Accurate estimation of microseismic events in time and space enables important characterization of hydraulic fracture networks. Determining the orientation of the fracture can help differentiate between reactivation of an in situ fracture and a hydraulically induced fracture, which is important in characterizing the effectiveness of the stimulation process. We consider the source as separable in time and space and invert for a complete description of the source in one of these two dimensions assuming the other is known. We recover the wavelet, which includes the source amplitude and time evolution. In space, we recover a description of the source that is distributed and includes an estimate of the moment tensor at every point in the domain. A change of variables applied to the velocity-stress form of the elastic-wave equation ensures that the system is self-adjoint. Thus, full-waveform inversion can be tailored to estimate microseismic events with limited modifications to the forward wave solver. The inversion does not use any a priori assumptions about the form of the source, does not require a good starting guess for accurate source recovery, and is robust in the presence of noise. Applying this technique to wavelet inversion correctly recovers a 30 Hz Ricker wavelet from a zero initial guess with and without noise. Furthermore, for a realistic microseismic event generated from coupled flow and deformation modeling, the algorithm recovers the peak time and approximate shape of the wavelet. In fact, our algorithm recovers as much of the true wavelet as possible given the energy in the observed data. Furthermore, experiments involving a distributed source in the shape of an ellipse illustrate that the inversion scheme can not only estimate the focal mechanism of failure events, but also the geometry of the failure plane.

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