Reverse-time-migration imaging for a crack in a thin plate by dispersed flexural waves

A modified reverse time migration (RTM) method for dispersed flexural waves is proposed to detect and image the crack in a thin plate. The key point of this method is to take the time reversal (TR) of the expanded flexural wave at a receiver as the backward wave in the cross-correlation imaging condition of RTM, thus the expanded backward wave will be re-compressed also due to dispersion of the flexural wave. The backward wave is time reversed again when arriving at a certain point on the plate, and then correlated to the forward wave derived from the original source. The image of the crack is obtained by superimposing the cross-correlation results of many source-receiver pairs in transmitting and receiving arrays. The experimental studies on imaging a crack in a thin plate are conducted, and the transmitted and received waveforms are recorded with high fidelity by a laser vibrometer, thus, ensuring that an accurate and clear image of the crack is obtained by this modified RTM process.A modified reverse time migration (RTM) method for dispersed flexural waves is proposed to detect and image the crack in a thin plate. The key point of this method is to take the time reversal (TR) of the expanded flexural wave at a receiver as the backward wave in the cross-correlation imaging condition of RTM, thus the expanded backward wave will be re-compressed also due to dispersion of the flexural wave. The backward wave is time reversed again when arriving at a certain point on the plate, and then correlated to the forward wave derived from the original source. The image of the crack is obtained by superimposing the cross-correlation results of many source-receiver pairs in transmitting and receiving arrays. The experimental studies on imaging a crack in a thin plate are conducted, and the transmitted and received waveforms are recorded with high fidelity by a laser vibrometer, thus, ensuring that an accurate and clear image of the crack is obtained by this modified RTM process.


I. INTRODUCTION
The detection and imaging of the cracks in a thin plate are of critical importance for the structural health monitoring of high-pressure vessels, aircrafts, reactors and other plate-like structures.Ultrasonic echo and thickness-measuring techniques used to detect the crack in the plates are powerless.Flexural wave, as one of the guided waves in the plate, is a main tool to detect the defects on the plate.However, detecting and further imaging cracks in the plate by flexural waves is difficult due to dispersion of the wave.It becomes urgent for acoustic detection and imaging of the cracks in thin plates.
][8][9][10] Wilcox et al. adopted the guided waves to accurately locate the defects in the rod by removing the effect of dispersion. 11,124][15] However, their results on the image of the crack was not good in numerical simulation, especially in the experiments. 15Similar result was also observed by Zhu et al. 16 As described in their articles, one of the main reasons for this result is the lack of a sufficient number of source-receiver pairs satisfying the imaging condition.In addition, since PZT piezoelectric transducers are used as actuators and sensors, we think that the frequency response, finite size a Xiang Gao and Jian Li contributed equally to this work.b Author to whom correspondence should be addressed: lijian212@mails.ucas.ac.cn and chwang@mail.ioa.ac.cn.(non-point source) of the piezoelectric transducers, and contact measurements via transducers adhered to the plate have an impact on the original flexural wave field, which may fail to extract the true flexural wave signals and leads to inaccurate imaging of the crack.Recently, He and Yuan used a hybrid system composed of a single piezoelectric actuator mounted onto the structure and a laser vibrometer for two-dimensional scan to image the multiple point defects. 17,18Besides, the synthetic aperture focusing technique (SAFT) 19,20 and the topological imaging 21 can be used to perform Lamb wave imaging of the multiple point defects in a plate.
In this paper, a modified RTM method for dispersed flexural waves is proposed to image a crack in a thin plate.The core of this approach is removing the effect of dispersion of the flexural wave.The principle, method, experiment and imaging process of this modified RTM method are given, and the results show that in situ accurate and clear image of the crack is obtained.

