A Time-Reversed Reciprocal Method for Detecting High-Frequency
Events in Civil Structures with Accelerometer Arrays
Monica D. Kohler
1
, Thomas H. Heaton
2
, and Vanessa M. Heckman
2
1
Center for Embedded Networked Sensing, Uni
versity of California at Los Angeles,
Los Angeles, CA 90095
2
Department of Civil Engineering, Cal
ifornia Institute of Technology,
Pasadena, CA 91125
kohler@ess.ucla.edu
,
heaton@caltech.edu
, heckman@caltech.edu
ABSTRACT
A high-frequency experimental method of detecting
a failure event in engineered structures is pre-
sented that uses the property of wave propagation r
eciprocity and time-reversed reciprocal Green’s
functions. The premise is that if a numerical datab
ase of pre-event, source-receiver Green’s functions
can be compiled for multiple locations of potential da
mage in a structure, that database can subse-
quently be used to identify the location and time of o
ccurrence of a real failure
event in the structure.
Once a fracture source emits a wavefield that is reco
rded on a distributed set of accelerometers in the
structure, time-reversed waves can be obtained by
convolving the displacements with the database of
time-reversed Green’s functions and stacking the results. The correct location and time of the fracture
source can be inferred from the subset of Green’s functi
ons that exhibits the best focus in the form of a
delta function. The 17-story, steel moment-frame UCLA Factor building contains a cutting-edge, con-
tinuously recording, 72-channel, seismic array. Th
e accelerometers’ 500 sample-per-second recordings
have been used to verify the ability to observe impulse-like sources in a full-
scale structure. Applica-
tion of an impulse-like source on the 3
rd
and 15
th
floors of the Factor building shows that the associated
displacements serve as useful approximations to the
building’s Green’s functions in the far field, and
can be used in investigations of scenario fracture location and timing.
INTRODUCTION
Understanding the data signature of building damage
due to earthquake shaking is fundamental to
designing structural health monitoring and damage detection tools. The 1994 Northridge, California
earthquake highlighted a common type of structural
failure, fractured welds in beam-column connec-
1
Professional Researcher
2
Professor, Graduate Student
tions, that is difficult to identify either visually
or through localized ultrasonic testing (Roeder, 2000).
Weld fracture significantly decreases the ductility of tall steel buildings (Hall et al., 1995). The preva-
lence of fractured welds in mid-rise and high-ri
se structures shows how new computational tools are
needed to immediately identify their occurrence after a large earthquake for the health of the structure
and the safety of the occupants.
Dense structural networks are producing orders of
magnitude more data than before; thus it is be-
coming more challenging to wade through the volumes
of data to determine whether something unique
has happened and where in the structure it has happene
d. There is a need for new computational tech-
niques that will enable end-users to identify and locate (spatially and temporally) significant damage
events. Since uncertainty is intrinsic in any analysis of this type, it
will also be necessary to develop
robust methodologies for error estimation. Predictive numerical simulations will help to determine just
how well structural array data can resolve the location of damage. Advances in SMART structure em-
bedded sensor networks and sophisticated in-network processing are making damage assessment tech-
niques, such as that introduced
here, application-realistic.
We present a new, high-frequency numerical method of detecting a failure event in engineered
structures that uses the property of wave propagation
reciprocity and time-reversed reciprocal Green’s
functions. The focus here is on a specific type of
structural failure event: brittle-fractured welds of
beam-column connections. In this st
udy, we postulate that if we can compile a numerical database of
pre-event, source-receiver Green’s functions for mult
iple locations of potential damage in a structure,
we should be able to use that database to identify
the location and time of occurrence of a subsequent,
real failure event in the structure. Once a fracture source emits a wavefield that is recorded at a distrib-
uted set of accelerometers in the structure, time-
reversed waves can be obt
ained by convolving the
recorded displacements with the database of time
-reversed impulse-source Green’s functions. The pre-
event impulse sources can be, for example, hammer blows applied adjacent to the welded connections,
ideally during construction when they are still accessi
ble. The correct location and time of each subse-
quent fracture source can be inferred by identifyi
ng which Green’s function most nearly produces a
delta function when it is time-reversed and convolve
d with a record from an individual station. Time-
reversed convolution of two time series is, alternatively, cross correlation. If an actual weld fracture is
seen in several stations, it should produce an impulse at the same time for the
cross correlation at each
station. By summing (stacking) the cross correlatio
ns from several stations, it should be possible to
significantly improve the resolution of the technique.
