This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 1 Ultrasonic Curved Coordinate Transform-RAPID With Bayesian Method for the Damage Localization of Pipeline Kai Tao , Member, IEEE, Geen Chen , Qiang Wang , Member, IEEE, and Dong Yue , Fellow, IEEE Abstract—Pipeline damage could lead to accidents such as the ground subsidence and rainwater backflow. It is critical to monitor and locate the damage in real-time for the safety of urban infrastructure. In this research, a curved coordinate transform- reconstruction algorithm for probabilistic inspection of damage (RAPID) with Bayesian method was proposed to locate the damage of pipeline. First, the coordinate transform algorithm was presented. The correlation-based damage probability density was calculated in the curved coordinate system. Then, the time of flight parameter was extracted using the Hilbert transform. The curved ellipse trajectory of multipath was constructed as the likelihood function of damage location. Finally, the posterior probability was calculated by Bayesian method to update the damage position. The excitation and propagation of ultrasonic guided waves were conducted through the finite element simulation and experiments. The experiment shows that the proposed method could locate the damage of pipeline effectively. Compared with the traditional RAPID and ellipse trajectory method, the average accuracy of this method is increased by 71.315% and 58.21%. Index Terms—Bayesian updating, curved coordinate transform, ellipse trajectory location, rapid, ultrasonic guided waves. I. INTRODUCTION IPELINE has been widely used in urban infrastructure such as the oil and gas transport, water supply, as well as drainage [1], [2], [3], [4]. During the long-term service, pipelines are impacted by high loads, corrosion, etc. The reliability of P Manuscript received 15 October 2023; revised 7 January 2024; accepted 6 February 2024. This work was supported in part by the National Natural Science Foundation of China under Grant 62103198, in part by the Natural Science Foundation of Jiangsu Province under Grant BK20210598, in part by China Postdoctoral Science Foundation under Grant 2022M711694, in part by the Fund of National Dam Safety Research Center under Grant CX2022B03, in part by the Fund of Chongqing Key Laboratory of Geomechanics and Geoenvironment Protection under Grant LQ22KFJJ07. (Corresponding author: Dong Yue.) The authors are with the College of Automation and College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing 210023, China (e-mail: kai.tao@njupt.edu.cn; 1222056528@ njupt.edu.cn; wangqiang@njupt.edu.cn; yued@njupt.edu.cn). Color versions of one or more figures in this article are available at https://doi.org/10.1109/TIE.2024.3366209. Digital Object Identifier 10.1109/TIE.2024.3366209 materials gradually decreases. Pipelines are prone to damage, such as cracking, leakage, etc. [5], [6], [7]. Thus, the real time monitoring and location of damage is critical to the pipeline safety [8], [9], [10]. Ultrasonic would be reflected at the discontinuous interface. This could cause the interference and geometrical dispersion, resulting in the formation of guided waves [11], [12], [13], [14]. Guided waves propagate in an elastic waveguide with a thickness similar to the wavelength. In pipeline structures, guided waves have great propagation rules and long distances [15], [16], [17], [18], [19]. Therefore, it has been widely used in the health monitoring of pipeline, for example, Quy and Kim [20] introduced an ultrasonic-based approach to crack detection and localization in the pipeline transporting fluid under high pressure. Zeng et al. [21] proposed a damage imaging method by the encoded information in multipath scattering Lamb wave signal. Yang et al. [22] presented a novel method for the real-time monitoring of pipeline inner wall corrosion using the piezoelectric active sensing technology combined with time reversal method. Reconstruction algorithm for probabilistic inspection of damage (RAPID) evaluates the damage by calculating the probability of each unit in the structure. In the field of damage location, RAPID method has extensive applications. Wei et al. [23] proposed a modified RAPID based on the damaged virtual sensing paths to improve the localization accuracy. Azuara et al. [24] presented a geometrical modification of RAPID to reduce the influence of path intersection points among transducers. Traditional RAPID has high efficiency and accuracy in the plate structures [25]. For curved structures (e.g., pipeline), the shortcomings are as follows. 1) The pipe wall is a 360° curved structure. Starting from the excitation source, the guided waves would have two paths to the receiving point, namely the path above the excitation and the path below it. During the probability calculation, with the difference of the location distribution of the excitation point and the receiving point, sometimes the correlation parameter is calculated based on the upper path, and sometimes it is calculated based on the lower path. The randomness of the propagation path of guided wave would affect the probability distribution, and lead to errors. 