# Improved Acoustic Emission Tomography Algorithm Based on Lasso Regression
## 4 Numerical Simulation  
### 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 \(\times\) 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 Figures 5-7. 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.  
Figure 4: The setting of the relay points for each cell.  
Figure 3: Different rates of the damaged area in the original distribution: (**a**) damage area with 10 cells (10%); (**b**) damage area with 20 cells (20%); (**c**) damage area with 30 cells (30%).  
Figure 5: Identified results of the 10% defected area: (**a**) LASSO algorithm; (**b**) SIRT algorithm.  
Figure 6: Identified results of the 20% defected area: (**a**) LASSO algorithm; (**b**) SIRT algorithm.  
Figure 7: Identified results of the 30% defected area: (**a**) LASSO algorithm; (**b**) SIRT algorithm.  
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 \(\ell\)2-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. 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.