Two types of multiple solutions for microseismic source location based on arrival-time-difference approach

The arrival-time-difference approach is the dominant source location approach used in the microseismic source location area. Multiple solutions problem is one of the major concerns in microseismic source location, which is closely related to the microseismic network. This paper categorizes the multiple solutions into two types based on the origin times when using the arrival-time-difference approach. Type I multiple solutions are those which have the same origin time; type II multiple solutions are those with different origin times. The sufficient and necessary conditions to produce type I multiple solutions are that all sensors are located in a straight line for two-dimensional cases and on a plane for three-dimensional cases. The sufficient and necessary conditions to produce type II multiple solutions are that all sensors are located on a hyperbola for two-dimensional cases and on a hyperboloid for three-dimensional cases. Furthermore, the proofs indicate that type I multiple solutions are preventable, while a microseismic network consisting of the minimum number of sensors can never be free of type II multiple solutions. It means, besides the minimum number of sensors, at least one more sensor which is not on this hyperbola or hyperboloid is needed to uniquely determine a source. The results from field tests and applications indicate that when the sensors of a network lie on a hyperbola, the type II multiple solutions may not be the necessary outcome under the influence of errors in real data. However, the accuracy of the microseismic source location is affected significantly by this kind of networks. The results also show that not only the multiple solutions problem can be avoided effectively, but more importantly, the accuracy of the source location will be greatly improved by the optimization of network based on the characteristics of the microseismic network and field conditions.