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Implications of the Recent Findings for Practical Calculations and Designing the Time-domain Finite-difference Schemes

Description: 

Since the beginning of the time-domain finite-difference (FD) modeling of seismic wave propagation, the FD scheme developers have been analyzing consistency and order of approximation, stability, and temporal and spatial grid dispersions. In most cases the analysis was performed for the homogeneous medium assuming a harmonic wave. The stability analysis of the space-time FD scheme gives a relation leading to a condition for the temporal discretization (maximum possible time step). The stability condition is then used to obtain a grid-dispersion relation: a relation for the phase velocity as a function of the time step and spatial grid spacing. A choice of admissible difference between the true and grid velocities then determines a value of the grid spacing.

The outlined (in general simplified) traditional analysis does not account for the consequences of spatial discretization in the wavenumber domain. Moczo et al. (2022) and Valovcan et al. (2023, 2024) showed that such an analysis is necessary and leads to truly fundamental consequences for discrete representation of the heterogeneous medium and spatial sampling of the simulated wavefield. A heterogeneous medium must be wavenumber limited by the Nyquist wavenumber in order to avoid aliasing in the wavenumber domain. Then, however, the wavenumber-limited medium imposes a spatial sampling condition for the simulated wavefield. The wavefield should not be sampled by less than 4 spatial grid spacings per shortest wavelength to obtain sufficiently accurate results. This applies to any FD scheme. We present numerical results to demonstrate fundamental conditions for constructing time-domain FD schemes.

Session: Numerical Modeling in Seismology: Theory, Algorithms and Applications - I

Type: Oral

Date: 4/17/2025

Presentation Time: 08:30 AM (local time)

Presenting Author: Peter

Student Presenter: No

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