⇦Chinese Physics Letters, 2019, Vol. 36, No. 11, Article code 114301 Detection and Location of a Target in Layered Media by Snapshot Time Reversal and Reverse Time Migration Mixed Method * Hong-Juan Yang (杨红娟)1,2**, Jian Li (李鉴)3,4, Xiang Gao (高翔)5, Jun Ma (马军)1, Jun-Hong Li (李俊红)1, Wen Wang (王文)1, Cheng-Hao Wang (汪承灏)1,2** Affiliations 1Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190 2University of Chinese Academy of Sciences, Beijing 100049 3School of Electronic and Electrical Engineering, Wuhan Textile University, Wuhan 430200 4State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190 5College of Mechanical Engineering and Application Electronics Technology, Beijing University of Technology, Beijing 100124 Received 15 July 2019, online 21 October 2019 *Supported by the Key Research Program of the Chinese Academy of Sciences under Grant No QYZDY-SSW-JSC007, and the National Natural Science Foundation of China under Grant Nos 11474304 and 11804256.
**Corresponding author. Email: yanghongjuan@mail.ioa.ac.cn; chwang@mail.ioa.ac.cn Citation Text: Yang H J, Li J, Gao X, Ma J and Li J H et al 2019 Chin. Phys. Lett. 36 114301    Abstract A target in layered medium can be located by the ridge-like distribution time reversal and reverse time migration (TR-RTM) mixed method. However, this method cannot distinguish between acoustic field distributions of the interface and target for the wider acoustic pulse signals, which may result in inaccurate location of the target. A snapshot TR-RTM mixed method is proposed to solve this problem. The principle of snapshot TR-RTM mixed method is first given. Experiments are then carried out, and a mountain-like acoustic field distribution is obtained by processing experimental data. The results show that the location of the peak is that of the target, and the ratio of the scattered signal and interface reflection signal (signal-to-interference ratio) is improved by about four times after processing. Furthermore, this method can effectively suppress the interface reflection signal and enhance the target scattering signal. Therefore, it can achieve effective detection and location of a target in a layered medium. DOI:10.1088/0256-307X/36/11/114301 PACS:43.20.+g, 43.35.+d, 43.60.+d © 2019 Chinese Physics Society Article Text In ultrasonic testing, the traditional ultrasonic pulse echo method cannot distinguish interface reflection signals from target scattering signals. Therefore, the time reversal (TR)[1–4] and reverse time migration (RTM)[5–7] mixed method for the detection and location of a target in a layered medium have been proposed to solve this problem.[8] However, this method is only suitable for narrow pulse signals and is unsuitable for wider acoustic pulse signals. In fact, due to the limitation of the transducer bandwidth, the emitted acoustic pulse is relatively wider, thus this method is inapplicable. To solve this problem, a snapshot time reversal and reverse time migration (TR-RTM) mixed method is proposed. To be distinguished from the snapshot TR-RTM mixed method, the TR-RTM mixed method in Ref. [8] is renamed as the ridge-like distribution TR-RTM mixed method. As shown in Fig. 1, the layered medium is composed of media 1 and 2, where $c_{1}$ and $c_{2}$ are their acoustic velocities. The interface is at $z=h$, and there is a target ${\boldsymbol O} (x_{0}, z_{0})$ in medium 2. Assume that the array element $i(x_{i}, 0)$ is used as the transmitter and an acoustic pulse signal $F_{i}(t)$ is excited. In the ray approximation, the acoustic beam is reflected by the interface and scattered by the target, and then