This approach assumes that we have a general id
ea of where the failure event (i.e., the fractured
weld) is likely to occur based on the known weld locations, and that a multi-channel seismic network
will record the event. The number of structures with
dense seismic networks embedded in them is in-
creasing; thus there is an opportunity for new approaches to identifying damage that take advantage of
high-density processing techniques such as stacking
and beamforming for significantly increased sig-
nal-to-noise ratios. The expense and permitting diffic
ulties associated with installing permanent seis-
mic arrays are rapidly becoming surmountable with the growing popularity of deploying cheap, easy-
to-install, MEMS-based USB sensors at desktop computer sites.
Whereas traditional modal analysis typically used in
system identification techniques theoretically
does not need input data from more than one sensor to
detect perturbations in modal frequencies, char-
acterization of the complete wavefield from an intern
al (e.g., weld fracture) or external (earthquake)
source of vibration requires spatial sampling on th
e scale of twice the smallest wavelength desired.
The development of robust designs in seismometer
hardware and software is making it more feasible
to densely instrument civil structures on a permanent
basis in order to study their states of health.
The 17-story UCLA Factor building (Fig. 1) contai
ns one of those cutting-edge structural arrays,
recording building vibrations at high sample rates.
The array consists of 72 single-channel accelerome-
ters recording continuously on 24-bit digitizers. It
is one of only a handful of buildings in the U.S.
permanently instrumented on every floor, providing
information about how a common class of urban
structures, mid-rise moment-frame steel buildings
, will respond to strong ground shaking (Kohler
et al
.,
2007). Structural stiffness undoubtedly decreased wh
en welded, beam-column connections fractured
extensively in numerous moment-frame steel buildi
ngs during the 1994 Northridge earthquake. Unfor-
tunately, there are very few seismic records from bu
ildings with this type of damage. Ample experi-
mental evidence motivates an in-depth study on
how changes may be observable through analysis of
vibration data for an instrumented building.
THEORY
The purpose of this paper is to develop a met
hod which illustrates how wave propagation proper-
ties could be used in a new approach
to identify structural damage events in a structure. We are moti-
vated by experimental evidence that such waves will be readily observable on strong-motion networks
by understanding what features to look for in high-samp
le-rate data. Our strategy is to detect character-
istic high-frequency waveforms produced at each station by fracture of any individual weld. Since we
cannot actually fracture all the welds in a building
to produce these waveforms, we explore the possi-
bility of using data from hammer blows adjacent to welds as proxy sources.
We first discuss the similarities and difference
s of high-frequency waves expected from two types
of sources: 1) an impulsive point force due to a ha
mmer blow, and 2) a near-instantaneous tensile fail-
ure of a welded connection. In particular, we are in
terested in the differences in the waveforms that
will be produced by these two sources. By definition, the displacement from the hammer blow,
u
H
, is
described by the Green’s function for the building, or
(1)
)0,()()(
ij
j
i
ξ
x
ξ
x
τ
;,tG,
τ
F,tu
H
−∗=
where the observed i
th
component of displacement occurs at receiver location
x
, and the source occurs
at location
ξ
at time
τ
(e.g., Aki and Richards, 1980).
G
ij
is the Green’s function for the i
th
component
of displacement due to an impulse source point force
F
in the j
th
direction for a specific source-receiver
path.
The weld fracture problem is more complex. In
principle it could be simulated using an elastic
model with a tensile crack that experiences a step
change in normal traction on the crack surface.
