A time-domain complex-valued wave equation for modelling visco-acoustic wave propagation

Abstract

In this paper, we derive a time-domain complex-valued wave equation for modelling viscoacoustic wave propagation in frequency-independent Q media. Starting from the frequencydomain visco-acoustic wave equation, we use a second-order polynomial to approximate the dispersion term and a pseudo-differential operator to approximate the dissipation term. These two approximations enable us to transformthe frequency-domain visco-acousticwave equation to the time domain. Due to the introduction of an imaginary unit in the dispersion approximation, the new wave equation is complex valued, which is similar to the time-dependent Schrödinger equation. The advantages of the proposed visco-acoustic wave equation include: (1) the dispersion and dissipation effects are naturally separated, which can be used to compensate amplitude loss in reverse-time migration by simply flipping the sign of the dissipation term; (2) the quality factor Q is explicitly incorporated in the wave equation, making it easier to derive the misfit gradient with respect to Q in full-waveform inversion in comparison with the traditional standard linear solid method; and (3) the new visco-acoustic wave equation can be numerically solved using finite-difference time marching and Fourier transform, which requires lower memory costs than frequency-domain visco-acoustic modelling solvers. The accuracy of the proposed equation is first verified by comparing with the analytical Green's function in a homogeneous medium. Then, two numerical examples are used to demonstrate that the new wave equation is capable of describing attenuation effects in heterogeneous frequency-independent Q media.