II. PRINCIPLE AND METHOD
A modified RTM method for dispersed flexural waves is presented for imaging cracks in a thin plate.In this method, the far-field ray approach is adopted to study the RTM, which will avoid numerous calculations of the finite difference method, [13][14][15] and the spreading factor 1/ √ r is neglected.As shown in Fig. 1, the flexural wave S i (t) emitted by source S i can be written as where S i (ω) is the Fourier transform of S i (t).This wave is reflected at point D of the crack under Snell's law and then propagated to receiver R j .Due to the dispersion of flexural wave, the reflected flexural wave R ij (t) recorded by receiver R j has expanded, and it can be expressed as where A is the reflection coefficient; k=ω/c(ω) is the wave number; c(ω) is the propagation velocity of flexural waves changing with the circular frequency ω, i.e., the frequency dispersion; and r 1 and r 2 are the propagation distances from S i to D and from D to R j , respectively.The general RTM image of the defects is based on travel time (or distance) of acoustic wave propagation in the medium with constant acoustic velocity, while it is difficult to obtain the travel time in dispersion media.To overcome these difficulties, the time reversal (TR) method proposed by Fink et al. [22][23][24] is applied to the backward wave since it can re-compress the expanded flexural wave due to dispersion.The flexural wave R ij (t) recorded by receiver R j is time reversed as backward wave and written as where T is a sufficiently large time.The backword wave is re-emitted from receiver R j , and the expanded waveform will be re-compressed also due to dispersion of the flexural wave during its back propagation.
FIG. 1. Sound wave propagation path: the incident wave propagating from the source is reflected by the crack, and then recorded by the receiver.
The backward wave arriving at a certain point X (see Fig. 1) on the plate can be represented as where r 2 (x) represents the distance between point X and receiver R j .The TR operation is conducted again, At point X, the forward wave S i (t, X) propagating from source S i can be expressed as where r 1 (x) represents the distance between point X and source S i , and it is cross-correlated to the backward wave R ij (t, X) at this point, where ⊗ represents the correlation.At the reflected point D, Eq. ( 7) reaches its maximum.The above mentioned cross-correlation imaging condition of RTM for a single source -receiver pair is described in the time domain, whereas these processes conserve time in the frequency domain for the calculation.
The forward wave, which propagates to point X in the plate from the source S i , can be expressed in the frequency domain as The received flexural wave R ij (t) at receiver R j can be expressed in the frequency domain as where R ij (ω) is the Fourier transform of R ij (t).Since the TR in the time domain is equivalent to the phase conjugate in the frequency domain, 25 the backward wave R * ij (ω) propagating to point X can be written as R * ij (ω)e −jkr 2 (x) , and its time reversal can be represented as where * represents the conjugate.Since the cross-correlation in the time domain means to multiply by the complex conjugate in the frequency domain, the cross-correlation imaging condition shown in Eq. ( 7) can be written in the frequency domain as Eq. ( 9) can be substituted into Eq.( 11) to obtain, If the condition is satisfied, Eq. ( 12) can attain its maximum, Eq. ( 13) defines an ellipse with foci of source S i and receiver R j , the so-called migration ellipse. 26,27The actual reflected point D is located at this ellipse, but its specific location is unknown at the moment.Eq. ( 11) or ( 12) produces an ellipse annulus around this migration ellipse shown with deep red in Fig. 2, among which the migration ellipse exhibits the maximum value from Eq. ( 14).
In the experiment, the signals of transmitted and received flexural wave, i.e., S i (t) and R ij (t) (see the following Fig. 4), are recorded, while the distance r 1 +r 2 in Eq. ( 12) cannot be directly extracted from these experimental data due to the dispersion of flexural wave.Thus, we adopt Eq. ( 11) rather than Eq. ( 12) in the process of the modified RTM imaging, and the ellipse annulus can be formed by the transmitted and received waveforms recorded in the experiment.
The flexural wave signal S i (t) excited by a source S i is reflected by the crack and then arrives at several receivers R j ( j=1, 2, . .., n), and it is recorded as R ij (t).In the frequency domain, the corresponding S i (ω) and R ij (ω) are obtain, and n ellipse annuli are formed based on Eq. ( 11).Similar operations are conducted for the other sources S i (i=1, 2, . .., m), therefore, m × n source -receiver pairs produce m × n ellipse annuli.The image of the whole crack will be achieved by coherence stacking m × n ellipse annuli; thus, the final cross-correlation imaging condition can be expressed as Three common used imaging conditions by taking the real part, the modulus and the envelope of Eq. ( 15) are listed as follows, To improve the resolution, the imaging conditions shown in Eq. ( 16) are modified by introducing a normalization factor, 28 Eqs. ( 15) - (17) show that the imaging quality of RTM mainly depends on the transmitted and received waveforms recorded with high fidelity in the experiment, and the distortion of these waveforms will lead to poor imaging quality or even no imaging.