However, this is a difficult problem to solve anal
ytically and we can gain important insight by ap-
proximating the weld fracture as a localized region that experiences very large elastic tensile strains,
thereby causing finite opening across
the strained region. This is a body-force equivalent source that is
often used in seismology. The source is characterized by the seismic moment tensor which is essen-
tially a point stress multiplied by the product of th
e crack area times the crack opening (Aki and Rich-
ards, 1980). Unlike a point force, the moment tensor
consists of combinations of linear force couples
(the diagonal components) and shear force couples (the off-diagonal components). The response of the
medium to these force couples is just the spatial
derivative of the point force Green’s functions, or
).0,;(*),()(
ij
k
jk
i
ξ
x
ξ
x
τ
,tG
ξ
τ
M,tu
−
∂
∂
=
(2)
The body force equivalent source for a point tensile crack is given by Burridge and Knopoff
(1964) as a point moment tensor whose amplitude is
(3)
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
=
λ
λ
μλ
DSt
00
00
002
)(
0
M
where
S
is the area of the tensile crack,
D
is the average opening,
λ
is the first Lamé constant and
μ
is
the rigidity. Since we are describing the size of a sing
ular strain that results from a singular point stress,
the solution only depends on the elastic constants at th
e point of application of the stress. We have as-
sumed that the crack opens instantaneously and hence the time history of crack opening is assumed to
be a Heaviside step function. Since everything in
the problem is linear, any opening time history can
be achieved by convolution with th
e appropriate time function.
We will show, however, that for a broad, applicable set of fracture recording conditions, the impul-
sive force Green’s functions serve as useful approxima
tions to their spatial derivatives, but with a step
function time history,
ij
k
0
jk
i
))(
G
ξ
H(
τ
M,tu
∂
∂
∗
=
x
(4)
(
∫
)
∞
∞−
−
∂
∂
=
τ
,
τ
tG
ξ
M
d0;
ij
k
0
jk
ξ
x,
. (5)
The point source is treated here as a general source. For example, a weld fracture at or near the steel
beam-column connection is likely to be a combina
tion explosive plus single force couple mechanism.
Bolt slippages would be analogous to
strike-slip double couple sources.
For any point source in an elastic medium, the re
sponse of the medium can be decomposed into
near-field and far-field terms. We will use this sepa
ration to show that for recording conditions in a
building, we can use
G
as an approximation for the time integral of its spatial derivative. The near-
field terms generally comprise the particular solution to the equation of motion. Near-field terms are
necessary to describe any static displacements that result from the source (i.e., a steady-state solution).
Although their time history can be complex, their tim
e dependence is generally given by that of the
point source, or definite time integrals of the tim
e history. The distance decay of near-field terms is
typically the same as, or faster than
, the decay of the static solution.
The rest of the solution is described as far-field terms. These far-field terms are solutions to the
homogeneous equation of motion and they can be viewed as the free vibrations of the system. The far-
field terms generally have a time history that is a
time derivative of the time history of the near-field
terms. In the absence of significant damping, they
typically persist indefinitely with equipartition of
potential and kinetic energy. The far-field terms can
always be decomposed as either a sum of normal
modes, or as a sum of traveling waves. If we consider the Green’s function to be a sum of rays, each
traveling at some velocity
c
ℓ
, then we can approximate the far-field portion of the Green’s function
(the large distance, high-frequency part of the solution) as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−δ≈
⎟
⎠
⎞
⎜
⎝
⎛
>−
∑
∞
=
l
l
l
l
c
r
τ
A
c
r
ττ
,tG
1
ij
ij
,;
ξ
x
(6)
where
A
are the amplitude coefficients corresponding to each ray
ℓ
(e.g., P- and S-waves traveling dif-
ferent paths through the frame) and
signifies the path length that the ray takes to get from the
source to the receiver. This approximation is an overs
implification of the dynamics of a real building
where various wave types may be dispersive (e.g., bending waves in a floor slab). Furthermore, there
may be parts of the solution that are difficult to re
present as rays. Nevertheless, this analysis shows
why weld fractures and hammer blows have far-field wavefields that are similar in nature. Considering
only the radial portion of the gradient operator, Eqs. (5) and (6) can be rewritten as
l
r
()
∫
∑
∞
∞
∞
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−δ
∂
∂
≈
-
1
ij
0
jk
i
d
τ
c
r
τ
A
ξ
M,tu
k
l
l
l
l
x
(7)
∫
∑
∞
∞
∞
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−δ
′
=
-
1
ij
0
jk
d
1
τ
ξ
r
cc
r
τ
A
M
k
l
ll
l
l
l
(8)
τ
c
r
τ
BM
d
1
ij
0
jk
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−δ
′
=
∫
∑
∞
∞−
∞
=
l
l
l
l
(9)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−δ
=
∑
∞
=
l
l
l
l
c
r
τ
BM
1
ij
0
jk
. (10)
We see that we expect similarities between the seismograms produced by a hammer blow point
source (Eq. 6) and the seismograms produced by a weld fracture (Eq. 10) since they are only different
by the ratio of the constants
A
and
B
. Ideally we would need to compute
∂
G
ij
/
∂
ξ
k
to obtain the appro-
priate displacement response for a fracture source.