recorded by each receiving array element $j(x_{j}, 0)$ ($j=1, 2, 3,\ldots, i,\ldots, n$). The signal received by the $j$th array element can be written as $$\begin{alignat}{1} P_{ij} (t)=a_{ij} F_{i} (t-t_{ij}^{\rm R})+b_{ij} F_{i} [t-(t_{i}^{\rm o}+t_{j}^{\rm o})],~~ \tag {1} \end{alignat} $$ where $t_{ij}^{\rm R}=2r_{ij 1}/c_{1}$, $t_{i}^{\rm o}=r_{i1}^{\rm o}/c_{1}+ r_{i2}^{\rm o}/c_{2}$, $t_{j}^{\rm o}=r_{j 1}^{\rm o}/c_{1}+ r_{j2}^{\rm o}/c_{2}$, $a_{ij}$ is the reflection coefficient, $b_{ij}$ is the scattering coefficient, $r_{i1}^{\rm o}$ and $r_{i2}^{\rm o}$ are the incident acoustic beam and refracted acoustic beam,[9] respectively, and they satisfy Fermat's shortest path principle, namely, Snell's law.[10] Let $$ \Delta T_{ij}^{\rm R} =(t_{i}^{\rm o} +t_{j}^{\rm o})-t_{ij}^{\rm R},~\Delta T_{ij}^{\rm o} =t_{j}^{\rm o} -t_{i}^{\rm o}.~~ \tag {2} $$ The forward acoustic beam $F_{i}(t)$ is emitted by the transmitting array element. The signal $P_{ij}(t)$ received by the $j$th array element is time reversed as the backward acoustic beam, and then re-emitted by array element $j$ with a time of $\Delta T_{ij}^{\rm o}$ ahead of the forward wave. The time reversal of signal $f(t)$ is denoted as $\overline f(t)$, thus the backward acoustic beam can be expressed as $$\begin{alignat}{1} \!\!\!\!\!\!R_{ij} (t)=a_{ij} \overline {F_{i}} (t\!-\!\Delta T_{ij}^{\rm R} \!+\!\Delta T_{ij}^{\rm o})\!+\!b_{ij} \overline {F_{i}} (t\!+\!\Delta T_{ij}^{\rm o}).~~ \tag {3} \end{alignat} $$
cpl-36-11-114301-fig1.png
Fig. 1. Propagation path of acoustic wave in the layered medium.
For any point ${\boldsymbol X}(x, z)$ in space, the convolution of forward acoustic beam and backward acoustic beam is $$\begin{align} M_{0} (t,\boldsymbol{X})=F_{i} (t,\boldsymbol{X})\otimes R_{ij} (t,\boldsymbol{X}),~~ \tag {4} \end{align} $$ where $\otimes$ represents the convolution, and the forward acoustic beam $F_{i}(t, {\boldsymbol X})$ propagating to point ${\boldsymbol X}$ is $$\begin{align} F_{i} (t,\boldsymbol{X})=\,&\sum\limits_\omega {F_{i} (\omega)} e^{j\omega (t-t_{i} (x))},\\ t_{i} (x)=\,&\frac{r_{i1} (x)}{c_{1}}+\frac{r_{i2} (x)}{c_{2}},~~ \tag {5} \end{align} $$ where $F_{i}(\omega)$ is the Fourier transform of $F_{i}(t)$. Since the time reversal in the time domain is corresponding to the conjugate in the frequency domain,[1] the backward acoustic beam propagating to point ${\boldsymbol X} $ can be expressed as $$\begin{align} R_{ij} (t,\boldsymbol{X})=\,&\sum\limits_\omega F_{i}^{\ast}(\omega)e^{j\omega (t+t_{j}^{\rm o} -t_{i}^{\rm o} -t_{j} (x))}\\ &\cdot (b_{ij} +a_{ij} e^{-j\omega \Delta T_{ij}^{\rm R}}),~~ \tag {6} \end{align} $$ where $t_{j}(x)=r_{j 1}(x)/c_{1}+r_{j2}(x)/c_{2}$, and $\ast$ represents the conjugation. Equation (6) is the conjugate of Fourier transform of the waveform received in the time domain, which can be derived directly from the received waveform without having to calculate each travel time, thus reducing the error. The convolution of two functions in the time domain corresponds to the product of the spectrum of two functions in the frequency domain,[11] thus the acoustic field value at point ${\boldsymbol X} (x, z)$ can be expressed in the frequency-space domain as $$\begin{alignat}{1} \!\!\!\!\!\!I_{j} ({\omega,\boldsymbol{X}})=\,&\sum\limits_\omega (F_{i} (\omega)e^{j\omega [ {t-t_{i} (x)} ]}) (F_{i}^{\ast}(\omega)\\ &\cdot e^{j\omega [t+(t_{j}^{\rm o} -t_{i}^{\rm o})-t_{j} (x)]}(b_{ij}\!+\!a_{ij} e^{-j\omega \Delta T_{ij}^{\rm R}})).