The waveform comparison of three fractional viscous acoustic wave equations

The forward simulation of the viscous acoustic wave equation is essential for understanding wave propagation and seismic inversion. The viscous acoustic seismic wave equations are diverse, even if we limit the study scope to the fractional viscous wave equations. In present study, we consider three Riesz fractional viscous wave equations: the Fractional Viscous Acoustic Wave (FVAW) equation, Dispersion-Dominated Wave (DDW) equation, and Attenuation-Dominated Wave (ADW) equation. The Acoustic Wave (AW) equation, as a special fractional wave equation, is used to compare with the three fractional viscous acoustic equations. The Asymptotic Local Finite Difference (ALFD) method is adopted to solve the fractional derivative term; while, the Lax–Wendroff Correction (LWC) scheme is used to solve the integer derivative term. The analysis results indicate that the numerical scheme of the ADW equation exhibits the most rigorous stability condition, and that of the DDW equation is the most flexible. When the product of complex wavenumber k and spatial step size h equal to \(\pi\) $\pi$ , the maximum phase velocity errors of the FVAW equation, DDW equation, ADW equation, and AW equation are 27.78%, 28.02%, 2.25%, and 3.04%, respectively. Numerical experiments demonstrate that the FVAW equation not only governs the arrival time but also influences the amplitude. The DDW equation governs the arrival time but not amplitude; while, the ADW equation controls the amplitude but not arrival time. As the quality factor Q decreases, the viscous features of these three wave equations become pronounced. The change of amplitude is more noticeable than that of arrival time, suggesting that arrival time is more robust than amplitude. Based on these findings, we suggest incorporating the step for selecting the governing equations when dealing with practical Full–Waveform Inversion, which is helpful to improve the accuracy and reliability of the inversion results. Our results not only emphasize the importance of understanding the behavior of viscous wave equations, but also provide waveform evidence for selecting seismic governing equations in Full–Waveform Inversion.