An adaptive sparse deconvolution method for distinguishing the overlapping echoes of ultrasonic guided waves for pipeline crack inspection

An obvious phenomenon caused by dispersion is that the width of the pulse will be extended in the time domain, and the amplitude of the pulse will decrease. The changed pulse can be treated as an energy-attenuated, time-shifted and frequency-dissipated version of the excitation pulse [19, 20]. In this paper, we utilize the Gaussian echo model to analyze the incident pulse. The model can be expressed as

$$\begin{eqnarray}&&f(\theta ;t)=\beta {{\text{e}}^{-\alpha {{t}^{2}}}}\cos (2\pi \omega t+\phi )\end{eqnarray} \tag{ 2 }$$

where $\theta =(\beta,\alpha,\omega,\phi )$ is the parameter vector, $\beta$ is the amplitude, $\alpha$ expresses the bandwidth factor, $\omega$ describes the center frequency, and $\phi$ is the phase of the pulse. Each parameter has an intuitive meaning for the incident pulse. The amplitude $\beta$ is related to the properties of the active and received sensors.

The main purpose of this work is to predict the parameter vector $\theta =(\beta,\alpha,\omega,\phi )$ according to the observed incident pulse. The Gauss Newton (GN) algorithm was employed to estimate the parameter of the parameter vector. According to the computing principle of the GN algorithm, the estimation of a parameter vector can be expressed as

$$\begin{eqnarray}&&{{\theta}^{(k+1)}}={{\theta}^{(k)}}+{{\left({{G}^{T}}\left({{\theta}^{(k)}}\right)G\left({{\theta}^{(k)}}\right)\right)}^{-1}}{{G}^{T}}\left({{\theta}^{(k)}}\right)\left(h-f\left({{\theta}^{(k)}}\right)\right)\end{eqnarray} \tag{ 3 }$$

where $G(\theta )$ is the gradient of the echo model for each parameter in the parameter vector $\theta =(\beta,\alpha,\omega,\phi )$. The whole iteration process of the GN algorithm has the following steps:

1. Choose the initial parameter vector ${{\theta}^{(0)}}$, and set the initial number $k=0$.

2. Compute the gradients $G\left({{\theta}^{(k)}}\right)$ and the model $s\left({{\theta}^{(k)}}\right)$.

3. Compute ${{\theta}^{(k+1)}}={{\theta}^{(k)}}+{{({{G}^{T}}({{\theta}^{(k)}})G({{\theta}^{(k)}}))}^{-1}}{{G}^{T}}({{\theta}^{(k)}})(h-f({{\theta}^{(k)}}))$.

4. If $\|{{\theta}^{(k+1)}}-{{\theta}^{(k)}}\|<\text{tolerance}$, then stop.

5. Set $k\to k+1$ and go to step 2.

According to the principle of mathematical convolution, the operation in equation (1) can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. Thus, the convolution model of the guided wave inspection (equation (1)) can be formulated as

$$\begin{eqnarray}&&s=Hx+n\end{eqnarray} \tag{ 4 }$$

where H is the convolution matrix. Meanwhile, it is also a Toeplitz matrix of size $M\times N$ with $M>N$. The columns of the Toeplitz matrix are the incident pulse $h(t)$ in the convolution model of the guided wave inspection.

Due to the sparse properties of the reflection sequence $x$ in equation (4), the desired reflection sequence $x$ can be intuitively recovered from the measured signal $s$ by solving

$$\begin{eqnarray}&&\underset{x}{{\min}}\,\|x{{\|}_{0}}\quad \text{subject}\,\text{to}\,\|s-Hx\|_{2}^{2}\leqslant \delta \,\end{eqnarray} \tag{ 5 }$$

where $\|x{{\|}_{0}}$ is the ${{l}_{0}}$-norm of the solution, and is defined as $\|x{{\|}_{0}}={\sum}_{m}|{{x}_{m}}{{|}_{0}}$, which is used to count the number of non-zero values in $x$, and the tolerance $\delta$ is the noise level. The problem of finding the optimization solution of ${{l}_{0}}$-norm regularization is NP-hard [21, 22]. Compared with ${{l}_{2}}$-norm regularization, which controls the energy of the unknown solution, the ${{l}_{1}}$-norm regularization typically yields a sparse solution, and the reason for this has been proofed in [23, 24]. Thus, Chen in [25] replaces the ${{l}_{0}}$-norm with the ${{l}_{1}}$-norm, because minimizing the ${{l}_{1}}$-norm can help us to find a sparse solution; meanwhile, the ${{l}_{1}}$-norm is a simple convex problem which can be solved by some classical optimization methods [26, 27]. This transforms the NP-hard problem into a convex optimization problem

