⇦Chinese Physics Letters, 2020, Vol. 37, No. 6, Article code 064301 Detection and Location of a Target in Layered Media without Prior Knowledge of Medium Parameters * Jian Li (李鉴)1,2†**, Hong-Juan Yang (杨红娟)3,4†, Jun Ma (马军)3, Xiang Gao (高翔)5, Jun-Hong Li (李俊红)3, Jian-Zheng Cheng (程建政)1, Wen Wang (王文)3, Cheng-Hao Wang (汪承灏)3,4** Affiliations 1School of Electronic and Electrical Engineering, Wuhan Textile University, Wuhan 430200, China 2State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China 3Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China 4University of Chinese Academy of Sciences, Beijing 100049, China 5College of Mechanical Engineering and Application Electronics Technology, Beijing University of Technology, Beijing 100124, China Received 19 January 2020, online 26 May 2020 *Supported by the Key Research Program of the Chinese Academy of Sciences (Grant No. QYZDY-SSW-JSC007), the National Natural Science Foundation of China (Grant No. 11804256), and the State Key Laboratory of Acoustics, Chinese Academy of Sciences (Grant No. SKLA201807).
†Jian Li and Hong-Juan Yang contributed equally to this work.
**Corresponding author. Email: lijian212@mails.ucas.ac.cn; chwang@mail.ioa.ac.cn Citation Text: Li J, Yang H J, Ma J, Gao X and Li J H et al 2020 Chin. Phys. Lett. 37 064301    Abstract Without prior knowledge of medium parameters, a method is proposed to detect and locate a target in layered media. Experiments were carried out for liquid/liquid and solid/liquid layered media, and the location of a target in them was obtained using three methods combined, i.e., the least-square method, the method of finding minimum dispersal degree of target distribution, and the snapshot time reversal and reverse time migration mixed method. The medium parameters, i.e., the acoustic velocities of upper and lower media as well as the thickness of the upper medium, were inversed simultaneously. The results show that the position of target is consistent with its actual position. Thus, the detection and location of a target in layered media are achieved without prior knowledge of medium parameters, and it overcomes the difficulty that the common time reversal method only detects the target, but cannot locate it. DOI:10.1088/0256-307X/37/6/064301 PACS:43.20.+g, 43.35.+d, 43.60.+d © 2020 Chinese Physics Society Article Text To locate the target in layered media, medium parameters, i.e., the material parameter (acoustic velocity in the medium) and structural parameter (thickness of the medium), are often required. The time reversal (TR) method is a self-adaptive focusing technique that can detect the target in inhomogeneous media without prior knowledge of medium parameters, but it cannot be used to locate the target.[1,2] For the target in layered media, the pulse signals scattered by the target and reflected by the interface may overlap after processing by TR operation. Therefore, they cannot be distinguished from each other to locate the target.[3,4] In ultrasonics and underwater acoustics, the TR method has many applications,[5–8] e.g., Kuperman et al. attempted to improve the echo-to-reverberation in shallow water using the TR method,[9] whereas their experiments were not successful.[10] In short, the shortcomings of the TR method have not yet been overcome. Our research group recently proposed a snapshot TR and reverse time migration (RTM)[11–13] mixed method to detect and locate a target in layered media.[14] However, this method still requires medium parameters, for example, the velocity and thickness of the upper medium as well as the velocity of the lower medium for a two-layer media. Can the target in layered media be located without prior knowledge of medium parameters? This problem is solved in this study. Liquid/liquid (silicone rubber/water) and solid/liquid (plexiglass/water) layered media including a target are investigated.
cpl-37-6-064301-fig1.png
Fig. 1. Propagation path of acoustic wave in a two-layer medium, where $D$ and $X$ represent the target and a certain point located in the lower medium, respectively.
A two-layer medium with a flat interface is shown in Fig. 1, where silicone rubber (or plexiglass) with thickness $h$ and water are used as the upper and lower media in the experiment, respectively. For silicone rubber, because the transverse wave (s wave) exhibits a much larger propagation attenuation compared with the longitudinal wave (p wave), the s wave can be ignored. Thus, only the p wave (compressional wave) exists, i.e., silicone rubber can be regarded as a quasi-fluid.[14,15] For plexiglass, the amplitude of the p wave is about 5.2 times larger than that of the s wave in directivity pattern.[16] In addition, the amplitude of the p wave or s wave decreases with propagation distance, and its amplitude attenuation in far field can be expressed as $1/\sqrt {kr}$, where $k$ and $r$ are the wave number and propagation distance. Thus, the amplitude of the p wave is about $\sqrt {{k_{\rm s} } / {k_{\rm p} }} =\sqrt {{c_{\rm p} } / {c_{\rm s} }} =1.45$ times larger than that of the s wave, where $c_{\rm p} = 2696$ m/s and $c_{\rm s} = 1285$ m/s for plexiglass. By superposing these two factors, the amplitude of the p wave is about 7.5 times larger than that of the s wave for plexiglass.
cpl-37-6-064301-fig2.png
Fig. 2. Signals recorded by several receivers: (a) silicone rubber/water, (b) plexiglass/water.
