Improved Acoustic Emission Tomography Algorithm Based on Lasso Regression

Abstract

This study developed a novel acoustic emission (AE) tomography algorithm for non-destructive testing (NDT) based on Lasso regression (LASSO). The conventional AE tomography method takes considerable measurement data to obtain the elastic velocity distribution for structure evaluation. However, the new algorithm in which the LASSO algorithm is applied to AE tomography eliminates these deficiencies and reconstructs equivalent velocity distribution with fewer event data to describe the defected range. Three numerical simulation models were studied to reveal the capacity of the proposed method, and the functional performance was verified by three different types of classical concrete damage numerical simulation models and compared to that of the conventional SIRT algorithm in the experiment. Finally, this study demonstrates that the LASSO algorithm can be applied in AE tomography, and the shadow parts are eliminated in resultant elastic velocity distributions with fewer measurement paths.

1. Introduction

There are many gradually aging infrastructures all over the world. However, always challenges to exposing internal damage are demanded, since these damages have a significant safety impact on structures. As a consequence, non-destructive testing is required as an evaluation technique. Non-destructive testing is a meaningful way for existing structures to repair in advance and ensure safety while protecting human safety and avoiding economic losses. In comparison with other non-destructive testing techniques, AE tomography can estimate the source locations of AE events and identify elastic wave velocity distribution to evaluate the structure while having the advantage of using only one side of the structure to evaluate the healthy internal condition [1].

Schubert proposed AE tomography in 2004 by combining elastic wave tomography and source localization technology [2]. Theoretically, the propagation path of elastic waves in the medium is assured as a straight line. Kobayashi proposed and applied AE tomography theoretically and numerically for two-dimensional and three-dimensional issues from 2012 to 2016 [3,4,5]. The AE tomography technique involves RC bridge decks for the bullet train and concrete plates [6,7,8]. The basic theory of AE tomography is that AE events are launched from an unknown transmitting position and the receiver at the corresponding position receive the elastic wave correspondingly. After measuring and collecting the arrival time, the travel time of each mesh cell is calculated based on the grid divided in advance. Finally, the results assess the extent and area of the defect by velocity distribution, which employs a reconstructing algorithm. The well-known and oldest methodology is algebraic reconstruction (ART). Other advanced methods include the simultaneous algebraic reconstruction technique (SART) and the simultaneous iterative reconstruction technique (SIRT) [9,10,11,12]. The existing mainstream inversion algorithms, such as SIRT, obtain the velocity distribution and visualize the internal damage of the structure, which is conventional and mature to apply in practice [1]. However, to construct the velocity distribution, the traditional algorithm requires a sufficient number of event data to maintain the quality of figures, which is a vital deficiency. For example, the SIRT utilizes a certain amount of event data to obtain the velocity distribution by iteration in each cell. Meanwhile, the characteristics of the defective area are local in comparison with the extent of the structures, and it satisfies the conditions for applying Lasso regression to obtain accurate velocity distribution while requiring a fewer amount of event data.

In this paper, the second section introduces the basic theory of AE tomography. The third section introduces the fundamental hypothesis of Lasso regression, including the theoretical calculation process and conditions. Then, this section combines Lasso regression with AE tomography and determines the relevant parameters to ensure its feasibility in AE tomography systems. The capability of the proposed method was verified with three numerical models that simulated general damage patterns of concrete structures in the fourth section. In the fifth section, experiments were conducted to validate the proposed method by comparing the reconstructed velocity distribution of Lasso regression with fewer uniformly selected event data and the traditional SIRT algorithm in AE tomography. The discussion in the sixth section demonstrates different scenarios of AE tomography in theory and practice. In addition, the mechanism of the Lasso regression algorithm applied to AE tomography was illustrated in detail, and the principle of the Lasso regression algorithm was explained to be accurate and economical for velocity reconstruction. The conclusions are described in the final section.

