Obtaining hypocenters of microseismic events, a primary task in mining, geothermal, and hydraulic-fracturing applications of induced seismicity, requires a velocity model for computing those hypocenters. In our paper, relying on a notion that information provided by microseismic events themselves enables one to construct a velocity model and calculate the event hypocenters in that model, we derive exact analytic solutions to the joint velocity-estimation/event-location problem for downhole microseismic data acquired in homogeneous isotropic media. We show that traveltimes and polarization vectors of the direct P- and S-waves excited by a microseismic event and recorded by a string of receivers placed in one or two vertical wells not only uniquely constrain the event location and the medium velocities but also entail a straightforward analysis of the uncertainties of those estimates caused by the presence of noise in the data. Although the P- and S-wave velocities calculated analytically under the assumption of the medium homogeneity cannot fully absorb the complexity of heterogeneous subsurface models, they become the proper effective velocities for a given microseismic event, and the corresponding event location—which is no longer exact even for noise-free data—might serve as a useful initial guess for more sophisticated event-location techniques that account for the velocity heterogeneity.

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