Seismic moment tensors provide a concise mathematical representation of point sources that can be used to characterize microseismic focal mechanisms. After correction for propagation effects, the six independent components of a moment tensor can be found by least-squares inversion based on P- and/or S-waveform (or spectral) amplitudes observed at different directions from the source. Using synthetic waveform data, we investigated geometrical factors that affect the reliability of such inversions. We demonstrated that the solid angle subtended by the receiver array, as viewed from the source location, plays a fundamental role in the stability of the inversion. In particular, the condition number of the generalized inverse scales approximately inversely with the solid angle, implying that for a solid angle of zero (as is the case for a single vertical borehole) the inversion is ill-conditioned. The presence of random noise alsohas a significant effect on the inversion results; our results showed that the signal-to-noise ratio (S/N) for reliable inversion scales approximately as the square root of the condition number. Taken together with geometrical considerations, we found that a [Formula: see text] is generally needed to obtain reliable inversion results for the full moment tensor under certain microseismic acquisition scenarios that include dual observation wells or surface star pattern. Our numerical tests indicated that least-squares moment-tensor solutions obtained under nonideal conditions are biased toward limited regions of the full parameter space. In particular, random noise introduces a bias toward volumetric source types, whereas ill-conditioned inversions may exhibit bias toward poorly resolved eigenvector(s) of the inversion matrix. Possible strategies to improve the reliability of moment-tensor inversion include ensuring a nonzero solid-angle aperture by using multiple observation wells, and/or incorporating other types of data such as a priori knowledge of fracture orientation.
A RESTRAINED CONDITION NUMBER LEAST SQUARES TECHNIQUE WITH ITS APPLICATIONS TO AVOIDING RANK DEFICIENCY