III. EXPERIMENT AND RESULTS
A precise experiment for imaging a crack located at the centre area of a thin aluminium plate with dimensions of 1 m × 1 m × 1 mm is conducted.To reduce the boundary reflection, the edges of the plate are wrapped with sound absorbing plasticine.A PZT piezoelectric transducer with a centre frequency of 200 kHz is used as the transmitter.The signals generated by the arbitrary waveform generator are amplified by the power amplifier, and then fed into the transducer to excite the flexural waves in the plate.Compared with the previous receiving signals using the general piezoelectric transducer, we adopt the laser vibrometer as the receiver to record the reflected flexural wave signals with high fidelity because of its point receiving measurement characteristics with noncontact and a large bandwidth up to 2 MHz.The experimental configuration is shown in Fig. 3, where the transmitters denoted by solid circles and receivers denoted by the solid triangles are placed along y=-10 mm and y=0 mm, respectively.A horizontal crack with length of 100 mm is located on the back of the plate with a centre coordinate of (0 mm, 141 mm).The transmitter is launched every 8 mm from x=-72 mm to x=72 mm, which produces a total of 19 excitations.For each excitation, a total of 81 receivers are placed every 2.5 mm from x=-100 mm to x=100 mm to record the reflected flexural waves, and they form a receiving linear array similar to RTM imaging in geophysics.
When transducer T 3 emits flexural wave signals, Before performing the processes of RTM imaging, the dispersion curve of the propagation velocity c(ω) of flexural wave in the plate is required, and it can be obtained by fitting the measured values in the vicinity of 100 kHz, 200 kHz and 400 kHz, 29 as shown in Fig. 5.
For a single transmitter T 3 -receiver R 63 pair, the correlation of the forward wave and backward wave forms a pattern of the ellipse annulus in the plane of the plate, as shown in Fig. 6.Among the ellipse annulus, the trajectory of the maximum values of cross-correlation is an ellipse with FIG. 3. Location of the horizontal crack and the source-receiver configuration.deep red, i.e., the so-called migration ellipse, and the reflection point D falls on this migration ellipse.
The image of the crack is the coherent superposition of 19×81=1539 ellipse annuli obtained by performing the modified RTM operation, as shown in Fig. 7, where Figs.7(a-c) and 7(d-f) are acquired by the imaging conditions exhibited in Eqs. ( 16) (before normalization) and ( 17) (after normalization), respectively.The black dashed line in Fig. 7 denotes the actual location of the crack.The location, shape and length of the crack on the back of the plate can be accurately confirmed.The image of the crack is enhanced by taking the envelope (see Figs.  (a-c) with Figs.7(d-f), the imaging quality can be greatly improved after normalization, and a clearer image is obtained.
Fig. 8 shows the longitudinal profile of the image of the crack along the x=0 mm line, and the corresponding location of the crack, i.e., the position of the maximum value in Fig. 8, and 3 dB bandwidth are shown in Table I.The results show that the experimental location of the crack is consistent with the actual location and the 3 dB bandwidth is narrower after normalization.

IV. CONCLUSIONS
In this paper, a modified RTM method for the dispersed flexural waves is first presented to image the crack in a thin plate.In order to image the crack in the plate, we just need to know the waveforms of the transmitted flexural wave signals and the received flexural wave signals reflected by the crack, and they can be directly obtained from the signals recorded with high fidelity by the laser vibrometer.Thus, an accurate and clear image of the crack is obtained by the modified RTM method.Although a horizontal crack in a thin plate is only studied in the work, we believe that the cracks with other shape, such as circular arc shapes and titled cracks, can also be imaged by the same method, which will be studied in the next work.Besides, this method can also be applied to other dispersive media.

FIG. 2 .
FIG.2.For a single source -receiver pair, the ellipse annulus is formed based on the cross-correlation imaging condition.

Fig. 4 (
a) shows the signals recorded by all receivers.The signals recorded by each receiver contain the direct flexural wave signals and the reflected flexural wave signals by the crack.The flexural waves excited by the transducer T 3 can be approximated by the direct flexural wave signals (see Fig. 4(b)) at the receiver R 19 closest to the transducer T 3 , denoted by the source S 3 , and similar processes are conducted for the other transducers.

FIG. 4 .FIG. 7 .
FIG. 4. (a) When transducer T 3 emits flexural wave signals, the signals are recorded by several receivers.(b) For the receiver R 19 closest to the transducer T 3 , the direct flexural wave approximately represents the flexural waves excited by the transducer T 3 , denoted by the source S 3 .
7(c) and 7(f)) or real part (see Figs. 7(a) and 7(d)) of the imaging condition, but a larger fictitious shadow exists by taking the modulus operation (see Figs. 7(b) and 7(e)).By comparing Figs.7

FIG. 8 . 8 Gao
FIG.8.Profile of the crack along the x = 0 mm line.Red dashed lines and blue solid lines correspond to the images shown in Figs.7(a-c) and Figs.7(d-f), respectively.The shadows denoted by the light red and blue colors represent the 3 dB bandwidth.

TABLE I .
Experimental location of the crack and 3 dB bandwidth.
a (N) represents the values after normalization.