
.
4.7.4. Find
 
k
T
T
ck θψTWθˆ
00
ˆ1
2/1
,
and
 
TT
ck TψWψTWP2/12
1
2/1
,)1,,1,(diag
ˆ
.
4. 52-Story Structural Model Case Study
The benchmark building used for this case study is a 52-story building structure,
instrumented as a part of the Community Seismic Network (CSN) project. In addition to 52
stories above ground, the building has five underground parking stories. The lateral load resisting
system for the building is a steel structure composed of a braced frame core with outrigger
moment frames in both north-south (NS) and east-west (EW) directions for the above-ground
structure, and reinforced concrete basement shear walls for the underground stories. The floor
plans include various set-backs and notches along the building height. A detailed three-
dimensional linear-elastic model of the structural system has been developed in the structural
analysis software ETABS [33] based on the available structural drawings [34] (Figure 3). The
model is used to generate simulated structural response data for the purpose of validating the
wave model parameter estimation method presented in this study.
The beams, columns, and braces in the ETABS model are modeled using structural frame
elements. The floor slabs and shear walls are modeled using linear-elastic shell elements with no
rigid diaphragm constraint at the floor levels. The building mass is modeled using the self-mass
of the structural members in addition to the mass of a uniformly distributed load on the floor
slabs. The floor distributed loads are estimated based on design code requirements. The story
masses of the structural model are listed in Table 2. The floor absolute acceleration responses of
the structural model to a uniform (or rigid) base acceleration are computed using a modal
superposition method [35]. The first 100 structural modes (with natural periods ranging from
sec02.6
1T
to
sec13.0
100 T
) with a uniform 2% modal damping ratio are used for the dynamic
time history analysis.
As mentioned before, each layer of the multilayer Kelvin-Voigt shear beam model represents
a single story or group of stories of the building structures. In this case study, we consider two
layering configurations, which are presented in Table 2. For the first, referred to as Case #1
model, the building structure is modeled as a 51-layer shear beam model. The basement and
penthouse stories are combined together and modeled as a single layer. Every two layers
between layers #2 and #49 have the same phase velocity and damping ratio parameterization (see
12
Table 2). Therefore, 27 sets of phase velocity and damping ratio parameters, or
 
1352327
unknown parameters, characterize Case #1 model. In contrast, Case #2 model consists of a 51-
layer shear beam model, in which each layer has an independent parameterization to characterize
the phase velocity and damping ratio. Therefore, Case #2 model has
 