2. Conventional Algorithm of AE Tomography

AE tomography is a non-destructive testing technology that identifies the AE source locations and the velocity distribution by using the arrival time of elastic waves. The source locations and the velocity distribution are evaluated simultaneously and affect each other while the evaluation. The AE source locations are calculated on the assumed elastic wave velocity distribution that is from the corresponding relay point to the receiver, and then, the velocity is reconstructed on the basis of the travel time between the AE sources and the receiver on the assumed elastic wave velocity distribution. The arrival time is obtained by the emission time and the first travel time. Then, AE source locations are revised based on the meshed cells to update the velocity. The homogeneous velocity of the material in each meshed cell determined as the initial velocity distribution is assumed to be the location of the AE source. After the emission time is determined, the tomography technique will be performed [8]. The flowchart of AE tomography is shown in Figure 1. It is noted that this process uses the straight-line ray path based on the homogeneity of the elastic wave velocity distribution. The travel time spanning from the nodal point to the receiver $\Delta T_{ik}$ is determined by the following hypotheses:

$$\Delta T_{ik} ={{\displaystyle \sum }}_{j=1}^n{S}_j{l}_j,$$

(1)

where $n$ is the number of the meshed elements, $j$ is the number of elements, I represents the ith receiver, k depicts the kth nodal and relay point to be accessed, slowness is the reciprocal of elastic wave velocity represented as $S_j$, and $l_j$ is the ray-path length in cell $j$.

In the first stage, the source location and the emission time of AE events are calculated by using the arrival time according to the initial elastic wave velocity distribution and the receiver’s position. In order to acquire the theoretical travel time, the second stage is to calculate the sum of the estimated emission time and the travel time from the estimated source location to the receiver [13]. Finally, the velocity distribution is reconstructed and updated by minimizing the difference between the theoretical and measured travel times. These two stages are performed iteratively, until the convergence criteria are satisfied and the source position and the elastic wave velocity are determined simultaneously. In the following subsections, the procedure of each stage is briefly introduced.

The source location is estimated from the velocity distribution, the location of the receiver, and the arrival time of the AE in AE tomography [1]. The source location estimation starts with calculating the potential emission time. The idea of the computation of the potential emission time is shown in Figure 2. The potential emission time is calculated from the results of the ray trace between the receiver to each nodal and relay point as follows:

$$P_{ik}^{\prime }=A_i-\Delta T_{ik},$$

(2)

where i represents the ith receiver, k depicts the kth nodal and relay point to be accessed, A_i is the observed arrival time at receiver i, and $P_{ik}^{\prime }$ is the potential emission time at point k. The potential emission time denotes the time at which an AE event is expected to be emitted at the point k if the observed arrival time at the receiver i is A_i on the present elastic-wave velocity structure. This work is performed for each receiver associated with a single measurement event. For example, each node has n potential emission times for one event, when n receivers are installed. The potential emission time should be able to track the correct source location, when the elastic wave velocity distribution and ray path are accurate. However, it is usually selected based on the nodal and relay point with the minimum potential emission time variance due to the insufficient velocity distribution in this source location algorithm. AE tomography can raise the accuracy of the source location and emission time by increasing node and relay points. The relay points in each cell are described in Figure 2. An orthogonal straight line to the edge is drawn at each division point. Relay points are constructed at the intersections of the straight lines and the edges. It is essential to determine the appropriate density of relay points before calculation, since the density of relay points and calculation cost are proportional. The average of the potential emission time$O_k^{\prime }$ at the source location is shown as:

$$O_k^{\prime } =\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{P}_{ik}^{\prime }.$$

(3)

This value is adopted for the reconstruction algorithm in the following part. The velocity distribution is identified using the difference between the theoretical and observed travel times. Typically, it is determined that the averaged potential emission time at the nodal or relay point of the minimum variance is chosen as the source location. The theoretical arrival time $A_i^{\prime }$ of receiver i shown as follows:

$$A_i^{\prime }=O_k^{\prime }+\Delta T_{ik}.$$

(4)