Figure 2 shows the signals recorded using 19 receiving array elements (receivers) when the acoustic signal $F(t)$ is emitted by the 10th element (emitter), including interface reflection signals and target scattering signals. For the plexiglass/water layered media shown in Fig. 2(b), the first arrival is p–p reflection wave, i.e., the incident p wave and reflected p wave in the upper layer, followed by p–s and s–p reflection waves arrived at the same time and s–s reflection wave. It can be seen that for the 1st and 19th receivers furthest from the emitter, the amplitude of p–s and s–p reflection waves is about 8 times lower than that of p–p reflection wave, which is consistent with the above-mentioned theoretical estimate. The p–s and s–p reflection waves exhibit a lower amplitude as the distance between the source and receiver decreases, and for the 9th, 10th and 11th receivers nearest to the emitter, they are shown with a very small amplitude and almost disappear in Fig. 2(b) since the amplitude of the s wave is zero at vertical direction of transducer array in directivity pattern. For p–p reflection wave and p–p scattering wave, the changes of their amplitude with the distance between the emitter and receiver are insignificant. Therefore, the p–s and s–p reflection waves as interferences can be easily distinguished from p–p reflection wave and p–p scattering wave and ignored in the following process. For s–s reflection wave, p–s, s–p and s–s scattering waves, their amplitudes are very small and almost disappear in Fig. 2(b). The signals recorded using the $i$th receiver can be expressed as follows: $$ P_{i} (t)=a_{i} F(t-t_{i,{\rm r}})+b_{i} F[t-(t_{i,{\rm s}} +t_{i,{\rm so}})].~~ \tag {1} $$ The first term in Eq. (1) is the interface reflection signal, in which $a_{i}$ is the reflection coefficient and $t_{i,{\rm r}}$ is the travel time of acoustic pulse signal emitted by the emitter arriving at the $i$th receiver through the interface reflection. The second term in Eq. (1) is the target scattering signal, in which $b_{i}$ is the scattering coefficient, $t_{i,{\rm s}}$ is the travel time of acoustic pulse signal emitted by the emitter arriving at the target through the interface refraction, and $t_{i,{\rm so}}$ is the travel time of target scattering acoustic pulse signal arriving at the $i$th receiver through the interface refraction. The far field ray approach is used in our study, and Snell's law should be satisfied for the acoustic beam when reflected by the interface and refracted through the interface.[17]
cpl-37-6-064301-fig3.png
Fig. 3. Parameter inversions of silicone rubber (a) and plexiglass (b) using the least-square method.
For the interface reflection signal recorded using the $i$th receiver, using the geometric relationship (Fig. 1) we can obtain $$ \left({x_{i} -x_{\rm s} } \right)^{2}+4h^{2}=c_{1}^{2}t_{i,{\rm r}}^{2},~~ \tag {2} $$ where $x_{\rm s}$ and $x_{i}$ are the coordinates of emitter and $i$th receiver, respectively; $x_{\rm s}$ and $x_{i}$ are known when the transducer array is placed, and $t_{i,{\rm r}}$ can be measured by experiment. The unknown parameters $c_{1}$ and $h$ of the upper medium can be evaluated using the least-squares method. As shown in Fig. 3, based on Eq. (2), a straight line is fitted with $x_{\rm s}$, $x_{i}$, and $t_{i,{\rm r}}$ for several receivers, which exhibits root-mean-squared errors of $2.386\times 10^{-5}$ and $1.332\times 10^{-5}$, and from the slope and intercept of the fitting line, $c_{1} = 1011$ m/s and $h = 48.9$ mm for silicone rubber and $c_{1} = 2595$ m/s and $h = 36.9$ mm for plexiglass can be evaluated. The acoustic velocity of the lower medium is assumed to be $c_{2 x}$ in the following discussion. Furthermore, $t_{i,{\rm s}}$ and $t_{i,{\rm so}}$ can be measured experimentally. Based on the proposed snapshot TR-RTM mixed method,[14] the acoustic pulse signal $F(t)$ emitted by the emitter is used as the forward wave, and the signal $P_{i}(t)$ recorded by the $i$th receiver is time reversed as the backward wave $R_{i}(t)=\overline {P_{i} } (t)$ and then re-emitted from the $i$th receiver with a time $\Delta T=t_{i,{\rm so}}-t_{i,{\rm s}}$ prior to the forward wave. The forward wave propagates in all directions with a travel time of $t_{i,{\rm s}}$. Thus, an arc-shaped wavefront $S_{1}$ is formed based on some refraction equations[15] with the inversed parameters $c_{1}$, $h$ of the upper medium and the tried acoustic velocity $c_{2 x}$ of the lower medium, as shown in Fig. 4. Similarly, an arc-shaped wavefront $S_{2}$ is formed as the backward wave propagates reversely from the $i$th receiver with the same travel time ($t_{i,{\rm so}}-\Delta T=t_{i,{\rm s}}$). An intersection point of two arc-shaped wavefronts $S_{1}$ and $S_{2}$ can be deduced. Similar operations were conducted for the other emitter-receiver pairs, and an intersection distribution was developed under the tried acoustic velocity $c_{2 x}$ of the lower medium.
