Prediction of Wave Propagation in Buildings Using Data
from a Single Seismometer
by Ming Hei Cheng, Monica D. Kohler, and Thomas H. Heaton
Abstract
Crowd-sourced seismic networks in buildings collect important scientific
data, in addition to allowing a diverse audience to visualize the vibrations of buildings.
Visualization of a building
’
s deformation requires spatiotemporal interpolation of mo-
tions from seismometers that are located wherever the crowd places them. In many
cases, a crowd-sourced building network may actually be just a single seismometer. A
method to rapidly estimate the total displacement response of a building based on
limited observational data, in some cases from only a single seismometer, is presented.
In general, the earliest part of the response is simulated by assuming a vertically pro-
pagating shear wave. Later motions are simulated using mode shapes derived from a
beam model (a shear beam, or more generally a Timoshenko beam), the parameters of
which are determined from the ratios of the modal frequencies and the building
’
s
exterior dimensions. The method is verified by (1) comparing predicted and actual re-
cords from a 54-story building in downtown Los Angeles, California, and (2) comparing
finite-element simulations of the 17-story University of California, Los Angeles
(UCLA) Factor building. The response of each of these buildings can be simulated
with a simple shear beam. The importance of including the traveling wave part of the
solution depends on the characteristics of the base ground shaking; the traveling wave
becomes more apparent as the excitation becomes more impulsive. The method can be
straightforwardly applied to multiple instrumented buildings, resulting in a tool to
visualize linear elastic motions of those buildings.
Introduction
Crowd-sourced seismic networks that enable rapid ex-
pansion of sensor deployments in multiple buildings are cur-
rently under development in the United States, including the
Community Seismic Network (
Clayton
et al.
, 2011
) and the
Quake-Catcher Network (
Cochran
et al.
, 2009
,
2011
). These
networks can potentially provide many new observations that
might not be obtained by traditional networks. Installation
and maintenance of stations is largely the responsibility of
volunteers; it is important to help them understand the sci-
ence they are facilitating. In the case of a building seismic
array, the data can be used to produce movies of the build-
ing
’
s deformation in earthquakes. In a crowd-sourced net-
work, station locations are typically chosen for reasons
that are unrelated to the optimal design of a seismic network.
In particular, there may be instances in which a single seis-
mometer is the only data source that is available from a build-
ing. Furthermore, it may not be practical to construct a
detailed finite-element model of the building; even if there
were sufficient resources to construct a dynamic model, the
structural design of the building may not be available. Trans-
fer functions can be used to relate the input and output of a
time-invariant system (e.g.,
Snieder and Safak, 2006
;
Kohler
et al.
, 2007
). Simulated time series from a computational
model or seismic records from both locations are needed to
obtain the corresponding transfer function. In this study, we
present a new computational tool to estimate and visualize
the linear elastic motions of existing buildings that have re-
corded earthquake shaking, given in some instances only a
single seismometer deployed in the building. The method is
verified with the observations from dense structural array
data and numerical simulations.
Using the time series from a single station in a building,
we show how the linear elastic displacement response pre-
dicted for all floors based solely on modal decomposition of
the first few modes does not adequately capture the initial
impulsive response. This can lead to an underestimation of
floor displacement, especially if maximum floor response
occurs during the coherent, impulsive ground motion before
the resonant mode response dominates (
Iwan, 1997
). To
address this issue, we decompose building motion due to
earthquake excitation into the broadband traveling wave
component and the low-frequency resonant mode compo-
nent. Computed mode shapes of the building are used to re-
late the resonant mode response from an observation on one
107
Bulletin of the Seismological Society of America, Vol. 105, No. 1, pp. 107
–
119, February 2015, doi: 10.1785/0120140037
floor to calculated predictions for response on the other
floors.
The application of fixed-base shear beam mode shapes
to tall buildings is often a useful first-order approximation
(e.g.,
Jennings, 2003
), particularly if the observed frequency
ratios match those of the shear beam (i.e.,
f
1
,
3
f
1
,
5
f
1
, etc.,
in which
f
1
is the fundamental translational frequency in a
particular direction). The building response of the other
floors can then be assembled. The method presented here
is suitable if building response remains linear elastic.
