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INSTITUTE OF PHYSICS PUBLISHING INVERSE PROB LEMS
Inverse Problems 18 (2002) 1761–1773 PII: S0266-5611(02)38007-9
Time reversal techniques in ultrasonic nondestructive
testing of scattering media
Claire Prada, Estelle Kerbrat, Didier Cassereau and Mathias Fink
Laboratoire Ondes et Acoustique, ESPCI, Universit´
eParis 7, UMR 7587, 10 rue Vauquelin,
75352 Paris Cedex 05, France
Received 11 June 2002, in final form 4 October 2002
Published 8 November 2002
Onlineatstacks.iop.org/IP/18/1761
Abstract
Time reversal techniques are adaptive methods that can be used in
nondestructive evaluation to improve flaw detection through inhomogeneous
and scattering media. Two techniques are presented: the iterative time reversal
process and the DORT (French acronym for decomposition of the time reversal
operator) method. In pulse echo mode, iterative time reversal mirrors allow
one to accurately control wave propagation and focus selectively on a defect
reducing the speckle noise due to the microstructure contribution. The DORT
method derives from the mathematical analysis of the iterative time reversal
process. Unlike time reversal mirrors, it does not require programmable
generators and allows the simultaneous detection and separation of several
defects. These two procedures are presented and applied to detection in titanium
billets where the grain structure renders detection difficult. Then, they are
combined with the simulation code PASS (phased array simulation software)
to form images of the samples.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Detection and imaging of small defects in heterogeneous solids of complex geometry are very
difficult problems. The development of new ultrasonic techniques is still an active domain of
research and several approaches have been studied in order to achieve focusing in complex
media.
In the usual approach, the beam focusing is obtainedwith a single transducer whose geom-
etry is matched to the liquid–solid interface and to the desired focal point. The front face of the
transducer is designed to equalize all the propagation times between the transducer surface and
the focal point in the solid, resulting in a Fermat surface transducer adapted to the specific focal
point. In most applications, the solid sample is immersed in water and the transducer is moved
to scan the area of interest. Since Fermat surface transducers are adapted to one specific focal
0266-5611/02/061761+13$30.00 © 2002 IOP Publishing Ltd Printed in the UK 1761
1762 CPradaet al
point,thistechnique is problematic for thick samples for which multiple transducers of differ-
ent shapes may be required to scan through the volume of interest. Furthermore, large focusing
apertures are necessary to detect small defects with good resolution and step-by-step scanning
of a sample using these transducers is increasingly time-consuming as focusing is improved.
Multi-element transducer arrays with electronic focusing and steering capabilities appear
as an interesting alternative for real-time inspection. Indeed, a greater flexibility is obtained by
using a transducer array to generate a focusedbeam at any specified angle and range. One- or
two-dimensional transducer arrays are connected to a set of electronic delay lines calculated
to achieve focusing in transmit and receive modes. A combination of transmit and receive
focusing provides high resolution.
However, these techniques suffer important limitations. They are all based on an apriori
knowledge of the geometry and acoustic properties of the sample and assume that the ultrasound
velocity is known and constant in each medium. Furthermore, they require highly precise
positioning of the transducer array with respect to the interface. The larger the probe aperture
is, the more precise the positioning needs to be and these constraints are not always compatible
with fast NDE inspections. Because of these various limitations and in order to improve the
flexibility of the focusing process, self-focusing techniques have recently been proposed.
In a first attempt to solve these problems, adaptive focusing techniques have been
implemented. After a first illumination of the region of interest, the echoes from the defect
are received by the transducer array. In a technique proposed by O’Donnell for medical
applications [1] and Achenbach for NDT [2], the received signals are cross-correlated and
the timedelays are determined by the time shift corresponding to the maximum of the cross-
correlation between signals from neighbouring transducer elements. This technique yields
good results when the ultrasound field backscattered by the defect dominates other noise
sources received by the array. However, the first illumination may not be efficient enough
to get a significant echo from the defect. Furthermore, when several defects are located
in the insonified region, the interference between the individual echoes gives a resulting
backscattered signal with a poorspatial coherence and the small degree of correlation between
adjacent transducers limits the focusing process. Finally, in heterogeneous materials, the
microstructure yields a strong scattering noise, which may hide the echo from the defect. This
is the case for titanium alloys where a strong ultrasonic speckle is induced by the polycrystalline
microstructure. This is also the case for all grain materials such as steel and ceramics. Fibres
in composite materials also generate a strong speckle noise.Formostof these materials, the
average distance between scatterers is smaller than the ultrasonic wavelength (∼mm), such
that the speckle noise received by the array makes the target detection difficult [3–6].
