Causal velocity dispersion for general frequency-dependent Q: An iterative Kramers–Kronig method

Chen Ji, & Yuehua Zeng

Submitted August 30, 2026, SCEC Contribution #15494, 2026 SCEC Annual Meeting Poster #TBD

To accurately simulate seismic-wave movement in viscoelastic media, it is essential to use phase velocity dispersion that is both causal and matches the intended frequency-dependent attenuation model, Q(f), where f denotes the frequency. This is especially crucial when studying wavefield changes caused by varying crustal attenuation, as errors in approximating Q(f) could distort those findings. Analytical solutions are available for simple attenuation models, such as constant-Q media, but they usually cannot handle more complex, piecewise functions, such as models that stay constant at low frequencies and follow a power law at high frequencies. We introduce an iterative numerical algorithm designed to calculate the causal minimum-phase velocity dispersion for any piecewise Q(f) model using the subtracted Kramers–Kronig relations. In each step, the chosen Q(f) model adjusts the imaginary part of slowness, while the Kramers–Kronig relation updates the real part. This process continues until both parts are fully consistent with each other. Since velocity dispersions in Earth’s materials tend to be minimal, convergence is typically achieved after just two or three iterations. The resulting dispersion curves and the associated time-domain, ray-theoretical responses generally match those produced with the memory-variable formulation of Withers et al. (2015). We have implemented this algorithm as a lightweight preprocessing step for frequency-wavenumber domain synthetic seismogram calculations. Because the method precisely replicates the specified Q(f) model, any variations in synthetic wavefields can be directly linked to changes in the attenuation model, rather than errors from attempting to approximate it.

Key Words
Causal velocity dispersion, frequency-dependent Q

Citation
Ji, C., & Zeng, Y. (2026, 08). Causal velocity dispersion for general frequency-dependent Q: An iterative Kramers–Kronig method. Poster Presentation at 2026 SCEC Annual Meeting.

Related Projects & Working Groups
Seismology