2) The traditional time of flight (ToF) is the time from the excitation source to the damage point and then to the 0278-0046 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS A. Tomographic Imaging of Damage of Traditional RAPID In the tomographic imaging, the probability of damage on each excitation-receiving path was evaluated based on the correlation between the monitoring signal and the baseline signal [26]. The correlation coefficient ρ is K Yk − Y k=1 Xk − X (1) ρ = 2 K 2 K X Y − X × − Y k k k=1 k=1 Fig. 1. Diagram of the proposed method. receiving point. The ellipse trajectory method locates the damage through the ToF parameter. When the damage is on the signal propagation path, the damaged guided wave overlaps with the direct wave, resulting in the inability to acquire the scattered signal. Thus, there is possibility of missed detection in the pipeline structure. In order to overcome the errors and missed detection caused by the randomness of the propagation path of guided waves in pipe structure, in this article, a curved coordinate transform algorithm was proposed. The randomness of the propagation path of guided wave in the pipe surface was mathematically analyzed to calculate the probabilistic correlation. Then, the probability distribution was calculated based on the ToF parameter of guided wave in the curved coordinate system, which serves as the damage probability likelihood function. Finally, the posterior probability was calculated through Bayesian method, so that the damage in pipe could be updated and located probabilistically. The proposed method considers the damage probability function obtained from traditional RAPID as a prior probability. Then, the ellipse trajectory’s damage probability function was used as the likelihood function and the posterior probability was calculated. Thus, the proposed method could overcome the uncertainty of propagation path of the ultrasonic, so that the accuracy of damage location could be improved. II. METHODOLOGY First, the curved coordinate system was established to mathematically describe the randomness of dual propagation paths of guided waves on the arc surface. The damage probability distribution was calculated by the correlation coefficient between the monitoring signal and the undamaged signal. Then, the ToF parameter of scattered signal was extracted using Hilbert method. The location calculated by ellipse trajectory method was used as likelihood function. Further, the probability distribution was updated by means of the Bayesian method as the final damage location. The diagram of the proposed method was shown in Fig. 1. where Xk and Yk are the baseline signal and the monitoring signal in the kth path, respectively. X̄ and Ȳ are the average of Xk and Yk . Correlation coefficient represents the similarity between the monitoring signal and the baseline signal. The smaller ρ is, the lower the similarity is, so the damage is more likely to be in this path. For an excitation-receiving path, the distribution of damage is a linearly decreasing elliptical weighing function whose two foci are located at the excitation and the receiving sensor. When there are N paths, the probability of damage P(x, y) at location (x, y) is P (x, y) = N pn (x, y) = n=1 where Rn (x, y) = √ = n 1 − RD β 0 N ρRn (x, y) RDn ≤ β RDn > β RDn = (Dsn + √Drn ) /Dn (x−xsn )2 +(y−ysn )2 + (x−xrn )2 +(y−yrn )2 √ (2) n=1 (3) (4) (xsn −xrn )2 +(ysn −yrn )2 where pn (x, y) is the damage probability distribution at location (x, y) on the nth path. Dsn is the distance from (x, y) to the excitation transducer on the n-th path. Drn is the distance from (x, y) to the receiving transducer on the nth path. Dn is the distance from the excitation transducer to the receiving transducer. (xsn , ysn ) is the coordinate of the excitation transducer. (xrn , yrn ) is the coordinate of the receiving transducer. β is the shape factor of ellipse. B. Curved Coordinate Transform Pipe is a hollow cylinder. There would be two paths for the excitation guided waves to the receiving point. The monitoring area is divided into uniform grids. Different propagation paths would affect the calculation accuracy of the damage probability on each grid. In order to describe the randomness of the propagation path of guided waves on the pipeline surface, the curved coordinate system represented by z and α was proposed. The transform process from the original plane coordinate to curved coordinate was shown in Fig. 2. Geometrically, planes could be folded into pipelines. The two sides of the unfolding of pipe correspond to the x-axis and y-axis in the original coordinate, respectively. Assume that there are m guided waves excitation sensors and receiving sensors in the pipeline, the layout of sensors was shown in Fig. 2(a). Take the line connecting S0 (i.e., the origin of the plane coordinate Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TAO et al.: ULTRASONIC CURVED COORDINATE TRANSFORM-RAPID WITH BAYESIAN METHOD Fig. 2. (a) Layout of sensors in the original plane coordinate. (b) Unfolding of pipeline. (c) Curved coordinate system in the pipeline. (d) Two coordinate systems in the pipeline. system) and the pipe section as the o-axis. The variable α is the angle between the line connecting the projection and the center of the circle and the o-axis. Pipe was unfolded in Fig. 2(b). In this way, (x, y) in the original coordinate was transformed into (z, α) in the new coordinate. S0 is (0, 0) in the original plane coordinate, and (0, 0°) in the curved coordinate. Take S0 as the origin, the line connecting the centers of the front and rear circles of the pipe is the z-axis. The curved coordinate system was shown in Fig. 2(c), where 0 ≤ α ≤ 360◦ (5) (z, α) could express all positions on the pipe surface. The spatial relationship between the two coordinate systems in the pipeline was shown in Fig. 2(d). 