## 5 Conclusions and Discussion
### Outlier Tolerance  
To investigate the outlier-tolerant ability of PCFWE under different outlier proportions, simulating tests are used because of the admirable repeatability and flexibility. Simulating tests with controllable errors in the input data such as TDOA measurements and the velocity system are described in this section. Figure 9 shows an assumed cubic locating system with a side length of 300 mm, where an AE source \(O\) (150, 100, 200) surrounded by 21 sensors is set to generate AE signals. It is assumed that the trigger time is 0 \(\upmu\)s, and the average wave velocity of the media is 5000 m/s. In addition, to simulate minor systemic errors in TDOA measurements, the extra random errors from the normal distribution with a mean of zero and a standard deviation of 2% of measurements are added. To simulate the uncertainty of the velocity along different paths, a minor error of 5% of velocity for each path is generated. For measurements with abnormal errors (outliers), dramatic errors of \(\pm\)30% of TDOA measurements, which are much larger than the systemic error, are added with different proportions. Thus, the simulated location consists of velocity uncertainty, systemic minor errors, and different outlier proportions. To obtain a reliable statistical result, the AE source location is repeated 100 times by random changes of TDOA measurements and the velocity errors, and then their location coordinates are calculated. The location results of the 100 simulating tests with different outlier proportions are shown in Figure 10. It can be seen that all of the sizes of solid circles are small and close to true sources, when there is no outlier in the input data. However, they have obvious differences as shown in Figure 11, where the average absolute distance error of the PCFWE is smaller than that of traditional methods, due to the use of preconditioning for the linear equations. In other words, the PCFWE performs better than traditional location methods without outliers. The location performance of the PCFWE differs more substantially from traditional methods in the presence of outliers. Solid circles of traditional methods always appear more discrete and have a larger diameter with the increase of outlier proportions, which indicates that the location errors increase with outlier proportions. While the solid circles of the PCFWE always appear more compact and darker under different outlier proportions, which illustrate the PCFWE location method has a higher accuracy and stability than traditional methods. Figure 11 shows the average absolute distance errors of 100 location results under different outlier proportions determined by three methods. The average absolute distance errors for traditional methods increase dramatically with the increasing of outlier proportions. Whereas, the average absolute distance errors of PCFWE method always keep low and stable under different outlier proportions, due to the filtering of outliers in calculation, which further illustrates the good performance of the PCFWE method. Moreover, the relationship between valid location ratios and outlier proportions is displayed in Figure 12. The valid location ratio is the location proportion whose location errors are smaller than 2% of the distance between the two farthest sensors in the sensor array. Clearly, the two curves of traditional methods decline dramatically with the increase of outliers, the valid locations of which are less than 30% when there are 25% outliers. Compared with traditional methods, the curve of the PCFWE always keeps stable and achieves valid locations of nearly 100% when the outliers are less than or equal to 25%. Then, the curve falls to 73% at the outlier proportion of 45%, which is still far above the other two curves that have valid locations of less than 10%. Therefore, the outlier-tolerant ability of the proposed method is far higher than those of traditional methods, which can realize valid locations of more than 90%, even when the outlier proportion reaches 35%.