Energy Current Imaging Method for Time Reversal in Elastic Media
An energy current imaging method is presented for use in locating sources of wave energy during the back propagation stage of the time reversal process. During the back propagation phase of an ideal time reversal experiment, wave energy coalesces from all angles of incidence to recreate the source event; after the recreation, wave energy diverges in every direction. An energy current imaging method based on this convergence/divergence behavior has been developed. The energy current imaging method yields a smaller spatial distribution for source reconstruction than is possible with traditional energy imaging methods.
Energy current imaging method for time reversal in elastic media
Brian E. Anderson,1,2, Robert A. Guyer,1 Timothy J. Ulrich,1 Pierre-Yves Le Bas,1
Carène Larmat,1 Michele Griffa,1,3 and Paul A. Johnson1
1Geophysics Group EES-17, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
2Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA
3Laboratory for Building Technologies, Swiss Federal Laboratories for Materials Testing
and Research (EMPA), Überlandstrasse 129, Dübendorf 8600, Switzerland
Received 29 January 2009; accepted 25 June 2009; published online 16 July 2009
An energy current imaging method is presented for use in locating sources of wave energy during
the back propagation stage of the time reversal process. During the back propagation phase of an
ideal time reversal experiment, wave energy coalesces from all angles of incidence to recreate the
source event; after the recreation, wave energy diverges in every direction. An energy current
imaging method based on this convergence/divergence behavior has been developed. The energy
current imaging method yields a smaller spatial distribution for source reconstruction than is
possible with traditional energy imaging methods. © 2009 American Institute of Physics.
DOI: 10.1063/1.3180811
Time reversal TR is a robust method that provides the
means to reconstruct and therefore localize a previously unknown
source of wave energy both in space and time.1,2 TR
is applicable to all types of classical waves, either acoustic,
elastodynamic, or electromagnetic. The TR process consists
of two phases, the forward propagation and the back propagation.
During the forward propagation, the characteristics of
the source need not be known; rather one or more detectors
record the wave disturbances they experience. These signals
are then reversed in time and broadcast from the original
detector locations. The wave energy retraces the original
forward propagation paths resulting in a focus of wave energy
at the original source location; this constitutes the backward
propagation phase when TR focusing occurs.
The backward propagation phase of a TR experiment
may be done experimentally or numerically particularly for
problems where the source is internal to a three-dimensional
solid and one cannot physically probe the interior. TR has
been used in seismology applications to locate earthquake
sources.3,4 We are currently working on applying a similar
imaging method to the one proposed in this letter. TR has
also been used in nondestructive evaluation to locate surficial
cracks,5–7 disbonding features just below the sample
surface,8 and we are currently exploring its use for internal
features where the backward propagation is done numerically.
Each of these applications, as well as others for fluid
media, could benefit from using a higher resolution imaging
method, as the one proposed in this work.
An important aspect regarding the TR focusing is the
fact that during the back propagation stage of a TR experiment
there is an inflow of energy prior to the focal time and
an outflow of energy after the focal time, with the focal time
defined as the time of maximum energy at the focal location.
In order to characterize the energy coalescence that corresponds
to the TR focus, one selects a method of processing
the wave field data measured during the TR back propagation,
defined as the imaging condition. In the following we
describe a measurement and processing method that employs
the energy current to isolate the focal time and spatial location.
In the experiment, described in detail below, a laser vibrometer
is used to measure the out-of-plane particle velocity
wave field vzx ; t as a function of position x =x; y on
the sample surface, during the backward propagation phase,
at each time step of a finite discrete interval about the time of
focus. Since information about the displacement field is limited
to a single component we form an energy current density
vector wave field appropriate to this circumstance
S x ;t = -
uz
t
x ;t xuzx ;t, 1
where vzx ; t=uz /tx ; t, x= /x; /y, and have
units of force per unit length. This equation is analogous to
that for the energy current density on a membrane.9 In the
experiments described below, the Fourier spectrum of the
displacement field is centered around 0 /2=200 kHz so
that we may use uzvz /0 to write
S x ;t = -
0
2
vz
t
x ;t xvzx ;t = - k
vz
t
x ;t xvzx ;t,
2
where /0
2=k. We form a scalar measure of the energy
flow by considering the flux of S x ; t through a closed rectangular
path x I surrounding a point x I on the surface
x I ;t = x
I
drx · S x ;t, 3
where drx =dlnˆ x and nˆ x is the unit vector normal to the
path, pointing outward. The quantity is called the energy
flux. In practical applications this flux field is calculated at
each pixel of an image grid. The energy flux, calculated at a
specific time, is the imaging condition. Below we compare it
with a standard maximum-in-time imaging condition.
