Modeling Frequency-Independent Q Viscoacoustic Wave Propagation in Heterogeneous Media

Quantifying the attenuation of seismic waves propagating in the Earth interior is critical to study the subsurface structure. Previous studies have proposed fractional anelastic wave equations to model the frequency-independent Q seismic wave propagation. Such wave equations involve fractional derivatives that pose computational challenges for the numerical schemes in terms of accuracy and efficiency when dealing with heterogeneous Earth media. To tackle these challenges, here we derive a new viscoacoustic wave equation, where the power terms of the fractional Laplacian operators are spatially independent, thus accurate and efficient methods (e.g., the Fourier pseudospectral method) can be adopted. Our derivation enables the resultant equation to capture both amplitude and phase signatures of the anelastic wave propagation by matching the complex wave numbers for all the frequencies of interest. We verify the derivation by comparing the dispersion curves of both the attenuation factor and the phase velocity produced by the new wave equation with their theoretical values as well as the Pierre Shale in situ measurements. Following that, we use a synthetic attenuating gas chimney model to demonstrate the attenuation effects on seismic waveforms and then construct the Q-compensated reverse time migration to undo these effects for seismic image enhancement. Finally, we find that our forward modeling results can characterize the spatiotemporal attenuation effects revealed in the Frio-II CO₂ injection time-lapse seismic monitoring data. We expect this proposed equation to be useful to quantify the attenuation in seismic data to push the resolution limits of seismic imaging and inversion.