Building Response during Earthquakes
Given a particular ground motion, the linear elastic build-
ing response due to earthquake excitation can be computed in
two complementary bases: the traveling wave representation
and the resonant mode representation. As is common in wave
problems, the choice of an appropriate coordinate frame can
greatly simplify the description of the wave. In particular, the
use of a modal coordinate frame is especially helpful in prob-
lems that are primarily composed of standing waves; reso-
nance phenomena are typically easier to analyze with a
modal coordinate frame. On the other hand, traveling waves
are usually easier to represent using characteristic coordinate
frames (e.g., Ray theory). This is especially true for waves
having an intrinsic velocity that is independent of the fre-
quency (nondispersive). Given no observational limitations,
either the modal or the characteristic coordinate frame can be
used to describe the entire motion of a building, but it is easi-
est to simulate the earliest parts of the motion as traveling
waves (characteristic coordinates) and the later parts as
standing waves (modes).
Kanai (1965)
suggested that multi-
ple reflections of waves in an elastic layer could be used to
study seismic vibration of a structure, as well as to design a
seismic resistant structure. Equation
(1)
shows the wave pro-
pagation solution of a continuous fixed-base shear beam
model (e.g.,
Iwan, 1997
;
Sasani
et al.
, 2006
):
u
t; z
X
∞
k
0
−
1
k
u
g
t
−
2
kH
z
c
ξ
2
kH
z
c
;t
−
X
∞
k
1
−
1
k
u
g
t
−
2
kH
−
z
c
ξ
2
kH
−
z
c
;t
;
1
in which
u
t; z
is the (total) displacement response,
t
is time,
z
is height from the ground,
H
is the total building height,
c
is
the traveling wave velocity,
ξ
(distance traveled,
t
) is the
damping function,
is the convolution operator,
u
g
t
is the
horizontal ground motion, and
k
is the index of summation
indicating transit leg. This closed-form solution is true for a
shear beam in which the traveling wave velocity is indepen-
dent of the wavelength, that is, wave velocity
c
4
H
=T
1
.
To include wave dispersion, convolution with functions that
include the phase information would be necessary, but the
solution becomes complicated. Nevertheless, the wave solution
for a shear beam is a good approximation for displacement
response in buildings when the dispersion effect is small
(i.e., when the initial impulse begins its first transit leg up
the building). The damping function is typically simulated with
a minimum-phase causal filter for which the amplitude spec-
trum decays exponentially with the product of frequency and
travel time. This introduces a frequency-dependent group delay
(lower frequencies have larger group delays); that is, damping
introduces some degree of dispersion.
Alternatively, displacement response in buildings can be
obtained by summing an infinite number of resonant modes,
each with a specific phase (e.g.,
Roberts and Lutes, 2003
):
u
rel
t; z
X
∞
i
1
u
rel
i
t; z
X
∞
i
1
φ
i
z
q
i
t
;
2
in which
u
rel
t; z
is the (relative) building response of the
i
th
mode,
φ
i
z
is the
i
th mode shape,
q
i
t
is the damped modal
displacement at the
i
th modal coordinate, and
i
is the index
of summation. Because
u
rel
t; z
is displacement in a noni-
nertial frame, the displacement measured by a seismometer is
u
t; z
u
rel
t; z
u
g
t
:
3
Dispersion effects are implicitly incorporated into this
solution. If viscous damping is proportional to some linear
combination of the masses and stiffnesses (Rayleigh damp-
ing), then the mode shapes are simple functions of
z
, and the
time histories solve linearly damped single-degree-of-freedom
linear oscillator problems. Using a single record to determine
the appropriate transient response of each mode is problem-
atic, especially during the initial upward traveling impulse.
To determine the motion in the modal coordinate frame, we
would need to sum the records from each floor weighted by
the mode shape function. To get around this limitation, we
present a fast and robust method that captures the initial
broadband impulse traveling along the building height by
the wave solution (equation
1
) for the early parts of the mo-
tion and transitions to subsequent low-frequency building
displacement response using the resonant mode solution
(equation
3
). If the duration of the ground motion is short
compared with the travel time up the building, then the wave-
form of the upward traveling wave can be identified because
it is separated in time from later reflected phases.