$$\begin{eqnarray}&&\underset{x}{{\min}}\,\|x{{\|}_{1}}\quad \text{subject}\,\text{to}\,\|s-Hx\|_{2}^{2}\leqslant \delta \,\end{eqnarray} \tag{ 6 }$$

where $\|x{{\|}_{1}}={\sum}_{i=1}^{m}|{{x}_{i}}|$ is the ${{l}_{1}}$-norm of the solution. The unconstrained version of equation (6) is given by

$$\begin{eqnarray}&&\arg \underset{x}{{\min}}\,\frac{1}{2}\|s-Hx\|_{2}^{2}+\lambda \|x{{\|}_{1}}\end{eqnarray} \tag{ 7 }$$

where $\lambda$ is the regularization parameter, which is an important factor for controlling the sparsity of the solution. The method for finding the optimum value of the regularization parameter has been discussed in [12]. The ${{l}_{1}}$-norm regularization defined in equation (7) is also known as basis pursuit denoising (BPD) [25].

The ${{l}_{1}}$-norm regularization must be solved by iterative methods due to the lack of an analytic solution; thus, ${{l}_{1}}$-norm regularization problems can be solved by convex optimization algorithms. Actually, many researchers have made a lot of effort to develop many effective algorithms for solving ${{l}_{1}}$-norm regularization problems, such as the two-step iterative shrinkage/thresholding algorithm (TwIST) [30], sparse reconstruction by separable approximation (SpaRSA) [31] and SALSA [18, 30]. In this paper the latter is employed to solve the deconvolution problem, because its convergence properties are better than TwIST and SpaRSA in practice [18, 30, 31].

A deconvolution problem like equation (7) is an unconstrained optimization problem, which must be transformed into a constrained optimization problem. Equation (7) can be decomposed into two functions: ${{f}_{1}}(x)$ and ${{f}_{2}}(x)$

$$\begin{eqnarray}&&\underset{x}{{\min}}\,{{f}_{1}}(x)+{{f}_{2}}(x)\end{eqnarray} \tag{ 8 }$$

where ${{f}_{1}}(x)=\frac{1}{2}\|s-Hx\|_{2}^{2}$, ${{f}_{2}}(x)=\lambda \|x{{\|}_{1}}$. Under the constrain $x=c$, equation (8) is equivalent to

$$\begin{eqnarray}&&\underset{x,c}{{\min}}\,{{f}_{1}}(x)+{{f}_{2}}(c)\quad \text{subject}\,\text{to}\quad \,x=c\end{eqnarray} \tag{ 9 }$$

which is the so-called variable splitting method. Using the augmented Lagrangian problem to represent this problem as follows

$$\begin{eqnarray}&&\underset{z}{{\min}}\,E(z)\quad \text{subject}\,\text{to}\quad \,Az-b=0\end{eqnarray} \tag{ 10 }$$

where $E(z)={{f}_{1}}(x)+{{f}_{2}}(c)$, $z=\left[\begin{array}{c} x \\ c \end{array}\right]$, $b=0$, $A=\left[\begin{array}{c} I & -I \end{array}\right]$. The solution of equation (10) can be written as

$$\begin{eqnarray}&&{{z}_{k+1}}=\arg \underset{z}{{\min}}\,E(z)+\frac{\mu}{2}\|Az-{{v}_{k}}\|_{2}^{2}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{{v}_{k+1}}={{v}_{k}}-\left(A{{z}_{k+1}}-b\right)\end{eqnarray} \tag{ 12 }$$

where $k$ denotes the iteration index and $\mu$ is the penalty parameter. Alternating between minimization with respect to $x$ and $c$, equation (11) can be written as

$$\begin{eqnarray}&&{{x}_{k+1}}=\arg \underset{x}{{\min}}\,{{f}_{1}}(x)+\frac{\mu}{2}\|x-{{c}_{k}}-{{v}_{k}}\|_{2}^{2}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{{c}_{k+1}}=\arg \underset{c}{{\min}}\,{{f}_{2}}(c)+\frac{\mu}{2}\|{{x}_{k+1}}-c-{{v}_{k}}\|_{2}^{2}.\end{eqnarray} \tag{ 14 }$$