~~ \tag {7} \end{alignat} $$ At the moment $t=t_{i}^{\rm o}=r_{1}/c=r_{i1}^{\rm o}/c_{1}+r_{i2}^{\rm o}/c_{2}$, considering that $\Delta T_{ij}^{\rm R}=t_{i}+t_{j} - T_{ij}^{\rm R}$, the radio-frequency coherent superposition of Eq. (7) for all transmitter-receiver pairs can be written as $$\begin{alignat}{1} I(\omega,\boldsymbol{X})=\,&\sum\limits_j I_{j} (\omega,\boldsymbol{X})\\ =\,&\sum\limits_j b_{ij} \sum\limits_\omega {|{F_{i} (\omega)}|}^{2}e^{j\omega ((t_{i}^{\rm o} +t_{j}^{\rm o})-(t_{i} (x)+t_{j} (x)))} \\ &+\sum\limits_j {a_{ij} \sum\limits_\omega {|{F_{i} (\omega)}|}}^{2}e^{j\omega (t_{ij}^{\rm R} -t_{i} (x)-t_{j} (x))}.~~ \tag {8} \end{alignat} $$ At the moment $t=t_{i}^{\rm o}=r_{1}/c_{1}+r_{2}/c_{2}$, the snapshot should be carried out. The snapshot diagram of acoustic field distribution is shown in Fig. 2, where the forward acoustic beam is denoted by an arc band with waveform width $c_{2}\Delta t$ ($\Delta t$ is duration of pulse) and the backward acoustic beam is denoted by two arc bands with the same width corresponding to the target scattering part and interface reflection part. There are intersecting sections between the forward acoustic beam and backward acoustic beam, which are correlated regions (see Fig. 2). The target is located at the center of the correlation region of the forward wave and at the target scattering part of the backward wave. When point ${\boldsymbol X}$ and target $O$ are the same location, the first term of Eq. (8) can be written as $$\begin{align} I(\omega,O)=\big(\sum\limits_j {b_{ij}}\big)\sum\limits_\omega {|{F_i(\omega)}|}^{2}.~~ \tag {9} \end{align} $$ At this point, the acoustic field value reaches a maximum, which is complete correlation. The other points are partially correlated with a smaller value in the correlation region. Thus, a peak distribution is formed. The position of the summit is that of the target $O$, thus achieving the detection and location of the target in the layered medium.
cpl-36-11-114301-fig2.png
Fig. 2. Schematic diagram of forward acoustic beam and backward acoustic beam in the snapshot TR-RTM mixed method.
In addition, the forward wave and interface reflection part of backward wave also form a cross correlation region (see Fig. 2). Due to the difference of $t_{ij}^{\rm R}$, the acoustic field value at each point in the corresponding region is different for different transmitter-receiver pairs. Therefore, the phases $\omega (t_{ij}^{\rm R}-(t_{i}(x)+t_{j}(x)))$ in the second term of Eq. (8) are not the same, or even exist a difference of $\pi$ in phase, and they may cancel each other out. Consequently, the acoustic field distribution of interface reflection part cannot be a single peak distribution but is a messy and non-uniform distribution with a more small amplitude. At this time, the interference of interface reflection is suppressed. In the experiment, we selected silicone rubber and water as the upper medium and the lower medium, respectively, and their parameters are listed in Table 1. The transducer array consists of 15 elements, where the first element is used as the transmitter and several array elements are used as the receivers. A rigid round steel bar with a radius of 2.9 mm is placed in the water as the target, and it is placed at three different positions (the targets $a$, $b$ and $c$), $(x, z)$=(18 mm, 60 mm), (18 mm, 75 mm) and (18 mm, 90 mm). Figure 3(a) shows the signals received by several array elements, where the former is the interface reflection signal and the latter is the target scattering signal. Figure 3(b) shows the ratio ($P_{\rm st}/P_{\rm r}$) of scattered acoustic pressure $P_{\rm st}$ and reflected acoustic pressure $P_{\rm r}$ for different transmitter-receiver pairs.
