Mathematical Problems in Engineering

Volume 2016, Article ID 8540303, 14 pages

http://dx.doi.org/10.1155/2016/8540303

## Two-Dimensional Far Field Source Locating Method with Nonprior Velocity

^{1}IOT Perception Mine Research Center, China University of Mining and Technology, Xuzhou 221008, China^{2}School of Mathematics & Physics Science, Xuzhou Institute of Technology, Xuzhou 221008, China^{3}School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou 221008, China^{4}School of Mechanical Engineering, Nanjing University of Science & Technology, Nanjing 221000, China

Received 22 December 2015; Accepted 30 March 2016

Academic Editor: Anna Vila

Copyright © 2016 Qing Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Relative position of seismic source and sensors has great influence on locating accuracy, particularly in far field conditions, and the accuracy will decrease seriously due to limited calculation precision and prior velocity error. In order to improve the locating accuracy of far field sources by isometric placed sensors in a straight line, a new locating method with nonprior velocity is proposed. After exhaustive research, this paper states that the hyperbola which is used for locating will be very close to its asymptote when seismic source locates in far field of sensors; therefore, the locating problem with prior velocity is equivalent to solving linear equations and the problem with nonprior velocity is equivalent to a nonlinear optimization problem with respect to the unknown velocity. And then, this paper proposed a new locating method based on a one-variable objective function with respect to the unknown velocity. Numerical experiments show that the proposed method has faster convergence speed, higher accuracy, and better stability.

#### 1. Introduction

Microseismic monitoring, which was first studied by Obert [1], has been successfully applied in mining [2–5], oil exploitation [6–10], water conservancy construction [11–13], and so forth. One of the core technologies of microseismic monitoring is source locating method, which can be reduced to (1) in homogeneous medium:where , , and are coordinates of seismic source and is the initial time of source; , , and are coordinates of the th sensor; is the average wave velocity measured by the th sensor; is arrival time measured by th sensor. It is obvious that 4 or more sensors at least are needed to solve for if is known. A set of nonlinear equations as expressed by (1) can be solved either iteratively or noniteratively.

The well-known noniterative methods, such as Inglada method [14, 15] and USBM (United States Bureau of Mines) method [15–17], may not be applicable for current microseismic monitoring system. The Inglada method uses only a minimum number of sensors that are mathematically required for source locating, and because of this requirement, no optimization method can be applied to the algorithm. The USBM method introduces least squares method so that all available arrival time information can be used simultaneously for source locating calculation. However, for its matrix inversion, the method will be unstable while the measured values have gross errors, which is common in a real-world situation.

The main iterative methods for source locating include Geiger’s method [18, 19], Simplex Algorithm [20, 21], and Genetic Algorithm [22]. Geiger’s method has heavy computing burden and may be divergent because of the foundation of the method-first-order Taylor expansion and least square method. Many scholars studied these problems and have proposed some solutions [23, 24]. Simplex Algorithm [25], a robust geometry search algorithm, is one of the dominant methods for source locating [26–28]. Simplex is a geometric figure which contains one more vertex than the solution space’s dimension, and the optimization process is simple which is moving the worst vertex till the optimal solution meets the preset conditions. GA (Genetic Algorithm), an optimization method, simulates nature selection in which only the “fittest” solutions survive so that they can create even better answers in the process of reproduction. Although the theory of GA is imperfect, the algorithm was still introduced in earthquake source locations in 1992 [29–32], and it has shown attractive prospects in parallel computing, such as SPARK [33], because of its intrinsic parallelism. According to the research from Yun and Xi [34] and He et al. [35], EGA (Elitist preserved GA) is global convergent, which provides guidance for GA based algorithms.

All of the algorithms mentioned above have the same hypothesis, which is that the velocity structure is known before location calculation, which is obviously impossible in practice. Therefore, the location accuracy of algorithms mentioned above will be seriously affected by prior velocity. According to Dong et al.’s research [36], the accuracy of methods with prior velocity in homogeneous medium will decrease seriously when the velocity error reaches 1%–5%, while the accuracy of methods with nonprior velocity will not be affected. Consequently, Dong and Li [37] proposed a new location method with nonprior velocity based on arrival time of PS waves; however, the method was not used when P-wave and S-wave could not be distinguished clearly. Li et al. [38] proposed a location method with nonprior velocity based on Simplex Algorithm and the essence of its method is a new objective function without velocity, which could also be used by other optimization methods.

Actually, Li’s method is a concrete realization of Prugger’s method [21], which will give a correct result when the seismic source is surrounded by sensors and the measured arrival times have no error. However, in practice, the seismic source is not always surrounded by sensors for a variety of reasons, and Li’s objective function changes very little near true value which will decrease the accuracy of location by reason of finite word length effect. To solve this problem, this paper proposed a new objective function, which can locate faster and more stable.

#### 2. Two-Dimensional Source Locating Theory

In this paper, we will discuss only one scenario that frequently happened during mining monitoring as shown in Figure 1.