It is noted that $\Delta T_{ik}$ is the travel time from the receiver to the nodal and relay point selected as the source location correspondingly. The traditional algorithm SIRT adopts iterates through comparative measurement data to generate a velocity distribution representation of an interior physical structure based on the analysis in AE tomography. The equation of the SIRT algorithm is shown as follows:

$$S_j^{\left(I\right)}=S_j^{\left(I-1\right)}+{{\displaystyle \sum }}_i\left(T_{oi}-\Delta T_i^{\left(I-1\right)}\Delta l_{ij}/L_i\right)/{{\displaystyle \sum }}_i{l}_{ij},$$

(5)

where I is the iteration times and j is the cell number, i is the number of ray, $S_j^{\left(I\right)}$ means the slowness of cell j in iteration I, $T_{oi}$ is the observed travel time, $\Delta T_i^{\left(I-1\right)}$ is the theoretical travel time of ray i for the (I − 1)th iteration, $l_{ij}$ is the ray length of i in j cell, and $L_i$ is the ray length between the source location to the receiver [14]. It is obvious that the SIRT algorithm needs lots of event data for iterate calculation. However, these algorithms have their own restrictions. For instance, they need sufficient event data for velocity reconstructions. This research utilized Lasso regression for AE tomography to eliminate these deficiencies.

3. LASSO Algorithm Combined with AE Tomography

In 1995, Tibshirani proposed Lasso (least absolute shrinkage and selection operator), a regression model that uses L1 regularization techniques, also referred to as Lasso regression. Regularization is a general term for a class of methods that introduces additional information to the original loss function in order to prevent overfitting and improve model generalization performance. There are generally two commonly used extra items called L1 regularization (ℓ1−norm) and L2 regularization (ℓ2−norm). L1 regularization and L2 regularization can be regarded as penalty terms of the loss function, and penalty refers to some restrictions on some parameters in the loss function [15]. The algorithm also performs to solve convex-cardinality problems as an essential reconstruction algorithm (also called the L1 heuristic) rebuilt from the L1-norm [16,17,18]. It is an efficient approach to reconstructing sparest solutions to specific underdetermined systems of linear equations. More specifically, there are an unknown vector x∈R^N and an underdetermined matrix A∈R^M×N and y = A · x. The problem can be solved as follows:

$$\min \parallel \mathrm{x}{\parallel }_1 \mathrm{subject} \mathrm{to} \parallel \mathrm{A}·\mathrm{x} - \mathrm{y}{\parallel }_{2 }< \mathsf{\epsilon },$$

(6)

where$\parallel ·{\parallel }_1$and $\parallel ·{\parallel }_2$represent the ℓ1−norm and the ℓ2−norm, respectively, and ε represents the noise boundary [19]. According to the feature of this algorithm, LASSO regularization can make parameters sparse, and obtained parameters are a sparse matrix, which can be provided for feature selection. The characteristic of this theory can contribute to shrinking the number of coefficient matrices and obtaining local optimal solutions simultaneously. In words, the LASSO algorithm requires a fewer number of measurement paths than the other algorithm, such as the SIRT, for velocity distribution according to the number of coefficients depending on the number of paths in elastic velocity tomography. The following demonstrates the theory of AE tomography combined with the LASSO algorithm:

$$\mathrm{T}=\mathrm{A} · \left({\mathrm{s}}_0+\mathsf{\Delta }\mathrm{s}\right),$$

(7)

where T is the vector of the measured travel time, S₀ is the standard slowness in a completely healthy state, and Δs is the slowness difference vector generated by subtracting the actual slowness from the standard slowness. All of the ray-path matrices are used to construct A, a coefficient matrix that is generated from all of the travel lengths. A vector representing the results of the deviation between the measured travel time and the travel time in totally healthy conditions, y, is defined as:

$$\mathrm{y}=\mathrm{T}-{\mathrm{T}}_0=\mathrm{A} · \mathsf{\Delta }\mathrm{s}, \mathrm{where} {\mathrm{T}}_0=\mathrm{A} · \mathrm{s}{0}_{,}$$

