## 1 Introduction  
The acoustic emission (AE) source location method based on sensor array has been a hot research topic with widespread applications in many fields of structural health, underground tunneling, and deep mining [1; 2; 3; 4; 5; 6; 7]. Nevertheless, locating the source is not a simple task, because there are two major challenges, namely: measurement errors and real-time implementation [8; 9; 10; 11; 12]. The least squares principle, which is employed in most methods, can achieve a favorable location accuracy on the condition that all of the measurements are accurate or only have minor errors [13; 14; 15; 16; 17; 18]. However, adding an outlier will cause the residual value to dramatically increase, which is enlarged due to the square nature of the least squares principle, and the location performance to significantly deteriorate [19; 20]. Accurate estimation for the arrival times of acquired signals associated with each sensor is the underlying challenge for the most picking-based location method. Researchers have proposed varying algorithms, trying to pick arrivals accurately and in real-time, such as a short and long-time average ratio [21], higher order statistics method [22]. However, these methods only have good performances in clean or high signal-to-noise ratio environments. [PERSON] et al. [23; 24] proposed the joint autoregressive modeling of noise and the signal in windows using Akaike information criterion with an automatic procedure for signal data processing under a low signal-to-noise ratio environment. This method, which can eliminate certain false or doubtful arrivals before the source location, can be utilized to obtain the real-time arrivals and reduce the probability of an outlier. However, it does not mean that all of outliers will be eliminated, because there are other factors that can generate outliers besides the background noise. Practical applications of monitoring systems have reported that the sensors are easy to be triggered incorrectly by interference signals due to the complex engineering environment, which brings external outlier signals [25; 26]. In addition, outliers can also occur in these cases, such as signal interference, sensor location errors, weak and ambiguous of arrivals, interchannel crosstalk, or simply hardware failure. Therefore, all the above cases may bring outliers in measurements, especially in the use of automated picking methods. Considering the difficult acquisition of an accurate measurement and the deterioration effect of the outliers on the source location, the error-tolerant location methods have been developed by scholars to reduce the influence of outliers [27; 28]. A large amount of error-tolerant methods of time difference of arrival (TDOA) have been proposed, which can be classified into nonlinear methods and linear methods [29; 30; 31; 32; 33]. Nonlinear location methods, including maximum likelihood [34; 35], the virtual field optimization method [36], and the least absolute deviation method [37], have been suggested to improve the error-tolerance. To gain the optimum solution of these nonlinear methods, iterative techniques should be used because of nonlinear TDOA equations. Alternative choices include the simplex method [38], Newton iterative method [39], and the Gauss-Newton method [40]. However, these methods not only require an appropriate guess of initial location near the true solution, which is difficult to obtain in practice, but also probably suffer from convergence problems and a large computational burden caused by the iterative nature. Compared with nonlinear methods, linear methods in the close form guarantee the convergence and real-time implementation. The weighted least squares algorithm, which gives different weights to different equations, is a generic form of the error-tolerant closed-form solution. Two-step weighted least squares (TSWLS), which linearizes the nonlinear TDOA equations by introducing an additional variable [41; 42; 43; 44], is one of the prevalent weighted least squares methods.