A Regularized Approach for Estimation of a Composite Focal Mechanism from a Set of Microearthquakes

We have developed a novel regularized approach to estimate a composite focal mechanism for microseismic events that share a similar source mechanism. The method operates by minimizing the weighted misfits of the SH/P amplitude ratios (in absolute sense and logarithmic scale) and P-wave polarities, using a regularization parameter determined from the trade-off curve for these values. This approach overcomes the low signal-to-noise ratio (S/N) and single-event azimuthal gaps that may otherwise limit the effectiveness of sparse surface arrays. Compared with focal mechanisms derived from P-wave polarity or amplitude-based methods, our regularized approach reduces the multiplicity of solutions and avoids the use of signed amplitude ratios, which may be ambiguous for data with low S/N. We apply our method to a set of 13 microseismic events recorded during hydraulic-fracture stimulation of the Marcellus Shale in West Virginia and Pennsylvania, USA, yielding a strike-slip focal mechanism accompanied by a minor normal component. Our solution is similar to previously reported focal mechanisms in this area. Jackknife analysis, which tests stability of the inversion based on random sampling of the observation, indicates 95% confidence intervals of 1° and 2°, respectively, for the plunge and azimuth of the P and T axes. By analyzing the event subsets, outliers are identified and the assumption of a single dominant focal mechanism is validated. Numerical modeling demonstrates that our approach is robust in the presence of variations of up to 0°–10° and 0°–35°, respectively, for the plunge and azimuth of P and T axes of the focal mechanisms of these events. Sensitivity analysis using synthetic data also indicates that the algorithm is tolerant to mispicks as well as errors in polarity and amplitude ratio. In the presence of some dissimilar focal mechanisms, the dominant focal mechanism can be reliably estimated if at least 70% of the events have similar source mechanisms.