To overcomethese difficulties, a different approach has been developed in our laboratory,
which relies on the time reversal (TR) invariance of acoustic wave propagation in lossless
media. We have implemented successively two time reversal techniques and these adaptive
methods have proved their efficiency in the field of nondestructive evaluation. The first
technique is the iterative TR mirror (TRM) which allows one to focus selectively on the
strongest defect, reducing the speckle noise due to the microstructure [7–11]. The second one is
the DORT method [12, 13] (French acronym for decomposition of the time reversal operator).
This detection technique, developed since 1993, derives from the mathematical analysis of
the iterative time reversal process. In contrast to TRMs, it does not require programmable
generators and allows the simultaneous detection and separation of several defects. These two
procedures have been presented in a topical review ‘Acoustic time reversal mirrors’ [14] where
several applications are shown. They have also been applied to detection in plates and hollow
cylinders with Lamb waves [15–17]. In this paper, we explain both techniques and present
their application to the detection of flaws in titanium billets, where the grain structure renders
Time reversal techniques in ultrasonic nondestructive testing of scattering media 1763
detection difficult [18–21]. After the detection process, the data are used to calculate an image
of the sample using the code PASS (phased array simulation software) that assumes a known
and constant sound speed in titanium.
2. The time reversal method
TR techniques rely on the property of TR invariance of the acoustic wave propagation in a
lossless medium. A consequence of this TR invariance is that for any burst of sound diverging
from a source—and possibly reflected, refracted or scattered by the medium that may be
heterogeneous—a second wave exists theoretically, that precisely retraces all of these complex
propagation events in the reverse order, and converges in synchrony at the original source, as
if time was going backwards.
2.1. Time reversal mirrors
In an ideal TR experiment, an acoustic source (seismic source) located in a solid radiates a
short elastic pulse (with longitudinal and transverse components of different velocities) that
propagates through the solid and is transmittedinthe surrounding fluid through the interface.
Mode conversions yield a pressure field in the liquid that is measured on a closed surface
surrounding the fluid. This surface is covered with reversible transducers (the TR cavity) that
sense the field during a time Tthat is long enough to ensure the vanishing of the wave. Once
this field is memorized, the surface re-emits the time-reversed signals. The time-reversed field
converges towards the initial source and mode conversions occur in the reverse way. Both the
longitudinal and the transverse waves are recreated in the solid and they exactly converge in
synchrony at the initial source location.
If the source has small dimensions, this technique allows one to generate a wave that
propagates through any complex interface and focuses back on the source. But some
information on the source may be lost. Spatial details of the source smaller than the shortest
wavelength are also lost, resulting from classical diffraction limits. If a short pulse (broadband
spectrum) is emitted by a pointlike source, the acoustic field generated by the ideal TR method
focuses on the source with a spot whose dimensionsare of the order of the smallest wavelength
contained in the original spectrum.
In practice, a closed TR cavity is difficult to realize. The TR operation is usually performed
on a limited angular aperture (the TRM). Each transducer has its own electronics: detection
amplifier, A/D and D/A converters, digital memory, and a programmable generator able to
synthesize the temporally inverted signal stored in memory.
