*5.2. Velocity Sensibility*

The premeasured velocity is necessary for most location methods, and measurement errors in velocity are unavoidable due to the complex construction environment and manual operation. Moreover, the wave velocity of the propagation medium always changes with material testing or engineering construction. An effective location method should have a lower velocity sensitivity, i.e., a higher tolerance of velocity error. Therefore, the analysis of the velocity sensitivity of different methods is conducted in this section. This simulating test is still based on the above location system. To obtain a reliable statistical result, 100 AE sources are selected randomly inside the sensor array to generate AE signals. Besides, the 2% minor systemic errors in measurements are applied to simulate minor systemic errors, and 5% to 25% errors are added to velocity, in turn to investigate the velocity sensitivity. Hence, the simulated location consists of systemic minor errors in TDOA measurements and different error scales in velocity.

**Figure 10.** Three-dimensional location maps determined by three methods with outlier proportions of 0%, 5%, 15%, and 25% respectively. Diameters of solid circles indicate the location error, and smaller diameters indicate a higher location accuracy. To further distinguish the location accuracy of different sources, colors in the solid circles also indicate the location accuracy in millimeters.

**Figure 11.** The average absolute errors of the AE sources determined by three methods under different outlier proportions.

**Figure 12.** Valid location ratios of three methods and the average location error of the PCFWE under different outlier proportions.

The average absolute distance errors under different error scales in velocity are calculated, which are shown in Figure 13. All three curves show a rising trend with the increase of error scales in velocity, but the curve of the PCFWE is significantly lower than the other two methods. Therefore, the PCFWE has the lowest velocity sensitivity, which indicates that the PCFWE can bear larger errors in velocity than traditional methods.

During material testing, the wave velocity changes with the loading stress. Through the analysis of the velocity sensitivity of different methods, conclusions are gained, as the proposed method has a stronger tolerance to velocity errors, which is suitable for a material test where the wave velocity changes along with the test. However, in the later stage of a test, the wave velocity changes dramatically due to the propagation of cracks and voids. The proposed method also fails to achieve an ideal location result. A possible solution is the dynamic inversion for real-time wave velocity by taking the velocity as an unknown parameter or modifying the velocity data according to the real-time velocity test, which still needs to be investigated in future research [46].

**Figure 13.** The average absolute distance error determined using three methods under different error scales in velocity.

In practical engineering applications, the proposed method still has the following two limitations. This method is established based on a uniform medium, while most of the materials in engineering practice are anisotropic and heterogeneous, such as rock and concrete. In addition, although the sensitivity of the wave velocity is low, the proposed method is still affected by the velocity error. Besides, it is troublesome to measure the wave velocity in advance or real-time. Therefore, further studies should extend the method to address these issues.

**Author Contributions:** Z.Z. and Y.R. proposed the location method and performed the experiments; Z.Z., Y.R., J.Z. and R.C. analysed the data and wrote the paper; and L.D., L.C. and X.C. supervised the research and revised the manuscript.

**Acknowledgments:** The authors would like to acknowledge financial support from the National Basic Research Program of China (973 Program) (No. 2015CB060200), the National Natural Science Foundation of China (Nos. 41772313 and 51478479), and the Fundamental Research Funds for the Central Universities of Central South University (No. 2018zzts716).

**Conflicts of Interest:** The authors declare no conflict of interest.
