Modeling Frequency‐Independent Q Viscoacoustic Wave Propagation in Heterogeneous Media

Abstract

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).

Propagating Seismic Waves in VTI Attenuating Media Using Fractional Viscoelastic Wave Equation

“…Thus, the approximate dispersion relation (Equation 2) can well capture the wavefield behavior described by the Kjartansson model (Equation 1) in terms of dynamic dissipation (imaginary part of k ) and kinematic dispersion (real part of k ). The detailed derivation of Equation 3 can be found in Xing and Zhu (2019). We have to emphasize that the wavenumber k has a constant power (i.e., spatially independent) in Equation 2, which leads to fractional Laplacian operators with spatially independent‐order in the time‐space domain (Section 2.2) and thus guarantees the accuracy when handling heterogeneous Q media.…”

“…To improve the efficiency, several intermediate approaches (e.g., Chen et al., 2016; Sun et al., 2015; Wang et al., 2018; Wang et al., 2020; Zhang et al., 2020) have been proposed. Recently, Xing and Zhu (2019) developed a novel spatial‐independent‐order DFL viscoacoustic wave equation that fully avoids the difficulty of computing mixed‐domain operators.…”

Spatial Filter for the Pseudo-spectral Implementation of Fractional Derivative Wave Equation

“…When forming the FSDs for a general viscoelastic case, bðxÞ is a spatial function. Based on the small perturbation assumption, FSDs is still approximated valid for a general viscoelastic case (Xing & Zhu, 2019;Zhu & Harris, 2014).…”

“…The spatially varying order of FSD is a function of bðxÞ. Chen et al (2016) and Xing and Zhu (2019) use either the Taylor expansion or a polynomial approximation to transfer the spatial-varying order FSD into the constant order FSDs, and then implement the pseudo-spectral method directly. Yao et al (2017) apply the Hermite distributed approximation to transfer the spatial-varying FSD to an integral of the fractional derivative of delta function, and this fractional derivative can be solved locally.…”

“…But all these schemes aforementioned scarify the efficiency for accuracy to some extent. For example, when using the polynomial approximation to derive the constant-order wave equation (Xing & Zhu, 2019), the resultant wave equation includes six fractional Laplacian operators. This scheme greatly lowers the efficiency, if compared with the original two-operator equation.…”