In NDT, one has to work in pulse echo mode and the sources are not active as in the
previous experiment: the only available sources are defects that behave passively and reflect
an incident acoustic field. For this purpose, a TRM array may be used according to the
following multi-step sequence, illustrated by figure 1. A portion of the array generates a
brief ultrasonic pulse to illuminate the region of interest in the solid. If this region contains a
reflector, the reflected wavefront is detected by the array that now works in the receive mode,
and converted into electrical signals that arerecorded. Then a temporal window is used to
select from the received wavefront the portion of signals that are time reversed and stored in
electronic memories. The time-reversed signals are finally transmitted to the transducers, thus
resulting in an ultrasonic wavefront that refocusesonthe target through the interfaces. This
process also compensates unknown deformation of the array and it can be iterated.
Another very attractive feature of TR processing is its capability to reduce the speckle
noise. Indeed, if the speckle noise results from a random microstructure, whose spatial scale
is less than the wavelength, the TR process cannot refocus on the speckle noise sources. Each
1764 CPradaet al
Transmission 1
defect
Transmission 2 : after time reversal
transducers array solid sample
time
Reception 1
time
time
Figure 1. Adaptive focusing by theTRprocess.
individual scatterer yields a time-reversed wave which focuses back on it with a diffraction spot,
but the interference between all these spots yields a complex pattern that does not match the
exact scatterer distribution. Using this property, we h ave developed processing techniques that
allow one to distinguish, during the inspection, alow-contrast flaw from high-level incoherent
speckle noise.
In a joint programme with the French company SNECMA (Soci´
et´
eNationale d’Etude
et de Construction de Moteurs d’Aviation), we developed a 128 channel TRM to detect the
presence of low-contrast defects in titanium alloys used in jet engines [18, 19]. Titanium
billets are the primary source for the fabrication of rotating parts of aircraft engines. The
development of larger engines requires larger billets, and their inspection becomes more and
more challenging.
For this application, we built several Fermat surface arrays of piezo-composite transducers,
with a frequency ranging from 5 to 7.5 MHz. For the measurements presented in this paper, we
used an array of 121 transducers with a central frequency of 5 MHz, as illustrated by figure 2.
The inspected medium is a Ti6-4 titanium alloy cylindrical sample of 125 mm radius with
three flat bottom holes (FBHs).Thesedefects of 0.8, 0.4 and 0.5 mm diameter are located
at a depth of 140 mm and spaced 15 mm apart, as shown in figure 3. To ensure a coupling,
the array and the sampleareboth immersed in a water tank. The longitudinal wavelength in
the titanium sample is 1.2 mm at 5 MHz and the transverse resolution at 140 mm inside the
material is 3.3 mm at −6dB.Consequently, the defects are well resolved by the array.
The titanium sample is first illuminated using the 25 central elements of the transducer
array and the reflected echoes are recorded by the whole array. In this experiment, the incident
beam is mostly converted into a longitudinal wave propagating in titanium.
The TR window is delimited by the vertical lines. The echoes coming from a region
containing the 0.8 mm defect, after emission by the central elements of the array, are shown
Time reversal techniques in ultrasonic nondestructive testing of scattering media 1765
Figure 2. Fermat surface array of transducers.
0.8 0.4 0.5
y
Figure 3. Inspected solid sample.
in figure 4. Each horizontal line represents the acoustic signal received by one element as a
function of time. Along the vertical axis is the code number of each array element. One can
observe the speckle noise due to the titanium microstructure, and the echo from the defect is
hidden in the high-level speckle noise. The signals contained in a 0.8 µstemporal window
are then selected, time reversed, and re-emitted from the whole array. The recorded data at
iteration 1 are completely different and show a well contrasted wavefront. Although the defect
is 5 mm away from the geometrical focus of the array, the TR procedure is able to refocus
the acoustic energy on the defect, resulting in an undulating wavefront. A second iteration of
the TR process gives a similar wavefront with higher signal-to-noise ratio. In this case, the
iterative TR process has a rapid convergence.
The same process, conducted in a defect-free zone, produces completelydifferent echoes:
the echo patterns originating from the microstrucure are very different after iterations 1 and 2.
