Collocated versus staggered grid Finite Difference modeling in isotropic elastic media

Seismic modeling simulates wave propagation to understand the Earth’s subsurface and
predict ground motion. The finite difference (FD) method is a key numerical approach for
its efficiency in seismic modeling. Among the FD methods, the staggered-grid FD (SG-FD,
also called velocity-stress finite-difference) method is widely used for its accuracy in seismic
wave simulations. However, the wavefield interpolation in anisotropic media significantly
increases computational time. The wavefield interpolation is performed in every timestep,
which requires more runtime and memory usage. The collocated-grid FD (CG-FD) method,
avoiding interpolation, offers a potential alternative approach.
This thesis presents a systematic comparison between CG-FD and SG-FD methods for 2D
wave propagation in isotropic media. The CG-FD schemes are separated into acoustic and
elastic cases, while the SG-FD scheme works for both cases. The simplified versions for
homogeneous media are also provided.
The comparison covers stability and numerical dispersion. Based on plane-wave analysis,
the derivation of the formulas for the stability limit and numerical dispersion relation is
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provided in this thesis. For the CG-FD scheme, the stability limit formulas for arbitrary
orders of spatial accuracy are proposed in this thesis. The grid scan method is used to find
the global stability limit boundary, and the results are verified. Furthermore, the stability
boundary of the acoustic CG-FD is explicitly compared with the traditional stability limit.
For elastic CG-FD, the stability limit and numerical dispersion depend on the S-wave to
P-wave velocity ratio. In addition, an accuracy analysis of numerical dispersion is provided.
A homogeneous model and a two-layer model are used for simulating the snapshots and
seismograms of the wavefield. In the results of the homogeneous model, the seismograms
are compared with the analytical solution. In the results of the heterogeneous model, a
comparison between CG-FD and SG-FD is provided.
The results show that, in the acoustic case, CG-FD and SG-FD exhibit nearly identical
dispersion behavior and comparable numerical accuracy. In the elastic case, CG-FD
provides a less restrictive stability limit than SG-FD but suffers from stronger directional
numerical dispersion, especially for diagonal wave propagation and smaller S-wave to P-
wave velocity ratios. These findings indicate that CG-FD can serve as a feasible alternative
to SG-FD in isotropic media, although a higher order of spatial accuracy is generally
required to achieve comparable dispersion performance. The study provides a basis for
future extensions to anisotropic media