Three Dimensional Comprehensive Analytical Solutions for Locating Sources of Sensor Networks in Unknown Velocity Mining System

The accuracy of localization methods based on the arrival time difference is usually affected by the iterative algorithm, the initial value, and the pre-measured wave velocity. The analytical solutions are non-unique because of the square root operations in the calculating process for source coordinates. To solve these significant problems, the nonlinear equations were simplified to the linear equations. An analytical localization method without the pre-measured velocity or the square root operations was developed. The explicit formulas for analytical solutions were resolved for the six sensors network. The source coordinates can be solved for real time by substituting the arrival times and coordinates of sensors. Focusing on the practical engineering where the number of sensors is greater than six, the comprehensive analytical solutions were proposed on account of sensor networks, which formed through the combination of different sensors, and the logistic probability density function. The blasting tests in two mines verified its effectiveness and accuracy. Results show that the locating accuracy of three dimensional comprehensive analytical solutions is superior to the traditional methods. The assumed examples proved that the proposed method performs well under different scales of arrival time errors. This proposed method highlights four advantages: without iterative algorithm, without pre-measured velocity, without initial value, and without square root operations.