2552351
unknown
phase velocity and damping ratio parameters. Both Case #1 and Case #2 models have the same
parameterization for mass density, as listed in the last column of Table 2. Since the mass
distribution along the height of the building structure is piecewise uniform, many layers are
expected to have similar mass densities. Furthermore, following equation (4), it can be concluded
that the analytical FRFs depend on the relative mass density of the layers. To simplify the
estimation process, the mass density of the top layer (i.e., layer #51) is set as a unit constant (i.e.,
1
7
). Thus, six unknown mass density parameters (i.e.,
1
to
6
) are used to characterize the
mass distribution for both Case #1 and Case #2 models. In fact, each mass density parameter
represents the relative mass density of the corresponding layer with respect to layer #51. The last
three columns in Table 2 list the labels for parameter sets that characterize phase velocity (and
damping ratio) and mass density.
Figure 3, top: 3D linear-elastic
ETABS model, bottom: typical
floor model.
Table 2: ETABS model parameters and layering configurations used for multilayer Kelvin-
Voigt shear beam model identification.
13
Case #1 Case #2 Case #1&#2
Roof 5.0 240.4
52 5.5 301.6
51 5.8 314.4
50 7.6 1024.2 50 26 50 6
49 4.0 770.5 49 25 49 5
48 4.0 765.5 48 25 48 5
47 4.0 775.2 47 24 47 5
46 4.0 817.0 46 24 46 5
45 4.0 838.0 45 23 45 5
44 4.0 836.1 44 23 44 5
43 4.0 838.9 43 22 43 5
42 4.0 840.1 42 22 42 5
41 4.0 894.1 41 21 41 5
40 4.0 895.8 40 21 40 5
39 4.0 895.9 39 20 39 5
38 4.0 897.1 38 20 38 5
37 4.0 898.2 37 19 37 5
36 4.0 961.3 36 19 36 4
35 4.0 937.2 35 18 35 4
34 4.0 938.8 34 18 34 4
33 4.0 940.4 33 17 33 4
32 4.0 942.1 32 17 32 4
31 4.0 945.9 31 16 31 4
30 4.0 941.0 30 16 30 4
29 4.0 941.2 29 15 29 4
28 4.0 942.8 28 15 28 4
27 4.0 944.4 27 14 27 4
26 4.0 945.8 26 14 26 4
25 4.0 948.0 25 13 25 4
24 4.0 950.9 24 13 24 4
23 4.0 953.4 23 12 23 4
22 4.0 955.2 22 12 22 4
21 4.0 957.5 21 11 21 4
20 4.0 966.2 20 11 20 4
19 4.0 961.8 19 10 19 4
18 4.0 966.4 18 10 18 4
17 4.0 968.4 17 917 4
16 4.0 973.5 16 916 4
15 4.0 976.9 15 815 4
14 4.0 982.4 14 814 4
13 4.0 988.4 13 713 4
12 4.0 993.3 12 712 4
11 4.0 996.1 11 611 4
10 4.0 1000.9 10 610 4
9 4.0 1005.3 9 5 9 4
8 4.0 1010.1 8 5 8 4
7 4.0 1012.3 7 4 7 4
6 4.0 1032.5 6 4 6 4
5 4.6 1072.3 5 3 5 3
4 4.6 1103.1 4 3 4 3
3 4.3 1009.8 3 2 3 3
2 6.1 1100.8 2 2 2 2
1 4.6 3524.7
A 3.6 5830.9
B 3.0 3472.5
C 3.0 3363.5
D 3.0 3362.3
1
1
1
1
51
27
51
7
ETABS Structural Model
Multilayer Shear Model
Level
Height (m)
Story M ass (x 103 kg)
Layer Label
Parameter Set Label
,V
,V
14
In this case study, first, the structural system is identified by estimating the wave model
parameters using the simulated response of the building structure to a low-amplitude earthquake
base excitation. Then, a structural damage scenario is introduced in the structural model by
reducing the axial stiffness of the brace members, which results in reduction of the lateral
stiffness of the structure. Next, the wave model parameters for the damaged building model are
estimated. The objective is to validate the capability of the proposed system identification
method to identify the location and extent of damage by comparing the undamaged (i.e.,
baseline) with “damaged” wave model parameters (i.e., the wave model parameters estimated
from the response of the damaged structure).
4.1. Structural system identification
Two horizontal components of the 1992 Big Bear earthquake ground acceleration, recorded
at the basement of a tall building located in downtown Los Angeles, are selected from the
CESMD database [36] for this study. The ground acceleration time histories, sampled at 100 Hz,
are shown in Figure 4. Since the simulated structural responses are obtained from a linear-elastic
ETABS model in this validation study, the response behavior of the structural model will remain
linear-elastic regardless of the intensity of the earthquake considered. Nevertheless, this is not
the case in a real-world scenario and therefore, a low-intensity earthquake record is selected to
ensure that the response behavior of the structure will remain only weakly nonlinear. Since the
proposed system identification method in this study is based on a linear-elastic response behavior
assumption for the building, the earthquake intensity and therefore, structural response
nonlinearity may have non-negligible effects in a real-world application.
The uniform base acceleration records are applied in the NS and EW directions, and the time
histories of the floor absolute acceleration response are computed at the geometric center of the
floors to reduce the torsional response effects. The simulated structural response and the base
acceleration time histories are polluted by 0.1% g root-mean-square (RMS) independent and
identically distributed zero-mean Gaussian white noise to mimic the measurement noise effects.
The spectral analysis method based on Welch’s averaging method, as described in Section 3.2, is
used to find the NS and EW FRF of each floor’s absolute acceleration response with respect to
the input base accelerations. The acceleration time histories are divided into five estimation
windows with 50% overlap. Each window is weighted by a Hann function. The auto/cross
spectra densities are estimated across each window and averaged over the five estimation
windows. The FRFs are then estimated using equation (15) in the frequency range of 0 to 6 Hz (
6
max f
Hz), with a frequency spacing of
0244.0f
Hz. It should be noted that the presented
method in Section 3.2 to estimate the FRF is only applicable to stationary white noise input
excitations. This is not the case for the earthquake ground acceleration time histories in this case
study, since they are non-stationary and non-white processes and therefore, correlated with the
measurement noise. Consequently, the estimated FRFs might be inaccurate and/or noisy.
Two multilayer shear beam models with independent parameterization are fit separately to
the FRFs estimated in NS and EW directions. Therefore, the structural system is separately
identified in the NS and EW direction. The sequential Bayesian estimation algorithm, as detailed
in Table 1, is used to estimate the wave model parameters in the NS and EW direction. The
initial values for the phase velocity and damping ratio parameters are the same for all layers in
both Case #1 and Case #2 models and in both NS and EW directions. The initial value for the
phase velocity parameter vector is selected as
 
46,7,12000
,
Vinitialm
θ
for each layer. These
15
values result in a maximum shear wave velocity of
m/s300
(
 
m/s300Hz6
max fVV
), an
initial slope for the phase velocity function that is 40 times the linear slope (
max
max
0
40 f
V
df
dV
f
),
and an end slope for the phase velocity function that is 20% the linear slope (
max
max
max
2.0 f
V
df
dV
ff
)
See Figure 2. The initial value for the damping parameter vector is taken as
 
02.0,002.0
,
initialm
θ
for each layer. These values result in a frequency dependent damping ratio
that starts with 2% at
0f
and increases linearly to 2.2% at
Hz6f
. The initial shear wave
phase velocity and damping ratio as a function of frequency are shown in Figure 7. The initial
values for the relative mass densities can be selected based on the prior information about the
mass distribution along the building height. Here, they are initialized based on the story masses
defined in the model as
 
