We evaluated a time-domain wave equation for modeling acoustic wave propagation in attenuating media. The wave equation was derived from Kjartansson's constant-Q constitutive stress-strain relation in combination with the mass and momentum conservation equations. Our wave equation, expressed by a second-order temporal derivative and two fractional Laplacian operators, described very nearly constant-Q attenuation and dispersion effects. The advantage of using our formulation of two fractional Laplacians over the traditional fractional time derivative approach was the avoidance of time history memory variables and thus it offered more economic computations. In numerical simulations, we formulated the first-order constitutive equations with the perfectly matched layer absorbing boundaries. The temporal derivative was calculated with a staggered-grid finite-difference approach. The fractional Laplacians are calculated in the spatial frequency domain using a Fourier pseudospectral implementation. We validated our numerical results through comparisons with theoretical constant-Q attenuation and dispersion solutions, field measurements from the Pierre Shale, and results from 2D viscoacoustic analytical modeling for the homogeneous Pierre Shale. We also evaluated different formulations to show separated amplitude loss and dispersion effects on wavefields. Furthermore, we generalized our rigorous formulation for homogeneous media to an approximate equation for viscoacoustic waves in heterogeneous media. We then investigated the accuracy of numerical modeling in attenuating media with different Q-values and its stability in largecontrast heterogeneous media. Finally, we tested the applicability of our time-domain formulation in a heterogeneous medium with high attenuation.

Reduced amplitude and distorted dispersion of seismic waves caused by attenuation, especially strong attenuation, always degrades the resolution of migrated images. To improve image resolution, we evaluated a methodology of compensating for attenuation (∼1∕Q) effects in reverse-time migration (Q-RTM). The Q-RTM approach worked by mitigating the amplitude attenuation and phase dispersion effects in source and receiver wavefields. Source and receiver wavefields were extrapolated using a previously published time-domain viscoacoustic wave equation that offered separated amplitude attenuation and phase dispersion operators. In our Q-RTM implementation, therefore, attenuation-and dispersion-compensated operators were constructed by reversing the sign of attenuation operator and leaving the sign of dispersion operator unchanged, respectively. Further, we designed a low-pass filter for attenuation and dispersion operators to stabilize the compensating procedure. Finally, we tested the Q-RTM approach on a simple layer model and the more realistic BP gas chimney model. Numerical results demonstrated that the Q-RTM approach produced higher resolution images with improved amplitude and phase compared to the noncompensated RTM, particularly beneath high-attenuation zones.

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We report for the first time on a distributed acoustic sensing (DAS) array using preexisting underground fiber optics beneath the Penn State campus for detecting and characterizing thunder‐induced ground motions. During a half‐hour interval from 03:20–03:50 UTC on 15 April 2019 in State College, PA, we identify 18 thunder‐induced seismic events in the DAS array data. The high‐fidelity DAS data show that the thunder‐induced seismics are very broadband, with their peak frequency ranging from 20 to 130 Hz. We use arrival times of the 18 events to estimate the phase velocity of the near surface, the back azimuth, and location of thunder‐seismic sources that are verified with lightning locations from the National Lightning Detection Network. Furthermore, the dense DAS data enable us to simulate thunder‐seismic wave propagation and full waveform synthetics and further locate the thunder‐seismic source by time‐reversal migration. Interestingly, we found that thunder‐seismic power recorded by DAS is positively correlated with National Lightning Detection Network lightning current power. These findings suggest that fiber‐optic DAS observations may offer a new avenue of studying thunder‐induced seismics, characterizing the near‐surface velocity structure, and probing the thunder‐ground coupling process.

Distributed sensor networks are designed to provide computation in-situ and in real-time. The conventional time-reversal imaging approach for microseismic event location may not be optimal for such an environment. To address this challenge, we develop a methodology of locating multiple microseismic events with unknown start times based on the cross-correlation imaging condition borrowed from active-source seismic imaging. The imaging principle states that a true microseismic source must correspond to the location where all the backwardpropagated events coincide in both space and time. Instead of simply stacking the backward-propagated seismic wavefields, as suggested by time-reversal imaging, we perform multiplication reduction to compute a high-resolution microseismicity map. The map has an extra dimension of time, indicating the start times of different events. Combined with a distributed sensor network, our method is designed for monitoring microseismic activities and mapping fracture development during hydraulic fracturing in-situ and in real-time. We use numerical examples to test the ability of the proposed technique to produce high-resolution images of microseismic locations.