Equations (13) and (14) should be substituted with the explicit form for the two functions ${{f}_{1}}$ and ${{f}_{2}}$; then, the SALSA algorithm can be obtained as follows:

$$\begin{eqnarray}&&{{x}_{k+1}}=\arg \underset{x}{{\min}}\,\frac{1}{2}\|s-Hx\|_{2}^{2}+\frac{\mu}{2}\|x-{{c}_{k}}-{{v}_{k}}\|_{2}^{2}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{{c}_{k+1}}=\arg \underset{c}{{\min \lambda}}\,\|c{{\|}_{1}}+\frac{\mu}{2}\|{{x}_{k+1}}-c-{{v}_{k}}\|_{2}^{2}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{{v}_{k+1}}={{v}_{k}}-\left({{x}_{k+1}}-{{c}_{k+1}}-b\right).\end{eqnarray} \tag{ 17 }$$

By running the iterative SALSA until the stopping criterion is satisfied, the solution of equation (7) can be found.

The procedure of the proposed ASD method is illustrated in figure 4. In theory, the columns of the Toeplitz matrix in equation (7) are the input signal. However, the input signal is substantially changed due to the dispersion properties of the guided wave, and the specific changes in the input signals are barely known. Thus, the Gaussian echo model is employed in this paper to estimate the incident pulse in the measured signal. This model is sensitive to the signal characteristics: center frequency, amplitude, bandwidth, and the phase of the incident pulse. Then, the estimation results are used as the columns of the Toeplitz matrix. Compared to the ASD method, the ${{l}_{1}}$-norm deconvolution method does not have a model estimation procedure (step 2 in figure 4), which is the main difference between the two deconvolution methods. Because of this difference, the time-varying pulse problem caused by dispersion is solved well by the ASD method, whereas it cannot be solved by the conventional ${{l}_{1}}$-norm deconvolution method.

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The dispersive propagation model of the guided wave presented in [32, 33] is employed in this paper. A field quantity $f(x,t)$ is used to describe the propagation of the mode of interest in time and space, where $x$ is the propagation distance and $t$ is time. At position $x=0$, where the transducers are located, $f(x,t)=f(t)$ is the input signal. If the propagation characteristics of the model of interest are known, then $f(x,t)$ can be evaluated at any other point:

$$\begin{eqnarray}&&f(x,t)={\int}_{-\infty}^{\infty}F(\omega ){{\text{e}}^{\text{i}\left(k(\omega )x-\omega t\right)}}\text{d}\omega \end{eqnarray} \tag{ 18 }$$

where $F(\omega )$ is the Fourier transform of the input signal $f(t)$, $\omega$ is the angular frequency, the wavenumber $k(\omega )$ is a function of the angular frequency $\omega$, by the simple relationship:

$$\begin{eqnarray}&&k(\omega )=\frac{\omega}{{{v}_{\text{p}}}(\omega )}.\end{eqnarray} \tag{ 19 }$$

Substituting equation (19) into equation (18) yields

$$\begin{eqnarray}&&f(x,t)={\int}_{-\infty}^{\infty}F(\omega ){{\text{e}}^{\text{i}\left(\frac{\omega}{{{v}_{\text{p}}}(\omega )}x-\omega t\right)}}\text{d}\omega \end{eqnarray} \tag{ 20 }$$

where ${{v}_{\text{p}}}$ is the phase velocity of the guided wave mode of interest. Knowing the phase velocity dispersion curve of the pipeline means that $f(x,t)$ can be calculated at any point in time and space.

In practical situations, the received time-trace $y(t)$ usually contains numerous echoes from different features at different positions. In ideal conditions, $y(t)$ can be formulated as the superposition of (18):

$$\begin{eqnarray}&&y(t)=\underset{j}{\sum}\,{\int}_{-\infty}^{\infty}{{A}_{j}}F(\omega ){{\text{e}}^{\text{i}\left(\frac{\omega}{{{v}_{\text{p}}}(\omega )}x-\omega t\right)}}\text{d}\omega +n(t)\end{eqnarray} \tag{ 21 }$$

where ${{A}_{j}}$ is the reflection coefficient of each reflector and $n(t)$ is noise.

