This paper presents two dimensional close-form solutions for acoustic emission source location using Time Difference Of Arrival (TDOA) measurements from N receivers, for known velocity system (N ≥ 4) and unknown velocity system (N ≥ 5). We simplify the nonlinear location equations for TDOA to linear equations, and then to obtain the analytical solution directly from the linear equations. Due to free of calculations of square roots in solution equations, it not only solved the problems of the existence and multiplicity induced by the calculations of square roots in existed close-form methods, but also can locate the source in unknown velocity systems.


The solution of the problem of locating a signal source using Time Difference Of Arrival (TDOA) measurements has numerous applications in aerospace, surveillance, structural health, navigation, industrial process, speaker location, machine condition, the monitoring of nuclear explosions, and mining induced areal seismology (Aki K 1980, Rens et al. 1997, Spencer 2007, Li et al. 2011, Liu et al. 2005, Spencer 2010, Dong et al. 2014, Dong et al. 2013).

Many authors have discussed and faced numerous problems connected with the location of acoustic emission. The time difference of arrival TDOA method, based on estimates of time delay for a correlated signal as detected by spatially distributed sensor elements in an array, remains a commonly used technique for source location (Dong et al. 2014).

The TDOAs are proportional to the differences in sensor-source range, called Range Differences (RDs). Conventionally, the source location is estimated from the intersection of a set of hyperboloids defined by the RD measurements and the known sensor locations (Mellen et al. 2003). The inverse problem for TDOA source location is usually solved by an iterative technique such as nonlinear least-squares (ILS), minimum error, or an optimization method in recognition that the equations are nonlinear with respect to source location (Belchamber et al. 1990, Landis et al. 1991, Chan et al. 1994, Brandstein et al. 1997. Dibiase et al. 2001). Kalman filtering has also been used to iteratively solve the TDOA source location problem for microphone speaker location (Gannot et al. 2006, Ulrich et al. 2006). Although these iterative algorithms are resilient to varying extents to errors in arrival time data, they may be computationally expensive. This is a key consideration in some realtime applications (Chan et al. 1994, Brandstein et al. 1997).

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