However, Eq. (10) shows that the impulse source
G
ij
serves as a useful approximation since it is different from
∂
G
ij
/
∂
ξ
k
only by a constant of propor-
tionality in this approximation. Of course, the probl
em is more complex in detail and a careful com-
parison requires numerical simula
tion of hypothetical buildings experiencing impulse force and open-
ing crack source mechanisms.
OBSERVATIONS OF POINT FORCES AND FRACTURES
If the excitation is an impulse that can
described by a point force in the j
th
direction
F
j
(
ξ
,
τ
)=F
0
j
δ
(
ξ
)
δ
(
τ
), what might the Green’s functions look like recorded on an actual building net-
work? In April, 2007, we conducted an experiment
to record responses to an impulse-like source on
the 3
rd
and 15
th
floors of the UCLA Factor building. The goal was to test whether a single, high-
frequency impulse source would be observed throughout th
e building; the data illustrate that, indeed, it
is. We applied a hammer blow to a steel plate
on the concrete floor of a corner of the 15
th
floor and in a
stairwell in the SW corner of the 3
rd
floor. The data recorded from the 3
rd
floor hammer blow is shown
in Fig. 1. The 15
th
floor hammer blow waveforms are comparable, with the top floors exhibiting the
earliest arrivals and highest signa
l-to-noise ratios. The blow was nearest the accelerometers on the
south and west walls of the 2
nd
and 3
rd
floors. These waveforms are the responses recorded on the 500
sps streams of the Factor array; however they ha
ve been filtered for freque
ncies between 10 and 95 Hz
in order to compare with 200 sps fracture data pres
ented in the next secti
on. Maximum acceleration
was 0.006
g
. Fig. 1 shows the resulting traveling wave throughout much of the building. Top floors
typically have higher background noise, eventually
obscuring the signal. The travel time from the 3
rd
floor to the top of the building for this frequency range is ~0.05 s, corresponding to an average wave
speed of 1200 m/s. Opposite floor arrivals are de
layed due to horizontal wave propagation through
slower concrete floor slabs.
Fig. 1.
Left
: The UCLA Factor building and its seismic arra
y. Arrows show locations and polarities of
sensors on each floor.
Right
: Accelerations (top) and displacements (bottom) recorded from force
hammer impulse source applied near 3
rd
floor SW corner in the Factor building.
To demonstrate what an actual fracture would look like on typical strong-motion instrumentation,
we recorded data from an induced fracture that occurred during a beam-column steel connection test
for a commercial steel manufacturer. The test was conducted at UC San Diego’s Department of Civil
Engineering. Cyclical loading was applied to a full-s
cale, steel beam-column to test its brittle-plastic
response. Prior to the test we installed a ±2
g
Episensor at one end of the beam and connected it to a
24-bit digitizer. The initial sensor orientations were
horizontal along the length of the beam (X direc-
tion), horizontal and perpendicular to the beam (Y di
rection), and vertical (Z direction). We recorded
waveform data at 200 sps throughout the test. Cyc
lical loading corresponding to story drifts between
1% and 7% was applied at one end of the beam by a
hydraulic piston. The test ended at 7% story drift
when a crack occurred at the bolts
near the welded beam-column conn
ection. Interestingly, the crack
did not actually occur in the weld but at the bolts which experienced large plastic strains and eventual
embrittlement due to strain hardening.
