**4. Conclusions**

In this study, in order to solve the instability of the absorbing boundary conditions caused by the frequency-domain algorithm for viscoelastic anisotropic wavefields, wave propagation was simulated by the finite-difference frequency-domain, and the reliability of the results was verified by combining the analytical wavefront obtained from the anisotropic analytical group velocity. The multi-axis perfectly matched layer (M-PML) was introduced in the frequency domain. We compared the wavefield snapshots and energy attenuation curves of M-PML and PML in different elastic anisotropic media. Then, we gave the viscoelastic anisotropic wavefield to demonstrate the reliability of M-PML in viscous media. The results indicate that M-PML can still stably and effectively absorb reflections from the truncated boundaries in strongly anisotropic and viscous media. Simulations in a complex cross-well viscoelastic anisotropic model indicate the applicability of this algorithm to a heterogeneous medium.

Moreover, the implementation of M-PML only needs to combine the damping profiles of PML, which improves absorption performance and stability while maintaining computational efficiency. In practice, this algorithm can be used as a forward operator for seismic migration, waveform inversion, and wave equation travel-time tomography. The frequencydomain algorithm can also improve the inversion efficiency of multi-source problems. For complex subsurface media, the proposed method can be applied for inversion of medium anisotropy and attenuation properties. Although our proposed algorithm works well, it has some limitations. Bad choices of *p*(*z*/*x*) and *p*(*x*/*z*) may lead to artificial reflections.

**Author Contributions:** Conceptualization, J.Y. and X.H.; methodology, J.Y.; investigation, H.C.; writing—original draft preparation, J.Y.; writing—review and editing, J.Y. and X.H.; visualization, J.Y.; supervision, X.H. and H.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China under grant numbers 11774373 and 12174421.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We gratefully acknowledge financial support from the National Natural Science Foundation of China (11774373, 12174421).

**Conflicts of Interest:** The authors declare no conflict of interest.