(8)

where T₀ is the travel time vector for a healthy specimen. The velocity distribution imaging of AE tomography depends on reconstructed $\mathsf{\Delta }\mathrm{s}$ in Equation (9), which is defined by the LASSO algorithm after the sampling phase from Equation (6). It is noted that soluting the slowness vector Δs is a defect detection problem in physical understanding:

$$\min ⫽\mathsf{\Delta }\mathrm{s}{⫽}_1 \mathrm{subject} \mathrm{to} ⫽\mathrm{A}·\mathsf{\Delta }\mathrm{s} - \mathrm{y}{⫽}_{2 }< \mathsf{\epsilon },$$

(9)

where the $⫽·{⫽}_1$ and $⫽·{⫽}_2$are the ℓ1−norm and ℓ2−norm, respectively. It would be more forceful to adopt λ, which is the Lagrangian multiplier (regularization parameter) in identifying the velocity distribution stage, to provide a balance between ($⫽\mathrm{A} · \mathsf{\Delta }\mathrm{s} - \mathrm{y}{⫽}_2$) and the constraint ($⫽\mathsf{\Delta }\mathrm{s}{⫽}_1$) for data misfit, which in this case is 0.083 determined by numerical simulation [20,21]. Equation (9) is updated to become Equation (10):

$$\mathrm{argmin} (⫽\mathrm{A} · \mathsf{\Delta }\mathrm{s} - \mathrm{y}{⫽}_2 +\mathsf{\lambda }⫽\mathsf{\Delta }\mathrm{s}{⫽}_1).$$

(10)

It is noticed that the slowness distribution is updated during the iterative procedure. In accordance with Figure 1, it is necessary to evaluate whether the criterion is fulfilled or not after calculating the source location and updating the slowness distribution iteratively.

4. Numerical Simulation

4.1. Ability of the LASSO Algorithm

In order to study the capability and scope of application of the LASSO algorithm for the AE tomography, numerical models were adopted to detect fault areas and illustrate the implementation. There were three models that have different ratios of defect areas as illustrated in Figure 3. The blue area indicates that the concrete in healthy conditions has a velocity of 4000 m/s, and the red area denotes the defective status with a velocity of 3000 m/s. The size of each model was 1 m × 1 m, and the damaged areas occupied 10%, 20%, and 30% of the model, respectively. The initial elastic wave velocity was 4000 m/s, given as uniform distribution for AE tomography. The total number of sensors was 20, which was set around the model’s edge with an interval of 0.2 m. The number of mesh was 100, and each meshed cell had 12 relay points, as shown in Figure 4. One hundred source locations were randomly generated in the model described as the white spheres in Figure 3. The number of measurement paths for every model was 2000 in consequence.

The calculation steps are as follows: (1) the arrival time at receivers was computed using the straight path from the generated source to the individual receiver; (2) the velocity distribution was reconstructed by using the computed arrival time with the SIRT and LASSO to compare the target and resultant velocity distributions to investigate the capability of the proposed method.

The velocity reconstruction results are shown in Figure 5, Figure 6 and Figure 7. The red rectangle zone indicates the low-velocity regions that should be identified as the low velocity region. From the calculation results, the following two aspects were denoted: (1) two algorithms can accurately identify the low-velocity region when the damaged areas were 10%, 20%, and 30%. Although the value of the low velocity region identified by LASSO was slightly lower than the SIRT, the identification range for the low-velocity region was accurate; (2) for healthy region identification, LASSO performed remarkably better in reconstructing the velocity distribution with fewer shadows than the SIRT in different models.

The normalized errors of the LASSO and SIRT algorithms for the velocity identification of each corresponding cell in different damage percentages were elaborated as follows. The calculation method is the ratio of the calculated ℓ2-norm of the difference between the identified velocity and original velocity values and the calculated L2-norm of the difference between the initial velocity and original velocity values. The normalized L2-norm error trends and the calculated values for the two algorithms are shown in

Table 1. Normalized errors of the L2-norm.