To transition from traveling wave solution to resonant
mode solution, we use a linear transition function that begins
at the time when the initial impulse first reaches the roof of
the building (i.e.,
t
T
1
=
4
from the start of base excitation)
and ends at the time when the initial impulse finishes its first
reflection back to the building
’
s base (i.e.,
t
T
1
=
2
from the
start of base excitation), in which
T
1
is the fundamental
period of the building. The total estimated displacement re-
sponse
^
u
t; z
is defined as a function of weighted traveling
wave and weighted modal solution:
108
M. H. Cheng, M. D. Kohler, and T. H. Heaton
^
u
t; z
u
w
t; z
×
w
w
t; z
u
r
t; z
×
w
r
t; z
;
4
in which
u
w
t; z
and
w
w
t; z
are the wave solution and
weighting function for the wave solution;
u
r
t; z
and
w
r
t; z
are the resonant mode solution (see equation
2
) and weighting
function for the resonant mode solution. At subsequent times
when wave attenuation and dispersion effects become signifi-
cant, building motions cannot be easily modeled by the wave
solution. The time
t
discussed here is relative to the
S
-wave
arrival time, and
t
0
s is defined here when the initial
S
wave arrives at the base of the building.
We assume the initial impulse takes time
T
1
=
4
to travel
up the building height
H
(equal to the wave travel time in a
shear beam), that is, wave velocity
c
4
H
=T
1
, only in the
first upward leg vertically along the building. The subsequent
modal response solution capturing the low-frequency vibrat-
ing modes is identified either by fast Fourier transform analy-
sis or by system identification methods (e.g.,
Overschee and
Moor, 1994
;
Ljung, 1999
;
Clinton
et al.
, 2006
) applied to the
seismic records (equation
2
). Using the precalculated mode
shapes that assumed a beam representation, the modal dis-
placement responses on the other floors can be computed.
Seismic Records from a 54-Story Building
We present earthquake records of a 54-story office
building in downtown Los Angeles, California, (Fig.
1
)to
describe our response prediction method. The 54-story build-
ing, constructed in 1991, is rectangular with base dimensions
of
64
:
7
m×
41
:
5
m and founded on a concrete mat founda-
tion. The lateral resisting system is mainly composed of the
moment-resisting perimeter steel frame with 3 m column
spacing. There are Virendeel trusses and 14.6 m transfer
girders at the 36th and 46th floors to accommodate the dis-
locations of vertical structural elements. The building is in-
strumented by the California Strong Motion Instrumentation
Program, with accelerometers on the penthouse, 46th floor,
36th floor, 20th floor, ground floor, and the P4 underground
level. Structural details and seismic records of this building
can be found in the Center for Engineering Strong Motion
Data database (see
Data and Resources
).
Rahmani and Todorovska (2014)
suggested a layered
shear beam and torsional shaft model and
Ebrahimian and To-
dorovska (2014)
suggested a layered Timoshenko beam
model to represent the building. Records from several seismic
stations in the building are required to construct their proposed
64.7m
41.5m
7.9m
4.1m
23.8m
4.1m
35.7m
4.1m
126.8m
4.6m
7m
4.6m
6.1m
3.4m
Sensor Location
Total height = 210.2 m
Figure
1.
Photograph and diagrams of the 54-story building in downtown Los Angeles, California, showing building dimensions, floor
heights, and seismic network configuration (from the Center for Engineering Strong Motion Data database, see
Data and Resources
). The
color version of this figure is available only in the electronic edition.
Prediction of Wave Propagation in Buildings Using Data from a Single Seismometer
109
models based on deconvolution interferometry. In this study,
we will focus on building response prediction using only data
from as few as a single seismometer and without the knowl-
edge of a computational model for the building. We show that
although our proposed method neglects the influence of tor-
sion (which can be significant for tall buildings), our first-
order prediction agrees well with the measured data.
