The waveform comparison of three common-used fractional viscous acoustic wave equations

Abstract The forward simulation of the viscous acoustic wave equation is an essential part of geophysics and energy resources exploration research. The viscous acoustic seismic wave equations are diverse, even if we limit the study scope to the fractional viscous wave equations. In the present study, we consider three commonly used fractional-order 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 viscous acoustic equations. The asymptotic local finite difference (ALFD) method is adopted to solve the three fractional wave equations, while the Lax-Wendroff Correction (LWC) scheme is used to solve the integer wave equation. The analysis shows that the stability of the ADW equation is the most rigorous, and that of the DDW equation is the most flexible. When the numerical wave number \(\vartheta =\pi\), 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 show that the waveforms simulated by the four equations with the same parameters are distinct. Specifically, the FVAW equation, DDW equation, and quality factor Q are sensitive to the arrival time, while the FVAW equation, ADW equation, and quality factor Q are sensitive to the amplitude. Furthermore, the change of amplitude is more apparent than that of the arrival time, giving the results that the arrival time is more robust than the amplitude.