*4.1. The Filtering of Outliers*

TDOA measurements, the coordinates of triggered sensors, and average velocity are used to locate the four AE events by the method proposed in this paper. The outlier TDOA measurements are generated factitiously by adding ±30% large errors to the largest two measurements, which are used to simulate the picking deviation caused by automated programs in complex engineering practice. Location results with the additive outliers in measurements are determined by using the LLS method without the outliers filtering process, and the detailed data are listed in upper half of Table 1. It is clear to see that the location errors between the authentic coordinates and the location results are great, where the maximum absolute distance error reaches to 57.61 mm. Obviously, the location results are unreasonable and unreliable, because corrupted measurements dramatically affect the location accuracy. Therefore, it is significant to identify and filter the outliers for the AE source location.

Figure 5 illustrates the process of eliminating the outliers at the location of source *T*. The initial location using the LLS method with equal weights has a large location error due to the existence of the outliers shown in Step *a*. The corresponding residuals are obtained and shown in Step *b*; it can be seen that the absolute residuals of 1, 4, 5, 6, 7, 8, 9, 12, and 13 are less than the threshold *λm*, so the corresponding weights of the measurements *t*1, *t*4, *t*5, *t*6, *t*7, *t*8, *t*9, *t*12, and *t*<sup>13</sup> are set as equal according to Equation (7), while the weights of the other measurements are set as 1/Δ*i*. Moreover, it should be noted that no outliers can be found after the first location, because the residuals obtained by the initial location result can not reflect the deviations of the true model effectively, especially when outliers exist. Therefore, further iterations between the source location and weight

estimation are necessary in order to filter the outliers, which are shown from step *c* to Step *f*. The 14th and 15th residuals exceed the maximum allowable deviation *λ<sup>L</sup>* at step *c* and *d*, respectively. Therefore, they are regarded as outliers, and *t*<sup>14</sup> and *t*<sup>15</sup> are considered outlier measurements that are filtered by setting the weights to zeros. The change of residuals between steps *e* and *f* is small enough, so the iteration stops. After that, linear equations are reconstructed by using the remaining measurements. Finally, the location results can be obtained by solving the new linear equations using the LLS method. It can be seen in Step *g* that the calculated result and theoretical locations respectively denoted by the diamond and spot are close to each other. Therefore, the location accuracy is improved effectively after outlier filtering.

The lower half of Table 1 lists the location results and absolute distance errors of the four AE sources after filtering the outliers. Through the comparison between location results before and after filtering the outliers in Figure 6, it is obvious that the location accuracy is improved significantly. For example, the absolute distance error of the source *R* is reduced to 9.28 mm from 57.61 mm, and the absolute distance error of source *T* is reduced to 10.55 mm from 46.82 mm. Therefore, it can be concluded that the location accuracy is improved effectively through the filtering of outliers. However, it should be noticed that linear equations are always in or near ill condition, which still cause a large deviation with a minor disturbance in the coefficient. Location accuracy is improved using a preconditioning method by reducing the ill condition of linear equations. Location results in the following section, Section 4.2, prove the ability of solving source coordinates with higher accuracy by preconditioning for linear equations.


**Table 1.** Closed-form solutions before and after filtering using the linear least squares (LLS) method.