1) Curved Coordinate Transform-RAPID: When guided waves propagate on the pipe surface, there are always two paths from an excitation sensor to a receiving sensor. Only the path with shorter distance is valid, and the other one should not participate in the calculation of damage location. Therefore, the mathematical expression of the effective path is Dn = min(Dn1 , Dn2 ) (6) where Dn1 and Dn2 are the distances from the excitation sensor to the receiving sensor along the upper and lower directions of the pipeline respectively. In the curved coordinate system, the two distances are 2 αsn − αrn rπ (7) Dn1 = (zsn − zrn )2 + 180 2 360 − |αsn − αrn | rπ . (8) Dn2 = (zsn − zrn )2 + 180 For point (z, α), there are two propagation paths in the upper and lower directions to the excitation sensor on the nth path, and 3 Fig. 3. (a) Elliptical monitoring area in the pipeline. (b) Geometric distribution of monitoring area in the curved coordinate system. (c) Monitoring area in the unfolded pipeline. (d) Damage probability in the monitoring area. the distances Dsn1 and Dsn2 are 2 α − αsn rπ Dsn1 = (z − zsn )2 + 180 360 − |α − αsn | rπ Dsn2 = (z − zsn )2 + 180 (9) 2 . (10) Thus, the effective distance to the excitation sensor on the nth path is Dsn = min (Dsn1 , Dsn2 ) . (11) Similarly, there are two propagation paths in the upper and lower directions to the receiving sensor, and the distances Drn1 and Drn2 are 2 α − αrn Drn1 = (z − zrn )2 + rπ (12) 180 2 360 − |α − αrn | rπ . (13) Drn2 = (z − zrn )2 + 180 The effective distance to the receiving sensor is Drn = min (Drn1 , Drn2 ) . (14) In RAPID, the guided waves propagation and damage probability reconstruction area is an ellipse, as shown in Fig. 3(a), where the foci are located at the excitation and receiving sensors. The geometric distribution of monitoring area in the curved coordinate system was shown in Fig. 3(b). The monitoring area in the unfolded pipeline was shown in Fig. 3(c). The magnitude of the damage probability is reflected by the depth of the color, as shown in Fig. 3(d). The closer to the excitation-receiving path, the greater the probability of damage. The probability of damage outside the monitoring area is zero. 2) Curved Coordinate Ellipse Trajectory: Ellipse trajectory method calculates the ToF of guided waves on sensing paths and locates damage through the intersections of multiple Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS The guided wave signal on the pipe surface is x(t) = A0 2 1 − cos 2πfc t np sin (2πfc t) (18) where A0 is the amplitude of signal, np is the number of wave peaks, and fc is the center frequency. Express the guided wave as a real signal (i.e., (16)), that is x(t) = Fig. 4. (a) Ellipse trajectory in the pipeline. (b) Damage location in the curved ellipse trajectory system. ellipses. The original coordinate was transformed into the curved coordinate, so that the damage could be located by ellipse trajectory. As for the number of ellipses selected, since two ellipses can intersect with up to 4 intersections, more than four ellipses are needed to fully identify a common intersection point. Here, 5 is chosen as this number. The ellipse trajectory in the pipeline was shown in the Fig. 4(a). S is the excitation sensor, R1-R3 are the receiving sensors. Take the path S-R3 as an example, after being emitted from S, the ultrasonic wave propagates along the pipe surface in the form of a circular array [27]. It would scatter when it reaches the damage. Thus, the signal received by R3 is the superposition of the signal propagating along a straight line and the signal passing through the damage. The scattered signal is defined as the difference between the damaged signal and the healthy signal. The envelope of the scattered signal was extracted, and the first effective wave and time were detected to get the ToF parameter [28]. The propagation distance of guided waves in the pipeline is A0 − 8 −e H {x(t)} = A0 + 8 H {u (t)} = ei(ωt+ϕ− 2 ) . π (17) +e . (19) e e 2(np+1)πfc t π i −2 np 2(np−1)πfc t π i −2 np 2(np+1)πfc t π −i −2 np −e 2(np−1)πfc t π −i −2 np −e . (20) This signal could be rewritten as a combination of sine and cosine functions H {x (t)} = A0 A0 cos (2πfc t) − cos 2 4 2 (np + 1) πfc t n 2 (np − 1) πfc t np A0 cos 4 . (21) C. Damage Location by Bayesian Updating Ellipse trajectory has low sensitivity to the damage in a straight path, which could lead to the missed detection. Bayesian method can modify the probability of RAPID and ellipse trajectory, and is sensitive to damage. The probability density function of damage in the pipe calculated by RAPID was used as the prior distribution. The