Damage Percent (%)	LASSO	SIRT
10	0.32010	0.27366
20	0.38114	0.18681
30	0.52836	0.13941

. The results indicate that the error results of the LASSO algorithm tended to be stable and within an acceptable scope, and the boundary description of the healthy area and the damaged area in the velocity reconstruction image was distinct.

4.2. Different Types of Damage

Numerical models were created to demonstrate the efficacy and confirm the functionality of the proposed damage detection approach in the LASSO algorithm. Usually, the concrete structure loses structural integrity due to earthquake, fire, impact, or malicious damage and structural overloads (such as static overloading), and defects appear. As a method of inspection, the types of concrete failure suitable for this non-destructive testing technique by AE tomography have hole destruction, concentrated holes, and spalling or edge peeling [22].

The following numerical simulation was based on actual concrete damage sorts, and three different damage types were studied to simulate the situation in the fundamental structure. The velocity of elastic wave propagation in concrete was assumed as 4000 m/s, and the standard slowness was determined from the velocity. In the simulation, the blue part in each model is the healthy area with a velocity of 4000 m/s, and the red part is the damaged area with a velocity of 3000 m/s; the source locations that were randomly generated in the model are expressed as the white spheres. Case 1 was a simulation of the concrete slab that appeared with damaged holes, shown in Figure 8a. The simulation model in Figure 8b represented the size of the model and the location of the receivers. The low-velocity cell denoted the damaged area. The model information is displayed in

Table 2. Model information for case 1.

Case	Model Size	Damage Area Size	Initial Velocity	Sensor No.	Mesh No.	Events No.	Relay Points for Each Cell No.	Measurement Paths No.
1	1 m × 0.6 m	0.05 m × 0.05 m	4000 m/s	16	240	100	12	1600

. The damaged area of case 2 was a concentrated hole, shown in Figure 9. The damage was located at the center of the model, and it consisted of four cells. The precise information is described in

Table 3. Model information for case 2.

Case	Model Size	Damage Area Size	Initial Velocity	Sensor No.	Mesh No.	Events No.	Relay Points for Each Cell No.	Measurement Paths No.
2	1 m × 1 m	0.2 m × 0.2 m	4000 m/s	20	100	100	12	2000

. Case 3 was a simulation of the damage at the edge of the concrete frame, and the simulation model is shown in Figure 10; the damage location was in the upper right corner with a lower velocity, and it consisted of six cells for analysis. The exact information is shown in

Table 4. Model information for case 3.

Case	Model Size	Damage Area Size	Initial Velocity	Sensor No.	Mesh No.	Events No.	Relay Points for Each Cell No.	Measurement Paths No.
3	1 m × 2 m	0.2 m × 0.3 m	4000 m/s	30	200	100	12	3000

. Every simulation model owned a uniform initial elastic wave velocity distribution of 4000 m/s. The calculation steps remain the same in Section 4.1.

The results of three different simulation models with different damage types, different damage ranges, and different numbers of elastic wave events are illustrated in Figure 11, Figure 12 and Figure 13. According to the results, it is demonstrated that the defected area was correctly identified. In comparison with the reconstructed velocity distribution of the SIRT algorithm, the damaged classification of hole damage, concentrated hole, and edge damage, the LASSO algorithm could accurately identify the lower velocity region in the model. Notably, the LASSO algorithm described accurate identification in the high-velocity region of the healthy region in all of the numerical simulation models and few shadows exist in identified velocity distribution. The LASSO algorithm is suitable for AE tomography techniques to identify typical concrete damage types with relatively accurate reconstructed velocity distribution results.

5. Experiment

After numerical simulations, the lower velocity area described as defect can be accurately identified, and the effectiveness and ability of Lasso regression used in AE tomography for velocity distribution were verified. The following experiments were designed to validate the proposed method in practical works, whose results were compared with the results of the SIRT in the AE tomography technology.