cpl-37-6-064301-fig4.png
Fig. 4. Location of target confirmed by the intersection point of two arc-shaped wavefronts.
If the medium parameters $c_{1}$, $h$, and $c_{2 x}$ (= $c_{2}$) are their true values and the travel times $t_{i,{\rm r}}$, $t_{i,{\rm s}}$, and $t_{i,{\rm so}}$ are measured without errors, the intersections should be theoretically gathered in a point, i.e., the target position. However, the values of $c_{1}$, $h$, $t_{i,{\rm s}}$, and $t_{i,{\rm so}}$ obtained by experiment have some experimental errors, and the acoustic velocity $c_{2}$ of the lower medium cannot be exactly determined. Then, the experimental intersections should be a discrete distribution rather than gathered in a point. $D_{\rm dis}$ is the dispersal degree of intersection distribution: $$ D_{\rm dis} =\sum\limits_{a=1}^{18} {\sum\limits_{b=1}^{18} {\sqrt {(x_{a} -x_{b})^{2}+(z_{a} -z_{b})^{2}} } },~~ \tag {3} $$ where ($x_{a}, z_{a}$) and ($x_{b}, z_{b}$) are two coordinates in the intersection distribution. Figure 5 shows the variation curve of $D_{\rm dis}$ with the tried value $c_{2 x}$ in the experiment; the acoustic velocity corresponding to the minimum of $D_{\rm dis}$ is thought of as the acoustic velocity $c_{2}$ of the lower medium. It is shown that $D_{\rm dis}$ reaches its minimum when $c_{2 x}$ is equal to 1536 m/s for rubber/water and 1539 m/s for plexiglass/water (Fig. 5).
cpl-37-6-064301-fig5.png
Fig. 5. Variation curve of $D_{\rm dis}$ with $c_{2 x}$.
cpl-37-6-064301-fig6.png
Fig. 6. Acoustic field distribution obtained using the snapshot TR-RTM method: (a) silicone rubber/water, (b) plexiglass/water.
