Explicit Solutions to Microseismic Event-location Problem

Locating microseismic events is a basic task in mining, geothermal, and hydraulic-fracturing applications of induced seismicity. Our paper derives analytic solutions to the joint event-location/velocity-estimation problem for downhole data-acquisition geometries in homogeneous isotropic media. We show that traveltimes and polarizations of the direct P- and S-waves excited by a single microseismic event and recorded by a string of receivers placed in one or two vertical wells uniquely constrain the event location and the medium velocities; as an added benefit, those traveltimes allow a straightforward analysis of uncertainties of the estimates caused by noises in the data. For heterogeneous subsurface models, the analytically derived event locations might serve as initial guesses for more sophisticated iterative techniques that properly account for the velocity heterogeneity.