The specimen was an aluminum plate, and its size was 1 m × 1 m × 0.005 m. A slit of 0.5 m × 0.04 m was made in the lower right area in the specimen as an artificial defect. The specimen was modeled as a two-dimensional plate, since the thickness of the specimen was small in comparison with its size. At the same time, the source of acoustic wave emission and the sensors were placed on the same side of the specimen. The velocity of elastic wave propagation in the specimen was 5500 m/s. There were nine sensors arranged around the specimen at the interval of 0.4 m with one located in the center of the specimen, as shown in Figure 14. In each sensor, the threshold to start recording, the preamp amplifies, and the frequency range of the bandpass filter were set to 50 dB, 60 dB, and 1 kHz to 1 MHz, respectively. Therefore, the accuracy of the arrival time had six decimal places. Seventy-two source locations are shown as white points with an interval of 0.1 m and an even distribution in the model as illustrated in Figure 14b.

The inspection steps for AE tomography with LASSO are as follows: (1) The method is the application of the elastic wave generated by breaking a pencil lead at the individual source locations, as shown in Figure 15a. A 2H pencil lead with a length of 3 mm was broken on the surface of the 72 setting positions to emit the events [23], and nine sensors received the elastic wave separately, which were then transmitted to the acquisition system to obtain the detected measured AE wave after being amplified by a preamplifier. The noise and signal parts of the AE events wave were clearly distinguished in this case. (2) The arrival times were selected at the end of the noise part of each measurement path, shown in Figure 15b. It is worth noting that the relay point part divided each meshed cell edge into seven parts, meaning that 60 relay points were in each mesh. The model was meshed by squares with a length of about 0.13 m, meaning that there were 36 (6×6) cells in total. (3) To reconstruct the elastic velocity distribution, the travel time between source locations and sensors was collected from the corresponding time history, since the amplitudes of the signal and noise components varied considerably. (4) The internal velocity distribution of the model was described as the difference slowness vector Δs in Equation (10), and the slowness distribution was updated by using Δs.

The reconstructions of the velocity distribution by the SIRT and LASSO algorithms were conducted to verify the proposed algorithm with different numbers of events. The numbers of events of 9, 18, 27, 36, 45, 54, 63, and 72 were selected, and related occupations of the entire event and measurement paths are shown in Figure 16.

The red circle is the event selected on the model, and the selection was as uniform as possible to ensure for making the reconstructed velocity distributions were relatively accurate. In the reconstructed results, the red part is the damaged area with a velocity lower than 3000 m/s, and the higher velocity (5500 m/s) part is a healthy condition detected as the blue area for both algorithms. The reconstructed elastic wave velocity distribution results of the LASSO algorithm using 18 events (25% of the total) with 162 measurement paths began to stabilize according to Figure 16. Although events and measurement paths increased later, the results did not change much, and the velocity distribution can be clearly described; While the SIRT algorithm used 36 events (50% of the total) with 324 measurement paths, the results became stable as well. Although the events and paths increased, the results maintained the same trend, as shown in Figure 16.

According to Figure 17, the result of identified velocity distribution by LASSO was consistent with that by the SIRT algorithm, which validates that the proposed method identified critical damage results accurately through a smaller number of event data in AE tomography. It is essential that the LASSO algorithm could more accurately restore the range of defects using a smaller number of samples, which had 18 events data instead of 36 events data adopted by the SIRT algorithm. The flow chart and the image of measurement paths for these two kinds of algorithms are shown in Figure 18 and compare the characteristics in

Table 5. Comparison of the characteristics of different algorithms.

Traditional SIRT Algorithm	Proposed LASSO Algorithm
Request a sufficient number of events	Request few events
High rates of measurement paths	Lower rates of measurement paths
Velocity distribution with shadows	Few shadows in the velocity distribution

. A more important point is that the LASSO results can effectively eliminate the shadows part in the reconstructed velocity distribution image, catch a more comprehensive extensive low-velocity range and describe the edge of the damaged area with more precision, which means the result of the velocity tomography was more precise and distinguishable than the result of the SIRT algorithm.

