*4.2. Comparing Location Results with and without Outliers*

For each event, the mentioned TSWLS and LLS methods in Sections 1 and 2 are applied for comparison with the PCFWE with additive outliers and without additive outliers.

Figure 7 shows the location results in a three-dimensional space. It can be seen that the location results of the PCFWE method are approximate to the true sources, regardless of whether the additive outliers are contained in the TDOA measurements or not. Location results, which are determined by the TSWLS and LLS methods, also achieve desirable location results with minor deviations from true sources, but it deviates dramatically if outliers exist. Figure 8 shows the absolute distance errors of location results for three methods with and without outliers. When there are no outliers in the measurements, the absolute distance errors of the three methods are all small, while the PCFWE is the smallest, because of the applied preconditioning method. When the outliers are contained in measurements, the location errors of the traditional methods are dramatically large, while the location performance of the proposed method always remains stable with a higher location accuracy.

**Figure 5.** The process of locating the AE source *T* using the PCFWE method. The weights in this figure have been standardized by *wi*/sum(*wi*).

**Figure 6.** Comparison of location errors before and after filtering using the LLS method.

From Table 2, it is obvious that the biggest absolute distance errors of traditional methods can occasionally exceed 50 mm. There are two main reasons for the poor location results of traditional methods. Firstly, due to the influence of an outlier in the measurement, residuals are significantly intensified by the least squares principle, which results in a large deviation to the final location result for traditional methods. Secondly, traditional methods take no account of the problem of ill-conditioned linear equations, which also affect the location accuracy to some extent. It is worth mentioning that the proposed method in this paper can achieve an accurate location, where the best absolute distance error can reach about 1 mm. The good performance of the new method is attributed to the fact that, it not only reduces the ill condition of linear equations, it also eliminates outliers. Therefore, location accuracy is improved effectively.

**Figure 7.** Location results from three methods: (**a**) three-dimensional (3D) schematic diagram of location results and sensor layout; (**b**) projections of AE events on the *x*-*z*, *y*-*z*, and *x*-*y* planes.

**Figure 8.** The absolute distance error of the location results for three methods.

**Table 2.** Coordinates of AE sources solved by three methods.