In this second case, the iterative TR process has aslowconvergence. After one TR, the elements
used in the first illumination receive a high signal. This phenomenon can be understood as a
mirror effect. During the first insonification, the microstructure acts as a ‘delocalized’ mirror
1766 CPradaet al
Iteration 0
0 1 2 3 4 5 6 7
20
40
60
80
100
Iteration 1
0 1 2 3 4 5 6 7
20
40
60
80
100
Iteration 2
time in µs
0 1 2 3 4 5 6 7
20
40
60
80
100
Figure 4. Tw o iterations of the TR process; the 0.8 mm defect is located 5 mm away from the
geometrical array focus.
Iteration 0
0 1 2 3 4 5 6 7
20
40
60
80
100
Iteration 1
0 1 2 3 4 5 6 7
20
40
60
80
100
Iteration 2
time in µs
0 1 2 3 4 5 6 7
20
40
60
80
100
Figure 5. Two iterations of the TR process in a defect-free zone. The TR window is delimited by
the vertical lines.
Time reversal techniques in ultrasonic nondestructive testing of scattering media 1767
and produces a virtual image of the initial source; then after TR, the virtual image is reflected
as the real image and focuses on the transducersthathavebeen excited during the first step of
the process. Hopefully this phenomenon remains weak.
Although the presence of a defect is clearly visible on the bscan shown in figure 4,
some pertinent information has to be extracted from these signals to achieve automatic
detection. For this purpose, various techniques from standard signal processing methods
can be used. One natural quantity is the maximum of the summation of the signals obtained
after two TR operations. This quantity is called the incoherent summation and is expressed as
Inc =maxtN
k=1R2,k(t),whereR2,k(t)is the ultrasonic signal recorded on the transducer
kat the second iteration.
Although this technique gives spectacular results, it only consists of adaptive focusing in
transmit mode. A better signal-to-noise ratio can be obtained by focusing both in transmit
and receive modes. In the receive mode, we have to compensate the different propagation
time delays between the elements of the arrayandthedefect before summation. But such
acompensation is efficient only if the temporal window effectively contains the acoustic
signature of a defect in thematerial. Indeed, one has to be very careful in the presence of speckle
noise: the compensation before summation may seriously increase the noise contribution. To
avoidthiseffect, the trick consists of comparing two consecutive TR iterations. If the initial
temporal window selects the echo coming from a defect, the signals received after the second
iteration contain the same wavefront, with the same phase modulation (see figure 4). This
wavefront is an invariant of the TR process. We will see in the next section that it corresponds
to an eigenvector of the so-calledTRoperator. In contrast, if the temporal window only selects
speckle noise, the second iteration yields a different noise pattern (figure 5). An automatic
determination of the delay law is then made at the first iteration by measuring the arrival time of
the peak signal on each channel. Once this delay law is determined, it is usedto compensate the
signals received at the second iteration. These compensated signals are then summed, therefore
resulting in the coherent summation given by Coh =maxtN
k=1R2,k(t−τk),wherethe
delay τkis the time when the signal R1,k(t)recorded at the first iteration is maximum. This
procedure is illustrated by figure 6.
In the case of an effective defect, thetime-delaylawcomputed from the first iteration is then
optimal to compensate the signals measured during the second iteration and the summation of
the corresponding signals gives a high amplitude.Inthe case of speckle noise, the time-delay
law resulting from the first iteration does not match the signals of the second iteration and the
summation is inefficient. It is like a random phase screen introduced in the process before the
summation.
The results obtained by coherent and incoherent summations are presented in figure 7.
The 0.4 mm defect is clearly detected with both methods. Furthermore, it is clear that the
coherent summation allows the detection of each defect from multiple positions of the array,
thus allowing coarser sampling and faster control process. The presence of two peaks around
the 0.8 mm defect observed on the coherent summation isaconsequence of the size and shape
of this defect that provides a non-isotropic acoustic response. However, this curve is used for
detection and is not an image of the defect. To form an image of the medium, backpropagation
of the data is used, as will be shown below.
We h ave s h o w n t hat the iterative TR process allows autofocusing and detection of defects
as small as 0.4 mm at a depth of 140 mm inside a 250 mm diameter titanium billet. This
technique offers good signal-to-noise ratio and can detect small defects in the billet core where
ultrasonic beams are severely scattered. However, it requires programmable generators that
are expensive and may not be available. Parallel to the implementation of TRMs, we have
developed another detection technique called the DORT method that we now describe.