6. Discussion

In this experiment, because the defect of the specimen was exceptionally narrow and long instead of sparse and scattered distribution, the elastic emission wave bypassed the defected part and then reached the receivers during the transmission process according to the basic theory of the ray-trace technique in AE tomography. The velocity on the left side of the defected area was displayed as a significant high-velocity distribution, because the real ray paths through this area were crossed with a length close to that through the non-defected area, rather than theoretically through the defective area, as shown in Figure 19. On the contrary, the elastic emissions wave needed to bypass the entire defective part to reach the receivers with long-distance propagations on the right half of the model, resulting in a low-velocity range in the right half since the wave propagation time increased. In AE tomography technology, the exact elastic wave emission position is unknown, it needs to estimate time and the source location and then identify the velocity distribution. As a consequence, AE tomography is a technique that can detect the realistic condition of velocity distribution for defect area identification in the real world.

It is noted that the residual error between the observed and calculated travel times should be zero in intact microstructure theoretically, because the distance between the event point to receivers and the velocity on a homogeneity medium were identical. However, there was a discrepancy between the observed and theoretical travel times of the elastic wave propagating through the unbroken region in the experiment due to the existence of refraction and diffraction. The algorithms of the SIRT calculated each divided element iteratively, with the intact mesh cells affected during the iteration which is why the flawless part had shadows in elastic velocity distribution. In the LASSO algorithm, these entire areas were occupied by the majority of the model where the travel time residual is also described as infinitely close to zero. This means that the intact part was affected very little during the calculation; hence, the original velocity distribution in the intact part was almost maintained. In the calculation involving the defect area, which was described as low-velocity cells, due to the influence of λ in Equation (10), where λ is the regularization parameter and equal to the calculated extreme value under the constraints in the function. The damaged area described as low-velocity regions in the velocity reconstruction was sharpened and pronounced. Compared with the SIRT algorithm, the LASSO algorithm could identify a more distinguishable elastic velocity distribution, since the optimal local solution was solved subject to a minimum sum of residual errors that are not zero.

Last but not least, according to the simulation of the L1 algorithm capability in actual simulation and experimental results, the characteristics and applicability of the proposed theory in practice can be summarized as follows: (1) the damage location of the structure could be tracked when the defect range was already known to be less than about 30% of the total model in the condition. It is suggested to use the LASSO algorithm for detection if the damage is local, since the reconstructed velocity distribution by LASSO had slight shading and was more apparent than SIRT results. (2) in the case that the size and location of the damage range cannot be estimated, it is recommended to use both the LASSO and SIRT algorithms for comparing the velocity reconstruction and the expected results can be obtained.

7. Conclusions

This study proposed a new method of AE tomography based on Lasso regression, named LASSO algorithm. The proposed theory was validated by both simulations and experiments and compared with the traditional SIRT algorithm. The results are summarized as follows:

(1) Simulations were carried out by using the newly proposed theory for three models. The LASSO algorithm can obtain relatively acceptable identified results for structures with approximately less than 30% of the damage area in the model. Further, the velocity distribution of typical concrete damage types suitable for non-destructive testing of the hole, central concentration, and edge deterioration were identified with the conventional SIRT algorithm. LASSO could obtain a relatively accurate description of the damage range, and there was little impact in the healthy area with few shadows, thus obtaining accurate results. It was proven that the Lasso regression algorithm is feasible in AE tomography for velocity distribution.

(2) In comparison with the SIRT in the experiment, the Lasso regression algorithm obtained acceptable results of velocity distribution with fewer event data, and the approximate range of defects could be reconstructed. At the same time, the results of the velocity distribution with premium quality were more precise and distinguishable than the result of the SIRT, because the shadows in reconstructed image became invisible.