Several Problems On Stability Of Wave Simulation

Stable, efficient and high accuracy wave forward simulation method that can guarantee consistent accuracy in the whole inversion region is needed for wave inversion problem. The instability mechanism and stable implementation of high order artificial boundary in wave simulation is furtherly studied in this paper based on the former research work.1. The mechanism of high frequency oscillation instability caused by coupling the artificial boundary scheme and interior scheme could be clarified in the view of locality: that is both schemes support high frequency waves whose group velocity points to interior domain. Focusing on the lumped-mass finite element method(FEM) of SH wave simulation with the multi-transmitting formula(MTF), such high frequency waves are solely induced by the FEM, therefore a stable measure is presented by modifying the stiffness matrix of FEM. Theoretical analysis and numerical experiments have shown this method can ensure MTF to be implemented stably.2. The group velocity of P and SV wave motion surported by the artificial boundary scheme and interior scheme mustn’t point to interior domain for stable implementation of MTF in P-SV wave simulation. The stable method by modifying the stiffness matrix of FEM is still valid by theoretical analysis and numerical experiments.3. The error sources of high frequency oscillation and zero frequency drift local instability is clarified. Focusing on the lumped-mass FEM of SH wave simulation, based on MTF reflection coefficient that considers the discretization of interior equation, the round-off error of these plane waves that can be completely transmitted by MTF in wave simulation could raise the instability.4. Besides for the instability caused by local error in wave simulation, for high frequency instability, there also exists globale reflection-amplification instability whose instability mechanism is the amplification of reflection from the boundary and the multi-reflection of wave motion in a finite computational region. Focusing on the lumped-mass FEM of SH wave simulation, it is shown that such type of instability is independent with interior scheme, just induced by improper parameter of MTF. That means the instability is not caused by coupling the artificial boundary scheme and interior scheme.5. Energy dissipation of medium is needed to consider and instability can be suppressed by damping in wave simulation. Focusing on 1D wave simulation by lumped-mass FEM, the energy dissipation and effects on wave motion by Rayleigh damping in discrete grid is investigated by Fourier mode. It is shown that non-traveling wave whose wave number is zero or near zero exists and traveling wave attenuation is all the same for different wave number caused by alpha damping, while for beta damping, such non-traveling wave dose not exists and traveling wave attenuation increases with the increasing of wave number.