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 CO 2 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. Key Points: • We derive a new viscoacoustic wave equation that enables frequency-independent quality factor • This new equation is favored for having simplified fractional Laplacians and incorporating attenuation heterogeneity • This proposed equation directly benefits seismic modeling and imaging applications Figure 14. Calibration of modeled attenuation. (a) Synthetic seismograms recorded by 23 out of 151 receivers at different depths for time slices T1 (baseline, gray solid lines), T2 (12 hr, blue dashed lines), and T3 (48 hr, red dashed lines). (b) Frio-II field data at different depths for time slices T1 (gray solid lines), T2 (blue dashed lines), and T3 (red dashed lines). The amplitude of each trace in (a) and (b) is normalized according to the baseline recording. (c) Centroid frequency shift compared to T1 for T2 synthetic data (blue solid line), T3 synthetic data (red solid line), T2 field data (blue stars), and T3 field data (red stars).

Seismic velocity and attenuation anisotropy are ubiquitous in the crust and upper mantle, significantly modulating the characteristics of seismic wave propagation in the Earth's interior. Accurate seismic wave modeling of velocity and attenuation anisotropy is essential for the understanding of wave propagation in the Earth's interior as well as constructing global and region‐scale seismic full waveform tomography. Here, we derive a decoupled fractional Laplacian (DFL) viscoelastic wave equation to characterize the Earth's frequency‐independent Q behavior in the vertical transversely isotropic (VTI) media. We verify the accuracy of the proposed viscoelastic wave equation by 2D synthetic examples; to show its applicability in crustal‐scale seismic modeling, we present an example of 3D seismic wave propagation in the realistic Salton Trough model. Through extensive numerical tests, we conclude that the proposed viscoelastic wave equation is superior in four aspects. First, the viscoelastic wave equation takes VTI anisotropy of both velocity and attenuation into account and can describe the strong direction‐dependent attenuation. Second, our derivation contains spatially independent Laplacians, and thus the proposed wave equation enjoys higher simulation accuracy for heterogeneous Q media. Third, the new viscoelastic wave equation can decouple the amplitude decay and the phase distortion, which is appealing for improving the resolution in seismic imaging and inversion. Lastly, compared to viscoelastic wave equations with time‐fractional operators, our scheme has higher computational efficiency by avoiding substantial wavefield storage.

Surface wave tomography routinely uses empirically scaled density model in the inversion of dispersion curves for shear wave speeds of the crust and uppermost mantle. An improperly selected empirical scaling relationship between density and shear wave speed can lead to unrealistic density models beneath certain tectonic formations such as sedimentary basins. Taking the Sichuan basin east to the Tibetan plateau as an example, we investigate the differences between density profiles calculated from four scaling methods and their effects on Rayleigh wave phase velocities. Analytical equations for 1-D layered models and adjoint tomography for 3-D models are used to examine the trade-off between density and S-wave velocity structures at different depth ranges. We demonstrate that shallow density structure can significantly influence phase velocities at short periods, and thereby affect the shear wave speed inversion from phase velocity data. In particular, a deviation of 25 per cent in the initial density model can introduce an error up to 5 per cent in the inverted shear velocity at middle and lower crustal depths. Therefore one must pay enough attention in choosing a proper velocity-density scaling relationship in constructing initial density model in Rayleigh wave inversion for crustal shear velocity structure.

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We formulate the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that both the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we show that the adjoint wave propagator preserves the dispersion and compensates the amplitude, while the time-reversed adjoint wave propagator behaves identically as the forward propagator with the same dispersion and dissipation characters. Without introducing rheological mechanisms, this formulation adopts an explicit Q parameterization, which avoids the implicit Q in the conventional viscoacoustic/viscoelastic full waveform inversion ( Q-FWI). In addition, because of the decoupling of operators in the wave equation, the viscoacoustic Fréchet kernel is separated into three distinct contributions with clear physical meanings: lossless propagation, dispersion, and dissipation. We find that the lossless propagation kernel dominates the velocity kernel, while the dissipation kernel dominates the attenuation kernel over the dispersion kernel. After validating the Fréchet kernels using the finite-difference method, we conduct a numerical example to demonstrate the capability of the kernels to characterize both velocity and attenuation anomalies. The kernels of different misfit measurements are presented to investigate their different sensitivities. Our results suggest that rather than the traveltime, the amplitude and the waveform kernels are more suitable to capture attenuation anomalies. These kernels lay the foundation for the multiparameter inversion with the fractional formulation, and the decoupled nature of them promotes our understanding of the significance of different physical processes in the Q-FWI.