point and interval estimation of parameters were conducted based on the prior distribution, so that the uncertainty of location could be reduced. The damage probability calculated by RAPID is p (θ) = N ρRn (z, α) (22) n=1 where n 1 − RD β Rn (z, α) = 0 RDn = The expression after Hilbert transform is −e 2(np−1)πfc t i np π A0 i(2πfc t− π2 ) − e−i(2πfc t− 2 ) e 4 A0 − 8 (15) (16) 2(np+1)πfc t −i np Transform it using Hilbert algorithm, that is − where T is the ToF parameter. vg is the group velocity of guided waves. Assume that the number of excitation and receiving sensors is 8, take the S2, S4, and S8 as excitation sensors as an example, the principle of damage location using ellipse trajectory method in pipeline was shown in Fig. 4(b). The ultrasonic signal would be received by sensors R2, R3, R5, and R8 after propagating along the pipeline. The received signal is the superposition of the source signal and the damaged signal. The first effective wave of the scattered signal was identified. The ToF parameters of the five paths are calculated to obtain five ellipses. Theoretically, the damage is located at the intersection of the ellipses. The envelope was extracted by Hilbert algorithm to get ToF. Hilbert transform is a linear time-invariant integration algorithm. It can transform a real signal in the time domain (or space domain) into a complex signal represented by amplitude and phase in the frequency domain. For a real signal u(t) u (t) = ei(ωt+ϕ) . e 2(np+1)πfc t i np 2(np −1)πfc t −i np Dsn + Drn = min (Dsn1 , Dsn2 ) + min (Drn1 , Drn2 ) = T · vg A0 i(2πfc t) e − e−i(2πfc t) 4 RDn ≤ β RDn > β (23) Dsn +Drn min (Dsn1 , Dsn2 )+min (Drn1 , Drn2 ) = Dn min (Dn1 , Dn2 ) (24) Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TAO et al.: ULTRASONIC CURVED COORDINATE TRANSFORM-RAPID WITH BAYESIAN METHOD 5 TABLE II COORDINATES OF THE SENSOR Fig. 5. (a) Deployment of the excitation and receiving sensors. (b) Geometry of healthy pipe. (c) Geometry of damaged pipe. TABLE I PHYSICAL PARAMETERS USED IN THE SIMULATION where ρ is the correlation coefficient. For the area around the damage point (zd , αd ), the probability of damage is proportional to the number of ellipses passing through the area unit. The damage probability for areas far away from the damage point is 0. Thus, the damage probability function P(D|θ) (i.e., likelihood function) could be obtained based on the number of ellipses passing through each area unit. Further, the posterior probability density function P(θ|D) was fitted by the Bayesian mining of monitoring data, i.e., p(θ|D) = p(D|θ)p(θ). (25) First, the damage probability of each unit on the pipe wall was assessed by RAPID. The part with higher probability was the prior probability. Then, the area with high damage probability on the pipe wall was constructed using the curved coordinate ellipse trajectory method. Finally, the above two probability areas were superimposed using Bayesian method. The area with the highest probability was updated as the final damage location. III. NUMERICAL SIMULATION A. Simulation Setting 1) Geometry of Pipeline System: The propagation of guided waves on the surface of pipeline was simulated using finite element method. Geometry of the pipeline and the deployment of transducers were shown in Fig. 5(a). The length of the simulated pipe is 2 m. The inner radius of the section is 0.071 m, and the outer radius is 0.08 m. The pipe material is steel, and the physical parameters were given in Table I. Before the simulation of signal excitation, a circle was drawn and drilled through the pipe wall to simulate the damage. Simulate the excitation, propagation as well as the receiving of guided waves for the damaged and healthy pipelines respectively. Eight excitation sensors (S1-S8) were arranged at Fig. 6. Generation of meshes in the pipe. (a) Neutral axis method. (b) Meshes in the damaged pipe. (c) Hexahedron meshes. (d) Tetrahedral meshes. equal intervals around the 0.75 m section of the pipeline. Eight receiving sensors (R1-R8) were arranged at 1.25 m. The distance between the excitation and receiving sensors is 0.5 m. in the curved coordinate system. The coordinates of the 16 sensors were given in Table II. The geometry of the healthy and damaged pipe was shown in Fig. 5(b) and (c). 2) Meshes Generation: The pipe model needs to be defined as a deformable body in the simulation. The healthy pipe was divided using the neutral axis hexahedron meshes because it is a regular geometry [see Fig. 6(a)]. There are some irregular parts in the damaged pipe, which need to be divided by free tetrahedral meshes to improve the accuracy. Thus, the meshes near the damage are free tetrahedrons, and the meshes far away from the damage remain neutral axis hexahedrons, as shown in Fig. 6(b). Fig. 6(c) shows the detail of hexahedron meshes. Fig. 6(d) shows the detail of free tetrahedral meshes at the damage. B. Simulation of Guided Waves The distribution of the 16 sensors was shown in Fig. 7(a). In order to excite the ultrasonic in the vertical direction, a cylindrical coordinate system was set to replace the original coordinate in the simulation. The guided waves enter the pipe wall vertically in the direction of the arrow in Fig. 7(b). The excitation was the five-cycle signal modulated by the narrow-band Hanning window, that is x(t) = 2.5 1 − cos 2πfc t 5 sin (2πfc t) . (26) Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6 Fig. 7. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Cylindrical coordinate system in the simulation. Fig. 10. (a) Oscillograph. (b)Wideband amplifier. (c) Electric charge amplifier. (d) Experimental pipe. (e) Experiment system. (f) Flow of data acquisition system. IV. EXPERIMENT Fig. 8. Excitation wave. Fig. 9. Stress distribution during the propagation of guided waves. (a) Stress distribution at 6.0e-5s. (b) stress distribution at 9.0e-5s. (c) Stress distribution at 1.2e-5s. (d) Stress distribution at 1.5e-5s. The center frequency of the signal was 80 KHz and the amplitude was 5 V. The sampling frequency was 20 MHz. The excitation wave was shown in Fig. 8. Take the S1-R1 path as an example, in the healthy pipe, the stress distribution during the propagation of guided waves was shown in Fig. 9. At the beginning, the guided waves propagate outward on the pipe surface in the form of a circular array [see Fig. 9(a) and (b)]. At 1.2e-5s, the guided waves on both sides meet and then continue to propagate [see Fig. 9(c) and (d)]. The received signals have four peaks, including the direct waves and the superimposed waves of subsequent signals. The material of the experiment pipe is steel, and the length is 2 m. The excitation and receiving sensors are piezoelectric transducers, which were deployed at 0.5 and 1 m from the pipe section. The sensors were evenly pasted on the surface around the pipe section. A cylindrical iron block was pasted on the pipe surface to simulate damage. The excitation signal is an 80 KHz five-peak signal with a wave speed of 5880 m/s, which is the same as the simulation. The experiment system was shown in Fig. 10. There are 8 excitation and receiving sensors. The red wire is the positive pole, the black wire is the negative pole. The output of the signal generator was connected to the excitation sensor. The input of oscillograph was connected to the receiving sensor, and the output was connected to the signal generator. First, excitation signal was generated from S1. The guided waves were received by sensors R1-R8 after propagating on the healthy and damaged pipes. In this way, eight healthy signals and eight damaged signals could be acquired. Then, excitation signal was generated from S2, and the guided waves were received by sensors R1-R8. Atotal of eight healthy signals and eight damaged signals could be acquired as well. Perform the same operation for the other 6 excitations, and 128 signal could be acquired, including 64 healthy signals and 64 damaged signals. Each signal has 12 000 sample points. V. RESULTS AND DISCUSSION A. Damage Location in Numerical Simulation Take the damage position at (1, 155°) as an example, the guided waves in S1-R1 path in the simulation was shown in Fig. 11. First, the healthy signal [see Fig. 11(a)] and damaged signal [see Fig. 11(b)] were acquired. The comparison of two signals were shown in Fig. 11(c). Then, the correlation coefficient of S1-R1 path could be calculated from monitoring signal and the undamaged signal. The scattered signal was obtained by subtracting the healthy signal from the damaged signal. The ToF could be extracted by the first scattered wave [red dashed Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TAO et al.: ULTRASONIC CURVED COORDINATE TRANSFORM-RAPID WITH BAYESIAN METHOD 7 TABLE IV TOF FOR ALL SIMULATION PATHS (×10-4 S) Fig. 11. Guided waves in S1-R1 path in the simulation. (a) Signal in the healthy pipe. (b) Signal in the damaged pipe. (c) Comparison of two signals. (d) The scattered signal. TABLE III CORRELATION COEFFICIENTS OF ALL SIMULATION PATHS Fig. 13. (a) Damage location by RAPID in the simulation. (b) Comparison of the tomographic damage and the damage set in the simulation. Fig. 14. Ellipse trajectory and the distribution of the real damage in the simulation. Fig. 12. ToF in S1-R1 path in the simulation. line in Fig. 11(d)]. Calculate the correlation coefficient for all simulation paths, as given in Table III. The ToF parameter of scattered signal was extracted using the Hilbert transform. The ToF in S1-R1 path was shown in Fig. 12. The first peak of the envelope is the peak of the excitation signal. The second peak is the crest of the first scattered wave. The time difference between them is the ToF of the scattered wave. Extract the ToF for all simulation paths, as given in Table IV. The damage location by RAPID in the simulation was shown in Fig. 13(a). The comparison of the location of the tomographic damage and the damage set in the simulation was shown in Fig. 13(b). Fig. 13(a) shows that in the unfolded pipe, if the probabilistic images for each path were superimposed, there would be a yellow area where the probability is higher. Thresholding this area, and the center was used as the coordinate of the damage, which is (1, 158°). In Fig. 13(b), the yellow color is the damage imaging area, and the red circle is the set damage. The ellipse trajectory and the distribution of the real damage were shown in Fig. 14. In Fig. 14, the damage was located by five paths, namely S2R6 (purple), S2-R7 (black), S3-R5 (red), S4-R5 (blue), and S7R3 (green). The coordinate is (1.003, 155°). In Fig. 15, the figure in each cell indicates the number of ellipses passed through. There are 5 ellipses in total. For each area unit, an ellipse counts as a probability value of 0.2. Therefore, the number of ellipses multiplied by 0.2 is the damage probability p(D|θ) of that area unit. The posterior probability p(θ|D) was shown in Fig. 16(a), and the enlarged part was shown in Fig. 16(b). The yellow block is the maximum part of the posterior probability, and the coordinate is (0.999, 155°). The red block is the damage point set in the simulation. Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Fig. 15. Number of ellipses passing through the damage surrounding units in the simulation. Fig. 18. Fig. 16. (a) Posterior probability in damage location in the simulation. (b) Enlarged location. ToF in S2-R2 path in the experiment. Fig. 19. (a) Damage location by RAPID in the experiment. (b) Comparison of the tomographic damage and the real damage. Fig. 20. Ellipse trajectory and the distribution of the real damage in the experiment. Fig. 17. Guided waves in S2-R2 path in the experiment. (a) Signal in the healthy pipe. (b) Signal in the damaged pipe. (c) Comparison of two signals. (d) Scattered signal. B. Damage Location in Experiment The coordinate of the real damage is (1, 158°). The guided waves in S2-R2 path in the experiment was shown in Fig. 17. First, the healthy signal [see Fig. 17(a)] and damaged signal [see Fig. 17(b)] were acquired. Fig. 17(c) is the comparison of two signals. Fig. 17(d) is the scattered signal. Then, the correlation coefficient of S2-R2 path could be calculated from the monitoring signal and the undamaged signal. The ToF parameter could be calculated based on the scattered signal, which was obtained by subtracting the healthy signal from the damaged signal. After Hilbert transform, it can be shown in Fig. 18. The damage location by RAPID in the experiment was shown in Fig. 19(a). The center of the high probability area is the damage position, and the coordinate is (0.999, 155°), as shown in Fig. 19(b). The ellipse trajectory and the distribution of the real damage were shown in Fig. 20. In the experiment, damage was located by five paths, which were S2-R6 (purple), S2-R7 (black), S3-R5 (red), S4-R4 (blue), and S7-R2 (green). The coordinate is (1.001, 160°). In Fig. 21, the figure in each cell indicates the number of ellipses passed through. There are 5 ellipses in total. For each area unit, an ellipse counts as a probability value of 0.2. Therefore, the number of ellipses multiplied by 0.2 is the damage probability p(D|θ) of that area unit. The probability p(D|θ) based on the number of ellipses passed through was shown in Fig. 21. In Fig. 19(b), only a small section of the RAPID- probability has a valid value, and the rest of it is 0. Thus, the horizontal axis in Fig. 21 only needs to contain the fraction which has valid value. The part that less than 991 and more than 1008 would be 0 ultimately. Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TAO et al.: ULTRASONIC CURVED COORDINATE TRANSFORM-RAPID WITH BAYESIAN METHOD 9 TABLE V VALUE OF D USING DIFFERENT DAMAGE LOCATION ALGORITHMS(M) Fig. 21. units. Number of ellipses passing through the damage surrounding TABLE VI Npixel USING DIFFERENT DAMAGE LOCATION ALGORITHMS TABLE VII RUNNING TIME USING DIFFERENT DAMAGE LOCATION ALGORITHMS (MS) Fig. 22. (a) Posterior probability in damage location in the experiment. (b) Enlarged location. D. Algorithm Comparison In order to compare the proposed method with other traditional damage location methods, the distance d between the calculated value and the actual value was defined to assess the accuracy. In the simulation, the coordinate of damage was (1, 155°), the expression of d is 2 α − 155 × 0.08π . d1 = (z − 1)2 + (27) 180 Fig. 23. (a) Damage location by RAPID of two defects. (b) Posterior probability in damage location of two defects. The posterior probability p(θ|D) and the enlarged part were shown in Fig. 22(a) and (b). The coordinate of damage is (1.001, 157°). It can be seen that the consistency with the real damage (i.e., the red part) is high. C. Two Defects Location Based on the first damage, the damage with coordinates (1, 135°) was added. First, coordinate transform-RAPID was utilized to get the result shown in Fig. 23(a). Then, two points were located using coordinate ellipse trajectory. The probabilities were assigned to the region near the two points based on the number of the crossed ellipses. Finally, the posterior probability of damage was generated using Bayesian updating, as shown in Fig. 23(b). It can be seen that when two points are localized simultaneously, the first point results in (0.998, 155°) and the second point results in (0.999, 134°). In the experiment, the coordinate of real damage was (1, 158°), the expression of d is 2 α − 158 × 355 . (28) d2 = (z − 1)2 + 360 For the data from numerical simulation and experiment, the tomographic imaging method, ellipse trajectory method, as well as the proposed method were used to calculated d, as given in Table V. Table V gives that the proposed method has the highest accuracy. For the data from numerical simulation, the accuracy of the proposed method is 76.13% higher than the traditional tomographic imaging method, and 66.67% higher than the ellipse trajectory method. For the data from experiment, the accuracy of the proposed method is 66.5% higher than the traditional tomographic imaging method, and 49.75% higher than the ellipse trajectory method. It has applicability in the harsh pipeline working conditions. The number of the valid unit (whose value is not 0) in the probability of damage was defined as Npixel . The fewer the valid units, the lower the probability of errors. The results of Npixel of three methods were given in Table VI. Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS The running times of the three methods were given in Table VII. VI. CONCLUSION In order to overcome the uncertainty of propagation path of the guided waved in the damaged pipeline structure, a curved coordinate transform-RAPID with Bayesian method was proposed to locate the damage in this article. A mathematical description of the randomness of guided wave propagation in the pipe was established using the curved coordinate system. Furthermore, the probability density of tomographic imaging and ellipse trajectory method was corrected using the Bayesian algorithm. The damage was located by the posterior probability. The propagation of guided waves was simulated through the finite element. Damage was located for the data from the numerical simulation and experiment. The results show that the accuracy is much higher than traditional tomographic imaging method and ellipse trajectory method. REFERENCES [1] J. Xu et al., “Low-cost, tiny-sized MEMS hydrophone sensor for water pipeline leak detection,” IEEE Trans. Ind. Electron., vol. 66, no. 8, pp. 6374–6382, Aug. 2019, doi: 10.1109/TIE.2018.2874583. [2] H. Huan et al., “Real-time predictive temperature measurement in oil pipeline: Modeling and implementation on embedded wireless sensing devices,” IEEE Trans. Ind. Electron., vol. 68, no. 12, pp. 12689–12697, Dec. 2021, doi: 10.1109/TIE.2020.3040679. [3] H. Yu et al., “An online pipeline structural health monitoring method based on the spatial deformation fitting,” IEEE Trans. Ind. Electron., vol. 69, no. 7, pp. 7383–7393, Jul. 2022, doi: 10.1109/TIE.2021.3101003. [4] B. Zhou, A. Liu, X. Wang, Y. She, and V. Lau, “Compressive sensingbased multiple-leak identification for smart water supply systems,” IEEE Internet Things J., vol. 5, no. 2, pp. 1228–1241, Apr. 2018, doi: 10.1109/JIOT.2018.2812163. [5] J. Ling, K. Feng, T. Wang, M. Liao, C. Yang, and Z. Liu, “Data modeling techniques for pipeline integrity assessment: A State-of-the-art survey,” IEEE Trans. Instrum. Meas., vol. 72, Jun. 2023, Art. no. 3518117, doi: 10.1109/TIM.2023.3279910. [6] Z. Wang, S. Huang, S. Wang, S. Zhuang, Q. Wang, and W. Zhao, “Compressed sensing method for health monitoring of pipelines based on guided wave inspection,” IEEE Trans. Instrum. Meas., vol. 69, no. 7, pp. 4722–4731, Jul. 2020, doi: 10.1109/TIM.2019.2951891. [7] Z. Hu, S. Tariq, and T. Zayed, “A comprehensive review of acoustic based leak localization method in pressurized pipelines,” Mech. Syst. Signal Process., vol. 161, Dec. 2021, Art. no. 107994. [8] S. Shan and L. Cheng, “Two-dimensional scattering features of the mixed second harmonic A0 mode lamb waves for incipient damage localization,” Ultrasonics, vol. 119, Feb. 2022, Art. no. 106554. [9] L. Huang, X. Hong, Z. Yang, Y. Liu, and B. Zhang, “CNN-LSTM networkbased damage detection approach for copper pipeline using laser ultrasonic scanning,” Ultrasonics, vol. 121, Apr. 2022, Art. no. 106685. [10] K. Tao, Q. Wang, and D. Yue, “Data compression and damage evaluation of underground pipeline with musicalized Sonar GMM,” IEEE Trans. Ind. Electron., vol. 71, no. 3, pp. 3093–3102, Mar. 2024, doi: 10.1109/TIE.2023.3270519. [11] X. Zhu, P. Rizzo, A. Marzani, and J. Bruck, “Ultrasonic guided waves for nondestructive evaluation/structural health monitoring of trusses,” Meas. Sci. Technol., vol. 21, Mar. 2010, Art. no. 045701. [12] A. Rommeler et al., “Air coupled ultrasonic defect detection in polymer pipes,” NDT E Int., vol. 102, pp. 244–253, Mar. 2019. [13] H. Zhang, Q. Wu, C. Chen, Z. Liu, T. Bai, and K. Xiong, “Porosity evaluation of composites using the encapsulated cantilever fiber Bragg grating based acousto-ultrasonic method,” IEEE Sensors J., vol. 22, no. 16, pp. 15991–15998, Aug. 2022, doi: 10.1109/JSEN.2022.3188802. [14] Q. Bao, T. Xie, W. Hu, K. Tao, and Q. Wang, “Multi-type damage localization using the scattering coefficient-based RAPID algorithm with damage indexes separation and imaging fusion,” Struct. Health Monit., vol. 10, Aug. 2023, Art. no. 14759217231191267. [15] M. J. S. Lowe, D. N. Alleyne, and P. Cawley, “Defect detection in pipes using guided waves,” Ultrasonics, vol. 36, no. 1–5, pp. 147–154, Feb. 1998. [16] S. S. Bang, Y. H. Lee, and Y. J. Shin, “Defect detection in pipelines via guided wave-based time–frequency-domain reflectometry,” IEEE Trans. Instrum. Meas., vol. 70, Jan. 2021, Art. no. 9505811, doi: 10.1109/TIM.2021.3055277. [17] S. Livadiotis, A. Ebrahimkhanlou, and S. Salamone, “An algebraic