How accurate numerical simulation of seismic waves in a heterogeneous medium can be?

Analysis of equations of motion by Moczo et al. (2022) led to the conclusion that the discrete (grid) representation of the heterogeneous medium must be wavenumber bandlimited up to the Nyquist frequency. This is a consequence of the spatial discretization. Mittet (2021a) reported that if the discrete grid model of medium coincides with the true medium up to some wavenumber, the simulated wavefield is accurate only up to a half of this wavenumber. Here, we present results of the systematic and comprehensive analysis focused on the principal limits of accuracy of numerically simulated wavefields. First, we analyze wavenumber spectra of (1) exact wavefields in a heterogeneous elastic medium, (2) wavenumber bandlimited wavefields, and (3) spatially discretized wavefields. Then, we derive spatial dependence of the frequency spectrum of waves generated by a finite source, and perturbing wavefields due to a small perturbation of the medium and due to a small wavenumber bandlimited perturbation of the medium. We analyze an interaction of an incoming wave with the medium perturbation through a change of phase difference and through wavenumber spectra. We draw conclusions on the wavenumber limitation of wavefields in the wavenumber bandlimited heterogeneous medium. We numerically verify the fundamental finding using exact solutions. The main consequence for the finite-difference (FD) modeling based on spatial discretization of the computational domain is: Due to spatial sampling, the medium must be wavenumber limited up to the Nyquist frequency. Then, the wavefield should not be sampled by less than four spatial grid spacings per shortest wavelength to obtain sufficiently accurate results. This applies to any heterogeneous FD scheme.