Based on the inversed parameters $c_{1}$, $h$, and $c_{2}$, the location of target in the layered media can be confirmed using the snapshot TR-RTM mixed method.[14] At the moment $t=t_{i,{\rm s}}$, the snapshot should be taken. By coherence stacking the convolution result of forward wave and backward wave arriving at point ${\boldsymbol X}$ (Fig. 1) for each emitter-receiver pair, the so-called acoustic field distribution at point ${\boldsymbol X}$ can be described in frequency-space domain as follows: $$\begin{align} &I(\omega,{\boldsymbol X})\\ =\,&{\rm Env}\Big[\sum\limits_i {b_{i} \sum\limits_\omega {\left| {F(\omega)} \right|^{2}e^{j\omega \left[ {t_{i,{\rm s}} +t_{i,{\rm so}} -t_{i,{\rm s}} (x)-t_{i,{\rm so}} (x)} \right]}} } \\ &+\sum\limits_i {a_{i} \sum\limits_\omega {\left| {F(\omega)} \right|^{2}e^{j\omega [ {t_{i,{\rm r}} -t_{i,{\rm s}} (x)-t_{i,{\rm so}} (x)}]}} } \Big],~~ \tag {4} \end{align} $$ where $F(\omega)$ is the Fourier transform of $F(t)$; $t_{i,{\rm s}}(x)$ and $t_{i,{\rm so}}(x)$ are the travel times of forward wave and backward wave arriving at point ${\boldsymbol X}$; $a_{i}$ and $b_{i}$ are the reflection coefficient and scattering coefficient; Env represents the envelope. When point ${\boldsymbol X}$ coincides with target ${D}$, $t_{i,{\rm s}}(x) + t_{i,{\rm so}}(x)=t_{i,{\rm s}} + t_{i,{\rm so}}$, the first term (scattering part) in Eq. (4) reaches its maximum, $$ I(\omega,{\boldsymbol X})={\rm Env}\Big({\sum\limits_i {\Big[ {b_{i} \sum\limits_\omega {\left| {F(\omega)} \right|^{2}} } \Big]} } \Big),~~ \tag {5} $$ and the position of summit is the finding position of target ${D}$. A mountain-like acoustic field distribution around target ${D}$ is constructed in space, as shown in Fig. 6, and the target is located at (18.5 mm, 70.1 mm) for rubber/water and (19.6 mm, 69.7 mm) for plexiglass/water. The second term (reflection part) in Eq. (4) exhibits a nonuniform distribution with a low amplitude, and it has slight influence on target localization. It must be pointed out that the snapshot TR-RTM mixed method only requires that the waveforms in time are exactly recorded using a digital oscilloscope, so that the error of location is significantly reduced. Therefore, the detection and location of a target in layered media are achieved without prior knowledge of medium parameters. Meanwhile, the medium parameters, i.e., the acoustic velocity $c_{1}$ and thickness $h$ of the upper medium as well as the velocity $c_{2}$ of the lower medium, are also inversed together. The target positions and medium parameters $c_{1}$, $h$, and $c_{2}$ obtained simultaneously by the above processes are listed in Table 1. The locations of a target in layered media obtained without prior knowledge of medium parameters are consistent with their actual positions. In addition, the inversed medium parameters are consistent with their actual medium parameters.
Table 1. Target positions and medium parameters of silicone rubber/water (I) and plexiglass/water (II) layered media.
Media Parameters $D$ ($x_{D}, z_{D}$) (mm) $c_{1}$ (m/s) $h$ (mm) $c_{2}$ (m/s)
Liquid/liquid Actual parameters of I (19, 70) 1019 48.5 1500
Inversed parameters of I (18.5, 70.1) 1011 48.9 1536
Relative errors of I (2.63%, 0.14%) 0.79% 0.82% 2.40%
Solid/liquid Actual parameters of II (19, 70) 2696 36.4 1500
Inversed parameters of II (19.6, 69.7) 2595 36.9 1539
Relative errors of II (3.16%, 0.43%) 3.75% 1.37% 2.60%
In summary, we have presented a method to detect and locate a target in a two-layer medium without prior knowledge of medium parameters. The acoustic velocity and thickness of the upper medium are deduced using the least-square method by measuring the travel times of interface reflection signals experimentally. By combining these results with the measured travel times of target scattering signals, the acoustic velocity of the lower medium is evaluated by the minimum dispersal degree of target distribution. Finally, based on the inversed medium parameters, the location of target in a layered media is confirmed using the snapshot TR-RTM mixed method. At the time to locate the target in a layered medium, the medium parameters are inversed. It is demonstrated that for a simplest inhomogeneous medium, i.e., a two-layer medium with parallel interface, our proposed method can not only detect but also locate the target without prior knowledge of medium parameters, thus overcoming the drawbacks that the common time reversal method cannot locate the target. Furthermore, for the cases of layered media with a non-flat interface and inhomogeneity, it deserves further research. References Time‐reversal focusing through a plane interface separating two fluidsTime Reversed AcousticsTheoretical and Experimental Study of Time Reversal in Cubic CrystalsTime‐Reversal Acoustics in Biomedical EngineeringEcho-to-reverberation enhancement using a time reversal mirrorExperimental demonstration of adaptive reverberation nulling using time reversalAn overview of depth imaging in exploration geophysicsReverse Time Migration: A Seismic Imaging Technique Applied to Synthetic Ultrasonic DataReverse time migration: A prospect of seismic imaging methodologyDetection and Location of a Target in Layered Media by Snapshot Time Reversal and Reverse Time Migration Mixed Method *Experimental investigation of the detection and location of a target in layered media by using the TR-RTM mixed method
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