Stability conditions of multiple-relaxation-time lattice Boltzmann model for seismic wavefield modeling

In general, the wave equation can be taken to characterize the seismic wavefield propagation from a macroscopic perspective. The lattice Boltzmann method (LBM) is an alternative strategy to model seismic wavefields on mesoscopic scale. Due to its fully discrete nature and flexible boundary processing, LBM therefore has attracted increasing attention in seismology. The stability problem of LBM is a critical aspect in its seismic wavefield simulation. We give the steps of employing LBM for seismic wavefield modeling and some comparisons between LBM and the wave equation. Further, we derive theoretically the stability conditions of Bhatnagar-Groos-Krook (BGK) LBM and multiple-relaxation-time (MRT) LBM. Although the stability adjustment of MRT-LBM is more flexible, it is rather difficult to obtain an analytical solution concerning its relaxation parameters and other factors. In view of this, we first demonstrate that the stability and accuracy of MRT-LBM is superior to BGK-LBM to a certain extent by means of seismic wavefield modeling. Then, we investigate the individual effect of each relaxation parameter of MRT-LBM on the seismic waveforms. Most significantly, we construct the stability models for MRT-LBM only related to the relaxation parameters, based on the stability results of a large number of wavefield simulations in the homogeneous medium. Our stability models can well guide MRT-LBM for stable seismic wavefield modeling without considering other factors. Finally, we verify the correctness as well as the wide applicability of the proposed stability models by some simple layered media and modified Marmousi and BP models. The stability models may play a nice guiding role for further wavefield simulations based on MRT-LBM, especially for low-viscosity media.