The accelerometer recorded both bolt stick-slip and
actual fracture events on scale (Fig. 2), driving
home the fact that such failures in real structures
will be recorded on-scale on modern equipment, and
are observable because their amplitudes are much la
rger than other sources of vibration. Maximum
acceleration from the fracture was 1.5
g
. To predict what wave propagation might look like in the Fac-
tor building from a scenario fracture source suggested by the data, we convolved the hammer source
displacement pulses with the X-direction fracture
displacement. The predicted response exhibits the
dramatically increased signal-to-noise ratios observed in the beam fracture data (Fig. 2).
Fig. 2.
Left
: Accelerations recorded during
moment-frame connection test.
Middle
: Blow-up of dis-
placements at time of fracture.
Right
: Predicted response of Factor building to scenario fracture source.
RECIPROCITY AND SOURCE IMAGING
In order to better understand the specific scheme th
at we are proposing to detect and locate weld
fractures, it is helpful to first review the concept of
source-receiver reciprocity. Because the system is
linear, we can always exchange source and receiver locations (i.e., put our impulsive force at an accel-
erometer location and vice versa) and obtain the same Green’s function. Space-time reciprocity dic-
tates that
)
()0,()0(
ji
ji
ij
t,
τ
;,G
τ
;,tG,
τ
;,tG
−−=−=−
x
ξ
x
ξξ
x
(11)
where
G
is the time-reversed Green’s function for th
e source-receiver path (e.g., Aki and Richards,
1980). This also means that if the source and recei
ver positions are interchanged, identical seismo-
grams
u
and
v
would be recorded at the two sites, except
that waves would be traveling either forward
from the original source or backwards from the new “virtual” source (old receiver) (e.g., Derode et al.,
1995). We conclude that
),(),,()(
i
ji
j
tFt
τ
;G,
τ
v
xx
ξξ
∗−−=
(12)
where
v
is the displacement that corresponds to the ne
w receiver (original source) location and time.
In an ideal experimental set-up where a well-distr
ibuted structural network has recorded one or
more weld fractures and their locations are obtaine
d through waveform or travel-time inversion, we
could invert for
M
jk
given numerically or empirically determined
G
ij
. This type of waveform inversion
is commonly done for 3D source characterization of cracks in elastic media to identify the kinematic
source mechanisms. However, it does not allow for the
identification of the source location if the struc-
ture’s distribution of physical properties is not
known, for example through a finite-element stiff-
ness/mass model. For this we turn to the time-reve
rsed reciprocal Green’s method and stacking tech-
niques to determine the source location. We take a
dvantage of the fact that we can use the impulse
source
G
ij
instead of
∂
G
ij
/
∂
ξ
k
in the far-field approximation (Eqs. 6 and 10).
When a fracture emits a wavefield that is recorded
at a distributed set of sensors in the building,
those displacements can be treated
as secondary sources of energy that are retransmitted back to the
original true fracture source. If we know all the pre-damage-event
G
ij
corresponding to a complete set
of potential damage locations in a linear structure,
we can use forward modeling to compute a range of
v
to identify the correct
ξ
and
τ
, from the suite of
u
that recorded damage events. Each recorded dis-
placement acts as a secondary point source of wa
ves that, summed, will colla
pse back in time and
space to the correct original source
if the correct set of time-reversed Green’s functions is used. The
recorded displacements, acting as the distributed
source, are summed to define the excitation force
(13)
∑
=
=
W
w
ww
tuCtF
1
i
i
),(
),(
x
x
(Fink, 1997; Larmat
et al.
2006).
W
is the total number of receivers that recorded the fracture event,
x
w
are the receiver positions, and
C
w
is an amplitude coefficient that allows for receiver weighting.
Upon exchanging source and receiver locations, the
time-reversed reciprocal expression for dis-
placement in the far-field approximation is then
()
∫ ∫∫∫
−−
=
T
τ
V
Vt,
τ
;,GtFt
τ
v
d)(,d),(
ji
j
x
ξ
x
ξ
(14)
[
∑
=
∗
=
W
w
ww
GtuC
1
ji
i
),(
x
]
(15)
where
v
j
is the j
th
component of displacement at
ξ
that corresponds to the new receiver (original
source) location and time. The waveforms are window
ed and filtered for the finite time interval
τ
to
T
.
