**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 [1–7]. Nevertheless, locating the source is not a simple task, because there are two major challenges, namely: measurement errors and real-time implementation [8–12].

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 [13–18]. 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. Niccolini 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 [29–33]. 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 [41–44], is one of the prevalent weighted least squares methods. This method has been widely used in various fields as a benchmark for subsequent studies [9,31,45]. Firstly, source location and the additional variable are considered independent variables to conduct the estimation. Secondly, the estimation is developed by considering the relationship between the additional variable and the source location. However, the covariance matrix of measurement errors used in this location process is difficult to accurately estimate, which causes performance degradation at the source location [9]. Unlike the method of weight least squares, Dong et al. [46] recently proposed comprehensive closed-form solutions using a set of sensor networks and the logistic probability density function, so as to improve the location accuracy. However, location errors of the closed-form solution are still large, because only six sensors are considered in each calculation process, and the coefficient matrix of the linear equations is close to ill-conditioned. In addition, since many subsets of six sensors exist, it requires enormous computing power; thus, it has a poor performance in real-time implementation [47].

Aiming at improving the error-tolerance and real-time implementation of the AE source location, this paper proposes the preconditioned closed-form solution based on weight estimation (PCFWE) to seek the optimal source coordinates by filtering outliers and reducing the ill condition of linear equations. PCFWE is also compared with the TSWLS [41] and linear least squares (LLS) methods. The effectiveness and accuracy of the proposed PCFWE are verified by the pencil-lead break experiment and simulating tests. In Section 2, the ordinary closed-form method that fairly treats all of the measurements is stated. In Section 3, the theory of the proposed method is formulated mathematically. In Section 4, the location results of the pencil-lead break experiment are reported. Finally, outlier tolerance and the velocity sensibility of the proposed method are discussed and concluded in Section 5.

#### **2. Ordinary Closed-Form Method**

To achieve an accurate and real-time AE source location, many closed-form methods exploiting the LLS principle are proposed [15,16,48,49]. The process of the ordinary closed-form location method mainly contains two steps as follows. Firstly, nonlinear equations are transformed into linear forms to reduce the difficulty in solving nonlinear equations by iterative methods. Secondly, the LLS principle is incorporated to solve the overdetermined liner equations, which can use as many measurements as possible, and yield a better location accuracy than those that only use a minimum number of measurements [50,51], i.e., the number of equations is equal to the number of unknowns. From the viewpoint of error control, the dataset is statistically more reliable, and the array geometry is more reasonable when more measurements are used. The process of an ordinary closed-form location method with the LLS principle is stated as follows.

Assuming that an AE source is located at source *θ* (*x*, *y*, *z*), and the *n +* 1 sensors are located at *Si* (*xi*, *yi*, *zi*) (*i* = 0, 1, ···, *n*), for an arbitrary AE event, the source coordinates satisfy the following control equations:

$$\left(\left(x\_i - x\right)^2 + \left(y\_i - y\right)^2 + \left(z\_i - z\right)^2 = v^2 \left(\Delta t\_i + t\_0\right)^2\tag{1}$$

where *t*<sup>0</sup> is the propagation time of the acoustic wave from the source to the nearest sensor. The sensors are numbered *Si* (*i* = 0, 1, ···, *n*) according to the arrival orders of the AE signal. The nearest sensor *S*<sup>0</sup> that receives the AE signal first is regarded as the reference sensor. Δ*ti* is the time difference between reference sensor *S*<sup>0</sup> and sensor *Si* (*i* = 1, 2, ···, *n*), namely the time difference of arrival (TDOA). Besides, when *i* = 0, Δ*ti* = 0, nonlinear equations can be linearized by subtracting the equation *i* = 0 from *i* ≥ 1.

$$2(\mathbf{x}\_i - \mathbf{x}\_0)\mathbf{x} + 2(y\_i - y\_0)y + 2(z\_i - z\_0)z + 2\Delta t\_i v^2 t\_0 = L\_i, i = 1, 2, \dots, n \tag{2}$$

where *Li* = *xi* <sup>2</sup> − *<sup>x</sup>*<sup>0</sup> <sup>2</sup> + *yi* <sup>2</sup> − *<sup>y</sup>*<sup>0</sup> <sup>2</sup> + *zi* <sup>2</sup> − *<sup>z</sup>*<sup>0</sup> <sup>2</sup> − <sup>Δ</sup>*<sup>t</sup>* 2 *<sup>i</sup> <sup>v</sup>*2.

Equation (2) can be expressed in matrix form as:

$$A\theta = b \tag{3}$$

$$\text{where } A = 2 \begin{bmatrix} \begin{array}{cccc} \mathbf{x}\_1 - \mathbf{x}\_0 & \mathbf{y}\_1 - \mathbf{y}\_0 & \mathbf{z}\_1 - \mathbf{z}\_0 & \Delta t\_1 \mathbf{z}^2 \\\ \mathbf{x}\_2 - \mathbf{x}\_0 & \mathbf{y}\_2 - \mathbf{y}\_0 & \mathbf{z}\_2 - \mathbf{z}\_0 & \Delta t\_2 \mathbf{z}^2 \\\ \vdots & & \\\ \mathbf{x}\_n - \mathbf{x}\_0 & \mathbf{y}\_n - \mathbf{y}\_0 & \mathbf{z}\_n - \mathbf{z}\_0 & \Delta t\_n \mathbf{z}^2 \end{bmatrix}, \theta = \begin{bmatrix} \mathbf{x} \\\ \mathbf{y} \\\ \mathbf{z} \\\ \mathbf{t}\_0 \end{bmatrix}, b = \begin{bmatrix} L\_1 \\ L\_2 \\ \vdots \\ L\_n \end{bmatrix}.$$

To find the location parameters *θ* in Equation (3), which minimizes the sum of the residual square for the linear equation system:

$$\arg\min\_{\theta} ||b - A\theta|| = \arg\min\_{\theta} \sum\_{i=1}^{n} \left(b\_{i} - A\_{i}\theta\right)^{2} \tag{4}$$

The closed-from solution with the least squares principle can be obtained by:

$$\theta = \left(A^T A\right)^{-1} A^T b \tag{5}$$

where the symbol *T* denotes the matrix transpose, and *ATA* is the Hessian matrix.

The solution of LLS is the best linear unbiased estimator for the AE source location when the measurement errors are independent and identically distributed. However, it may produce dramatic location errors due to the equal treatments of all of the measurements, even the outliers. Therefore, outliers are considered to be filtered by weight estimation first; then, linear equations are reconstructed with the remaining measurements that contain no outliers to obtain more accurate location results. However, the linear equation system (3) is always close to ill-conditioned, which can cause a large

location error with a minor disturbance. Therefore, the preconditioning of linear equations is adopted to weaken the ill condition and further improve the location accuracy.