0.1,5.2,0.4,6.4,5.4,4.3,0.16,,,, 7621
initial
θ
, where
0.1
7
is constant. Using these initial values, the prior mean vector of the wave model parameters,
c,0
ˆ
θ
,
is set up according to equation (10). Other different initial values for the wave model parameters
have also been examined to check the consistency of the estimation results.
The prior wave model parameter covariance matrix is defined as a diagonal matrix, i.e.,
 
ic p
,0
ˆ
P
, where
i
p
is the ith diagonal entry of
c,0
ˆ
P
.
i
p
is selected by assuming a uniform 20%
prior coefficient of variation (COV) for each wave model parameter; therefore,
 
2
,,0
ˆ
2.0 ici
pθ
,
where
ic,,0
ˆ
θ
= the ith entry of the wave model parameter prior mean vector. The 20% COV
represents the uncertainty in the initial estimates. The larger the COV, the less confidence exists
about the prior model parameters. Very large values for the COV may result in instability of the
estimation algorithm at early estimation steps, since the estimation algorithm doesn’t rely on the
prior estimates and frenetically perturbs the parameters, which may result in the instability of the
estimation algorithm.
Likewise, the ith diagonal entry of the
Q
matrix is selected as
 
2
,,0
ˆ
01.0 ici θQ
. The
Q
matrix serves as a random disturbance to fuel the estimation algorithm search for optimum
parameter estimates. Large diagonal entries for this matrix may result in estimation algorithm
instability, while too small values may result in non-optimal solutions.
The constant
r
is used to characterize the variance of the simulation error vector (equation
(19)), which in the ideal case (i.e., when no model uncertainties exist), represents the variance of
the estimated FRF values. While an accurate assessment of the FRF estimation variance is
possible (e.g., [27]), it is assumed here that
0.1r
. Assigning large values to
r
means that the
estimated FRF values have large uncertainties; therefore, the estimation algorithm gives
relatively small weight to the discrepancies between the predicted and measured FRFs, which
may lead to incorrect estimation results. On the other hand, assigning small values to
r
serves to
increase the algorithm sensitivity to the discrepancies between the predicted and measured FRFs,
which may result in estimation algorithm instability.
In order to improve the performance of the estimation process, two extra sets of linear
constraints are defined for the wave model parameters in addition to those described in Section
3.1. The maximum phase velocity at each layer is limited to
m/s1000
(i.e.,
16
 
mfVm,m/s1000Hz6
), and the relative mass density parameter values are constrained
between 1 and 12 (i.e.,
61,121j
j
). As for any optimization problem, the estimation
parameters are scaled to improve the performance of the estimation process.
The wave model parameters are estimated considering both dispersive and nondispersive
models. For the nondispersive system identification, the dispersive relations in equations (7) and
(9) are simplified as
 
V
mm afV
and
 
m
af
, respectively, where
V
m
a
and
m
a
denote the
constant (nondispersive) phase velocity and damping ratio for layer m. The same estimation
procedure is used for the nondispersive identification. The initial values for the phase velocity
and damping ratio for the nondispersive model are selected as
m/s9.275
,
Vinitialm
θ
and
%1.2
,
initialm
θ
, respectively, which are equivalent to the initial phase velocity and damping ratio
of the dispersive model at
Hz3f
.
Figure 4: 1992 Big Bear earthquake ground
motion recorded at station #24602 [36]; top:
0° component applied in the NS direction;
bottom: 90° component applied in the EW
direction.
Figure 5 and Figure 6 compare the measured (i.e., obtained from the simulated acceleration
responses of the structural model), initial (i.e., predicted using the initial wave model
parameters), and final estimated (i.e., predicted using the final estimated wave model parameters)
FRFs in NS and EW directions, respectively, for the Case #1 model at four layers: layers #1,
#17, #33, and #51. The real and imaginary parts of the FRFs are plotted separately. As can be
observed, the initial FRFs are significantly different from the measured FRFs, while the final
estimated FRFs using the dispersive model match the measurements with good accuracy. Our
empirical dispersive model captures correctly all the mode shapes in the frequency range of
estimation, which also verifies that the estimation algorithm has successfully reduced the
prediction discrepancies between the initial model and the measurements. Figure 5 and Figure 6
show the estimated FRFs for both the dispersive and nondispersive models, and clearly show that
the dispersive model provides a better prediction of the measured FRFs than the nondispersive
model.
17
(a)
(b)
(c)
(d)
Figure 5: Comparison of the measured (Meas), initial (Init), and estimated FRFs for dispersive
(Est-D) and nondispersive (Est-ND) Case #1 model in the NS direction at four layers: (a) layer
#1, (b) layer #17, (c) layer #33, and (d) layer #51. The real (R) and imaginary (I) parts of the
FRFs are shown separately.
(a)
(b)
18
(c)
(d)
Figure 6: Comparison of the measured (Meas), initial (Init), and estimated FRFs for dispersive
(Est-D) and nondispersive (Est-