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.

The physics‐guided neural network framework combines the effectiveness of data‐driven and physics‐based models, and it is, therefore, becoming increasingly popular in geophysical applications. We present a physics‐guided neural network–based approach to calibrate velocity models for microseismic data. In our implementation, the physics‐guided neural network comprises of a user‐selected number of fully connected layers, a scaling and shifting layer and a forward modelling operator layer. We input the observed P‐ and S‐wave arrival times to the neural network. In the forward pass, the network's output layer produces normalized P‐ and S‐wave velocities for the subsurface model. The scaling and shifting layer converts the normalized output to realistic velocity values. The forward modelling operator (i.e. a ray‐shooting algorithm) layer computes traveltimes using the velocities from the preceding scaling and shifting layer and the known source–receiver locations. We then evaluate a loss function that compares the predicted traveltimes with the input observed arrival times, and update network's weights and bias parameters. We also use a weight‐based saliency measure to evaluate whether the selected network architecture (i.e. number of hidden layers and neurons) is optimal for the model calibration problem. Finally, using synthetic data examples, we demonstrate that our unsupervised physics‐guided neural network–based approach can provide robust velocity model and uncertainty estimates.

We present a novel physics-guided neural network to estimate shear-tensile focal mechanisms for microearthquakes using displacement amplitudes of direct P waves. Compared with conventional data-driven fully connected (FC) neural networks, our physics-guided neural network is implemented in an unsupervised fashion and avoids the use of training data, which may be incomplete or unavailable. We incorporate three FC layers and a scaling and shifting layer to estimate shear-tensile focal mechanisms for multiple events. Then, a forward-modeling layer, which generates synthetic amplitude data based on the source mechanisms emerging from the previous layer, is added. The neural network weights are iteratively updated to minimize the mean squared error between observed and modeled normalized P-wave amplitudes. We apply this machine-learning approach to a set of 530 induced events recorded during hydraulic-fracture simulation of Duvernay Shale west of Fox Creek, Alberta, yielding results that are consistent with previously reported source mechanisms for the same dataset. A distinct cluster characterized by more complex mechanisms exhibits relatively large Kagan angles (5°–25°) compared with the previously reported best double-couple solutions, mainly due to model simplification of the shear-tensile focal mechanism. Uncertainty tests demonstrate the robustness of the inversion results and high tolerance of our neural network to errors in event locations, the velocity model, and P-wave amplitudes. Compared with a single-event grid-search algorithm to estimate shear-tensile focal mechanisms, the proposed neural network approach exhibits significantly higher computational efficiency.