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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
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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
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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
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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)
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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)
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Fig. 7.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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