Table 1. Parameters of the layered medium.
Medium Material Velocity (m/s) Density (kg/m$^{3})$ Thickness (mm)
Upper medium Silicone rubber 1070 1096 48.5
Lower medium Water 1500 1000
cpl-36-11-114301-fig3.png
Fig. 3. (a) Signal diagram received by several array elements. (b) Amplitude ratios of scattered signals to reflected signals ($P_{\rm st}/P_{\rm r}$).
cpl-36-11-114301-fig4.png
Fig. 4. Acoustic field distribution formed by ridge-like distribution TR-RTM mixed method for the width emitting pulse (a) and the narrow emitting pulse (b).
Using the ridge-like distribution TR-RTM mixed method proposed in our previous work,[12] the acoustic field distribution is shown in Fig. 4(a) for target $O$. It can be seen that the acoustic field distributions of interface signal and target signal overlap and cannot be distinguished from each other, thus the target location in the layered medium is uncertain. If the narrow pulse which takes the half period with the maximum amplitude among the above-mentioned acoustic pulse signal is used as emitting acoustic pulse, then a prominent peak-like distribution around the target can be constructed as shown in Fig. 4(b), and the position of the summit (17 mm, 58.4 mm) is very close to the position of target $O$. However, a smaller and inhomogeneous distribution of the interface reflection is formed. It has little influence on target localization.
cpl-36-11-114301-fig5.png
Fig. 5. Snapshots of acoustic field distributions for three target positions.
By the snapshot TR-RTM mixed method, the snapshot of acoustic field distributions for the targets $a$, $b$ and $c$ are shown in Figs. 5(a)–5(c), respectively. It is obvious that a prominent peak-like distribution around the target is formed by suppressing reflection signals, and the position of the peak is that of the measured target. The actual position and its measured data are listed in Table 2, it can be seen that the measured target position is close to its actual target position. A messy and non-uniform acoustic field distribution with small amplitude is generated for the interface reflection signals, thus the interference of interface reflection is suppressed. To further estimate the effect of our method, we introduce the signal-to-interference ratio (the ratio of scattering signal to interfacial interference signal, SIR), and the target signals are enhanced about four times after processing by the snapshot TR-RTM mixed method (see Table 2).
Table 2. Actual position, measured position and SIR before and after processing.
Case Actual position (mm) Measured position (mm) SIR before processing$^{\rm a}$ SIR after processing Multiple of increase
$(x,z)$ $(x,z)$
Target $a$ (18, 60) (16.5, 58.1) 0.85 2.82 3.32
Target $b$ (18, 75) (17.6, 73.9) 0.81 4.17 5.15
Target $c$ (18, 90) (17.7, 90.5) 0.67 2.88 4.30
$^{\rm a}$These data represent the average ratios of scattering signals to reflection signals of receiving array elements in Fig. 3(b).
cpl-36-11-114301-fig6.png
Fig. 6. (a) Signals received by each array element for the target near the interface. (b) Snapshot of acoustic field distribution for the target near the interface.
We carefully study the case of a target near the interface. The target is located at (34 mm, 54 mm), which is just below the transmitter (the 17th array element), and the signals recorded by 32 receivers are shown in Fig. 6(a). It can be seen that there are overlaps between the target scattering signals and the interface reflection signals from the 12th array element to 22nd array element, thus the target scattering signal cannot be separated from the received signal, which indicates that the ordinary pulse echo method cannot distinguish the target from the interface. The peak-like acoustic field distribution can be formed by the snapshot TR-RTM mixed method, as shown in Fig. 6(b). The peak position (32.8 mm, 51.4 mm) is in good agreement with the actual target position. In summary, the snapshot TR-RTM mixed method can directly use the time waveform of the recorded signal rather than taking the travel time of signals, thus the error of location is greatly reduced. Therefore, it is obvious that the snapshot TR-RTM mixed method is a good candidate for detection and location of a target in a layered media. References Time-reversal of ultrasonic fields. III. Theory of the closed time-reversal cavityProgress in reverse time migration imagingAn overview of depth imaging in exploration geophysicsA two‐way nonreflecting wave equation
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