Ideally, a large number of azimuthally distributed receivers would produce the best time-reversed
wavefield. Earthquake source imaging studies have
shown that even a limited number of receivers
with associated receiver weights that account for un
even distribution can produce meaningful results
(Larmat
et al.
, 2006).
When the displacements containing the fracture signal (Fig. 3b) are individually convolved with
the time-reversed impulse source Green’s functions (F
ig. 3a) and then stacked, the result is the time-
reversed wavefield. We infer the source location a
nd time from those values corresponding to the stack
that exhibits the maximum amplitude or “best” focus. Stacking the
v
using the correct set of
G
from a
specific fracture location will result in an approximate delta function at
t
=
τ
, corresponding to maxi-
mum constructive interference (Fig. 3c). The stack adds most coherently for the Green’s functions that
correspond to the closest fracture location because they will be approximately in phase. The stack
should grow increasingly less focused moving away fro
m the location of radiation because the Green’s
functions are becoming less and less coherent (Fig. 3d).
a
)
b
)
c
)
d
)
Fig. 3. Schematic diagrams showing: (
a
) Source-receiver paths for impulse source produced at numer-
ous possible weld fracture locations for Green’s functions. (
b
) Fracture subsequently occurs at a con-
nection, recorded by all receivers. (
c
) Displacement at
correct
fracture location produced by back-
projecting Green’s functions for correct subset of source-receiver paths. (
d
) Displacement at
incorrect
fracture location by back-projecting Green’s functions for another subset of source-receiver paths.
This is more conveniently approached as a cr
oss-correlation problem in which maximum values
are obtained by cross correlating discrete
u
with the correct set of discrete
G
ij
. Whereas convolution of
discrete time series involves the cu
mulative product of the time-shifted
ji
G
and each
u
, cross correla-
tion can be found as the cumulative product of
G
ij
and
u
. Maximum correlation occurs when
G
ij
and
u
are least time-shifted relative to each other. The algorithm would involve systematically running
through the catalogued impulse source Green’s functi
ons, each with a known source location, to de-
termine if the cross correlation stack results in a
well-resolved delta function around an initially un-
known source time
t
=
τ
. Ideally, the stations should be evenly distributed and dense enough to fully
record the original wavefield. This will never happen
in reality, and the station amplitude factor can be
modified to weight stations differently depending
on their configuration, and to account for wave po-
larity changes.
Calculating the time-reversed Green’s functions
and convolving with a set of displacements is
straightforward but in the cross correlation computa
tions, time reversing is not
necessary. In practical
application to structures, it remains to be shown in
a numerical study if such a tool could be imple-
mented on structural networks for damage detecti
on. Showing this problem numerically requires mod-
eling frequencies up to at least 200
Hz. Showing how the time-reversed reciprocal Green’s functions
method works in a 3D finite-element model of a building is the subject of a future study.
CONCLUSIONS
Wavefield properties of linear elastic media provide
the basis for a method to determine the loca-
tion and time of the occurrence of a high-frequency fracture event in a civil structure. The method
makes use of long-established prin
ciples of wave propagation recipr
ocity and the properties of time-
reversed reciprocal Green’s functions. The development presented here has focused on both point
force sources and more generalized source mechanisms
. We show that for our application, the general
Green’s functions can be used as an approximation to
their spatial derivatives for receivers in the far
field. The viability of the technique will be determ
ined through tests using both numerical simulation
and experiments on a reduced-scale laboratory structure.
ACKNOWLEDGMENTS
We appreciate support from the U.S. Geological Survey for Factor array operation and mainte-
nance, and the NSF Center for Embedded Networke
d Sensing at UCLA for basic research. We are
grateful to Prof. C-M. Uang for access to the moment-frame testing facility at UCSD, and to S. Irvine,
M. Lukac, and I. Stubailo for processing the fracture data.
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