Figure
2
shows the recorded displacements in the north
–
south direction of the 54-story building on the 20th, 36th,
46th, and penthouse floors during the 29 July 2008
M
5.4
Chino Hills, California, earthquake (epicentral distance
47 km). The displacement spectra demonstrate that the build-
ing
’
s displacement responses are dominated by the second
north
–
south mode. The influence due to the first mode is
significant, and there are lesser contributions from the third
mode. The first, second, and third north
–
south modal fre-
quencies of the building are observed to occur at 0.18,
0.52, and 0.84 Hz. The natural frequency ratios are approx-
imately 1, 3, and 5, so we assume the mode shapes of this
building are close to those of a fixed-base shear beam. In
addition, an initial upward traveling impulse is observed at
t
0
s (Fig.
2
). The first crest of the impulse is reflected at
the roof at
t
1
:
5
s, which is close to one-fourth of the
building
’
s fundamental period.
To demonstrate our method, we use the single record
from the 20th floor to predict records from other floors. We
first investigate the building response estimate solely based on
resonant modes. As was previously mentioned, it is difficult to
decompose the 20th floor record into the modal coordinate
frame. To get around this limitation, we decompose the mo-
tions into the different modal coordinates by using a band-pass
filter that is centered on the natural frequency of each mode to
calculate the resonance response that is harmonic-like at the
natural frequencies of the building. The difference between
these harmonic resonances and the actual motions are ascribed
to traveling waves. In this example, the 2008 Chino Hills
earthquake displacement records are band-pass filtered with
second-order, zero-phase Butterworth filters for frequencies
0.1
–
0.4 Hz for the first resonant mode, 0.4
–
0.7 Hz for the sec-
ond resonant mode, and 0.7
–
1.0 Hz for the third resonant
mode. The resonant mode (total) response
u
r
t; z
at height
z
from the ground, constrained by data produced by a single
seismometer at height
z
s
, can be approximated as
u
r
t; z
j
seismometer at
z
s
≈
X
N
i
1
f
i
t; z
s
φ
i
z
φ
i
z
s
u
rm
t; z
s
;
5
0
1
2
Spectra
Frequency (Hz)
−1
1
Mode Shape
Normalized Deflection
0
5
10
15
NS Displacement Response
Time (s)
Ground
20th floor
36th floor
46th floor
Penthouse
observation
prediction using 3 modes
prediction using 5 modes
1st mode
2nd mode
3rd mode
Figure
2.
Simulated (based only on resonant mode solution) and observed displacement responses of the 54-story building in the north
–
south
(NS) direction during the 2008
M
5.4 Chino Hills, California, earthquake. The vertical locations of the records are shown by the floor levels in the
building. Displacement time series are normalized to the same scale by data from the penthouse, which has maximum absolute displacement of
1.45 cm and minimum absolute displacement of
−
2
:
34
cm. The color version of this figure is available only in the electronic edition.
110
M. H. Cheng, M. D. Kohler, and T. H. Heaton
in which
t
is time,
N
is the number of vibrating modes ob-
served in the seismic record,
f
i
t; z
s
is the filtered response
(which represents the damped modal response) for the
i
th
mode for the single record at height
z
s
,
φ
i
z
is the mode
shape deflection value as a function of height
z
for the
i
th
mode, and
u
rm
t; z
is the residual motion after the filtered
response component has been subtracted from the data at
height
z
s
. Comparing with equation
(3)
, we assume this
residual motion
u
rm
t; z
to be the part of the ground exci-
tation that will be experienced by all the floors. It is com-
puted as follows:
u
rm
t; z
s
u
r
t; z
s
−
X
number of modes
i
1
f
i
t; z
s
:
6
In other words, the filtered time series for different modes are
related to the other floors using the corresponding mode
shapes of the building, and damping is implicitly included
in the floor response prediction.
The time series comparisons between the estimation
from the 20th floor and the data are displayed in Figure
2
.Even
when we include two extra north
–
south modes (the fourth and
fifth north
–
south modes at 1.16 and 1.51 Hz;
Rahmani and
Todorovoska, 2014
) in equation
(5)
, the predicted floor dis-
placement is very similar to the response using only the first
three modes, as shown in Figure
2
. The prediction solely based
on modal decomposition using the first few modes does not
adequately capture the initial upward traveling impulse from
the ground, and it underestimates the peak floor response at
the penthouse by a factor of more than 2 at
t
≅
1
:
5
s. However,
this harmonic modal estimation matches the seismic records
for subsequent times after the initial impulse has reached the
top of the building.