To illustrate the merits of the energy current imaging
condition, a set of TR experiments is conducted. These experiments
utilize a source transducer and a laser vibrometer.
aElectronic mail: bea@byu.edu. The reciprocal TR method, described below, is used to create
APPLIED PHYSICS LETTERS 95, 021907 2009
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pointlike time reversed foci by performing a single channel
TR experiment.2,6,8,10–16 In the reciprocal TR method, a pulse
is emitted from a source transducer and the time dependent
response is recorded at a user defined position A. This recorded
signal is then reversed in time and broadcast from the
original source transducer. Due to spatial reciprocity, a time
reversed focus is created at A,13 which is akin to creating a
virtual source at A.12–16 The virtual source method need not
be used in every implementation of energy current imaging,
rather it is simply a convenient experimental method used for
this study. A laser vibrometer is then used to scan a region of
space that includes A to determine the quality of the TR
spatial focusing.
A glass block of dimensions 8989101 mm3 is used
to conduct the TR experiments. The glass sample has a shear
wave speed vS=2340 m/ s and a quality factor Q=1500. A
lead zirconate titanate disk transducer of 12.7 mm diameter
and 2 mm thickness is bonded onto the sample with epoxy.
The source pulse is a 200 kHz sine wave modulated by a sine
squared envelope with a pulse length of 40 s, yielding a
wavelength of 11.7 mm corresponding to a center frequency
of 200 kHz. To enhance the reflectivity of the surface of the
sample, reflective tape is placed on it. A rectangular 85 by 85
point scan grid spacing of 1.0 mm, corresponding to a
1.0 mm2 resolution for the laser vibrometer is used to image
the TR focusing. During the TR back propagation stage
of each experiment, the scanning laser vibrometer measures
the particle vibration velocity, along one direction, at each
point in the scanning grid, vx , t. In the case of these experiments,
only the out-of-plane velocity has been measured.
However, our TR measurement apparatus at Los Alamos allows
us to measure also the in-plane components of the vector
velocity wave field, which will be the subject of future
research.
Because the wave field measured by the laser vibrometer
varies in space, at each time, the total energy flux of Eq. 3
depends on the line chosen to enclose the evaluation point
pixel location of the final image. It is found empirically that
the best imaging results are obtained using the eight nearest
neighbor scan points about the evaluation point. The x and y
spatial derivates at the corner points are each weighted by
1/2.
One common method used for imaging TR foci is the
amplitude maximum-in-time imaging condition. For purposes
of comparison to the energy current imaging condition,
the maximum-in-time imaging condition is squared so that it
is proportional to energy
Ex = maxt0,T vzx ,t 2
, 4
where T is the total length of time for the recorded signals.
An example of the current calculation is given in Fig. 1.
Figure 1a displays the instantaneous energy current density
vector wave field, for the TR focusing onto a monopole
source, by vector plot; at each point the vector’s magnitude
and direction are represented by the length and direction of
the arrow, respectively. The time at which the snapshot is
calculated is just prior to the focal time tpre-focus when energy
is flowing toward the virtual source location. Two boxes are
displayed in the plot centered at 10; 10 and 13; 13 mm,
respectively. The box centered at 10; 10 mm is the point of
maximum influx of energy, corresponding to the virtual
source location. The second box shows that the flux of energy
is much less away from the focal location see below.
Figure 1b displays the instantaneous energy flux calculated
from the energy current density vector data displayed in Fig.
1a, using Eq. 3.
Figure 2 displays the maximum-in-time energy image
FIG. 1. Color Example energy current computation.
a The instantaneous energy current density vector
wave field S . The length of the arrows indicates the
amplitude while the direction of the arrows corresponds
to the vector direction. The boxes represent locations
where two examples of line integrals are computed,
centered at 10, 10 mm at the source position and
13, 13 mm away from the source. b The instantaneous
energy flux x I ; tpre-focus just before the focal
time. The color bar represents amplitude.
FIG. 2. Color Contrasting images of the normalized
maximum-in-time energy image a and the normalized
instantaneous energy current b for a single monopole
source located at position 10; 10 mm in Fig. 1. Color
in these plots represents the magnitude.
021907-2 Anderson et al. Appl. Phys. Lett. 95, 021907 2009
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from Eq. 4 a, and the absolute value of the instantaneous
energy current just prior to the focal time b, for a single
time reversed focus onto a monopole source, respectively.
