Accuracy of the Finite-difference Modeling of Seismic Motion – Wavenumber Limitation of Medium

Material interfaces play crucial role in forming seismic wavefield in local surface sedimentary structures and resulting free-surface motion. Multiple reverberations between the free surface and sediment-bedrock interface can lead to resonant amplifications and generation of local surface waves, and consequently to strong site effects of earthquakes.

It is therefore important to properly implement material interfaces in numerical modelling of seismic wave propagation and seismic motion. This has been well known for some time, and several approaches have been developed in variety of numerical methods.

The finite-difference (FD) method is still dominant method in numerical investigations of site effects of earthquakes. It applies relatively simple discretization in space to the material parameters and discretization in space and time to wavefield variables. Therefore, consequences of discretization must be analyzed in time, space, frequency and wavenumber domains.

Interestingly enough, the least attention has been paid to the wavenumber domain. Mittet (2017) and Moczo et al. (2022) recently demonstrated that, due to spatial discretization, a model of the medium must be wavenumber-limited by a wavenumber k smaller than the Nyquist wavenumber. Mittet (2021) and Valovcan et al. (2024) proved that the wavefield (numerically simulated or exact) in a medium limited by wavenumber k can only be accurate up to half this wavenumber. This has significant consequence for practical FD modelling of motion in realistic models of local structures.

We numerically demonstrate a perfect and unprecedented sub-cell resolution (capability to sense the position of interface within a grid cell) of FD modelling based on the wavenumber-limited medium using a finite spatial low-pass filter. The finding that it is possible to use a finite-length filter for wavenumber limitation of the medium is of key importance for the next development of the concept in terms of computational efficiency in modelling site effects.