In this paper, a stainless steel pipe is selected as the research object, and its geometric size and material properties are shown in table 1. According to the parameters in table 1, the group velocity dispersion curves of the stainless steel pipe are obtained, as shown in figure 5. In total, there are three types of modes in the stainless steel pipe—longitudinal, torsional and flexural—labeled as L(m, n), T(m, n) and F(m, n), respectively. Here the integers m and n denote the circumferential order and group order of a mode. According to the energy distribution in the circumferential direction, these modes can also be divided into two categories: axisymmetric modes (L(m, n) and T(m, n)), and non-axisymmetric modes (F(m, n)). An axisymmetric mode has the circumferential number m  =  0. Thus, the longitudinal and torsional modes can be represented as L(0, n) and T(0, n), respectively. In this paper, the L(0, n) modes are selected as our inspection mode, because the direction of particle motion in the L(0, n) modes is perpendicular to the cracks.

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Figure 6 shows the geometry of the pipe with three cracks, which are labeled as cracks 1–3. Crack 1 is smaller than the other two cracks. The distance between crack 2 and crack 3 is 10 mm, which is shorter than a wavelength. Two PZT arrays are used as the actuator and sensor, respectively. The input signal is shown in figure 7, where the center frequency is 500 kHz. To compare the stability of the ASD method and the ${{l}_{1}}$-norm deconvolution method, a simulation study is carried out under dispersion and non-dispersion conditions. In case 1, the L(0, 2) mode is used as the input signal, because the dispersion phenomenon does not occur at the center frequency. In case 2, the L(0, 1) mode is selected, which shows a clear dispersion phenomenon around the center frequency.

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The dispersive propagation model shown in equation (20) is employed to illustrate the test results of case 1 and case 2, as shown in figures 8(a) and (b), respectively. Comparing figure 8(b) with (a), in the former it is clear that there is a decrease in the amplitude of the wave-packets and an increase in the width of the pulse in the time domain, caused by the dispersion properties of the L(0, 1) mode. Meanwhile, the wave-packets in figure 8(b) obviously lag behind the wave-packets in figure 8(a), because the group velocity of L(0, 1) is slower than the L(0, 2) mode at the same center frequency.

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Theoretically, there should be three echoes in figure 8(a), except the incident pulse and the pipe end echo. However, in fact, there are only two echoes in the received signal, and the echo of crack 1 is unclear as a consequence of its small dimension and low noise level in the test. In addition, the distance between crack 2 and crack 3 is shorter than the half-bandwidth of the incident pulse, leading to the reflected echoes overlapping, resulting in a single common echo. Such a situation also happens in figure 8(b). It is hard to directly determine the position of the cracks from the original signals without further processing. Thus, the ASD and the ${{l}_{1}}$-norm deconvolution method are employed to process the test signals of case 1 and case 2, respectively, and the results are shown as follows.

4.3.1. Case 1.

From figure 9, it is obvious that the echoes overlap, and the drowning problems are well solved in the two deconvolution results. The echo of crack 1 that drowns in noise is restored, and the overlapping echoes of cracks 2 and 3 are well distinguished. Besides the necessary information about the structural features, there are also some interference components caused by noise in the deconvolution results. According to the time difference between the incident response and the cracks in figure 9(a), and the known group velocity of the L(0, 2) mode 5335 m s⁻¹, the distances between the sensor and the cracks are 69.2 mm, 120.6 mm and 131.2 mm, respectively. These results are highly consistent with the crack location shown in figure 5. Comparing figure 9(b) with (a), there is no significant difference in the former.

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As can be seen in table 2, the calculation results of the two methods are consistent. This is mainly because the input signal is the non-dispersion mode L(0, 2), which does not cause large waveform changes in the input signal. The prototype of the columns of the convolution matrix used in the ASD and the ${{l}_{1}}$-norm deconvolution method are shown in figure 10, and there is no more difference between the two prototypes. Thus, in the absence of dispersion, both the ASD method and the ${{l}_{1}}$-norm deconvolution method show good performance in improving the time resolution of the guided wave signal. Table 3 shows the CPU times of TwIST, SpaRSA and SALSA in case 1. As excepted, the SALSA is slightly faster than SpaRSA and clearly faster than TwIST when the same stopping criterion (the relative change in the objective function $\frac{1}{2}\|s-Hx\|_{2}^{2}\,+\lambda \|x{{\|}_{1}}$ falls below ${{10}^{-5}}$) is satisfied.