Iwan (1997)
mentioned that pulses travel
through the buildings as waves, and the conventional tech-
niques using modal superposition method and response spec-
trum analysis may not capture the effect of these pulses. They
indicated that maximum floor response might occur during the
coherent impulsive ground motions before the resonant mode
response dominates.
Building Response Prediction Using a Single
Seismometer
Because modal decomposition using the first few modes
does not capture the initial traveling impulse during earth-
quakes, we use the traveling wave solution to capture the ini-
tial upward traveling impulse along the building height. The
initial impulse is observed in the displacement record from
the 20th floor of the 54-story building for the 29 July 2008
Chino Hills, California, earthquake (Fig.
3
). This initial im-
pulse shape is replicated on other floors according to the trav-
eling shear-wavespeed estimated for this building, that is,
wave velocity
c
4
H
=T
1
4
×
210
=
5
:
83
144
m
=
s.
We simulate this impulse for a cycle that includes one up-
ward and one downward leg vertically along the building.
Using our approximated shear-wave velocity, we estimate
the first crest of the impulse will arrive at the roof at time
t
1
1
:
5
s, where it will be reflected and will reach the base
of the building at time
t
2
2
:
9
s. In this example, let TS
b
t
be the time series response before
t
1
on the 20th floor; the
value of TS
b
t
is zero for
t>t
1
. Similarly, let TS
a
t
be the
time series response after
t
1
on the 20th floor; thus the entire
time series on the 20th floor is TS
20
F
t
TS
b
t
TS
a
t
.
This representation allows us to simply shift the time series
TS
b
t
and TS
a
t
according to our estimated wave velocity
to simulate the wave propagation effect up the building. The
simulated wave solution for the initial impulse on the target
floor
u
w
t; z
can then be expressed as
u
w
t; z
≈
TS
b
t
−
z
−
z
s
c
TS
a
t
z
−
z
s
c
;
7
in which
z
s
is the height of the seismometer floor from the
ground and
c
is the wave velocity. In our example,
z
−
z
s
=c
is positive when the target floor is above the 20th floor (the
single seismometer floor), which implies TS
b
t
and TS
a
t
will merge;
z
−
z
s
=c
is negative when the target floor is
below the 20th floor, which implies TS
b
t
and TS
a
t
will
diverge. In theory, the complete wave solution should include
a damping function (like the one in equation
1
). In our pro-
posed method, this wave solution is only used between
t
0
to
t
2
. Given the short traveling distance, the damping func-
tion is excluded for simplicity, and the simulated building
response still matches the observation well. For the 54-story
example, the simulated impulse travels up the building. Then
we apply weighting functions according to the time
t
1
and
t
2
for the transition to the resonant mode responses (see equa-
tion
4
and Fig.
3
). For subsequent times, the observed dis-
placement record from the single seismometer is band-pass
filtered with a second-order, zero-phase Butterworth filter on
the first three resonant frequencies to get the damped modal
response time series. Responses for the other floors are re-
lated to the 20th floor using the mode shapes of a fixed-base
shear beam (equation
5
).
Figures
4a,d,g,j
and
5a,d,g,j
show the results of the
method applied to the 54-story building comparing the four
different single seismometer inputs (20th, 36th, 46th, and pent-
house floors) during the 29 July 2008 Chino Hills, California,
earthquake. Unlike the mode-only solution, this method also
models the initial traveling wave. Figures
4a
and
5a
show the
predicted floor response time series and spectra using the rec-
ord at the penthouse as input. The penthouse data successfully
capture the first three modes, so the subsequent modal re-
sponses are well estimated on th
e other floors. The same ap-
plies to the data recorded on the 46th floor (Figs.
4d
and
5d
).
The 36th floor is near a nodal point of the second mode, so the
data do not capture any contribution from the second mode
(Figs.
4g
and
5g
). With only combinations of the first and third
modes, the predicted responses on the other floors are some-
what degraded. Although the 20th floor is near a nodal point in
the third mode, the prediction agrees with the observation using
Prediction of Wave Propagation in Buildings Using Data from a Single Seismometer
111