## 3 Numerical Experiment and Analysis  
To demonstrate the effectiveness of the presented algorithm, we implemented it on a PC (3.3 GHz CPU, 8.0 G memory), and all algorithms were implemented in the MATLAB 2017b programming environment (MathWorks Inc., Natick, MA, USA, 2017). The numerical experimental results using the algorithm of AMPGA with constraints are illustrated in this section. According to the characteristics of the narrow working face of the coal mine, the scale of the model was \(600\times 200\) m in this numerical simulation test. As shown in Figure 3, assume that there are two abnormal structures with different shapes in the working face. The attenuation coefficient of the abnormal structures was \(\beta_{1}=1\) dB/m, and the attenuation coefficient of the background area was \(\beta_{2}=0.5\) dB/m.  
First, the detection area was discretized by meshes, and the layout of observation system was as shown in Figure 4. Among them, the length of each grid was \(10\times 10\) m. The distance between signal sources was 30 m and between acquisition stations was 10 m. The ray coverage angle of each signal source was \(20^{\circ}\), the total number of rays was A = 420, and the total number of grids was B = 1200. Therefore, the solved model is highly sparse in this observation system.  
Figure 4: Signal source and acquisition station arrangement in observing systems.  
Figure 3: Reconstructed model with anomaly area.  
To investigate the effectiveness of the proposed algorithm, the results of the image reconstructed based on the SGA and AMPGA (without constraint) are shown, respectively. In the numerical experiment, the attenuation coefficients near the upper and lower edges of the model were used as prior information. After 500 iterations of evolution for the different algorithms, the reconstructed results are shown in Figure 5.  
As indicated in Figure 5, the AMPGA with constraints has a very good performance, and the result is optimally close to the results of the model settings. The inversion image from the SGA has a large area of abnormal structure with a lower resolution, and the boundary is not obvious. The resolution of the reconstructed image has been significantly improved by introducing a variety of adaptive co-evolution strategies, which is shown in Figure 5b. After introducing a priori information as shown in Figure 5c, the quality of AMPGA of the reconstructed image is further improved with a clearer outline, and the distribution of abnormal areas is more concentrated.  
To test the stability of these three algorithms, multiple repetitive tests were performed on different algorithms, and the results are shown in Figure 6. Compared with the other two algorithms, curves obtained from the AMPGA with constraints are basically overlapping, so it has the best performance in stability. The comprehensive analysis of the repeated experiments of these three algorithms are shown in Figure 7. After 500 generations of evolution, the average value of \(RMSE\) in SGA was reduced  
Figure 5: Reconstructed results according to the different algorithms after 500 iterations. (**a**) Inversion result from the simple genetic algorithm (SGA); (**b**) inversion result from the AMPGA; (**c**) inversion result from the AMPGA with constraints.  
from 37.01 to 15.89 dB/m, the average value of _RMSE_ in AMPGA was reduced from 36.15 to 4.44 dB/m, and the average value of _RMSE_ in AMPGA with constraints was reduced from 36.15 to 3.77 dB/m. Therefore, compared with SGA, AMPGA can effectively improve the accuracy of tomographic inversion and achieve better imaging results by adding prior information.  
Figure 6: _Cont._