Table 2. The test results of case 1.

Crack	Distance between the sensor and crack (mm)	Calculation result of the ASD method (mm)	Calculation result of ${{l}_{1}}$-norm deconvolution method (mm)
Crack 1	70	69.4	70.4
Crack 2	120	120.6	120.6
Crack 2	130	131.2	131.2

Table 3. Computational cost of three algorithms in case 1.

Algorithm	TwIST	SpaRSA	SALSA
CPU time (s)	10.16	0.546	0.4524

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4.3.2. Case 2.

The deconvolution results of case 2 are shown in figure 11. It is clear that the result of the ASD method is better than the result of the ${{l}_{1}}$-norm deconvolution method. In figure 11(a), the echoes drown, the overlapping problems are well solved, and the result is sparser than that in figure 11(b). In figure 11(b), although the useful information is recovered from the received signal, there are still a lot of interference components which make the result more complex. The group velocity of the L(0,1) mode at 500 kHz is 2929 m s⁻¹, and the position of the cracks can be determined, as shown in table 4.

Table 4. Test results of case 2.

Crack	Distance between the sensor and crack (mm)	Calculation result of the ASD method (mm)	Calculation result of the ${{l}_{1}}$-norm deconvolution method (mm)
Crack 1	70	70.9	70.9
Crack 2	120	119.5	114.4
Crack 3	130	129.5	122.5

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From table 4, the crack position calculated by the ASD method is more accurate than that calculated by the ${{l}_{1}}$-norm deconvolution method, especially for the overlapping echoes. This is mainly because the ${{l}_{1}}$-norm deconvolution method uses the input signal as the prototype of the columns of the convolution matrix, without considering the waveform changes caused by dispersion. For the ASD method, the prototype of the convolution matrix is adaptively estimated from the received signal, and the difference between the two prototypes is shown in figure 12. Thus, when the dispersion phenomenon exists, the ASD method shows better performance than the ${{l}_{1}}$-norm deconvolution method. As can be seen from table 5, for the same convergence condition, the SALSA algorithm is faster than TwIST and SpaRSA.

Table 5. Computational cost of three algorithms in case 2.

Algorithm	TwIST	SpaRSA	SALSA
CPU time (s)	16.82	0.702	0.5616

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A damage identification system based on the guided waves consists of a signal excitation unit, a data acquisition unit and a certain number of actuators and sensors. The experimental apparatus employed in the test is shown in figure 13. The signal generation subsystem is mainly constructed by an arbitrary waveform generator (Agilent 33511B), a linear power amplifier (PIEZO EPA-104) and an exciter ring, which consists of 16 PZT strain gauges. The role of the signal generation subsystem is to activate the diagnostic guided waves in the pipeline structure. First, the Agilent 33511B arbitrary waveform generator delivers the excitation signal of the desired waveform and central frequency to the PIEZO EPA-104 linear power amplifier; the excitation signal is amplified by the power amplifier in a suitable voltage range. Then the amplified excitation signal is sent to the exciter ring to convert the electrical excitation into a mechanical drive to activate the waves that can travel along the structure. The data acquisition subsystem is composed of a digital oscilloscope (Tek TDS5032B) and a receiver ring whose elements and number of units are the same as the exciter ring. The main purpose of the data acquisition subsystem is to receive guided waves after their propagation in the test structure. The PZT elements of the receiver ring, which serve as sensors, receive the dynamic responses of the test structure. Then the signal from the receiver ring is transferred to the TDS5032B digital oscilloscope for capture.

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The stainless steel pipes in this experiment are the same as the pipe mentioned in section 4.2; the geometric size and material properties of the experimental pipes are shown in table 1. In total, there are two groups of test pipe samples with defects labeled as cases 1 and 2, as shown in figure 14. There is only one crack in the test pipe in case 1, and case 2 has three artificial cracks. The pipe samples are all 1200 mm, and the cracks are located at different positions in the test samples. The distance between the receiver rings and the cracks is shown in the figure. In case 2, the radial depths of the cracks are 0.3 mm (crack A), 0.6 mm (crack B) and 0.9 mm (crack C), respectively.