reconstruction imaging approach for corrosion damage monitoring of pipelines,” Smart Mater. Structures, vol. 28, no. 5, Apr. 2019, Art. no. 055036. [18] X. Wang, H. Gao, K. Zhao, and C. Wang, “Time-frequency characteristics of longitudinal modes in symmetric mode conversion for defect characterization in guided waves-based pipeline inspection,” NDT E Int., vol. 122, Sep. 2021, Art. no. 102490. [19] Z. Lu, Z. Liu, W. Jiang, B. Wu, and C. He, “Intelligent defect location of a U-shaped boom using helical guided waves,” Struct. Health Monit., vol. 22, pp. 2827–2855, 2023. [20] T. B. Quy and J. M. Kim, “Crack detection and localization in a fluid pipeline based on acoustic emission signals,” Mech. Syst. Signal Process., vol. 150, Mar. 2021, Art. no. 107254. [21] L. Zeng, L. Huang, and J. Lin, “Damage imaging of composite structures using multipath scattering Lamb waves,” Composite Struct., vol. 216, pp. 331–339, May 2019. [22] D. Yang, X. Zhang, T. Wang, G. Lu, and Y. Peng, “Study on pipeline corrosion monitoring based on piezoelectric active time reversal method,” Smart Mater. Struct., vol. 32, Apr. 2023, Art. no. 054003. [23] J. Wei et al., “Modified reconstruction algorithm for probabilistic inspection of damage based on damaged virtual sensing paths,” Measurement, vol. 218, Aug. 2023, Art. no. 113182. [24] G. Azuara, E. Barrera, M. Ruiz, and D. Bekas, “Damage detection and characterization in composites using a geometric modification of the RAPID algorithm,” IEEE Sensors J., vol. 20, no. 4, pp. 2084–2093, Feb. 2020, doi: 10.1109/JSEN.2019.2950748. [25] H. Zhang et al., “Multi-sensor network for industrial metal plate structure monitoring via timereversal ultrasonic guided wave,” Measurement, vol. 152, Feb. 2020, Art. no. 107345. [26] H. Huo, J. He, and X. Guan, “A bayesian fusion method for composite damage identification using Lamb wave,” Struct. Health Monit., vol. 20, no. 5, pp. 2337–2359, 2021. [27] P. S. Lowe, R. M. Sanderson, N. V. Boulgouris, A. G. Haig, and W. Balachandran, “Inspection of cylindrical structures using the first longitudinal guided wave mode in isolation for higher flaw sensitivity,” IEEE Sensors J., vol. 16, no. 3, pp. 706–714, Feb. 2016, doi: 10.1109/JSEN.2015.2487602. [28] R. C. Sriramadasu, S. Banerjee, and Y. Lu, “Detection and assessment of pitting corrosion in rebars using scattering of ultrasonic guided waves,” NDT E Int., vol. 101, pp. 53–61, Jan. 2019. Kai Tao (Member, IEEE) received the B.S. degree in electrical engineering from Qingdao University of Technology, Qingdao, China, in 2015, and the Ph.D. degree in instrument science and technology from Chongqing University, Chongqing, China, in 2020. He is currently an Assistant Professor with Nanjing University of Posts and Telecommunications, Nanjing, China. He was a University Associate Researcher with the University of Tasmania, Australia, from 2018 to 2019. His research interests mainly focus on structural health monitoring and advance signal processing. Geen Chen received the B.S. degree in electrical engineering from Nanjing University of Posts and Telecommunications, Nanjing, China, in 2022. He is currently working toward the master’s degree in electronic information with Nanjing University of Posts and Telecommunications, Nanjing, China. His research interests mainly focus on structural health monitoring. Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TAO et al.: ULTRASONIC CURVED COORDINATE TRANSFORM-RAPID WITH BAYESIAN METHOD Qiang Wang (Member, IEEE) received the B.Sc. degree in instrument science and technology from Yanshan University, Qinhuangdao, China, in 2002, and the M.Eng. and Ph.D. degrees in instrument science and technology from the Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2005 and 2009, respectively. He is currently a Professor with the School of Automation, Nanjing University of Posts and Telecommunications, Nanjing. From 2011 to 2012, he was a Research Associate with the Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong. He has authored or coauthored over 10 refereed international journal papers and 10 conference papers. His current research interests include structural health monitoring, sensors, and digital signal processing. 11 Dong Yue (Fellow, IEEE) received the Ph.D. degree in control science and engineering from the South China University of Technology, Guangzhou, China, in 1995. He is currently a Professor and the Dean of the College of Automation, Institute of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing, China, and a Changjiang Professor with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. He has authored or co-authored more than 100 papers in international journals. His current research interests include analysis and synthesis of networked control systems, signal processing, and Internet of Things. Dr. Yue is currently an Associate Editor for the IEEE Control Systems Society Conference Editorial Board, IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, the Journal of the Franklin Institute, and the International Journal of Systems Science. Authorized licensed use limited to: The University of British Columbia Library. Downloaded on April 16,2024 at 01:04:53 UTC from IEEE Xplore. Restrictions apply.