1768 CPradaet al
Figure 6. Illustration of the coherent summation.
Figure 7. Maximum of the coherent and incoherent summations of the signals obtained after two
iterations of the TR process.
3. The DORT method
The DORT method derives from the theoretical analysis of the iterative TR process. It is based
on the decomposition of the TR operator that describes the iterative TR process. Similar to
theTRprocess, at most only rough assumptions are made on the ultrasound velocity and the
geometry of the medium. The method involves the determination of the possible transmitted
waveforms that are invariant under the TR process. For these waveforms, an iteration of the TR
operation gives stationary results. Such waveforms can be determined through the calculation
of the eigenvectors of the so-called TR operator.
Unlike the iterative TR process, the DORTmethod does not require programmable
generators and it allows the simultaneous detection and separation of several defects.
The method is based on two steps: the first step consists of measurement of the L×Linter-
element impulse responses klm(t)of the array insonifying a solid sample (see figure 8). For
this measurement, the first element is excited by a short pulse e(t)and the received signals are
Time reversal techniques in ultrasonic nondestructive testing of scattering media 1769
Figure 8. Acquisition of the inter-element impulse response matrix.
measured by the Lelements of the array. This operation is repeated for each emitting element,
using the same emission e(t).Then, the transfer matrix K(ω) is obtained by Fourier transform
of the inter-element responses klm(t).TheTRoperator K(ω)∗K(ω) can be diagonalized and
it has been shown that the number of significant eigenvalues of this operator corresponds to
the number of well-resolved scatterers. Furthermore, each eigenvector is the response of the
corresponding scatterer to the array. Therefore, it provides the phase andamplitude information
that should be applied to the transducer array in order to focus on the corresponding scatterer.
The eigenvalues are functions of the apparent reflectivity of the scatterers, and the largest
eigenvalue corresponds to the strongest scatterer. In practice, and for simplicity, we use the
singular value decomposition (SVD) of the transfer matrix K(ω) and analyse its singular values
and singular vectors.
This decomposition has been applied to the same experimental situation as described
above. The SVD of K(ω) at ω=5MHz is calculated for each position of the array
(figure 9). For all positions of the array, most singular values are below 500: they are
associated with the speckle noise. The first singular value reaches a local maximum when
the array geometrical focus coincides with a defect. In this case, the level is more than twice
the singular value associated with noise. For positions ranging from y=24 to 30 mm, the
singular value associated with the 0.8 mm defect decreases while the one associated with the
0.4 mm defect increases: the two defects are simultaneouslydetected. We see that the singular
value associated with the 0.4 mm defect remains visible until the geometrical focus is moved
more than 5 mm from the defect. This is approximately 2 mm more than for the TR process.
The SVD of thematrix K(ω) can be computed in the whole frequency band of the
transducer array. One advantage of using the SVD is that the phase information is not lost,
thus if the scatterer’s response is stable in the frequency band of the transducer, the time
domain response from the defect to the array can be obtained by inverse Fourier transform.
The frequency-dependent first eigenvectors are then combined to produce by inverse Fourier
transform the ‘temporal eigenvector’ as illustrated in figure 10. This eigenvector can be
understood as an invariant of the iterative TR process. At each frequency, it correspondsto the
limit of the iterative TR process. When, at first illumination, the echo from the defect clearly
emerges from speckle noise, the iterative TR process immediately converges. In this case this
first temporal eigenvector is similar to the bscan measured after one TR sequence (figure 4).
But when the echo of the defect is lower, it may not emerge from speckle noise after two TR
processes and in this case the DORT method is more efficient.
1770 CPradaet al
Figure 9. Distribution of the singular values versus the position of the array.
1.25 2.5 3.75 56.25
Time ( µs)
Transducer index
0
20
40
60
10
1
0
80
Figure 10. First ‘temporal eigenvector’ v1(t)when the 0.4 mm flaw is located 5 mm out of the
geometrical focus of the array.