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To make the test signal easy to interpret, it is better to excite a single non-dispersive mode in the guided wave inspection. At the same time, it is also essential to utilize the fast mode because it is helpful to separate the signals of interest from the rest of the measured signals in the time domain [34, 35]. The group velocity dispersion curves for the stainless steel pipe (as shown in figure 6) show that the L(0, 2) is the fastest mode, and at frequencies of around 70 kHz it is practically non-dispersive. Therefore, this mode is well suited for our application and satisfactory results can be obtained in the frequency range 50–100 kHz. Meanwhile, to guarantee that the desired mode can only be excited in the interested frequency region, the incident pulse can be obtained by employing a sinusoidal signal modulated using a Hanning window [36], as shown in figure 7(a).

5.3.1. Case 1.

The test result of case 1 is shown in figure 15, and the fitting result of the incident pulse in case 1 is shown in figure 16. In the original test signal (figure 15(a)), the crack echo is partly drowned in noise, and thus it is hard to determine the position of the crack because of the lack of time resolution. The original test signal is processed by the ASD and the ${{l}_{1}}$-norm deconvolution method, respectively, and the result is shown in figures 15(b) and (c). Obviously, the result of the ASD method is better than that of the ${{l}_{1}}$-norm deconvolution method. In figure 15(b), there are only three spikes, namely the incident response, crack response and end response, respectively. The group velocity of the L(0, 2) mode is 5335 m s⁻¹, and according to the time difference shown in figure 15(b), the position of the crack and the length of the pipe can be calculated as 840.3 mm and 1203 mm, respectively. The calculation result is highly consistent with the actual situation. Moreover, the result also contains a lot of interference components, which make the deconvolution result more complex, as shown in figure 15(c). Thus, it is difficult to obtain useful information from the deconvolution result of the ${{l}_{1}}$-norm deconvolution method.

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5.3.2. Case 2.

In case 2, three artificial cracks are located at different positions in the test pipeline, as shown in figure 17. In theory, there should be three reflected echoes in the original test signal (figure 17(a)) correspondingly. In fact, there are only two crack echoes in the test signal, apart from the incident pulse and pipe end echo. The echo of crack A is hard to identify, because it completely overlaps the incident pulse. The echoes of cracks B and C are hazy because of noise. In order to distinguish the overlapping echoes and restore those that have been drowned from the noise, the ASD and the ${{l}_{1}}$-norm deconvolution method are used to process the original test signal; the results are shown in figures 17(b) and (c), respectively. The fitting result of the incident pulse in case 2 is shown in figure 18.

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In figure 17(b), the overlapping echoes are well distinguished and the echoes that have been drowned in noise are also restored. According to the time difference shown in figure 17(b), the position of the cracks can be determined: the positions are 336.1 mm, 546.3 mm and 821.6 mm, respectively. The spike next to crack C is the F(1,1) mode, which is caused by mode conversion when the L(0,2) mode arrives in crack A. The time difference between crack A and the interference spike is 0.000 158 s, the distance between crack A and the sensor is 250 mm, and then the velocity of the wave packet is 1582.3 m s⁻¹. At the excitation frequency, the group velocity of the F(1,1) mode is 1574 m s⁻¹, considering the error in the arrival time of the echo in crack A; thus, the spike next to crack C is the F(1,1) mode. As can be seen in figure 17(c), the deconvolution result of the ${{l}_{1}}$-norm deconvolution method is grossly distorted. The result contains too much interference component leading to it not being not sparse enough. Meanwhile, the necessary information for crack A and crack B is missed in figure 17(c).

As can be seen from table 7, except for the fact that the measurement error in crack A is relatively large, the measurement results of the other cracks are consistent with the actual situation. The main reason for the large error is probably that the cross-section of crack A is the smallest one of the three cracks in the same pipe, which makes the echo in crack A weak. As a result, the measurement can easily be influenced by noise. In general, the ASD method is better at solving the echo overlap and the drowning problem of the guided wave signal than the ${{l}_{1}}$-norm deconvolution method. Meanwhile, table 6 shows that the computing speed of the SALSA algorithm is faster than TwIST and SpaRSA for solving the deconvolution problem.