Two observations can be made regarding the contrasting images,
first the spot size in the energy current image is diffraction
limited with a 1/2 wavelength diameter, while the focal
diameter for the maximum-in-time energy image is 3/4
wavelength focal spot size is determined from the average
null to null distance measured from multiple cross sections
of the focal spot. Second the background fringes in the energy
current image are slightly higher in amplitude. There is
an apparent shift, one pixel down and one to the left, in the
peak of Fig. 2b with respect to the peak of Fig. 2a. This
shift is likely due to the fact that the flux is greater from that
direction than from the other directions, enough to shift it
slightly. However, despite this shift in the focal center, one
would generally determine the focal location, in a blind experiment,
by selecting the center of the two-dimensional focal
shape, which in this case is still at the true focal center.
The virtual source experiment can be performed for two
different selected locations A and B, where the forward
propagation portions of the experiments are performed separately.
The detected signals from each experiment may then
be summed and then time reversed. During the backward
propagation, the laser vibrometer scan will detect two individual,
simultaneously created time reversed foci, which are
closely spaced.12 In Fig. 3 the maximum-in-time energy image
a, and the absolute value of the instantaneous energy
current just prior to the focal time b, are displayed for two
closely spaced foci spaced 10 mm or 6/7 wavelength apart,
respectively. The energy current image more clearly resolves
the two foci than the maximum-in-time energy image.
In summary, when imaging foci during the back propagation
phase of a TR experiment, the time dependent energy
current imaging condition yields higher resolution imaging
of complex sources than does strictly viewing the instantaneous
energy. The tradeoff is that the current imaging does
result in slightly higher background noise fringes/speckle.
The proposed imaging method should also be applicable to
internal sources in solid and fluid media, where the backward
propagation portion of the TR experiment is done numerically
and the internal wave field in the media is determined.
This work was supported by Institutional Support
LDRD at the Los Alamos National Laboratory.
1M. Fink, Phys. Today 50, 34 1997.
2B. E. Anderson, M. Griffa, C. Larmat, T. J. Ulrich, and P. A. Johnson,
Acoust. Today 4, 5 2008.
3C. Larmat, J.-P. Montagner, M. Fink, Y. Capdeville, A. Tourin, and E.
Clévédé, Geophys. Res. Lett. 33, L19312 2006.
4C. Larmat, J. Tromp, Q. Liu, and J.-P. Montagner, J. Geophys. Res. 113,
B09314, DOI:10.1029/2008JB005607 2008.
5A. M. Sutin, J. A. TenCate, and P. A. Johnson, J. Acoust. Soc. Am. 116,
2779 2004.
6T. J. Ulrich, P. A. Johnson, and A. M. Sutin, J. Acoust. Soc. Am. 119,
1514 2006.
7T. J. Ulrich, P. A. Johnson, and R. A. Guyer, Phys. Rev. Lett. 98, 104301
2007.
8T. J. Ulrich, A. M. Sutin, T. Claytor, P. Papin, P.-Y. Le Bas, and J. A.
TenCate, Appl. Phys. Lett. 93, 151914 2008.
9A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and
Continua McGraw-Hill, New York, 1980.
10C. Draeger and M. Fink, Phys. Rev. Lett. 79, 407 1997.
11J. de Rosny and M. Fink, Phys. Rev. Lett. 89, 124301 2002.
12M. Scalerandi, A. S. Gliozzi, B. E. Anderson, M. Griffa, P. A. Johnson,
and T. J. Ulrich, J. Phys. D 41, 155504 2008.
13T. J. Ulrich, M. Griffa, and B. E. Anderson, J. Appl. Phys. 104, 064912
2008.
14J.-L. Robert and M. Fink, J. Acoust. Soc. Am. 124, 3659 2008.
15B. E. Anderson, T. J. Ulrich, M. Griffa, P.-Y. Le Bas, M. Scalerandi, A. S.
Gliozzi, and P. A. Johnson, J. Appl. Phys. 105, 083506 2009.
16B. E. Anderson, R. A. Guyer, T. J. Ulrich, and P. A. Johnson, Appl. Phys.
Lett. 94, 111908 2009.
FIG. 3. Color Contrasting images of the normalized
maximum-in-time energy image a and the normalized
instantaneous energy current b for two coherent
monopole sources located at position 10; 10 mm and
10; 16 mm. Color in these plots represents the
magnitude.
021907-3 Anderson et al. Appl. Phys. Lett. 95, 021907 2009
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