4. Imaging
Up to now, only rough assumptions have been made about the propagatingmedium. However,
to confirm the presence of the defect, and also for convenience, it is useful to form an image
of the sample. To this end, knowledge of the geometry and acoustic properties of the solid is
required in order to calculate the responses from the array elements to current points in the
image plane. These responses are calculated using the simulation code PASS [22] developed
by Didier Cassereau. This code takes into account the geometry of the transducers and the
solid–liquid interface and assumes that the sound speed in titanium is constant and equal to
6mmµs−1.IfG(P,m,t)is theresponse from the current point Pin the image plane to the
transducer number m,andifV(m,t)is the signal to bepropagated from transducer m,then
the image amplitude at point Pis defined as I(P)=maxt|mG(P,m,−t)⊗V(m,t)|.
With the DORTmethod, an image is obtained by backpropagation of the first temporal
eigenvector (figure 10). A similar treatment can be applied to the backscattered signal measured
Time reversal techniques in ultrasonic nondestructive testing of scattering media 1771
0 10 20 30 40 50
y-axis (mm)
0
10
x-axis (mm)
-10 0.8 mm0.4 mm0.5 mm
Figure 11. Synthetic image obtained by summation of the numerical backpropagations of the first
temporal eigenvector (DORT method).
0.5mm
0 10 20 30 40 50
y-axis (mm)
0
10
x-axis (mm)
-10 0.4 mm 0.8 mm
Figure 12. Synthetic image obtained by summation ofthenumerical backpropagations of the
bscan obtained after two TR sequences.
during a TR process. Furthermore, with the same defect being detected from different positions
of the array, the average of the images obtained for each position of the probe provides synthetic
images with high signal-to-noise ratio. We chose to backpropagate the bscan measured after
two TR sequences as shown in figures 4 and 5. The corresponding results are illustrated in
figures 11 and 12, respectively.
With the TR process, the 0.4 mm defect appears 7.5 dB over noise, while it is 15 dB over
noise for the DORT method (figure 13). The signal-to-noise ratio is higher for the DORT
method for two reasons: first, the first eigenvector provided by the DORT method is the
signal that would result from an infinite number of TR iterations, the reduction of the speckle
noise is thus more efficient; second, by measuring the complete array response function, the
DORT method takes advantage of all possible angles of incidence to insonify the sample,
whereas the issue of two iterations of the TR process strongly depends on the choice of the
first insonification.
With the DORTmethod, the first singular vector is backpropagated into an ideal medium.
Amore sophisticated treatment was proposed in arecent theoretical and numerical study by
Borcea et al [23]. Several singular vectors are used to form an image and, above all, the question
‘how large can the heterogeneities of the background be, before the imaging technique fails?’
is addressed through simulations in a 2D fluid. Comparison of these 2D simulations with
experimental results in real 3D solid media would be interesting but is not straightforward and
remains an unachieved task.
1772 CPradaet al
Figure 13. Projection along the x-axis of the images shown in figures 11 and 12.
5. Conclusion
Time reversal methods are robust adaptive techniques that are very efficient in nondestructive
evaluation of noisy samples. Two methods have been presented in this paper,both implemented
with an array of transducers: the first is the iterative TRM that requires programmable
generators and allows fast inspection with little computation; the second is the DORT
method which does not require programmable generators, but is more expensive in terms of
computation time. Both methodsallow one to reduce speckle noise while enhancing adaptively
the coherent echo of a defect. We have shown how both methods allow the detection of a 0.4 mm
diameter FBH at a depth of 140 mm inside a titanium billet with more than 6 dB signal-to-noise
ratio. Adding information on the geometry and acoustic properties of the medium, synthetic
images of the sample have been calculated by numerical backpropagation of the data with
the PASS code. In the image obtained from the TR process, the 0.4 mm defect is 7.5 dB
above noise, while it reaches 15 dB above noise using the DORT method. Thus, the DORT
method appears to be more robust. However, the implementations of these two methods are
very different and the choice of the best one should depend upon application.
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