Wave equation for underground viscoelastic media and wavefield numerical simulation

The energy of wavefield is gradually attenuated in all real materials, which is a fundamental feature and more obvious in the media containing liquid and gas. Because the viscosity effect is not considered in the classical wave theory, the actual wavefield is different from the simulated scenario based on the assumption of complete elasticity so that the application of wavefield does not meet the expectations in engineering technology, such as geophysical exploration. In the rock physics field, the well-known constant-<i>Q</i> theory gives a linear description of attenuation and <i>Q</i> is regarded as independent of the frequency. The quality factor <i>Q</i> is a parameter for calculating the phase difference between stress and strain of the media, which, as an index of wavefield attenuation behavior, is inversely proportional to the viscosity. Based on the constant-<i>Q</i> theory, a wave equation can be directly obtained by the Fourier transform of the dispersion relation, in which there is a fractional time differential operator. Therefore, it is difficult to perform the numerical simulation due to memory for all historical wavefields. In this paper, the dispersion relation is approximated by polynomial fitting and Taylor expansion method to eliminate the fractional power of frequency which is uncomfortably treated in the time domain. And then a complex-valued wave equation is derived to characterize the propagation law of wavefield in earth media. Besides the superiority of numerical simulation, the other advantage of this wave equation is that the dispersion and dissipation effects are decoupled. Next, a feasible numerical simulation strategy is proposed. The temporal derivative is solved by the finite-difference approach, moreover, the fractional spatial derivative is calculated in the spatial frequency domain by using the pseudo-spectral method. In the process of numerical simulation, only two-time slices, instead of the full-time wavefields, need to be saved, so the demand for data memory significantly slows down compared with solving the operator of the fractional time differential. Following that, the numerical examples prove that the novel wave equation is capable and efficient for the homogeneous model. The research work contributes to the understanding of complex wavefield phenomena and provides a basis for treating the seismology problems.